Theorem resrhm 14220

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 14128	. . 3 |- ( F e. ( S RingHom T ) -> T e. Ring )
2		resrhm.u	. . . 4 |- U = ( S ↾s X )
3	2	subrgring 14196	. . 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 14134	. . . 4 |- ( F e. ( S RingHom T ) -> F e. ( S GrpHom T ) )
6		subrgsubg 14199	. . . 4 |- ( X e. (SubRing ` S ) -> X e. (SubGrp ` S ) )
7	2	resghm 13805	. . . 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 14131	. . . . 5 |- ( F e. ( S RingHom T ) -> F e. ( (mulGrp ` S ) MndHom (mulGrp ` T ) ) )
12	9	subrgsubm 14206	. . . . 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 13528	. . . . 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 14127	. . . . . 6 |- ( F e. ( S RingHom T ) -> S e. Ring )
17	2, 9	mgpress 13902	. . . . . 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 6022	. . . 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 14130	. 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 4721   ` cfv 5318  (class class class)co 6007   ↾_s cress 13041   MndHom cmhm 13498  SubMndcsubmnd 13499  SubGrpcsubg 13712   GrpHom cghm 13785  mulGrpcmgp 13891   Ringcrg 13967   RingHom crh 14122  SubRingcsubrg 14189

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 8098  ax-resscn 8099  ax-1cn 8100  ax-1re 8101  ax-icn 8102  ax-addcl 8103  ax-addrcl 8104  ax-mulcl 8105  ax-addcom 8107  ax-addass 8109  ax-i2m1 8112  ax-0lt1 8113  ax-0id 8115  ax-rnegex 8116  ax-pre-ltirr 8119  ax-pre-lttrn 8121  ax-pre-ltadd 8123

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 8191  df-mnf 8192  df-ltxr 8194  df-inn 9119  df-2 9177  df-3 9178  df-ndx 13043  df-slot 13044  df-base 13046  df-sets 13047  df-iress 13048  df-plusg 13131  df-mulr 13132  df-0g 13299  df-mgm 13397  df-sgrp 13443  df-mnd 13458  df-mhm 13500  df-submnd 13501  df-grp 13544  df-subg 13715  df-ghm 13786  df-mgp 13892  df-ur 13931  df-ring 13969  df-rhm 14124  df-subrg 14191

This theorem is referenced by: (None)