:: XCMPLX_0 semantic presentation
Lemma1:
1 = succ 0
;
:: deftheorem Def1 defines <i> XCMPLX_0:def 1 :
:: deftheorem Def2 defines complex XCMPLX_0:def 2 :
:: deftheorem Def3 defines zero XCMPLX_0:def 3 :
definition
let c
1, c
2 be
complex number ;
c
1 in COMPLEX
by Def2;
then consider c
3, c
4 being
Element of
REAL such that E3:
c
1 = [*c3,c4*]
by ARYTM_0:11;
c
2 in COMPLEX
by Def2;
then consider c
5, c
6 being
Element of
REAL such that E4:
c
2 = [*c5,c6*]
by ARYTM_0:11;
func c
1 + c
2 -> set means :
Def4:
:: XCMPLX_0:def 4
ex b
1, b
2, b
3, b
4 being
Element of
REAL st
( a
1 = [*b1,b2*] & a
2 = [*b3,b4*] & a
3 = [*(+ b1,b3),(+ b2,b4)*] );
existence
ex b1 being set ex b2, b3, b4, b5 being Element of REAL st
( c1 = [*b2,b3*] & c2 = [*b4,b5*] & b1 = [*(+ b2,b4),(+ b3,b5)*] )
uniqueness
for b1, b2 being set holds
( ex b3, b4, b5, b6 being Element of REAL st
( c1 = [*b3,b4*] & c2 = [*b5,b6*] & b1 = [*(+ b3,b5),(+ b4,b6)*] ) & ex b3, b4, b5, b6 being Element of REAL st
( c1 = [*b3,b4*] & c2 = [*b5,b6*] & b2 = [*(+ b3,b5),(+ b4,b6)*] ) implies b1 = b2 )
commutativity
for b1 being set
for b2, b3 being complex number holds
not ( ex b4, b5, b6, b7 being Element of REAL st
( b2 = [*b4,b5*] & b3 = [*b6,b7*] & b1 = [*(+ b4,b6),(+ b5,b7)*] ) & ( for b4, b5, b6, b7 being Element of REAL holds
not ( b3 = [*b4,b5*] & b2 = [*b6,b7*] & b1 = [*(+ b4,b6),(+ b5,b7)*] ) ) )
;
func c
1 * c
2 -> set means :
Def5:
:: XCMPLX_0:def 5
ex b
1, b
2, b
3, b
4 being
Element of
REAL st
( a
1 = [*b1,b2*] & a
2 = [*b3,b4*] & a
3 = [*(+ (* b1,b3),(opp (* b2,b4))),(+ (* b1,b4),(* b2,b3))*] );
existence
ex b1 being set ex b2, b3, b4, b5 being Element of REAL st
( c1 = [*b2,b3*] & c2 = [*b4,b5*] & b1 = [*(+ (* b2,b4),(opp (* b3,b5))),(+ (* b2,b5),(* b3,b4))*] )
uniqueness
for b1, b2 being set holds
( ex b3, b4, b5, b6 being Element of REAL st
( c1 = [*b3,b4*] & c2 = [*b5,b6*] & b1 = [*(+ (* b3,b5),(opp (* b4,b6))),(+ (* b3,b6),(* b4,b5))*] ) & ex b3, b4, b5, b6 being Element of REAL st
( c1 = [*b3,b4*] & c2 = [*b5,b6*] & b2 = [*(+ (* b3,b5),(opp (* b4,b6))),(+ (* b3,b6),(* b4,b5))*] ) implies b1 = b2 )
commutativity
for b1 being set
for b2, b3 being complex number holds
not ( ex b4, b5, b6, b7 being Element of REAL st
( b2 = [*b4,b5*] & b3 = [*b6,b7*] & b1 = [*(+ (* b4,b6),(opp (* b5,b7))),(+ (* b4,b7),(* b5,b6))*] ) & ( for b4, b5, b6, b7 being Element of REAL holds
not ( b3 = [*b4,b5*] & b2 = [*b6,b7*] & b1 = [*(+ (* b4,b6),(opp (* b5,b7))),(+ (* b4,b7),(* b5,b6))*] ) ) )
;
end;
:: deftheorem Def4 defines + XCMPLX_0:def 4 :
:: deftheorem Def5 defines * XCMPLX_0:def 5 :
for b
1, b
2 being
complex number for b
3 being
set holds
( b
3 = b
1 * b
2 iff ex b
4, b
5, b
6, b
7 being
Element of
REAL st
( b
1 = [*b4,b5*] & b
2 = [*b6,b7*] & b
3 = [*(+ (* b4,b6),(opp (* b5,b7))),(+ (* b4,b7),(* b5,b6))*] ) );
Lemma5:
0 = [*0,0*]
by ARYTM_0:def 7;
reconsider c1 = 1 as Element of REAL ;
Lemma6:
for b1, b2, b3 being Element of REAL holds
( + b1,b2 = 0 & + b1,b3 = 0 implies b2 = b3 )
:: deftheorem Def6 defines - XCMPLX_0:def 6 :
:: deftheorem Def7 defines " XCMPLX_0:def 7 :
for b
1, b
2 being
complex number holds
( ( b
1 <> 0 implies ( b
2 = b
1 " iff b
1 * b
2 = 1 ) ) & ( not b
1 <> 0 implies ( b
2 = b
1 " iff b
2 = 0 ) ) );
:: deftheorem Def8 defines - XCMPLX_0:def 8 :
:: deftheorem Def9 defines / XCMPLX_0:def 9 :
Lemma9:
for b1, b2, b3 being complex number holds b1 * (b2 * b3) = (b1 * b2) * b3
Lemma10:
REAL c= COMPLEX
by NUMBERS:def 2, XBOOLE_1:7;