Theorem resrhm 14197

Description: Restriction of a ring homomorphism to a subring is a homomorphism. (Contributed by Mario Carneiro, 12-Mar-2015.)

Hypothesis

Ref	Expression
resrhm.u	|- U = ( S ↾s X )

Assertion

Ref	Expression
resrhm	|- ( ( F e. ( S RingHom T ) /\ X e. (SubRing ` S ) ) -> ( F |` X ) e. ( U RingHom T ) )

Proof of Theorem resrhm

Step	Hyp	Ref	Expression
1		rhmrcl2 14105	. . 3 |- ( F e. ( S RingHom T ) -> T e. Ring )
2		resrhm.u	. . . 4 |- U = ( S ↾s X )
3	2	subrgring 14173	. . 3 |- ( X e. (SubRing ` S ) -> U e. Ring )
4	1, 3	anim12ci 339	. 2 |- ( ( F e. ( S RingHom T ) /\ X e. (SubRing ` S ) ) -> ( U e. Ring /\ T e. Ring ) )
5		rhmghm 14111	. . . 4 |- ( F e. ( S RingHom T ) -> F e. ( S GrpHom T ) )
6		subrgsubg 14176	. . . 4 |- ( X e. (SubRing ` S ) -> X e. (SubGrp ` S ) )
7	2	resghm 13783	. . . 4 |- ( ( F e. ( S GrpHom T ) /\ X e. (SubGrp ` S ) ) -> ( F |` X ) e. ( U GrpHom T ) )
8	5, 6, 7	syl2an 289	. . 3 |- ( ( F e. ( S RingHom T ) /\ X e. (SubRing ` S ) ) -> ( F |` X ) e. ( U GrpHom T ) )
9		eqid 2229	. . . . . 6 |- (mulGrp ` S ) = (mulGrp ` S )
10		eqid 2229	. . . . . 6 |- (mulGrp ` T ) = (mulGrp ` T )
11	9, 10	rhmmhm 14108	. . . . 5 |- ( F e. ( S RingHom T ) -> F e. ( (mulGrp ` S ) MndHom (mulGrp ` T ) ) )
12	9	subrgsubm 14183	. . . . 5 |- ( X e. (SubRing ` S ) -> X e. (SubMnd ` (mulGrp ` S ) ) )
13		eqid 2229	. . . . . 6 |- ( (mulGrp ` S ) ↾s X ) = ( (mulGrp ` S ) ↾s X )
14	13	resmhm 13506	. . . . 5 |- ( ( F e. ( (mulGrp ` S ) MndHom (mulGrp ` T ) ) /\ X e. (SubMnd ` (mulGrp ` S ) ) ) -> ( F |` X ) e. ( ( (mulGrp ` S ) ↾s X ) MndHom (mulGrp ` T ) ) )
15	11, 12, 14	syl2an 289	. . . 4 |- ( ( F e. ( S RingHom T ) /\ X e. (SubRing ` S ) ) -> ( F |` X ) e. ( ( (mulGrp ` S ) ↾s X ) MndHom (mulGrp ` T ) ) )
16		rhmrcl1 14104	. . . . . 6 |- ( F e. ( S RingHom T ) -> S e. Ring )
17	2, 9	mgpress 13880	. . . . . 6 |- ( ( S e. Ring /\ X e. (SubRing ` S ) ) -> ( (mulGrp ` S ) ↾s X ) = (mulGrp ` U ) )
18	16, 17	sylan 283	. . . . 5 |- ( ( F e. ( S RingHom T ) /\ X e. (SubRing ` S ) ) -> ( (mulGrp ` S ) ↾s X ) = (mulGrp ` U ) )
19	18	oveq1d 6009	. . . 4 |- ( ( F e. ( S RingHom T ) /\ X e. (SubRing ` S ) ) -> ( ( (mulGrp ` S ) ↾s X ) MndHom (mulGrp ` T ) ) = ( (mulGrp ` U ) MndHom (mulGrp ` T ) ) )
20	15, 19	eleqtrd 2308	. . 3 |- ( ( F e. ( S RingHom T ) /\ X e. (SubRing ` S ) ) -> ( F |` X ) e. ( (mulGrp ` U ) MndHom (mulGrp ` T ) ) )
21	8, 20	jca 306	. 2 |- ( ( F e. ( S RingHom T ) /\ X e. (SubRing ` S ) ) -> ( ( F |` X ) e. ( U GrpHom T ) /\ ( F |` X ) e. ( (mulGrp ` U ) MndHom (mulGrp ` T ) ) ) )
22		eqid 2229	. . 3 |- (mulGrp ` U ) = (mulGrp ` U )
23	22, 10	isrhm 14107	. 2 |- ( ( F |` X ) e. ( U RingHom T ) <-> ( ( U e. Ring /\ T e. Ring ) /\ ( ( F |` X ) e. ( U GrpHom T ) /\ ( F |` X ) e. ( (mulGrp ` U ) MndHom (mulGrp ` T ) ) ) ) )
24	4, 21, 23	sylanbrc 417	1 |- ( ( F e. ( S RingHom T ) /\ X e. (SubRing ` S ) ) -> ( F |` X ) e. ( U RingHom T ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   |` cres 4718   ` cfv 5314  (class class class)co 5994   ↾_s cress 13019   MndHom cmhm 13476  SubMndcsubmnd 13477  SubGrpcsubg 13690   GrpHom cghm 13763  mulGrpcmgp 13869   Ringcrg 13945   RingHom crh 14099  SubRingcsubrg 14166

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 4198  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-setind 4626  ax-cnex 8078  ax-resscn 8079  ax-1cn 8080  ax-1re 8081  ax-icn 8082  ax-addcl 8083  ax-addrcl 8084  ax-mulcl 8085  ax-addcom 8087  ax-addass 8089  ax-i2m1 8092  ax-0lt1 8093  ax-0id 8095  ax-rnegex 8096  ax-pre-ltirr 8099  ax-pre-lttrn 8101  ax-pre-ltadd 8103

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 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4381  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-f1 5319  df-fo 5320  df-f1o 5321  df-fv 5322  df-riota 5947  df-ov 5997  df-oprab 5998  df-mpo 5999  df-1st 6276  df-2nd 6277  df-map 6787  df-pnf 8171  df-mnf 8172  df-ltxr 8174  df-inn 9099  df-2 9157  df-3 9158  df-ndx 13021  df-slot 13022  df-base 13024  df-sets 13025  df-iress 13026  df-plusg 13109  df-mulr 13110  df-0g 13277  df-mgm 13375  df-sgrp 13421  df-mnd 13436  df-mhm 13478  df-submnd 13479  df-grp 13522  df-subg 13693  df-ghm 13764  df-mgp 13870  df-ur 13909  df-ring 13947  df-rhm 14101  df-subrg 14168

This theorem is referenced by: (None)