Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  resghm Structured version   Visualization version   GIF version

Theorem resghm 18350
 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 2820 . 2 (Base‘𝑈) = (Base‘𝑈)
2 eqid 2820 . 2 (Base‘𝑇) = (Base‘𝑇)
3 eqid 2820 . 2 (+g𝑈) = (+g𝑈)
4 eqid 2820 . 2 (+g𝑇) = (+g𝑇)
5 resghm.u . . . 4 𝑈 = (𝑆s 𝑋)
65subggrp 18258 . . 3 (𝑋 ∈ (SubGrp‘𝑆) → 𝑈 ∈ Grp)
76adantl 484 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → 𝑈 ∈ Grp)
8 ghmgrp2 18337 . . 3 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp)
98adantr 483 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → 𝑇 ∈ Grp)
10 eqid 2820 . . . . 5 (Base‘𝑆) = (Base‘𝑆)
1110, 2ghmf 18338 . . . 4 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
1210subgss 18256 . . . 4 (𝑋 ∈ (SubGrp‘𝑆) → 𝑋 ⊆ (Base‘𝑆))
13 fssres 6518 . . . 4 ((𝐹:(Base‘𝑆)⟶(Base‘𝑇) ∧ 𝑋 ⊆ (Base‘𝑆)) → (𝐹𝑋):𝑋⟶(Base‘𝑇))
1411, 12, 13syl2an 597 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → (𝐹𝑋):𝑋⟶(Base‘𝑇))
1512adantl 484 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → 𝑋 ⊆ (Base‘𝑆))
165, 10ressbas2 16531 . . . . 5 (𝑋 ⊆ (Base‘𝑆) → 𝑋 = (Base‘𝑈))
1715, 16syl 17 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → 𝑋 = (Base‘𝑈))
1817feq2d 6474 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → ((𝐹𝑋):𝑋⟶(Base‘𝑇) ↔ (𝐹𝑋):(Base‘𝑈)⟶(Base‘𝑇)))
1914, 18mpbid 234 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → (𝐹𝑋):(Base‘𝑈)⟶(Base‘𝑇))
20 eleq2 2899 . . . . . 6 (𝑋 = (Base‘𝑈) → (𝑎𝑋𝑎 ∈ (Base‘𝑈)))
21 eleq2 2899 . . . . . 6 (𝑋 = (Base‘𝑈) → (𝑏𝑋𝑏 ∈ (Base‘𝑈)))
2220, 21anbi12d 632 . . . . 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 480 . . 3 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎 ∈ (Base‘𝑈) ∧ 𝑏 ∈ (Base‘𝑈))) → (𝑎𝑋𝑏𝑋))
25 simpll 765 . . . . 5 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎𝑋𝑏𝑋)) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
2615sselda 3943 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ 𝑎𝑋) → 𝑎 ∈ (Base‘𝑆))
2726adantrr 715 . . . . 5 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎𝑋𝑏𝑋)) → 𝑎 ∈ (Base‘𝑆))
2815sselda 3943 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ 𝑏𝑋) → 𝑏 ∈ (Base‘𝑆))
2928adantrl 714 . . . . 5 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎𝑋𝑏𝑋)) → 𝑏 ∈ (Base‘𝑆))
30 eqid 2820 . . . . . 6 (+g𝑆) = (+g𝑆)
3110, 30, 4ghmlin 18339 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆)) → (𝐹‘(𝑎(+g𝑆)𝑏)) = ((𝐹𝑎)(+g𝑇)(𝐹𝑏)))
3225, 27, 29, 31syl3anc 1367 . . . 4 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎𝑋𝑏𝑋)) → (𝐹‘(𝑎(+g𝑆)𝑏)) = ((𝐹𝑎)(+g𝑇)(𝐹𝑏)))
335, 30ressplusg 16588 . . . . . . . 8 (𝑋 ∈ (SubGrp‘𝑆) → (+g𝑆) = (+g𝑈))
3433ad2antlr 725 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎𝑋𝑏𝑋)) → (+g𝑆) = (+g𝑈))
3534oveqd 7148 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎𝑋𝑏𝑋)) → (𝑎(+g𝑆)𝑏) = (𝑎(+g𝑈)𝑏))
3635fveq2d 6648 . . . . 5 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎𝑋𝑏𝑋)) → ((𝐹𝑋)‘(𝑎(+g𝑆)𝑏)) = ((𝐹𝑋)‘(𝑎(+g𝑈)𝑏)))
3730subgcl 18265 . . . . . . . 8 ((𝑋 ∈ (SubGrp‘𝑆) ∧ 𝑎𝑋𝑏𝑋) → (𝑎(+g𝑆)𝑏) ∈ 𝑋)
38373expb 1116 . . . . . . 7 ((𝑋 ∈ (SubGrp‘𝑆) ∧ (𝑎𝑋𝑏𝑋)) → (𝑎(+g𝑆)𝑏) ∈ 𝑋)
