Theorem mulid1 7917

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 7916	. 2 |- ( A e. CC -> E. x e. RR E. y e. RR A = ( x + ( _i x. y ) ) )
2		recn 7907	. . . . . 6 |- ( x e. RR -> x e. CC )
3		ax-icn 7869	. . . . . . 7 |- _i e. CC
4		recn 7907	. . . . . . 7 |- ( y e. RR -> y e. CC )
5		mulcl 7901	. . . . . . 7 |- ( ( _i e. CC /\ y e. CC ) -> ( _i x. y ) e. CC )
6	3, 4, 5	sylancr 412	. . . . . 6 |- ( y e. RR -> ( _i x. y ) e. CC )
7		ax-1cn 7867	. . . . . . 7 |- 1 e. CC
8		adddir 7911	. . . . . . 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 1321	. . . . . 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 7881	. . . . . 6 |- ( x e. RR -> ( x x. 1 ) = x )
12		mulass 7905	. . . . . . . . 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 1323	. . . . . . . 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 7881	. . . . . . . 8 |- ( y e. RR -> ( y x. 1 ) = y )
16	15	oveq2d 5869	. . . . . . 7 |- ( y e. RR -> ( _i x. ( y x. 1 ) ) = ( _i x. y ) )
17	14, 16	eqtrd 2203	. . . . . 6 |- ( y e. RR -> ( ( _i x. y ) x. 1 ) = ( _i x. y ) )
18	11, 17	oveqan12d 5872	. . . . 5 |- ( ( x e. RR /\ y e. RR ) -> ( ( x x. 1 ) + ( ( _i x. y ) x. 1 ) ) = ( x + ( _i x. y ) ) )
19	10, 18	eqtrd 2203	. . . 4 |- ( ( x e. RR /\ y e. RR ) -> ( ( x + ( _i x. y ) ) x. 1 ) = ( x + ( _i x. y ) ) )
20		oveq1 5860	. . . . 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 2185	. . . 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 2593	. 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 1348   e. wcel 2141   E.wrex 2449  (class class class)co 5853   CCcc 7772   RRcr 7773   1c1 7775   _ici 7776   + caddc 7777   x. cmul 7779

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-resscn 7866  ax-1cn 7867  ax-icn 7869  ax-addcl 7870  ax-mulcl 7872  ax-mulcom 7875  ax-mulass 7877  ax-distr 7878  ax-1rid 7881  ax-cnre 7885

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-iota 5160  df-fv 5206  df-ov 5856

This theorem is referenced by:  mulid2  7918  mulid1i  7922  mulid1d  7937  muleqadd  8586  divdivap1  8640  conjmulap  8646  nnmulcl  8899  expmul  10521  binom21  10588  binom2sub1  10590  bernneq  10596  hashiun  11441  fproddccvg  11535  prodmodclem2a  11539  efexp  11645  ecxp  13616  lgsdilem2  13731