Theorem mulrid 8287

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 8286	. 2 |- ( A e. CC -> E. x e. RR E. y e. RR A = ( x + ( _i x. y ) ) )
2		recn 8276	. . . . . 6 |- ( x e. RR -> x e. CC )
3		ax-icn 8238	. . . . . . 7 |- _i e. CC
4		recn 8276	. . . . . . 7 |- ( y e. RR -> y e. CC )
5		mulcl 8270	. . . . . . 7 |- ( ( _i e. CC /\ y e. CC ) -> ( _i x. y ) e. CC )
6	3, 4, 5	sylancr 414	. . . . . 6 |- ( y e. RR -> ( _i x. y ) e. CC )
7		ax-1cn 8236	. . . . . . 7 |- 1 e. CC
8		adddir 8281	. . . . . . 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 1363	. . . . . 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 8250	. . . . . 6 |- ( x e. RR -> ( x x. 1 ) = x )
12		mulass 8274	. . . . . . . . 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 1365	. . . . . . . 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 8250	. . . . . . . 8 |- ( y e. RR -> ( y x. 1 ) = y )
16	15	oveq2d 6074	. . . . . . 7 |- ( y e. RR -> ( _i x. ( y x. 1 ) ) = ( _i x. y ) )
17	14, 16	eqtrd 2267	. . . . . 6 |- ( y e. RR -> ( ( _i x. y ) x. 1 ) = ( _i x. y ) )
18	11, 17	oveqan12d 6077	. . . . 5 |- ( ( x e. RR /\ y e. RR ) -> ( ( x x. 1 ) + ( ( _i x. y ) x. 1 ) ) = ( x + ( _i x. y ) ) )
19	10, 18	eqtrd 2267	. . . 4 |- ( ( x e. RR /\ y e. RR ) -> ( ( x + ( _i x. y ) ) x. 1 ) = ( x + ( _i x. y ) ) )
20		oveq1 6065	. . . . 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 2249	. . . 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 2668	. 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 104   = wceq 1398   e. wcel 2205   E.wrex 2523  (class class class)co 6058   CCcc 8141   RRcr 8142   1c1 8144   _ici 8145   + caddc 8146   x. cmul 8148

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216  ax-resscn 8235  ax-1cn 8236  ax-icn 8238  ax-addcl 8239  ax-mulcl 8241  ax-mulcom 8244  ax-mulass 8246  ax-distr 8247  ax-1rid 8250  ax-cnre 8254

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-iota 5317  df-fv 5365  df-ov 6061

This theorem is referenced by:  mullid  8288  mulridi  8292  mulridd  8307  muleqadd  8962  divdivap1  9017  conjmulap  9023  nnmulcl  9278  expmul  10973  binom21  11041  binom2sub1  11043  bernneq  11050  hashiun  12193  fproddccvg  12287  prodmodclem2a  12291  efexp  12397  cncrng  14847  cnfld1  14850  ecxp  15896  lgsdilem2  16039