Theorem rhmmul 13720

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 2196	. . . . 5 |- (mulGrp ` R ) = (mulGrp ` R )
2		eqid 2196	. . . . 5 |- (mulGrp ` S ) = (mulGrp ` S )
3	1, 2	rhmmhm 13715	. . . 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 13711	. . . . . . 7 |- ( F e. ( R RingHom S ) -> R e. Ring )
7		rhmmul.x	. . . . . . . 8 |- X = ( Base ` R )
8	1, 7	mgpbasg 13482	. . . . . . 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 2266	. . . . 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 2266	. . . . 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 2196	. . . 4 |- ( Base ` (mulGrp ` R ) ) = ( Base ` (mulGrp ` R ) )
18		eqid 2196	. . . 4 |- ( +g ` (mulGrp ` R ) ) = ( +g ` (mulGrp ` R ) )
19		eqid 2196	. . . 4 |- ( +g ` (mulGrp ` S ) ) = ( +g ` (mulGrp ` S ) )
20	17, 18, 19	mhmlin 13099	. . 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 13480	. . . . . . 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 5939	. . . . 5 |- ( F e. ( R RingHom S ) -> ( A .x. B ) = ( A ( +g ` (mulGrp ` R ) ) B ) )
26	25	fveq2d 5562	. . . 4 |- ( F e. ( R RingHom S ) -> ( F ` ( A .x. B ) ) = ( F ` ( A ( +g ` (mulGrp ` R ) ) B ) ) )
27		rhmrcl2 13712	. . . . . 6 |- ( F e. ( R RingHom S ) -> S e. Ring )
28		rhmmul.n	. . . . . . 7 |- .X. = ( .r ` S )
29	2, 28	mgpplusgg 13480	. . . . . 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 5939	. . . 4 |- ( F e. ( R RingHom S ) -> ( ( F ` A ) .X. ( F ` B ) ) = ( ( F ` A ) ( +g ` (mulGrp ` S ) ) ( F ` B ) ) )
32	26, 31	eqeq12d 2211	. . 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 2167   ` cfv 5258  (class class class)co 5922   Basecbs 12678   +g cplusg 12755   .rcmulr 12756   MndHom cmhm 13089  mulGrpcmgp 13476   Ringcrg 13552   RingHom crh 13706

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-pre-ltirr 7991  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-map 6709  df-pnf 8063  df-mnf 8064  df-ltxr 8066  df-inn 8991  df-2 9049  df-3 9050  df-ndx 12681  df-slot 12682  df-base 12684  df-sets 12685  df-plusg 12768  df-mulr 12769  df-0g 12929  df-mgm 12999  df-sgrp 13045  df-mnd 13058  df-mhm 13091  df-grp 13135  df-ghm 13371  df-mgp 13477  df-ur 13516  df-ring 13554  df-rhm 13708

This theorem is referenced by:  rhmdvdsr  13731  rhmopp  13732  rhmunitinv  13734  znidom  14213  znidomb  14214  znunit  14215  znrrg  14216  lgseisenlem3  15313  lgseisenlem4  15314