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Theorem resghm 17657
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 2620 . 2 (Base‘𝑈) = (Base‘𝑈)
2 eqid 2620 . 2 (Base‘𝑇) = (Base‘𝑇)
3 eqid 2620 . 2 (+g𝑈) = (+g𝑈)
4 eqid 2620 . 2 (+g𝑇) = (+g𝑇)
5 resghm.u . . . 4 𝑈 = (𝑆s 𝑋)
65subggrp 17578 . . 3 (𝑋 ∈ (SubGrp‘𝑆) → 𝑈 ∈ Grp)
76adantl 482 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → 𝑈 ∈ Grp)
8 ghmgrp2 17644 . . 3 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp)
98adantr 481 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → 𝑇 ∈ Grp)
10 eqid 2620 . . . . 5 (Base‘𝑆) = (Base‘𝑆)
1110, 2ghmf 17645 . . . 4 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
1210subgss 17576 . . . 4 (𝑋 ∈ (SubGrp‘𝑆) → 𝑋 ⊆ (Base‘𝑆))
13 fssres 6057 . . . 4 ((𝐹:(Base‘𝑆)⟶(Base‘𝑇) ∧ 𝑋 ⊆ (Base‘𝑆)) → (𝐹𝑋):𝑋⟶(Base‘𝑇))
1411, 12, 13syl2an 494 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → (𝐹𝑋):𝑋⟶(Base‘𝑇))
1512adantl 482 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → 𝑋 ⊆ (Base‘𝑆))
165, 10ressbas2 15912 . . . . 5 (𝑋 ⊆ (Base‘𝑆) → 𝑋 = (Base‘𝑈))
1715, 16syl 17 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → 𝑋 = (Base‘𝑈))
1817feq2d 6018 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → ((𝐹𝑋):𝑋⟶(Base‘𝑇) ↔ (𝐹𝑋):(Base‘𝑈)⟶(Base‘𝑇)))
1914, 18mpbid 222 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → (𝐹𝑋):(Base‘𝑈)⟶(Base‘𝑇))
20 eleq2 2688 . . . . . 6 (𝑋 = (Base‘𝑈) → (𝑎𝑋𝑎 ∈ (Base‘𝑈)))
21 eleq2 2688 . . . . . 6 (𝑋 = (Base‘𝑈) → (𝑏𝑋𝑏 ∈ (Base‘𝑈)))
2220, 21anbi12d 746 . . . . 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 502 . . 3 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎 ∈ (Base‘𝑈) ∧ 𝑏 ∈ (Base‘𝑈))) → (𝑎𝑋𝑏𝑋))
25 simpll 789 . . . . 5 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎𝑋𝑏𝑋)) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
2615sselda 3595 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ 𝑎𝑋) → 𝑎 ∈ (Base‘𝑆))
2726adantrr 752 . . . . 5 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎𝑋𝑏𝑋)) → 𝑎 ∈ (Base‘𝑆))
2815sselda 3595 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ 𝑏𝑋) → 𝑏 ∈ (Base‘𝑆))
2928adantrl 751 . . . . 5 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎𝑋𝑏𝑋)) → 𝑏 ∈ (Base‘𝑆))
30 eqid 2620 . . . . . 6 (+g𝑆) = (+g𝑆)
3110, 30, 4ghmlin 17646 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆)) → (𝐹‘(𝑎(+g𝑆)𝑏)) = ((𝐹𝑎)(+g𝑇)(𝐹𝑏)))
3225, 27, 29, 31syl3anc 1324 . . . 4 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎𝑋𝑏𝑋)) → (𝐹‘(𝑎(+g𝑆)𝑏)) = ((𝐹𝑎)(+g𝑇)(𝐹𝑏)))
335, 30ressplusg 15974 . . . . . . . 8 (𝑋 ∈ (SubGrp‘𝑆) → (+g𝑆) = (+g𝑈))
3433ad2antlr 762 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎𝑋𝑏𝑋)) → (+g𝑆) = (+g𝑈))
3534oveqd 6652 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎𝑋𝑏𝑋)) → (𝑎(+g𝑆)𝑏) = (𝑎(+g𝑈)𝑏))
3635fveq2d 6182 . . . . 5 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎𝑋𝑏𝑋)) → ((𝐹𝑋)‘(𝑎(+g𝑆)𝑏)) = ((𝐹𝑋)‘(𝑎(+g𝑈)𝑏)))
3730subgcl 17585 . . . . . . . 8 ((𝑋 ∈ (SubGrp‘𝑆) ∧ 𝑎𝑋𝑏𝑋) → (𝑎(+g𝑆)𝑏) ∈ 𝑋)
38373expb 1264 . . . . . . 7 ((𝑋 ∈ (SubGrp‘𝑆) ∧ (𝑎𝑋𝑏𝑋)) → (𝑎(+g𝑆)𝑏) ∈ 𝑋)
3938adantll 749 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎𝑋𝑏𝑋)) → (𝑎(+g𝑆)𝑏) ∈ 𝑋)
40 fvres 6194 . . . . . 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 2656 . . . 4 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎𝑋𝑏𝑋)) → ((𝐹𝑋)‘(𝑎(+g𝑈)𝑏)) = (𝐹‘(𝑎(+g𝑆)𝑏)))
43 fvres 6194 . . . . . 6 (𝑎𝑋 → ((𝐹𝑋)‘𝑎) = (𝐹𝑎))
44 fvres 6194 . . . . . 6 (𝑏𝑋 → ((𝐹𝑋)‘𝑏) = (𝐹𝑏))
4543, 44oveqan12d 6654 . . . . 5 ((𝑎𝑋𝑏𝑋) → (((𝐹𝑋)‘𝑎)(+g𝑇)((𝐹𝑋)‘𝑏)) = ((𝐹𝑎)(+g𝑇)(𝐹𝑏)))
4645adantl 482 . . . 4 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎𝑋𝑏𝑋)) → (((𝐹𝑋)‘𝑎)(+g𝑇)((𝐹𝑋)‘𝑏)) = ((𝐹𝑎)(+g𝑇)(𝐹𝑏)))
4732, 42, 463eqtr4d 2664 . . 3 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎𝑋𝑏𝑋)) → ((𝐹𝑋)‘(𝑎(+g𝑈)𝑏)) = (((𝐹𝑋)‘𝑎)(+g𝑇)((𝐹𝑋)‘𝑏)))
4824, 47syldan 487 . 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 17650 1 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → (𝐹𝑋) ∈ (𝑈 GrpHom 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1481  wcel 1988  wss 3567  cres 5106  wf 5872  cfv 5876  (class class class)co 6635  Basecbs 15838  s cress 15839  +gcplusg 15922  Grpcgrp 17403  SubGrpcsubg 17569   GrpHom cghm 17638
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-er 7727  df-en 7941  df-dom 7942  df-sdom 7943  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-nn 11006  df-2 11064  df-ndx 15841  df-slot 15842  df-base 15844  df-sets 15845  df-ress 15846  df-plusg 15935  df-mgm 17223  df-sgrp 17265  df-mnd 17276  df-grp 17406  df-subg 17572  df-ghm 17639
This theorem is referenced by:  ghmima  17662  resrhm  18790  reslmhm  19033
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