Theorem rhmmul 14144

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 2229	. . . . 5 |- (mulGrp ` R ) = (mulGrp ` R )
2		eqid 2229	. . . . 5 |- (mulGrp ` S ) = (mulGrp ` S )
3	1, 2	rhmmhm 14139	. . . 4 |- ( F e. ( R RingHom S ) -> F e. ( (mulGrp ` R ) MndHom (mulGrp ` S ) ) )
4	3	3ad2ant1 1042	. . 3 |- ( ( F e. ( R RingHom S ) /\ A e. X /\ B e. X ) -> F e. ( (mulGrp ` R ) MndHom (mulGrp ` S ) ) )
5		simp2 1022	. . . 4 |- ( ( F e. ( R RingHom S ) /\ A e. X /\ B e. X ) -> A e. X )
6		rhmrcl1 14135	. . . . . . 7 |- ( F e. ( R RingHom S ) -> R e. Ring )
7		rhmmul.x	. . . . . . . 8 |- X = ( Base ` R )
8	1, 7	mgpbasg 13905	. . . . . . 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 2299	. . . . 5 |- ( F e. ( R RingHom S ) -> ( A e. X <-> A e. ( Base ` (mulGrp ` R ) ) ) )
11	10	3ad2ant1 1042	. . . 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 1023	. . . 4 |- ( ( F e. ( R RingHom S ) /\ A e. X /\ B e. X ) -> B e. X )
14	9	eleq2d 2299	. . . . 5 |- ( F e. ( R RingHom S ) -> ( B e. X <-> B e. ( Base ` (mulGrp ` R ) ) ) )
15	14	3ad2ant1 1042	. . . 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 2229	. . . 4 |- ( Base ` (mulGrp ` R ) ) = ( Base ` (mulGrp ` R ) )
18		eqid 2229	. . . 4 |- ( +g ` (mulGrp ` R ) ) = ( +g ` (mulGrp ` R ) )
19		eqid 2229	. . . 4 |- ( +g ` (mulGrp ` S ) ) = ( +g ` (mulGrp ` S ) )
20	17, 18, 19	mhmlin 13516	. . 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 1271	. 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 13903	. . . . . . 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 6024	. . . . 5 |- ( F e. ( R RingHom S ) -> ( A .x. B ) = ( A ( +g ` (mulGrp ` R ) ) B ) )
26	25	fveq2d 5633	. . . 4 |- ( F e. ( R RingHom S ) -> ( F ` ( A .x. B ) ) = ( F ` ( A ( +g ` (mulGrp ` R ) ) B ) ) )
27		rhmrcl2 14136	. . . . . 6 |- ( F e. ( R RingHom S ) -> S e. Ring )
28		rhmmul.n	. . . . . . 7 |- .X. = ( .r ` S )
29	2, 28	mgpplusgg 13903	. . . . . 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 6024	. . . 4 |- ( F e. ( R RingHom S ) -> ( ( F ` A ) .X. ( F ` B ) ) = ( ( F ` A ) ( +g ` (mulGrp ` S ) ) ( F ` B ) ) )
32	26, 31	eqeq12d 2244	. . 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 1042	. 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 1002   = wceq 1395   e. wcel 2200   ` cfv 5318  (class class class)co 6007   Basecbs 13048   +g cplusg 13126   .rcmulr 13127   MndHom cmhm 13506  mulGrpcmgp 13899   Ringcrg 13975   RingHom crh 14130

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-addcom 8110  ax-addass 8112  ax-i2m1 8115  ax-0lt1 8116  ax-0id 8118  ax-rnegex 8119  ax-pre-ltirr 8122  ax-pre-ltadd 8126

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-map 6805  df-pnf 8194  df-mnf 8195  df-ltxr 8197  df-inn 9122  df-2 9180  df-3 9181  df-ndx 13051  df-slot 13052  df-base 13054  df-sets 13055  df-plusg 13139  df-mulr 13140  df-0g 13307  df-mgm 13405  df-sgrp 13451  df-mnd 13466  df-mhm 13508  df-grp 13552  df-ghm 13794  df-mgp 13900  df-ur 13939  df-ring 13977  df-rhm 14132

This theorem is referenced by:  rhmdvdsr  14155  rhmopp  14156  rhmunitinv  14158  znidom  14637  znidomb  14638  znunit  14639  znrrg  14640  lgseisenlem3  15767  lgseisenlem4  15768