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Theorem resrhm 18730
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 = (𝑆s 𝑋)
Assertion
Ref Expression
resrhm ((𝐹 ∈ (𝑆 RingHom 𝑇) ∧ 𝑋 ∈ (SubRing‘𝑆)) → (𝐹𝑋) ∈ (𝑈 RingHom 𝑇))

Proof of Theorem resrhm
StepHypRef Expression
1 rhmrcl2 18641 . . 3 (𝐹 ∈ (𝑆 RingHom 𝑇) → 𝑇 ∈ Ring)
2 resrhm.u . . . 4 𝑈 = (𝑆s 𝑋)
32subrgring 18704 . . 3 (𝑋 ∈ (SubRing‘𝑆) → 𝑈 ∈ Ring)
41, 3anim12ci 590 . 2 ((𝐹 ∈ (𝑆 RingHom 𝑇) ∧ 𝑋 ∈ (SubRing‘𝑆)) → (𝑈 ∈ Ring ∧ 𝑇 ∈ Ring))
5 rhmghm 18646 . . . 4 (𝐹 ∈ (𝑆 RingHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
6 subrgsubg 18707 . . . 4 (𝑋 ∈ (SubRing‘𝑆) → 𝑋 ∈ (SubGrp‘𝑆))
72resghm 17597 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → (𝐹𝑋) ∈ (𝑈 GrpHom 𝑇))
85, 6, 7syl2an 494 . . 3 ((𝐹 ∈ (𝑆 RingHom 𝑇) ∧ 𝑋 ∈ (SubRing‘𝑆)) → (𝐹𝑋) ∈ (𝑈 GrpHom 𝑇))
9 eqid 2621 . . . . . 6 (mulGrp‘𝑆) = (mulGrp‘𝑆)
10 eqid 2621 . . . . . 6 (mulGrp‘𝑇) = (mulGrp‘𝑇)
119, 10rhmmhm 18643 . . . . 5 (𝐹 ∈ (𝑆 RingHom 𝑇) → 𝐹 ∈ ((mulGrp‘𝑆) MndHom (mulGrp‘𝑇)))
129subrgsubm 18714 . . . . 5 (𝑋 ∈ (SubRing‘𝑆) → 𝑋 ∈ (SubMnd‘(mulGrp‘𝑆)))
13 eqid 2621 . . . . . 6 ((mulGrp‘𝑆) ↾s 𝑋) = ((mulGrp‘𝑆) ↾s 𝑋)
1413resmhm 17280 . . . . 5 ((𝐹 ∈ ((mulGrp‘𝑆) MndHom (mulGrp‘𝑇)) ∧ 𝑋 ∈ (SubMnd‘(mulGrp‘𝑆))) → (𝐹𝑋) ∈ (((mulGrp‘𝑆) ↾s 𝑋) MndHom (mulGrp‘𝑇)))
1511, 12, 14syl2an 494 . . . 4 ((𝐹 ∈ (𝑆 RingHom 𝑇) ∧ 𝑋 ∈ (SubRing‘𝑆)) → (𝐹𝑋) ∈ (((mulGrp‘𝑆) ↾s 𝑋) MndHom (mulGrp‘𝑇)))
16 rhmrcl1 18640 . . . . . 6 (𝐹 ∈ (𝑆 RingHom 𝑇) → 𝑆 ∈ Ring)
172, 9mgpress 18421 . . . . . 6 ((𝑆 ∈ Ring ∧ 𝑋 ∈ (SubRing‘𝑆)) → ((mulGrp‘𝑆) ↾s 𝑋) = (mulGrp‘𝑈))
1816, 17sylan 488 . . . . 5 ((𝐹 ∈ (𝑆 RingHom 𝑇) ∧ 𝑋 ∈ (SubRing‘𝑆)) → ((mulGrp‘𝑆) ↾s 𝑋) = (mulGrp‘𝑈))
1918oveq1d 6619 . . . 4 ((𝐹 ∈ (𝑆 RingHom 𝑇) ∧ 𝑋 ∈ (SubRing‘𝑆)) → (((mulGrp‘𝑆) ↾s 𝑋) MndHom (mulGrp‘𝑇)) = ((mulGrp‘𝑈) MndHom (mulGrp‘𝑇)))
2015, 19eleqtrd 2700 . . 3 ((𝐹 ∈ (𝑆 RingHom 𝑇) ∧ 𝑋 ∈ (SubRing‘𝑆)) → (𝐹𝑋) ∈ ((mulGrp‘𝑈) MndHom (mulGrp‘𝑇)))
218, 20jca 554 . 2 ((𝐹 ∈ (𝑆 RingHom 𝑇) ∧ 𝑋 ∈ (SubRing‘𝑆)) → ((𝐹𝑋) ∈ (𝑈 GrpHom 𝑇) ∧ (𝐹𝑋) ∈ ((mulGrp‘𝑈) MndHom (mulGrp‘𝑇))))
22 eqid 2621 . . 3 (mulGrp‘𝑈) = (mulGrp‘𝑈)
2322, 10isrhm 18642 . 2 ((𝐹𝑋) ∈ (𝑈 RingHom 𝑇) ↔ ((𝑈 ∈ Ring ∧ 𝑇 ∈ Ring) ∧ ((𝐹𝑋) ∈ (𝑈 GrpHom 𝑇) ∧ (𝐹𝑋) ∈ ((mulGrp‘𝑈) MndHom (mulGrp‘𝑇)))))
244, 21, 23sylanbrc 697 1 ((𝐹 ∈ (𝑆 RingHom 𝑇) ∧ 𝑋 ∈ (SubRing‘𝑆)) → (𝐹𝑋) ∈ (𝑈 RingHom 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  cres 5076  cfv 5847  (class class class)co 6604  s cress 15782   MndHom cmhm 17254  SubMndcsubmnd 17255  SubGrpcsubg 17509   GrpHom cghm 17578  mulGrpcmgp 18410  Ringcrg 18468   RingHom crh 18633  SubRingcsubrg 18697
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-er 7687  df-map 7804  df-en 7900  df-dom 7901  df-sdom 7902  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-3 11024  df-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-ress 15788  df-plusg 15875  df-mulr 15876  df-0g 16023  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-mhm 17256  df-submnd 17257  df-grp 17346  df-subg 17512  df-ghm 17579  df-mgp 18411  df-ur 18423  df-ring 18470  df-rnghom 18636  df-subrg 18699
This theorem is referenced by:  evlsval2  19439
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