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Theorem resghm 17499
Description: Restriction of a homomorphism to a subgroup. (Contributed by Stefan O'Rear, 31-Dec-2014.)
Hypothesis
Ref Expression
resghm.u 𝑈 = (𝑆s 𝑋)
Assertion
Ref Expression
resghm ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → (𝐹𝑋) ∈ (𝑈 GrpHom 𝑇))

Proof of Theorem resghm
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2610 . 2 (Base‘𝑈) = (Base‘𝑈)
2 eqid 2610 . 2 (Base‘𝑇) = (Base‘𝑇)
3 eqid 2610 . 2 (+g𝑈) = (+g𝑈)
4 eqid 2610 . 2 (+g𝑇) = (+g𝑇)
5 resghm.u . . . 4 𝑈 = (𝑆s 𝑋)
65subggrp 17420 . . 3 (𝑋 ∈ (SubGrp‘𝑆) → 𝑈 ∈ Grp)
76adantl 481 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → 𝑈 ∈ Grp)
8 ghmgrp2 17486 . . 3 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp)
98adantr 480 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → 𝑇 ∈ Grp)
10 eqid 2610 . . . . 5 (Base‘𝑆) = (Base‘𝑆)
1110, 2ghmf 17487 . . . 4 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
1210subgss 17418 . . . 4 (𝑋 ∈ (SubGrp‘𝑆) → 𝑋 ⊆ (Base‘𝑆))
13 fssres 5983 . . . 4 ((𝐹:(Base‘𝑆)⟶(Base‘𝑇) ∧ 𝑋 ⊆ (Base‘𝑆)) → (𝐹𝑋):𝑋⟶(Base‘𝑇))
1411, 12, 13syl2an 493 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → (𝐹𝑋):𝑋⟶(Base‘𝑇))
1512adantl 481 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → 𝑋 ⊆ (Base‘𝑆))
165, 10ressbas2 15758 . . . . 5 (𝑋 ⊆ (Base‘𝑆) → 𝑋 = (Base‘𝑈))
1715, 16syl 17 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → 𝑋 = (Base‘𝑈))
1817feq2d 5944 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → ((𝐹𝑋):𝑋⟶(Base‘𝑇) ↔ (𝐹𝑋):(Base‘𝑈)⟶(Base‘𝑇)))
1914, 18mpbid 221 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → (𝐹𝑋):(Base‘𝑈)⟶(Base‘𝑇))
20 eleq2 2677 . . . . . 6 (𝑋 = (Base‘𝑈) → (𝑎𝑋𝑎 ∈ (Base‘𝑈)))
21 eleq2 2677 . . . . . 6 (𝑋 = (Base‘𝑈) → (𝑏𝑋𝑏 ∈ (Base‘𝑈)))
2220, 21anbi12d 743 . . . . 5 (𝑋 = (Base‘𝑈) → ((𝑎𝑋𝑏𝑋) ↔ (𝑎 ∈ (Base‘𝑈) ∧ 𝑏 ∈ (Base‘𝑈))))
2317, 22syl 17 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → ((𝑎𝑋𝑏𝑋) ↔ (𝑎 ∈ (Base‘𝑈) ∧ 𝑏 ∈ (Base‘𝑈))))
2423biimpar 501 . . 3 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎 ∈ (Base‘𝑈) ∧ 𝑏 ∈ (Base‘𝑈))) → (𝑎𝑋𝑏𝑋))
25 simpll 786 . . . . 5 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎𝑋𝑏𝑋)) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
2615sselda 3568 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ 𝑎𝑋) → 𝑎 ∈ (Base‘𝑆))
2726adantrr 749 . . . . 5 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎𝑋𝑏𝑋)) → 𝑎 ∈ (Base‘𝑆))
2815sselda 3568 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ 𝑏𝑋) → 𝑏 ∈ (Base‘𝑆))
2928adantrl 748 . . . . 5 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎𝑋𝑏𝑋)) → 𝑏 ∈ (Base‘𝑆))
