Theorem rhmmul 13926

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 2205	. . . . 5 |- (mulGrp ` R ) = (mulGrp ` R )
2		eqid 2205	. . . . 5 |- (mulGrp ` S ) = (mulGrp ` S )
3	1, 2	rhmmhm 13921	. . . 4 |- ( F e. ( R RingHom S ) -> F e. ( (mulGrp ` R ) MndHom (mulGrp ` S ) ) )
4	3	3ad2ant1 1021	. . 3 |- ( ( F e. ( R RingHom S ) /\ A e. X /\ B e. X ) -> F e. ( (mulGrp ` R ) MndHom (mulGrp ` S ) ) )
5		simp2 1001	. . . 4 |- ( ( F e. ( R RingHom S ) /\ A e. X /\ B e. X ) -> A e. X )
6		rhmrcl1 13917	. . . . . . 7 |- ( F e. ( R RingHom S ) -> R e. Ring )
7		rhmmul.x	. . . . . . . 8 |- X = ( Base ` R )
8	1, 7	mgpbasg 13688	. . . . . . 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 2275	. . . . 5 |- ( F e. ( R RingHom S ) -> ( A e. X <-> A e. ( Base ` (mulGrp ` R ) ) ) )
11	10	3ad2ant1 1021	. . . 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 1002	. . . 4 |- ( ( F e. ( R RingHom S ) /\ A e. X /\ B e. X ) -> B e. X )
14	9	eleq2d 2275	. . . . 5 |- ( F e. ( R RingHom S ) -> ( B e. X <-> B e. ( Base ` (mulGrp ` R ) ) ) )
15	14	3ad2ant1 1021	. . . 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 2205	. . . 4 |- ( Base ` (mulGrp ` R ) ) = ( Base ` (mulGrp ` R ) )
18		eqid 2205	. . . 4 |- ( +g ` (mulGrp ` R ) ) = ( +g ` (mulGrp ` R ) )
19		eqid 2205	. . . 4 |- ( +g ` (mulGrp ` S ) ) = ( +g ` (mulGrp ` S ) )
20	17, 18, 19	mhmlin 13299	. . 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 1250	. 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 13686	. . . . . . 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 5961	. . . . 5 |- ( F e. ( R RingHom S ) -> ( A .x. B ) = ( A ( +g ` (mulGrp ` R ) ) B ) )
26	25	fveq2d 5580	. . . 4 |- ( F e. ( R RingHom S ) -> ( F ` ( A .x. B ) ) = ( F ` ( A ( +g ` (mulGrp ` R ) ) B ) ) )
27		rhmrcl2 13918	. . . . . 6 |- ( F e. ( R RingHom S ) -> S e. Ring )
28		rhmmul.n	. . . . . . 7 |- .X. = ( .r ` S )
29	2, 28	mgpplusgg 13686	. . . . . 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 5961	. . . 4 |- ( F e. ( R RingHom S ) -> ( ( F ` A ) .X. ( F ` B ) ) = ( ( F ` A ) ( +g ` (mulGrp ` S ) ) ( F ` B ) ) )
32	26, 31	eqeq12d 2220	. . 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 1021	. 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 981   = wceq 1373   e. wcel 2176   ` cfv 5271  (class class class)co 5944   Basecbs 12832   +g cplusg 12909   .rcmulr 12910   MndHom cmhm 13289  mulGrpcmgp 13682   Ringcrg 13758   RingHom crh 13912

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-addcom 8025  ax-addass 8027  ax-i2m1 8030  ax-0lt1 8031  ax-0id 8033  ax-rnegex 8034  ax-pre-ltirr 8037  ax-pre-ltadd 8041

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-map 6737  df-pnf 8109  df-mnf 8110  df-ltxr 8112  df-inn 9037  df-2 9095  df-3 9096  df-ndx 12835  df-slot 12836  df-base 12838  df-sets 12839  df-plusg 12922  df-mulr 12923  df-0g 13090  df-mgm 13188  df-sgrp 13234  df-mnd 13249  df-mhm 13291  df-grp 13335  df-ghm 13577  df-mgp 13683  df-ur 13722  df-ring 13760  df-rhm 13914

This theorem is referenced by:  rhmdvdsr  13937  rhmopp  13938  rhmunitinv  13940  znidom  14419  znidomb  14420  znunit  14421  znrrg  14422  lgseisenlem3  15549  lgseisenlem4  15550