Theorem mulid1 7756

Description: 1 is an identity element for multiplication. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.)

Assertion

Ref	Expression
mulid1	|- ( A e. CC -> ( A x. 1 ) = A )

Proof of Theorem mulid1

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnre 7755	. 2 |- ( A e. CC -> E. x e. RR E. y e. RR A = ( x + ( _i x. y ) ) )
2		recn 7746	. . . . . 6 |- ( x e. RR -> x e. CC )
3		ax-icn 7708	. . . . . . 7 |- _i e. CC
4		recn 7746	. . . . . . 7 |- ( y e. RR -> y e. CC )
5		mulcl 7740	. . . . . . 7 |- ( ( _i e. CC /\ y e. CC ) -> ( _i x. y ) e. CC )
6	3, 4, 5	sylancr 410	. . . . . 6 |- ( y e. RR -> ( _i x. y ) e. CC )
7		ax-1cn 7706	. . . . . . 7 |- 1 e. CC
8		adddir 7750	. . . . . . 7 |- ( ( x e. CC /\ ( _i x. y ) e. CC /\ 1 e. CC ) -> ( ( x + ( _i x. y ) ) x. 1 ) = ( ( x x. 1 ) + ( ( _i x. y ) x. 1 ) ) )
9	7, 8	mp3an3 1304	. . . . . 6 |- ( ( x e. CC /\ ( _i x. y ) e. CC ) -> ( ( x + ( _i x. y ) ) x. 1 ) = ( ( x x. 1 ) + ( ( _i x. y ) x. 1 ) ) )
10	2, 6, 9	syl2an 287	. . . . 5 |- ( ( x e. RR /\ y e. RR ) -> ( ( x + ( _i x. y ) ) x. 1 ) = ( ( x x. 1 ) + ( ( _i x. y ) x. 1 ) ) )
11		ax-1rid 7720	. . . . . 6 |- ( x e. RR -> ( x x. 1 ) = x )
12		mulass 7744	. . . . . . . . 9 |- ( ( _i e. CC /\ y e. CC /\ 1 e. CC ) -> ( ( _i x. y ) x. 1 ) = ( _i x. ( y x. 1 ) ) )
13	3, 7, 12	mp3an13 1306	. . . . . . . 8 |- ( y e. CC -> ( ( _i x. y ) x. 1 ) = ( _i x. ( y x. 1 ) ) )
14	4, 13	syl 14	. . . . . . 7 |- ( y e. RR -> ( ( _i x. y ) x. 1 ) = ( _i x. ( y x. 1 ) ) )
15		ax-1rid 7720	. . . . . . . 8 |- ( y e. RR -> ( y x. 1 ) = y )
16	15	oveq2d 5783	. . . . . . 7 |- ( y e. RR -> ( _i x. ( y x. 1 ) ) = ( _i x. y ) )
17	14, 16	eqtrd 2170	. . . . . 6 |- ( y e. RR -> ( ( _i x. y ) x. 1 ) = ( _i x. y ) )
18	11, 17	oveqan12d 5786	. . . . 5 |- ( ( x e. RR /\ y e. RR ) -> ( ( x x. 1 ) + ( ( _i x. y ) x. 1 ) ) = ( x + ( _i x. y ) ) )
19	10, 18	eqtrd 2170	. . . 4 |- ( ( x e. RR /\ y e. RR ) -> ( ( x + ( _i x. y ) ) x. 1 ) = ( x + ( _i x. y ) ) )
20		oveq1 5774	. . . . 5 |- ( A = ( x + ( _i x. y ) ) -> ( A x. 1 ) = ( ( x + ( _i x. y ) ) x. 1 ) )
21		id 19	. . . . 5 |- ( A = ( x + ( _i x. y ) ) -> A = ( x + ( _i x. y ) ) )
22	20, 21	eqeq12d 2152	. . . 4 |- ( A = ( x + ( _i x. y ) ) -> ( ( A x. 1 ) = A <-> ( ( x + ( _i x. y ) ) x. 1 ) = ( x + ( _i x. y ) ) ) )
23	19, 22	syl5ibrcom 156	. . 3 |- ( ( x e. RR /\ y e. RR ) -> ( A = ( x + ( _i x. y ) ) -> ( A x. 1 ) = A ) )
24	23	rexlimivv 2553	. 2 |- ( E. x e. RR E. y e. RR A = ( x + ( _i x. y ) ) -> ( A x. 1 ) = A )
25	1, 24	syl 14	1 |- ( A e. CC -> ( A x. 1 ) = A )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   e. wcel 1480   E.wrex 2415  (class class class)co 5767   CCcc 7611   RRcr 7612   1c1 7614   _ici 7615   + caddc 7616   x. cmul 7618

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-resscn 7705  ax-1cn 7706  ax-icn 7708  ax-addcl 7709  ax-mulcl 7711  ax-mulcom 7714  ax-mulass 7716  ax-distr 7717  ax-1rid 7720  ax-cnre 7724

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-iota 5083  df-fv 5126  df-ov 5770

This theorem is referenced by:  mulid2  7757  mulid1i  7761  mulid1d  7776  muleqadd  8422  divdivap1  8476  conjmulap  8482  nnmulcl  8734  expmul  10331  binom21  10397  binom2sub1  10399  bernneq  10405  hashiun  11240  fproddccvg  11334  efexp  11377