3938adantll 712 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎𝑋𝑏𝑋)) → (𝑎(+g𝑆)𝑏) ∈ 𝑋)
4039fvresd 6664 . . . . 5 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎𝑋𝑏𝑋)) → ((𝐹𝑋)‘(𝑎(+g𝑆)𝑏)) = (𝐹‘(𝑎(+g𝑆)𝑏)))
4136, 40eqtr3d 2857 . . . 4 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎𝑋𝑏𝑋)) → ((𝐹𝑋)‘(𝑎(+g𝑈)𝑏)) = (𝐹‘(𝑎(+g𝑆)𝑏)))
42 fvres 6663 . . . . . 6 (𝑎𝑋 → ((𝐹𝑋)‘𝑎) = (𝐹𝑎))
43 fvres 6663 . . . . . 6 (𝑏𝑋 → ((𝐹𝑋)‘𝑏) = (𝐹𝑏))
4442, 43oveqan12d 7150 . . . . 5 ((𝑎𝑋𝑏𝑋) → (((𝐹𝑋)‘𝑎)(+g𝑇)((𝐹𝑋)‘𝑏)) = ((𝐹𝑎)(+g𝑇)(𝐹𝑏)))
4544adantl 484 . . . 4 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎𝑋𝑏𝑋)) → (((𝐹𝑋)‘𝑎)(+g𝑇)((𝐹𝑋)‘𝑏)) = ((𝐹𝑎)(+g𝑇)(𝐹𝑏)))
4632, 41, 453eqtr4d 2865 . . 3 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎𝑋𝑏𝑋)) → ((𝐹𝑋)‘(𝑎(+g𝑈)𝑏)) = (((𝐹𝑋)‘𝑎)(+g𝑇)((𝐹𝑋)‘𝑏)))
4724, 46syldan 593 . 2 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎 ∈ (Base‘𝑈) ∧ 𝑏 ∈ (Base‘𝑈))) → ((𝐹𝑋)‘(𝑎(+g𝑈)𝑏)) = (((𝐹𝑋)‘𝑎)(+g𝑇)((𝐹𝑋)‘𝑏)))
481, 2, 3, 4, 7, 9, 19, 47isghmd 18343 1 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → (𝐹𝑋) ∈ (𝑈 GrpHom 𝑇))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114   ⊆ wss 3912   ↾ cres 5531  ⟶wf 6325  ‘cfv 6329  (class class class)co 7131  Basecbs 16459   ↾s cress 16460  +gcplusg 16541  Grpcgrp 18079  SubGrpcsubg 18249   GrpHom cghm 18331 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pow 5240  ax-pr 5304  ax-un 7437  ax-cnex 10569  ax-resscn 10570  ax-1cn 10571  ax-icn 10572  ax-addcl 10573  ax-addrcl 10574  ax-mulcl 10575  ax-mulrcl 10576  ax-mulcom 10577  ax-addass 10578  ax-mulass 10579  ax-distr 10580  ax-i2m1 10581  ax-1ne0 10582  ax-1rid 10583  ax-rnegex 10584  ax-rrecex 10585  ax-cnre 10586  ax-pre-lttri 10587  ax-pre-lttrn 10588  ax-pre-ltadd 10589  ax-pre-mulgt0 10590 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-nel 3111  df-ral 3130  df-rex 3131  df-reu 3132  df-rab 3134  df-v 3475  df-sbc 3752  df-csb 3860  df-dif 3915  df-un 3917  df-in 3919  df-ss 3928  df-pss 3930  df-nul 4268  df-if 4442  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4813  df-iun 4895  df-br 5041  df-opab 5103  df-mpt 5121  df-tr 5147  df-id 5434  df-eprel 5439  df-po 5448  df-so 5449  df-fr 5488  df-we 5490  df-xp 5535  df-rel 5536  df-cnv 5537  df-co 5538  df-dm 5539  df-rn 5540  df-res 5541  df-ima 5542  df-pred 6122  df-ord 6168  df-on 6169  df-lim 6170  df-suc 6171  df-iota 6288  df-fun 6331  df-fn 6332  df-f 6333  df-f1 6334  df-fo 6335  df-f1o 6336  df-fv 6337  df-riota 7089  df-ov 7134  df-oprab 7135  df-mpo 7136  df-om 7557  df-wrecs 7923  df-recs 7984  df-rdg 8022  df-er 8265  df-en 8486  df-dom 8487  df-sdom 8488  df-pnf 10653  df-mnf 10654  df-xr 10655  df-ltxr 10656  df-le 10657  df-sub 10848  df-neg 10849  df-nn 11615  df-2 11677  df-ndx 16462  df-slot 16463  df-base 16465  df-sets 16466  df-ress 16467  df-plusg 16554  df-mgm 17828  df-sgrp 17877  df-mnd 17888  df-grp 18082  df-subg 18252  df-ghm 18332 This theorem is referenced by:  ghmima  18355  resrhm  19537  reslmhm  19797  dimkerim  31031
 Copyright terms: Public domain W3C validator