Theorem rhmmul 13660

Description: A homomorphism of rings preserves multiplication. (Contributed by Mario Carneiro, 12-Jun-2015.)

Hypotheses

Ref	Expression
rhmmul.x	|- X = ( Base ` R )
rhmmul.m	|- .x. = ( .r ` R )
rhmmul.n	|- .X. = ( .r ` S )

Assertion

Ref	Expression
rhmmul	|- ( ( F e. ( R RingHom S ) /\ A e. X /\ B e. X ) -> ( F ` ( A .x. B ) ) = ( ( F ` A ) .X. ( F ` B ) ) )

Proof of Theorem rhmmul

Step	Hyp	Ref	Expression
1		eqid 2193	. . . . 5 |- (mulGrp ` R ) = (mulGrp ` R )
2		eqid 2193	. . . . 5 |- (mulGrp ` S ) = (mulGrp ` S )
3	1, 2	rhmmhm 13655	. . . 4 |- ( F e. ( R RingHom S ) -> F e. ( (mulGrp ` R ) MndHom (mulGrp ` S ) ) )
4	3	3ad2ant1 1020	. . 3 |- ( ( F e. ( R RingHom S ) /\ A e. X /\ B e. X ) -> F e. ( (mulGrp ` R ) MndHom (mulGrp ` S ) ) )
5		simp2 1000	. . . 4 |- ( ( F e. ( R RingHom S ) /\ A e. X /\ B e. X ) -> A e. X )
6		rhmrcl1 13651	. . . . . . 7 |- ( F e. ( R RingHom S ) -> R e. Ring )
7		rhmmul.x	. . . . . . . 8 |- X = ( Base ` R )
8	1, 7	mgpbasg 13422	. . . . . . 7 |- ( R e. Ring -> X = ( Base ` (mulGrp ` R ) ) )
9	6, 8	syl 14	. . . . . 6 |- ( F e. ( R RingHom S ) -> X = ( Base ` (mulGrp ` R ) ) )
10	9	eleq2d 2263	. . . . 5 |- ( F e. ( R RingHom S ) -> ( A e. X <-> A e. ( Base ` (mulGrp ` R ) ) ) )
11	10	3ad2ant1 1020	. . . 4 |- ( ( F e. ( R RingHom S ) /\ A e. X /\ B e. X ) -> ( A e. X <-> A e. ( Base ` (mulGrp ` R ) ) ) )
12	5, 11	mpbid 147	. . 3 |- ( ( F e. ( R RingHom S ) /\ A e. X /\ B e. X ) -> A e. ( Base ` (mulGrp ` R ) ) )
13		simp3 1001	. . . 4 |- ( ( F e. ( R RingHom S ) /\ A e. X /\ B e. X ) -> B e. X )
14	9	eleq2d 2263	. . . . 5 |- ( F e. ( R RingHom S ) -> ( B e. X <-> B e. ( Base ` (mulGrp ` R ) ) ) )
15	14	3ad2ant1 1020	. . . 4 |- ( ( F e. ( R RingHom S ) /\ A e. X /\ B e. X ) -> ( B e. X <-> B e. ( Base ` (mulGrp ` R ) ) ) )
16	13, 15	mpbid 147	. . 3 |- ( ( F e. ( R RingHom S ) /\ A e. X /\ B e. X ) -> B e. ( Base ` (mulGrp ` R ) ) )
17		eqid 2193	. . . 4 |- ( Base ` (mulGrp ` R ) ) = ( Base ` (mulGrp ` R ) )
18		eqid 2193	. . . 4 |- ( +g ` (mulGrp ` R ) ) = ( +g ` (mulGrp ` R ) )
19		eqid 2193	. . . 4 |- ( +g ` (mulGrp ` S ) ) = ( +g ` (mulGrp ` S ) )
20	17, 18, 19	mhmlin 13039	. . 3 |- ( ( F e. ( (mulGrp ` R ) MndHom (mulGrp ` S ) ) /\ A e. ( Base ` (mulGrp ` R ) ) /\ B e. ( Base ` (mulGrp ` R ) ) ) -> ( F ` ( A ( +g ` (mulGrp ` R ) ) B ) ) = ( ( F ` A ) ( +g ` (mulGrp ` S ) ) ( F ` B ) ) )
