Theorem resrhm 13744

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 13652	. . 3 ⊢ (𝐹 ∈ (𝑆 RingHom 𝑇) → 𝑇 ∈ Ring)
2		resrhm.u	. . . 4 ⊢ 𝑈 = (𝑆 ↾s 𝑋)
3	2	subrgring 13720	. . 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 13658	. . . 4 ⊢ (𝐹 ∈ (𝑆 RingHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
6		subrgsubg 13723	. . . 4 ⊢ (𝑋 ∈ (SubRing‘𝑆) → 𝑋 ∈ (SubGrp‘𝑆))
7	2	resghm 13330	. . . 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 2193	. . . . . 6 ⊢ (mulGrp‘𝑆) = (mulGrp‘𝑆)
10		eqid 2193	. . . . . 6 ⊢ (mulGrp‘𝑇) = (mulGrp‘𝑇)
11	9, 10	rhmmhm 13655	. . . . 5 ⊢ (𝐹 ∈ (𝑆 RingHom 𝑇) → 𝐹 ∈ ((mulGrp‘𝑆) MndHom (mulGrp‘𝑇)))
12	9	subrgsubm 13730	. . . . 5 ⊢ (𝑋 ∈ (SubRing‘𝑆) → 𝑋 ∈ (SubMnd‘(mulGrp‘𝑆)))
13		eqid 2193	. . . . . 6 ⊢ ((mulGrp‘𝑆) ↾s 𝑋) = ((mulGrp‘𝑆) ↾s 𝑋)
14	13	resmhm 13059	. . . . 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 13651	. . . . . 6 ⊢ (𝐹 ∈ (𝑆 RingHom 𝑇) → 𝑆 ∈ Ring)
17	2, 9	mgpress 13427	. . . . . 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 5933	. . . 4 ⊢ ((𝐹 ∈ (𝑆 RingHom 𝑇) ∧ 𝑋 ∈ (SubRing‘𝑆)) → (((mulGrp‘𝑆) ↾s 𝑋) MndHom (mulGrp‘𝑇)) = ((mulGrp‘𝑈) MndHom (mulGrp‘𝑇)))
20	15, 19	eleqtrd 2272	. . 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 2193	. . 3 ⊢ (mulGrp‘𝑈) = (mulGrp‘𝑈)
23	22, 10	isrhm 13654	. 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 1364   ∈ wcel 2164   ↾ cres 4661  ‘cfv 5254  (class class class)co 5918   ↾_s cress 12619   MndHom cmhm 13029  SubMndcsubmnd 13030  SubGrpcsubg 13237   GrpHom cghm 13310  mulGrpcmgp 13416  Ringcrg 13492   RingHom crh 13646  SubRingcsubrg 13713

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-pre-ltirr 7984  ax-pre-lttrn 7986  ax-pre-ltadd 7988

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-map 6704  df-pnf 8056  df-mnf 8057  df-ltxr 8059  df-inn 8983  df-2 9041  df-3 9042  df-ndx 12621  df-slot 12622  df-base 12624  df-sets 12625  df-iress 12626  df-plusg 12708  df-mulr 12709  df-0g 12869  df-mgm 12939  df-sgrp 12985  df-mnd 12998  df-mhm 13031  df-submnd 13032  df-grp 13075  df-subg 13240  df-ghm 13311  df-mgp 13417  df-ur 13456  df-ring 13494  df-rhm 13648  df-subrg 13715

This theorem is referenced by: (None)