Theorem resrhm 14640

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 14547	. . 3 |- ( F e. ( S RingHom T ) -> T e. Ring )
2		resrhm.u	. . . 4 |- U = ( S ↾s X )
3	2	subrgring 14616	. . 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 14553	. . . 4 |- ( F e. ( S RingHom T ) -> F e. ( S GrpHom T ) )
6		subrgsubg 14619	. . . 4 |- ( X e. (SubRing ` S ) -> X e. (SubGrp ` S ) )
7	2	resghm 14116	. . . 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 2238	. . . . . 6 |- (mulGrp ` S ) = (mulGrp ` S )
10		eqid 2238	. . . . . 6 |- (mulGrp ` T ) = (mulGrp ` T )
11	9, 10	rhmmhm 14550	. . . . 5 |- ( F e. ( S RingHom T ) -> F e. ( (mulGrp ` S ) MndHom (mulGrp ` T ) ) )
12	9	subrgsubm 14626	. . . . 5 |- ( X e. (SubRing ` S ) -> X e. (SubMnd ` (mulGrp ` S ) ) )
13		eqid 2238	. . . . . 6 |- ( (mulGrp ` S ) ↾s X ) = ( (mulGrp ` S ) ↾s X )
14	13	resmhm 13847	. . . . 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 14546	. . . . . 6 |- ( F e. ( S RingHom T ) -> S e. Ring )
17	2, 9	mgpress 14314	. . . . . 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 6100	. . . 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 2317	. . 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 2238	. . 3 |- (mulGrp ` U ) = (mulGrp ` U )
23	22, 10	isrhm 14549	. 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 421	1 |- ( ( F e. ( S RingHom T ) /\ X e. (SubRing ` S ) ) -> ( F |` X ) e. ( U RingHom T ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   |` cres 4776   ` cfv 5377  (class class class)co 6085   ↾_s cress 13405   MndHom cmhm 13817  SubMndcsubmnd 13818  SubGrpcsubg 14023   GrpHom cghm 14096  mulGrpcmgp 14301   Ringcrg 14384   RingHom crh 14541  SubRingcsubrg 14609

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8271  ax-resscn 8272  ax-1cn 8273  ax-1re 8274  ax-icn 8275  ax-addcl 8276  ax-addrcl 8277  ax-mulcl 8278  ax-addcom 8280  ax-addass 8282  ax-i2m1 8285  ax-0lt1 8286  ax-0id 8288  ax-rnegex 8289  ax-pre-ltirr 8292  ax-pre-lttrn 8294  ax-pre-ltadd 8296

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-map 6924  df-pnf 8363  df-mnf 8364  df-ltxr 8366  df-inn 9308  df-2 9366  df-3 9367  df-ndx 13407  df-slot 13408  df-base 13410  df-sets 13411  df-iress 13412  df-plusg 13497  df-mulr 13498  df-0g 13665  df-mgm 13729  df-sgrp 13770  df-mnd 13783  df-mhm 13819  df-submnd 13820  df-grp 13861  df-subg 14026  df-ghm 14097  df-mgp 14302  df-ur 14347  df-ring 14386  df-rhm 14543  df-subrg 14611

This theorem is used by: (None)