Theorem rngohommul 33769

Description: Ring homomorphisms preserve multiplication. (Contributed by Jeff Madsen, 3-Jan-2011.)

Hypotheses

Ref	Expression
rnghommul.1	|- G = ( 1st ` R )
rnghommul.2	|- X = ran G
rnghommul.3	|- H = ( 2nd ` R )
rnghommul.4	|- K = ( 2nd ` S )

Assertion

Ref	Expression
rngohommul	|- ( ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) /\ ( A e. X /\ B e. X ) ) -> ( F ` ( A H B ) ) = ( ( F ` A ) K ( F ` B ) ) )

Proof of Theorem rngohommul

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

Step	Hyp	Ref	Expression
1		rnghommul.1	. . . . . . 7 |- G = ( 1st ` R )
2		rnghommul.3	. . . . . . 7 |- H = ( 2nd ` R )
3		rnghommul.2	. . . . . . 7 |- X = ran G
4		eqid 2622	. . . . . . 7 |- (GId ` H ) = (GId ` H )
5		eqid 2622	. . . . . . 7 |- ( 1st ` S ) = ( 1st ` S )
6		rnghommul.4	. . . . . . 7 |- K = ( 2nd ` S )
7		eqid 2622	. . . . . . 7 |- ran ( 1st ` S ) = ran ( 1st ` S )
8		eqid 2622	. . . . . . 7 |- (GId ` K ) = (GId ` K )
9	1, 2, 3, 4, 5, 6, 7, 8	isrngohom 33764	. . . . . 6 |- ( ( R e. RingOps /\ S e. RingOps ) -> ( F e. ( R RngHom S ) <-> ( F : X --> ran ( 1st ` S ) /\ ( F ` (GId ` H ) ) = (GId ` K ) /\ A. x e. X A. y e. X ( ( F ` ( x G y ) ) = ( ( F ` x ) ( 1st ` S ) ( F ` y ) ) /\ ( F ` ( x H y ) ) = ( ( F ` x ) K ( F ` y ) ) ) ) ) )
10	9	biimpa 501	. . . . 5 |- ( ( ( R e. RingOps /\ S e. RingOps ) /\ F e. ( R RngHom S ) ) -> ( F : X --> ran ( 1st ` S ) /\ ( F ` (GId ` H ) ) = (GId ` K ) /\ A. x e. X A. y e. X ( ( F ` ( x G y ) ) = ( ( F ` x ) ( 1st ` S ) ( F ` y ) ) /\ ( F ` ( x H y ) ) = ( ( F ` x ) K ( F ` y ) ) ) ) )
11	10	simp3d 1075	. . . 4 |- ( ( ( R e. RingOps /\ S e. RingOps ) /\ F e. ( R RngHom S ) ) -> A. x e. X A. y e. X ( ( F ` ( x G y ) ) = ( ( F ` x ) ( 1st ` S ) ( F ` y ) ) /\ ( F ` ( x H y ) ) = ( ( F ` x ) K ( F ` y ) ) ) )
12	11	3impa 1259	. . 3 |- ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) -> A. x e. X A. y e. X ( ( F ` ( x G y ) ) = ( ( F ` x ) ( 1st ` S ) ( F ` y ) ) /\ ( F ` ( x H y ) ) = ( ( F ` x ) K ( F ` y ) ) ) )
13		simpr 477	. . . 4 |- ( ( ( F ` ( x G y ) ) = ( ( F ` x ) ( 1st ` S ) ( F ` y ) ) /\ ( F ` ( x H y ) ) = ( ( F ` x ) K ( F ` y ) ) ) -> ( F ` ( x H y ) ) = ( ( F ` x ) K ( F ` y ) ) )
14	13	2ralimi 2953	. . 3 |- ( A. x e. X A. y e. X ( ( F ` ( x G y ) ) = ( ( F ` x ) ( 1st ` S ) ( F ` y ) ) /\ ( F ` ( x H y ) ) = ( ( F ` x ) K ( F ` y ) ) ) -> A. x e. X A. y e. X ( F ` ( x H y ) ) = ( ( F ` x ) K ( F ` y ) ) )
15	12, 14	syl 17	. 2 |- ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) -> A. x e. X A. y e. X ( F ` ( x H y ) ) = ( ( F ` x ) K ( F ` y ) ) )
16		oveq1 6657	. . . . 5 |- ( x = A -> ( x H y ) = ( A H y ) )
17	16	fveq2d 6195	. . . 4 |- ( x = A -> ( F ` ( x H y ) ) = ( F ` ( A H y ) ) )
18		fveq2 6191	. . . . 5 |- ( x = A -> ( F ` x ) = ( F ` A ) )
19	18	oveq1d 6665	. . . 4 |- ( x = A -> ( ( F ` x ) K ( F ` y ) ) = ( ( F ` A ) K ( F ` y ) ) )
20	17, 19	eqeq12d 2637	. . 3 |- ( x = A -> ( ( F ` ( x H y ) ) = ( ( F ` x ) K ( F ` y ) ) <-> ( F ` ( A H y ) ) = ( ( F ` A ) K ( F ` y ) ) ) )
21		oveq2 6658	. . . . 5 |- ( y = B -> ( A H y ) = ( A H B ) )
22	21	fveq2d 6195	. . . 4 |- ( y = B -> ( F ` ( A H y ) ) = ( F ` ( A H B ) ) )
23		fveq2 6191	. . . . 5 |- ( y = B -> ( F ` y ) = ( F ` B ) )
24	23	oveq2d 6666	. . . 4 |- ( y = B -> ( ( F ` A ) K ( F ` y ) ) = ( ( F ` A ) K ( F ` B ) ) )
25	22, 24	eqeq12d 2637	. . 3 |- ( y = B -> ( ( F ` ( A H y ) ) = ( ( F ` A ) K ( F ` y ) ) <-> ( F ` ( A H B ) ) = ( ( F ` A ) K ( F ` B ) ) ) )
26	20, 25	rspc2v 3322	. 2 |- ( ( A e. X /\ B e. X ) -> ( A. x e. X A. y e. X ( F ` ( x H y ) ) = ( ( F ` x ) K ( F ` y ) ) -> ( F ` ( A H B ) ) = ( ( F ` A ) K ( F ` B ) ) ) )
27	15, 26	mpan9 486	1 |- ( ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) /\ ( A e. X /\ B e. X ) ) -> ( F ` ( A H B ) ) = ( ( F ` A ) K ( F ` B ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   ran crn 5115   -->wf 5884   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167  GIdcgi 27344   RingOpscrngo 33693   RngHom crnghom 33759

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-rngohom 33762

This theorem is referenced by:  rngohomco  33773  rngoisocnv  33780  crngohomfo  33805  keridl  33831