Theorem mulrid 8329

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

Assertion

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

Proof of Theorem mulrid

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

Step	Hyp	Ref	Expression
1		cnre 8328	. 2 |- ( A e. CC -> E. x e. RR E. y e. RR A = ( x + ( _i x. y ) ) )
2		recn 8318	. . . . . 6 |- ( x e. RR -> x e. CC )
3		ax-icn 8280	. . . . . . 7 |- _i e. CC
4		recn 8318	. . . . . . 7 |- ( y e. RR -> y e. CC )
5		mulcl 8312	. . . . . . 7 |- ( ( _i e. CC /\ y e. CC ) -> ( _i x. y ) e. CC )
6	3, 4, 5	sylancr 418	. . . . . 6 |- ( y e. RR -> ( _i x. y ) e. CC )
7		ax-1cn 8278	. . . . . . 7 |- 1 e. CC
8		adddir 8323	. . . . . . 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 1367	. . . . . 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 289	. . . . 5 |- ( ( x e. RR /\ y e. RR ) -> ( ( x + ( _i x. y ) ) x. 1 ) = ( ( x x. 1 ) + ( ( _i x. y ) x. 1 ) ) )
11		ax-1rid 8292	. . . . . 6 |- ( x e. RR -> ( x x. 1 ) = x )
12		mulass 8316	. . . . . . . . 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 1369	. . . . . . . 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 8292	. . . . . . . 8 |- ( y e. RR -> ( y x. 1 ) = y )
16	15	oveq2d 6104	. . . . . . 7 |- ( y e. RR -> ( _i x. ( y x. 1 ) ) = ( _i x. y ) )
17	14, 16	eqtrd 2271	. . . . . 6 |- ( y e. RR -> ( ( _i x. y ) x. 1 ) = ( _i x. y ) )
18	11, 17	oveqan12d 6107	. . . . 5 |- ( ( x e. RR /\ y e. RR ) -> ( ( x x. 1 ) + ( ( _i x. y ) x. 1 ) ) = ( x + ( _i x. y ) ) )
19	10, 18	eqtrd 2271	. . . 4 |- ( ( x e. RR /\ y e. RR ) -> ( ( x + ( _i x. y ) ) x. 1 ) = ( x + ( _i x. y ) ) )
20		oveq1 6095	. . . . 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 2253	. . . 4 |- ( A = ( x + ( _i x. y ) ) -> ( ( A x. 1 ) = A <-> ( ( x + ( _i x. y ) ) x. 1 ) = ( x + ( _i x. y ) ) ) )
23	19, 22	syl5ibrcom 157	. . 3 |- ( ( x e. RR /\ y e. RR ) -> ( A = ( x + ( _i x. y ) ) -> ( A x. 1 ) = A ) )
24	23	rexlimivv 2674	. 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 )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   E.wrex 2529  (class class class)co 6088   CCcc 8183   RRcr 8184   1c1 8186   _ici 8187   + caddc 8188   x. cmul 8190

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-resscn 8277  ax-1cn 8278  ax-icn 8280  ax-addcl 8281  ax-mulcl 8283  ax-mulcom 8286  ax-mulass 8288  ax-distr 8289  ax-1rid 8292  ax-cnre 8296

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4132  df-iota 5340  df-fv 5388  df-ov 6091

This theorem is used by:  mullid  8330  mulridi  8334  mulridd  8349  muleqadd  9006  divdivap1  9061  conjmulap  9067  nnmulcl  9334  expmul  11042  binom21  11110  binom2sub1  11112  bernneq  11119  hashiun  12270  fproddccvg  12364  prodmodclem2a  12368  efexp  12474  cncrng  15015  cnfld1  15018  ecxp  16134  lgsdilem2  16360