Theorem rhmmul 14409

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 2234	. . . . 5 |- (mulGrp ` R ) = (mulGrp ` R )
2		eqid 2234	. . . . 5 |- (mulGrp ` S ) = (mulGrp ` S )
3	1, 2	rhmmhm 14404	. . . 4 |- ( F e. ( R RingHom S ) -> F e. ( (mulGrp ` R ) MndHom (mulGrp ` S ) ) )
4	3	3ad2ant1 1045	. . 3 |- ( ( F e. ( R RingHom S ) /\ A e. X /\ B e. X ) -> F e. ( (mulGrp ` R ) MndHom (mulGrp ` S ) ) )
5		simp2 1025	. . . 4 |- ( ( F e. ( R RingHom S ) /\ A e. X /\ B e. X ) -> A e. X )
6		rhmrcl1 14400	. . . . . . 7 |- ( F e. ( R RingHom S ) -> R e. Ring )
7		rhmmul.x	. . . . . . . 8 |- X = ( Base ` R )
8	1, 7	mgpbasg 14165	. . . . . . 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 2304	. . . . 5 |- ( F e. ( R RingHom S ) -> ( A e. X <-> A e. ( Base ` (mulGrp ` R ) ) ) )
11	10	3ad2ant1 1045	. . . 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 1026	. . . 4 |- ( ( F e. ( R RingHom S ) /\ A e. X /\ B e. X ) -> B e. X )
14	9	eleq2d 2304	. . . . 5 |- ( F e. ( R RingHom S ) -> ( B e. X <-> B e. ( Base ` (mulGrp ` R ) ) ) )
15	14	3ad2ant1 1045	. . . 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 2234	. . . 4 |- ( Base ` (mulGrp ` R ) ) = ( Base ` (mulGrp ` R ) )
18		eqid 2234	. . . 4 |- ( +g ` (mulGrp ` R ) ) = ( +g ` (mulGrp ` R ) )
19		eqid 2234	. . . 4 |- ( +g ` (mulGrp ` S ) ) = ( +g ` (mulGrp ` S ) )
20	17, 18, 19	mhmlin 13722	. . 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 1274	. 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 14163	. . . . . . 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 6075	. . . . 5 |- ( F e. ( R RingHom S ) -> ( A .x. B ) = ( A ( +g ` (mulGrp ` R ) ) B ) )
26	25	fveq2d 5679	. . . 4 |- ( F e. ( R RingHom S ) -> ( F ` ( A .x. B ) ) = ( F ` ( A ( +g ` (mulGrp ` R ) ) B ) ) )
27		rhmrcl2 14401	. . . . . 6 |- ( F e. ( R RingHom S ) -> S e. Ring )
28		rhmmul.n	. . . . . . 7 |- .X. = ( .r ` S )
29	2, 28	mgpplusgg 14163	. . . . . 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 6075	. . . 4 |- ( F e. ( R RingHom S ) -> ( ( F ` A ) .X. ( F ` B ) ) = ( ( F ` A ) ( +g ` (mulGrp ` S ) ) ( F ` B ) ) )
32	26, 31	eqeq12d 2249	. . 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 1045	. 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 1005   = wceq 1398   e. wcel 2205   ` cfv 5357  (class class class)co 6058   Basecbs 13296   +g cplusg 13374   .rcmulr 13375   MndHom cmhm 13712  mulGrpcmgp 14159   Ringcrg 14239   RingHom crh 14395

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-pre-ltirr 8255  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-map 6897  df-pnf 8326  df-mnf 8327  df-ltxr 8329  df-inn 9255  df-2 9313  df-3 9314  df-ndx 13299  df-slot 13300  df-base 13302  df-sets 13303  df-plusg 13387  df-mulr 13388  df-0g 13555  df-mgm 13619  df-sgrp 13665  df-mnd 13678  df-mhm 13714  df-grp 13758  df-ghm 13994  df-mgp 14160  df-ur 14203  df-ring 14241  df-rhm 14397

This theorem is referenced by:  rhmdvdsr  14420  rhmopp  14421  rhmunitinv  14423  znidom  14931  znidomb  14932  znunit  14933  znrrg  14934  lgseisenlem3  16071  lgseisenlem4  16072