21	4, 12, 16, 20	syl3anc 1249	. 2 |- ( ( F e. ( R RingHom S ) /\ A e. X /\ B e. X ) -> ( F ` ( A ( +g ` (mulGrp ` R ) ) B ) ) = ( ( F ` A ) ( +g ` (mulGrp ` S ) ) ( F ` B ) ) )
22		rhmmul.m	. . . . . . . 8 |- .x. = ( .r ` R )
23	1, 22	mgpplusgg 13420	. . . . . . 7 |- ( R e. Ring -> .x. = ( +g ` (mulGrp ` R ) ) )
24	6, 23	syl 14	. . . . . 6 |- ( F e. ( R RingHom S ) -> .x. = ( +g ` (mulGrp ` R ) ) )
25	24	oveqd 5935	. . . . 5 |- ( F e. ( R RingHom S ) -> ( A .x. B ) = ( A ( +g ` (mulGrp ` R ) ) B ) )
26	25	fveq2d 5558	. . . 4 |- ( F e. ( R RingHom S ) -> ( F ` ( A .x. B ) ) = ( F ` ( A ( +g ` (mulGrp ` R ) ) B ) ) )
27		rhmrcl2 13652	. . . . . 6 |- ( F e. ( R RingHom S ) -> S e. Ring )
28		rhmmul.n	. . . . . . 7 |- .X. = ( .r ` S )
29	2, 28	mgpplusgg 13420	. . . . . 6 |- ( S e. Ring -> .X. = ( +g ` (mulGrp ` S ) ) )
30	27, 29	syl 14	. . . . 5 |- ( F e. ( R RingHom S ) -> .X. = ( +g ` (mulGrp ` S ) ) )
31	30	oveqd 5935	. . . 4 |- ( F e. ( R RingHom S ) -> ( ( F ` A ) .X. ( F ` B ) ) = ( ( F ` A ) ( +g ` (mulGrp ` S ) ) ( F ` B ) ) )
32	26, 31	eqeq12d 2208	. . 3 |- ( F e. ( R RingHom S ) -> ( ( F ` ( A .x. B ) ) = ( ( F ` A ) .X. ( F ` B ) ) <-> ( F ` ( A ( +g ` (mulGrp ` R ) ) B ) ) = ( ( F ` A ) ( +g ` (mulGrp ` S ) ) ( F ` B ) ) ) )
33	32	3ad2ant1 1020	. 2 |- ( ( F e. ( R RingHom S ) /\ A e. X /\ B e. X ) -> ( ( F ` ( A .x. B ) ) = ( ( F ` A ) .X. ( F ` B ) ) <-> ( F ` ( A ( +g ` (mulGrp ` R ) ) B ) ) = ( ( F ` A ) ( +g ` (mulGrp ` S ) ) ( F ` B ) ) ) )
34	21, 33	mpbird 167	1 |- ( ( F e. ( R RingHom S ) /\ A e. X /\ B e. X ) -> ( F ` ( A .x. B ) ) = ( ( F ` A ) .X. ( F ` B ) ) )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2164   ` cfv 5254  (class class class)co 5918   Basecbs 12618   +g cplusg 12695   .rcmulr 12696   MndHom cmhm 13029  mulGrpcmgp 13416   Ringcrg 13492   RingHom crh 13646

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-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-pre-ltirr 7984  ax-pre-ltadd 7988

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-map 6704  df-pnf 8056  df-mnf 8057  df-ltxr 8059  df-inn 8983  df-2 9041  df-3 9042  df-ndx 12621  df-slot 12622  df-base 12624  df-sets 12625  df-plusg 12708  df-mulr 12709  df-0g 12869  df-mgm 12939  df-sgrp 12985  df-mnd 12998  df-mhm 13031  df-grp 13075  df-ghm 13311  df-mgp 13417  df-ur 13456  df-ring 13494  df-rhm 13648

This theorem is referenced by:  rhmdvdsr  13671  rhmopp  13672  rhmunitinv  13674  znidom  14145  znidomb  14146  znunit  14147  znrrg  14148  lgseisenlem3  15188  lgseisenlem4  15189