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 ∈ (𝑆 RingHom 𝑇) ∧ 𝑋 ∈ (SubRing‘𝑆)) → (𝐹 ↾ 𝑋) ∈ (𝑈 RingHom 𝑇))

Proof of Theorem resrhm

Step	Hyp	Ref	Expression
1		rhmrcl2 14128	. . 3 ⊢ (𝐹 ∈ (𝑆 RingHom 𝑇) → 𝑇 ∈ Ring)
2		resrhm.u	. . . 4 ⊢ 𝑈 = (𝑆 ↾s 𝑋)
3	2	subrgring 14196	. . 3 ⊢ (𝑋 ∈ (SubRing‘𝑆) → 𝑈 ∈ Ring)
4	1, 3	anim12ci 339	. 2 ⊢ ((𝐹 ∈ (𝑆 RingHom 𝑇) ∧ 𝑋 ∈ (SubRing‘𝑆)) → (𝑈 ∈ Ring ∧ 𝑇 ∈ Ring))
5		rhmghm 14134	. . . 4 ⊢ (𝐹 ∈ (𝑆 RingHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
6		subrgsubg 14199	. . . 4 ⊢ (𝑋 ∈ (SubRing‘𝑆) → 𝑋 ∈ (SubGrp‘𝑆))
7	2	resghm 13805	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → (𝐹 ↾ 𝑋) ∈ (𝑈 GrpHom 𝑇))
8	5, 6, 7	syl2an 289	. . 3 ⊢ ((𝐹 ∈ (𝑆 RingHom 𝑇) ∧ 𝑋 ∈ (SubRing‘𝑆)) → (𝐹 ↾ 𝑋) ∈ (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 ∈ (𝑆 RingHom 𝑇) → 𝐹 ∈ ((mulGrp‘𝑆) MndHom (mulGrp‘𝑇)))
12	9	subrgsubm 14206	. . . . 5 ⊢ (𝑋 ∈ (SubRing‘𝑆) → 𝑋 ∈ (SubMnd‘(mulGrp‘𝑆)))
13		eqid 2229	. . . . . 6 ⊢ ((mulGrp‘𝑆) ↾s 𝑋) = ((mulGrp‘𝑆) ↾s 𝑋)
14	13	resmhm 13528	. . . . 5 ⊢ ((𝐹 ∈ ((mulGrp‘𝑆) MndHom (mulGrp‘𝑇)) ∧ 𝑋 ∈ (SubMnd‘(mulGrp‘𝑆))) → (𝐹 ↾ 𝑋) ∈ (((mulGrp‘𝑆) ↾s 𝑋) MndHom (mulGrp‘𝑇)))
15	11, 12, 14	syl2an 289	. . . 4 ⊢ ((𝐹 ∈ (𝑆 RingHom 𝑇) ∧ 𝑋 ∈ (SubRing‘𝑆)) → (𝐹 ↾ 𝑋) ∈ (((mulGrp‘𝑆) ↾s 𝑋) MndHom (mulGrp‘𝑇)))
16		rhmrcl1 14127	. . . . . 6 ⊢ (𝐹 ∈ (𝑆 RingHom 𝑇) → 𝑆 ∈ Ring)
17	2, 9	mgpress 13902	. . . . . 6 ⊢ ((𝑆 ∈ Ring ∧ 𝑋 ∈ (SubRing‘𝑆)) → ((mulGrp‘𝑆) ↾s 𝑋) = (mulGrp‘𝑈))
18	16, 17	sylan 283	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 RingHom 𝑇) ∧ 𝑋 ∈ (SubRing‘𝑆)) → ((mulGrp‘𝑆) ↾s 𝑋) = (mulGrp‘𝑈))
19	18	oveq1d 6022	. . . 4 ⊢ ((𝐹 ∈ (𝑆 RingHom 𝑇) ∧ 𝑋 ∈ (SubRing‘𝑆)) → (((mulGrp‘𝑆) ↾s 𝑋) MndHom (mulGrp‘𝑇)) = ((mulGrp‘𝑈) MndHom (mulGrp‘𝑇)))
20	15, 19	eleqtrd 2308	. . 3 ⊢ ((𝐹 ∈ (𝑆 RingHom 𝑇) ∧ 𝑋 ∈ (SubRing‘𝑆)) → (𝐹 ↾ 𝑋) ∈ ((mulGrp‘𝑈) MndHom (mulGrp‘𝑇)))
21	8, 20	jca 306	. 2 ⊢ ((𝐹 ∈ (𝑆 RingHom 𝑇) ∧ 𝑋 ∈ (SubRing‘𝑆)) → ((𝐹 ↾ 𝑋) ∈ (𝑈 GrpHom 𝑇) ∧ (𝐹 ↾ 𝑋) ∈ ((mulGrp‘𝑈) MndHom (mulGrp‘𝑇))))
22		eqid 2229	. . 3 ⊢ (mulGrp‘𝑈) = (mulGrp‘𝑈)
23	22, 10	isrhm 14130	. 2 ⊢ ((𝐹 ↾ 𝑋) ∈ (𝑈 RingHom 𝑇) ↔ ((𝑈 ∈ Ring ∧ 𝑇 ∈ Ring) ∧ ((𝐹 ↾ 𝑋) ∈ (𝑈 GrpHom 𝑇) ∧ (𝐹 ↾ 𝑋) ∈ ((mulGrp‘𝑈) MndHom (mulGrp‘𝑇)))))
24	4, 21, 23	sylanbrc 417	1 ⊢ ((𝐹 ∈ (𝑆 RingHom 𝑇) ∧ 𝑋 ∈ (SubRing‘𝑆)) → (𝐹 ↾ 𝑋) ∈ (𝑈 RingHom 𝑇))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1395   ∈ 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)