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Theorem resrhm 18632
 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 18543 . . 3 (𝐹 ∈ (𝑆 RingHom 𝑇) → 𝑇 ∈ Ring)
2 resrhm.u . . . 4 𝑈 = (𝑆s 𝑋)
32subrgring 18606 . . 3 (𝑋 ∈ (SubRing‘𝑆) → 𝑈 ∈ Ring)
41, 3anim12ci 589 . 2 ((𝐹 ∈ (𝑆 RingHom 𝑇) ∧ 𝑋 ∈ (SubRing‘𝑆)) → (𝑈 ∈ Ring ∧ 𝑇 ∈ Ring))
5 rhmghm 18548 . . . 4 (𝐹 ∈ (𝑆 RingHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
6 subrgsubg 18609 . . . 4 (𝑋 ∈ (SubRing‘𝑆) → 𝑋 ∈ (SubGrp‘𝑆))
72resghm 17499 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → (𝐹𝑋) ∈ (𝑈 GrpHom 𝑇))
85, 6, 7syl2an 493 . . 3 ((𝐹 ∈ (𝑆 RingHom 𝑇) ∧ 𝑋 ∈ (SubRing‘𝑆)) → (𝐹𝑋) ∈ (𝑈 GrpHom 𝑇))
9 eqid 2610 . . . . . 6 (mulGrp‘𝑆) = (mulGrp‘𝑆)
10 eqid 2610 . . . . . 6 (mulGrp‘𝑇) = (mulGrp‘𝑇)
119, 10rhmmhm 18545 . . . . 5 (𝐹 ∈ (𝑆 RingHom 𝑇) → 𝐹 ∈ ((mulGrp‘𝑆) MndHom (mulGrp‘𝑇)))
129subrgsubm 18616 . . . . 5 (𝑋 ∈ (SubRing‘𝑆) → 𝑋 ∈ (SubMnd‘(mulGrp‘𝑆)))
13 eqid 2610 . . . . . 6 ((mulGrp‘𝑆) ↾s 𝑋) = ((mulGrp‘𝑆) ↾s 𝑋)
1413resmhm 17182 . . . . 5 ((𝐹 ∈ ((mulGrp‘𝑆) MndHom (mulGrp‘𝑇)) ∧ 𝑋 ∈ (SubMnd‘(mulGrp‘𝑆))) → (𝐹𝑋) ∈ (((mulGrp‘𝑆) ↾s 𝑋) MndHom (mulGrp‘𝑇)))
1511, 12, 14syl2an 493 . . . 4 ((𝐹 ∈ (𝑆 RingHom 𝑇) ∧ 𝑋 ∈ (SubRing‘𝑆)) → (𝐹𝑋) ∈ (((mulGrp‘𝑆) ↾s 𝑋) MndHom (mulGrp‘𝑇)))
16 rhmrcl1 18542 . . . . . 6 (𝐹 ∈ (𝑆 RingHom 𝑇) → 𝑆 ∈ Ring)
172, 9mgpress 18323 . . . . . 6 ((𝑆 ∈ Ring ∧ 𝑋 ∈ (SubRing‘𝑆)) → ((mulGrp‘𝑆) ↾s 𝑋) = (mulGrp‘𝑈))
1816, 17sylan 487 . . . . 5 ((𝐹 ∈ (𝑆 RingHom 𝑇) ∧ 𝑋 ∈ (SubRing‘𝑆)) → ((mulGrp‘𝑆) ↾s 𝑋) = (mulGrp‘𝑈))
1918oveq1d 6564 . . . 4 ((𝐹 ∈ (𝑆 RingHom 𝑇) ∧ 𝑋 ∈ (SubRing‘𝑆)) → (((mulGrp‘𝑆) ↾s 𝑋) MndHom (mulGrp‘𝑇)) = ((mulGrp‘𝑈) MndHom (mulGrp‘𝑇)))
2015, 19eleqtrd 2690 . . 3 ((𝐹 ∈ (𝑆 RingHom 𝑇) ∧ 𝑋 ∈ (SubRing‘𝑆)) → (𝐹𝑋) ∈ ((mulGrp‘𝑈) MndHom (mulGrp‘𝑇)))
218, 20jca 553 . 2 ((𝐹 ∈ (𝑆 RingHom 𝑇) ∧ 𝑋 ∈ (SubRing‘𝑆)) → ((𝐹𝑋) ∈ (𝑈 GrpHom 𝑇) ∧ (𝐹𝑋) ∈ ((mulGrp‘𝑈) MndHom (mulGrp‘𝑇))))
22 eqid 2610 . . 3 (mulGrp‘𝑈) = (mulGrp‘𝑈)
2322, 10isrhm 18544 . 2 ((𝐹𝑋) ∈ (𝑈 RingHom 𝑇) ↔ ((𝑈 ∈ Ring ∧ 𝑇 ∈ Ring) ∧ ((𝐹𝑋) ∈ (𝑈 GrpHom 𝑇) ∧ (𝐹𝑋) ∈ ((mulGrp‘𝑈) MndHom (mulGrp‘𝑇)))))
244, 21, 23sylanbrc 695 1 ((𝐹 ∈ (𝑆 RingHom 𝑇) ∧ 𝑋 ∈ (SubRing‘𝑆)) → (𝐹𝑋) ∈ (𝑈 RingHom 𝑇))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ↾ cres 5040  ‘cfv 5804  (class class class)co 6549   ↾s cress 15696   MndHom cmhm 17156  SubMndcsubmnd 17157  SubGrpcsubg 17411   GrpHom cghm 17480  mulGrpcmgp 18312  Ringcrg 18370   RingHom crh 18535  SubRingcsubrg 18599 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-mhm 17158  df-submnd 17159  df-grp 17248  df-subg 17414  df-ghm 17481  df-mgp 18313  df-ur 18325  df-ring 18372  df-rnghom 18538  df-subrg 18601 This theorem is referenced by:  evlsval2  19341
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