Theorem resrhm 14497

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 14404	. . 3 ⊢ (𝐹 ∈ (𝑆 RingHom 𝑇) → 𝑇 ∈ Ring)
2		resrhm.u	. . . 4 ⊢ 𝑈 = (𝑆 ↾s 𝑋)
3	2	subrgring 14473	. . 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 14410	. . . 4 ⊢ (𝐹 ∈ (𝑆 RingHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
6		subrgsubg 14476	. . . 4 ⊢ (𝑋 ∈ (SubRing‘𝑆) → 𝑋 ∈ (SubGrp‘𝑆))
7	2	resghm 14016	. . . 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 2234	. . . . . 6 ⊢ (mulGrp‘𝑆) = (mulGrp‘𝑆)
10		eqid 2234	. . . . . 6 ⊢ (mulGrp‘𝑇) = (mulGrp‘𝑇)
11	9, 10	rhmmhm 14407	. . . . 5 ⊢ (𝐹 ∈ (𝑆 RingHom 𝑇) → 𝐹 ∈ ((mulGrp‘𝑆) MndHom (mulGrp‘𝑇)))
12	9	subrgsubm 14483	. . . . 5 ⊢ (𝑋 ∈ (SubRing‘𝑆) → 𝑋 ∈ (SubMnd‘(mulGrp‘𝑆)))
13		eqid 2234	. . . . . 6 ⊢ ((mulGrp‘𝑆) ↾s 𝑋) = ((mulGrp‘𝑆) ↾s 𝑋)
14	13	resmhm 13745	. . . . 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 14403	. . . . . 6 ⊢ (𝐹 ∈ (𝑆 RingHom 𝑇) → 𝑆 ∈ Ring)
17	2, 9	mgpress 14173	. . . . . 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 6073	. . . 4 ⊢ ((𝐹 ∈ (𝑆 RingHom 𝑇) ∧ 𝑋 ∈ (SubRing‘𝑆)) → (((mulGrp‘𝑆) ↾s 𝑋) MndHom (mulGrp‘𝑇)) = ((mulGrp‘𝑈) MndHom (mulGrp‘𝑇)))
20	15, 19	eleqtrd 2313	. . 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 2234	. . 3 ⊢ (mulGrp‘𝑈) = (mulGrp‘𝑈)
23	22, 10	isrhm 14406	. 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 1398   ∈ wcel 2205   ↾ cres 4756  ‘cfv 5357  (class class class)co 6058   ↾_s cress 13300   MndHom cmhm 13715  SubMndcsubmnd 13716  SubGrpcsubg 13923   GrpHom cghm 13996  mulGrpcmgp 14162  Ringcrg 14242   RingHom crh 14398  SubRingcsubrg 14466

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-pre-ltirr 8255  ax-pre-lttrn 8257  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-map 6897  df-pnf 8326  df-mnf 8327  df-ltxr 8329  df-inn 9258  df-2 9316  df-3 9317  df-ndx 13302  df-slot 13303  df-base 13305  df-sets 13306  df-iress 13307  df-plusg 13390  df-mulr 13391  df-0g 13558  df-mgm 13622  df-sgrp 13668  df-mnd 13681  df-mhm 13717  df-submnd 13718  df-grp 13761  df-subg 13926  df-ghm 13997  df-mgp 14163  df-ur 14206  df-ring 14244  df-rhm 14400  df-subrg 14468

This theorem is referenced by: (None)