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Theorem resmgmhm2b 46180
Description: Restriction of the codomain of a homomorphism. (Contributed by AV, 26-Feb-2020.)
Hypothesis
Ref Expression
resmgmhm2.u 𝑈 = (𝑇s 𝑋)
Assertion
Ref Expression
resmgmhm2b ((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹𝑋) → (𝐹 ∈ (𝑆 MgmHom 𝑇) ↔ 𝐹 ∈ (𝑆 MgmHom 𝑈)))

Proof of Theorem resmgmhm2b
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mgmhmrcl 46161 . . . . . 6 (𝐹 ∈ (𝑆 MgmHom 𝑇) → (𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm))
21simpld 496 . . . . 5 (𝐹 ∈ (𝑆 MgmHom 𝑇) → 𝑆 ∈ Mgm)
32adantl 483 . . . 4 (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → 𝑆 ∈ Mgm)
4 resmgmhm2.u . . . . . 6 𝑈 = (𝑇s 𝑋)
54submgmmgm 46175 . . . . 5 (𝑋 ∈ (SubMgm‘𝑇) → 𝑈 ∈ Mgm)
65ad2antrr 725 . . . 4 (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → 𝑈 ∈ Mgm)
73, 6jca 513 . . 3 (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → (𝑆 ∈ Mgm ∧ 𝑈 ∈ Mgm))
8 eqid 2733 . . . . . . . . 9 (Base‘𝑆) = (Base‘𝑆)
9 eqid 2733 . . . . . . . . 9 (Base‘𝑇) = (Base‘𝑇)
108, 9mgmhmf 46164 . . . . . . . 8 (𝐹 ∈ (𝑆 MgmHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
1110adantl 483 . . . . . . 7 (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
1211ffnd 6670 . . . . . 6 (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → 𝐹 Fn (Base‘𝑆))
13 simplr 768 . . . . . 6 (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → ran 𝐹𝑋)
14 df-f 6501 . . . . . 6 (𝐹:(Base‘𝑆)⟶𝑋 ↔ (𝐹 Fn (Base‘𝑆) ∧ ran 𝐹𝑋))
1512, 13, 14sylanbrc 584 . . . . 5 (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → 𝐹:(Base‘𝑆)⟶𝑋)
164submgmbas 46176 . . . . . . 7 (𝑋 ∈ (SubMgm‘𝑇) → 𝑋 = (Base‘𝑈))
1716ad2antrr 725 . . . . . 6 (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → 𝑋 = (Base‘𝑈))
1817feq3d 6656 . . . . 5 (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → (𝐹:(Base‘𝑆)⟶𝑋𝐹:(Base‘𝑆)⟶(Base‘𝑈)))
1915, 18mpbid 231 . . . 4 (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → 𝐹:(Base‘𝑆)⟶(Base‘𝑈))
20 eqid 2733 . . . . . . . . 9 (+g𝑆) = (+g𝑆)
21 eqid 2733 . . . . . . . . 9 (+g𝑇) = (+g𝑇)
228, 20, 21mgmhmlin 46166 . . . . . . . 8 ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
23223expb 1121 . . . . . . 7 ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
2423adantll 713 . . . . . 6 ((((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
254, 21ressplusg 17176 . . . . . . . 8 (𝑋 ∈ (SubMgm‘𝑇) → (+g𝑇) = (+g𝑈))
2625ad3antrrr 729 . . . . . . 7 ((((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (+g𝑇) = (+g𝑈))
2726oveqd 7375 . . . . . 6 ((((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → ((𝐹𝑥)(+g𝑇)(𝐹𝑦)) = ((𝐹𝑥)(+g𝑈)(𝐹𝑦)))
2824, 27eqtrd 2773 . . . . 5 ((((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑈)(𝐹𝑦)))
2928ralrimivva 3194 . . . 4 (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)(𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑈)(𝐹𝑦)))
3019, 29jca 513 . . 3 (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → (𝐹:(Base‘𝑆)⟶(Base‘𝑈) ∧ ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)(𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑈)(𝐹𝑦))))
31 eqid 2733 . . . 4 (Base‘𝑈) = (Base‘𝑈)
32 eqid 2733 . . . 4 (+g𝑈) = (+g𝑈)
338, 31, 20, 32ismgmhm 46163 . . 3 (𝐹 ∈ (𝑆 MgmHom 𝑈) ↔ ((𝑆 ∈ Mgm ∧ 𝑈 ∈ Mgm) ∧ (𝐹:(Base‘𝑆)⟶(Base‘𝑈) ∧ ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)(𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑈)(𝐹𝑦)))))
347, 30, 33sylanbrc 584 . 2 (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → 𝐹 ∈ (𝑆 MgmHom 𝑈))
354resmgmhm2 46179 . . . 4 ((𝐹 ∈ (𝑆 MgmHom 𝑈) ∧ 𝑋 ∈ (SubMgm‘𝑇)) → 𝐹 ∈ (𝑆 MgmHom 𝑇))
3635ancoms 460 . . 3 ((𝑋 ∈ (SubMgm‘𝑇) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑈)) → 𝐹 ∈ (𝑆 MgmHom 𝑇))
3736adantlr 714 . 2 (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑈)) → 𝐹 ∈ (𝑆 MgmHom 𝑇))
3834, 37impbida 800 1 ((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹𝑋) → (𝐹 ∈ (𝑆 MgmHom 𝑇) ↔ 𝐹 ∈ (𝑆 MgmHom 𝑈)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wral 3061  wss 3911  ran crn 5635   Fn wfn 6492  wf 6493  cfv 6497  (class class class)co 7358  Basecbs 17088  s cress 17117  +gcplusg 17138  Mgmcmgm 18500   MgmHom cmgmhm 46157  SubMgmcsubmgm 46158
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-er 8651  df-map 8770  df-en 8887  df-dom 8888  df-sdom 8889  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-nn 12159  df-2 12221  df-sets 17041  df-slot 17059  df-ndx 17071  df-base 17089  df-ress 17118  df-plusg 17151  df-mgm 18502  df-mgmhm 46159  df-submgm 46160
This theorem is referenced by: (None)
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