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Theorem resghm 17942
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 2765 . 2 (Base‘𝑈) = (Base‘𝑈)
2 eqid 2765 . 2 (Base‘𝑇) = (Base‘𝑇)
3 eqid 2765 . 2 (+g𝑈) = (+g𝑈)
4 eqid 2765 . 2 (+g𝑇) = (+g𝑇)
5 resghm.u . . . 4 𝑈 = (𝑆s 𝑋)
65subggrp 17863 . . 3 (𝑋 ∈ (SubGrp‘𝑆) → 𝑈 ∈ Grp)
76adantl 473 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → 𝑈 ∈ Grp)
8 ghmgrp2 17929 . . 3 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp)
98adantr 472 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → 𝑇 ∈ Grp)
10 eqid 2765 . . . . 5 (Base‘𝑆) = (Base‘𝑆)
1110, 2ghmf 17930 . . . 4 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
1210subgss 17861 . . . 4 (𝑋 ∈ (SubGrp‘𝑆) → 𝑋 ⊆ (Base‘𝑆))
13 fssres 6252 . . . 4 ((𝐹:(Base‘𝑆)⟶(Base‘𝑇) ∧ 𝑋 ⊆ (Base‘𝑆)) → (𝐹𝑋):𝑋⟶(Base‘𝑇))
1411, 12, 13syl2an 589 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → (𝐹𝑋):𝑋⟶(Base‘𝑇))
1512adantl 473 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → 𝑋 ⊆ (Base‘𝑆))
165, 10ressbas2 16205 . . . . 5 (𝑋 ⊆ (Base‘𝑆) → 𝑋 = (Base‘𝑈))
1715, 16syl 17 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → 𝑋 = (Base‘𝑈))
1817feq2d 6209 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → ((𝐹𝑋):𝑋⟶(Base‘𝑇) ↔ (𝐹𝑋):(Base‘𝑈)⟶(Base‘𝑇)))
1914, 18mpbid 223 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → (𝐹𝑋):(Base‘𝑈)⟶(Base‘𝑇))
20 eleq2 2833 . . . . . 6 (𝑋 = (Base‘𝑈) → (𝑎𝑋𝑎 ∈ (Base‘𝑈)))
21 eleq2 2833 . . . . . 6 (𝑋 = (Base‘𝑈) → (𝑏𝑋𝑏 ∈ (Base‘𝑈)))
2220, 21anbi12d 624 . . . . 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 469 . . 3 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎 ∈ (Base‘𝑈) ∧ 𝑏 ∈ (Base‘𝑈))) → (𝑎𝑋𝑏𝑋))
25 simpll 783 . . . . 5 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎𝑋𝑏𝑋)) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
2615sselda 3761 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ 𝑎𝑋) → 𝑎 ∈ (Base‘𝑆))
2726adantrr 708 . . . . 5 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎𝑋𝑏𝑋)) → 𝑎 ∈ (Base‘𝑆))
2815sselda 3761 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ 𝑏𝑋) → 𝑏 ∈ (Base‘𝑆))
2928adantrl 707 . . . . 5 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎𝑋𝑏𝑋)) → 𝑏 ∈ (Base‘𝑆))
30 eqid 2765 . . . . . 6 (+g𝑆) = (+g𝑆)
3110, 30, 4ghmlin 17931 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆)) → (𝐹‘(𝑎(+g𝑆)𝑏)) = ((𝐹𝑎)(+g𝑇)(𝐹𝑏)))
3225, 27, 29, 31syl3anc 1490 . . . 4 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎𝑋𝑏𝑋)) → (𝐹‘(𝑎(+g𝑆)𝑏)) = ((𝐹𝑎)(+g𝑇)(𝐹𝑏)))
335, 30ressplusg 16267 . . . . . . . 8 (𝑋 ∈ (SubGrp‘𝑆) → (+g𝑆) = (+g𝑈))
3433ad2antlr 718 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎𝑋𝑏𝑋)) → (+g𝑆) = (+g𝑈))
3534oveqd 6859 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎𝑋𝑏𝑋)) → (𝑎(+g𝑆)𝑏) = (𝑎(+g𝑈)𝑏))
3635fveq2d 6379 . . . . 5 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎𝑋𝑏𝑋)) → ((𝐹𝑋)‘(𝑎(+g𝑆)𝑏)) = ((𝐹𝑋)‘(𝑎(+g𝑈)𝑏)))
3730subgcl 17870 . . . . . . . 8 ((𝑋 ∈ (SubGrp‘𝑆) ∧ 𝑎𝑋𝑏𝑋) → (𝑎(+g𝑆)𝑏) ∈ 𝑋)
38373expb 1149 . . . . . . 7 ((𝑋 ∈ (SubGrp‘𝑆) ∧ (𝑎𝑋𝑏𝑋)) → (𝑎(+g𝑆)𝑏) ∈ 𝑋)
3938adantll 705 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎𝑋𝑏𝑋)) → (𝑎(+g𝑆)𝑏) ∈ 𝑋)
40 fvres 6394 . . . . . 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 2801 . . . 4 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎𝑋𝑏𝑋)) → ((𝐹𝑋)‘(𝑎(+g𝑈)𝑏)) = (𝐹‘(𝑎(+g𝑆)𝑏)))
43 fvres 6394 . . . . . 6 (𝑎𝑋 → ((𝐹𝑋)‘𝑎) = (𝐹𝑎))
44 fvres 6394 . . . . . 6 (𝑏𝑋 → ((𝐹𝑋)‘𝑏) = (𝐹𝑏))
4543, 44oveqan12d 6861 . . . . 5 ((𝑎𝑋𝑏𝑋) → (((𝐹𝑋)‘𝑎)(+g𝑇)((𝐹𝑋)‘𝑏)) = ((𝐹𝑎)(+g𝑇)(𝐹𝑏)))
4645adantl 473 . . . 4 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎𝑋𝑏𝑋)) → (((𝐹𝑋)‘𝑎)(+g𝑇)((𝐹𝑋)‘𝑏)) = ((𝐹𝑎)(+g𝑇)(𝐹𝑏)))
4732, 42, 463eqtr4d 2809 . . 3 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎𝑋𝑏𝑋)) → ((𝐹𝑋)‘(𝑎(+g𝑈)𝑏)) = (((𝐹𝑋)‘𝑎)(+g𝑇)((𝐹𝑋)‘𝑏)))
4824, 47syldan 585 . 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 17935 1 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → (𝐹𝑋) ∈ (𝑈 GrpHom 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  wss 3732  cres 5279  wf 6064  cfv 6068  (class class class)co 6842  Basecbs 16132  s cress 16133  +gcplusg 16216  Grpcgrp 17691  SubGrpcsubg 17854   GrpHom cghm 17923
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-er 7947  df-en 8161  df-dom 8162  df-sdom 8163  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-nn 11275  df-2 11335  df-ndx 16135  df-slot 16136  df-base 16138  df-sets 16139  df-ress 16140  df-plusg 16229  df-mgm 17510  df-sgrp 17552  df-mnd 17563  df-grp 17694  df-subg 17857  df-ghm 17924
This theorem is referenced by:  ghmima  17947  resrhm  19078  reslmhm  19324
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