30 eqid 2610 . . . . . 6 (+g𝑆) = (+g𝑆)
3110, 30, 4ghmlin 17488 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆)) → (𝐹‘(𝑎(+g𝑆)𝑏)) = ((𝐹𝑎)(+g𝑇)(𝐹𝑏)))
3225, 27, 29, 31syl3anc 1318 . . . 4 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎𝑋𝑏𝑋)) → (𝐹‘(𝑎(+g𝑆)𝑏)) = ((𝐹𝑎)(+g𝑇)(𝐹𝑏)))
335, 30ressplusg 15818 . . . . . . . 8 (𝑋 ∈ (SubGrp‘𝑆) → (+g𝑆) = (+g𝑈))
3433ad2antlr 759 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎𝑋𝑏𝑋)) → (+g𝑆) = (+g𝑈))
3534oveqd 6566 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎𝑋𝑏𝑋)) → (𝑎(+g𝑆)𝑏) = (𝑎(+g𝑈)𝑏))
3635fveq2d 6107 . . . . 5 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎𝑋𝑏𝑋)) → ((𝐹𝑋)‘(𝑎(+g𝑆)𝑏)) = ((𝐹𝑋)‘(𝑎(+g𝑈)𝑏)))
3730subgcl 17427 . . . . . . . 8 ((𝑋 ∈ (SubGrp‘𝑆) ∧ 𝑎𝑋𝑏𝑋) → (𝑎(+g𝑆)𝑏) ∈ 𝑋)
38373expb 1258 . . . . . . 7 ((𝑋 ∈ (SubGrp‘𝑆) ∧ (𝑎𝑋𝑏𝑋)) → (𝑎(+g𝑆)𝑏) ∈ 𝑋)
3938adantll 746 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎𝑋𝑏𝑋)) → (𝑎(+g𝑆)𝑏) ∈ 𝑋)
40 fvres 6117 . . . . . 6 ((𝑎(+g𝑆)𝑏) ∈ 𝑋 → ((𝐹𝑋)‘(𝑎(+g𝑆)𝑏)) = (𝐹‘(𝑎(+g𝑆)𝑏)))
4139, 40syl 17 . . . . 5 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎𝑋𝑏𝑋)) → ((𝐹𝑋)‘(𝑎(+g𝑆)𝑏)) = (𝐹‘(𝑎(+g𝑆)𝑏)))
4236, 41eqtr3d 2646 . . . 4 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎𝑋𝑏𝑋)) → ((𝐹𝑋)‘(𝑎(+g𝑈)𝑏)) = (𝐹‘(𝑎(+g𝑆)𝑏)))
43 fvres 6117 . . . . . 6 (𝑎𝑋 → ((𝐹𝑋)‘𝑎) = (𝐹𝑎))
44 fvres 6117 . . . . . 6 (𝑏𝑋 → ((𝐹𝑋)‘𝑏) = (𝐹𝑏))
4543, 44oveqan12d 6568 . . . . 5 ((𝑎𝑋𝑏𝑋) → (((𝐹𝑋)‘𝑎)(+g𝑇)((𝐹𝑋)‘𝑏)) = ((𝐹𝑎)(+g𝑇)(𝐹𝑏)))
4645adantl 481 . . . 4 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎𝑋𝑏𝑋)) → (((𝐹𝑋)‘𝑎)(+g𝑇)((𝐹𝑋)‘𝑏)) = ((𝐹𝑎)(+g𝑇)(𝐹𝑏)))
4732, 42, 463eqtr4d 2654 . . 3 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎𝑋𝑏𝑋)) → ((𝐹𝑋)‘(𝑎(+g𝑈)𝑏)) = (((𝐹𝑋)‘𝑎)(+g𝑇)((𝐹𝑋)‘𝑏)))
4824, 47syldan 486 . 2 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎 ∈ (Base‘𝑈) ∧ 𝑏 ∈ (Base‘𝑈))) → ((𝐹𝑋)‘(𝑎(+g𝑈)𝑏)) = (((𝐹𝑋)‘𝑎)(+g𝑇)((𝐹𝑋)‘𝑏)))
491, 2, 3, 4, 7, 9, 19, 48isghmd 17492 1 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → (𝐹𝑋) ∈ (𝑈 GrpHom 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  wss 3540  cres 5040  wf 5800  cfv 5804  (class class class)co 6549  Basecbs 15695  s cress 15696  +gcplusg 15768  Grpcgrp 17245  SubGrpcsubg 17411   GrpHom cghm 17480
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-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-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-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-subg 17414  df-ghm 17481
This theorem is referenced by:  ghmima  17504  resrhm  18632  reslmhm  18873
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