Theorem resrhm 18809

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 18720	. . 3 |- ( F e. ( S RingHom T ) -> T e. Ring )
2		resrhm.u	. . . 4 |- U = ( S ↾s X )
3	2	subrgring 18783	. . 3 |- ( X e. (SubRing ` S ) -> U e. Ring )
4	1, 3	anim12ci 591	. 2 |- ( ( F e. ( S RingHom T ) /\ X e. (SubRing ` S ) ) -> ( U e. Ring /\ T e. Ring ) )
5		rhmghm 18725	. . . 4 |- ( F e. ( S RingHom T ) -> F e. ( S GrpHom T ) )
6		subrgsubg 18786	. . . 4 |- ( X e. (SubRing ` S ) -> X e. (SubGrp ` S ) )
7	2	resghm 17676	. . . 4 |- ( ( F e. ( S GrpHom T ) /\ X e. (SubGrp ` S ) ) -> ( F |` X ) e. ( U GrpHom T ) )
8	5, 6, 7	syl2an 494	. . 3 |- ( ( F e. ( S RingHom T ) /\ X e. (SubRing ` S ) ) -> ( F |` X ) e. ( U GrpHom T ) )
9		eqid 2622	. . . . . 6 |- (mulGrp ` S ) = (mulGrp ` S )
10		eqid 2622	. . . . . 6 |- (mulGrp ` T ) = (mulGrp ` T )
11	9, 10	rhmmhm 18722	. . . . 5 |- ( F e. ( S RingHom T ) -> F e. ( (mulGrp ` S ) MndHom (mulGrp ` T ) ) )
12	9	subrgsubm 18793	. . . . 5 |- ( X e. (SubRing ` S ) -> X e. (SubMnd ` (mulGrp ` S ) ) )
13		eqid 2622	. . . . . 6 |- ( (mulGrp ` S ) ↾s X ) = ( (mulGrp ` S ) ↾s X )
14	13	resmhm 17359	. . . . 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 494	. . . 4 |- ( ( F e. ( S RingHom T ) /\ X e. (SubRing ` S ) ) -> ( F |` X ) e. ( ( (mulGrp ` S ) ↾s X ) MndHom (mulGrp ` T ) ) )
16		rhmrcl1 18719	. . . . . 6 |- ( F e. ( S RingHom T ) -> S e. Ring )
17	2, 9	mgpress 18500	. . . . . 6 |- ( ( S e. Ring /\ X e. (SubRing ` S ) ) -> ( (mulGrp ` S ) ↾s X ) = (mulGrp ` U ) )
18	16, 17	sylan 488	. . . . 5 |- ( ( F e. ( S RingHom T ) /\ X e. (SubRing ` S ) ) -> ( (mulGrp ` S ) ↾s X ) = (mulGrp ` U ) )
19	18	oveq1d 6665	. . . 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 2703	. . 3 |- ( ( F e. ( S RingHom T ) /\ X e. (SubRing ` S ) ) -> ( F |` X ) e. ( (mulGrp ` U ) MndHom (mulGrp ` T ) ) )
21	8, 20	jca 554	. 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 2622	. . 3 |- (mulGrp ` U ) = (mulGrp ` U )
23	22, 10	isrhm 18721	. 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 698	1 |- ( ( F e. ( S RingHom T ) /\ X e. (SubRing ` S ) ) -> ( F |` X ) e. ( U RingHom T ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   |` cres 5116   ` cfv 5888  (class class class)co 6650   ↾_s cress 15858   MndHom cmhm 17333  SubMndcsubmnd 17334  SubGrpcsubg 17588   GrpHom cghm 17657  mulGrpcmgp 18489   Ringcrg 18547   RingHom crh 18712  SubRingcsubrg 18776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336  df-grp 17425  df-subg 17591  df-ghm 17658  df-mgp 18490  df-ur 18502  df-ring 18549  df-rnghom 18715  df-subrg 18778

This theorem is referenced by:  evlsval2  19520