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Theorem resmgmhm2b 41585
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 41566 . . . . . 6 (𝐹 ∈ (𝑆 MgmHom 𝑇) → (𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm))
21simpld 473 . . . . 5 (𝐹 ∈ (𝑆 MgmHom 𝑇) → 𝑆 ∈ Mgm)
32adantl 480 . . . 4 (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → 𝑆 ∈ Mgm)
4 resmgmhm2.u . . . . . 6 𝑈 = (𝑇s 𝑋)
54submgmmgm 41580 . . . . 5 (𝑋 ∈ (SubMgm‘𝑇) → 𝑈 ∈ Mgm)
65ad2antrr 757 . . . 4 (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → 𝑈 ∈ Mgm)
73, 6jca 552 . . 3 (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → (𝑆 ∈ Mgm ∧ 𝑈 ∈ Mgm))
8 eqid 2609 . . . . . . . . 9 (Base‘𝑆) = (Base‘𝑆)
9 eqid 2609 . . . . . . . . 9 (Base‘𝑇) = (Base‘𝑇)
108, 9mgmhmf 41569 . . . . . . . 8 (𝐹 ∈ (𝑆 MgmHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
1110adantl 480 . . . . . . 7 (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
12 ffn 5944 . . . . . . 7 (𝐹:(Base‘𝑆)⟶(Base‘𝑇) → 𝐹 Fn (Base‘𝑆))
1311, 12syl 17 . . . . . 6 (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → 𝐹 Fn (Base‘𝑆))
14 simplr 787 . . . . . 6 (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → ran 𝐹𝑋)
15 df-f 5794 . . . . . 6 (𝐹:(Base‘𝑆)⟶𝑋 ↔ (𝐹 Fn (Base‘𝑆) ∧ ran 𝐹𝑋))
1613, 14, 15sylanbrc 694 . . . . 5 (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → 𝐹:(Base‘𝑆)⟶𝑋)
174submgmbas 41581 . . . . . . 7 (𝑋 ∈ (SubMgm‘𝑇) → 𝑋 = (Base‘𝑈))
1817ad2antrr 757 . . . . . 6 (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → 𝑋 = (Base‘𝑈))
1918feq3d 5931 . . . . 5 (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → (𝐹:(Base‘𝑆)⟶𝑋𝐹:(Base‘𝑆)⟶(Base‘𝑈)))
2016, 19mpbid 220 . . . 4 (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → 𝐹:(Base‘𝑆)⟶(Base‘𝑈))
21 eqid 2609 . . . . . . . . 9 (+g𝑆) = (+g𝑆)
22 eqid 2609 . . . . . . . . 9 (+g𝑇) = (+g𝑇)
238, 21, 22mgmhmlin 41571 . . . . . . . 8 ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
24233expb 1257 . . . . . . 7 ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
2524adantll 745 . . . . . 6 ((((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
264, 22ressplusg 15764 . . . . . . . 8 (𝑋 ∈ (SubMgm‘𝑇) → (+g𝑇) = (+g𝑈))
2726ad3antrrr 761 . . . . . . 7 ((((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (+g𝑇) = (+g𝑈))
2827oveqd 6544 . . . . . 6 ((((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → ((𝐹𝑥)(+g𝑇)(𝐹𝑦)) = ((𝐹𝑥)(+g𝑈)(𝐹𝑦)))
2925, 28eqtrd 2643 . . . . 5 ((((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑈)(𝐹𝑦)))
3029ralrimivva 2953 . . . 4 (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)(𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑈)(𝐹𝑦)))
3120, 30jca 552 . . 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))))
32 eqid 2609 . . . 4 (Base‘𝑈) = (Base‘𝑈)
33 eqid 2609 . . . 4 (+g𝑈) = (+g𝑈)
348, 32, 21, 33ismgmhm 41568 . . 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)))))
357, 31, 34sylanbrc 694 . 2 (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → 𝐹 ∈ (𝑆 MgmHom 𝑈))
364resmgmhm2 41584 . . . 4 ((𝐹 ∈ (𝑆 MgmHom 𝑈) ∧ 𝑋 ∈ (SubMgm‘𝑇)) → 𝐹 ∈ (𝑆 MgmHom 𝑇))
3736ancoms 467 . . 3 ((𝑋 ∈ (SubMgm‘𝑇) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑈)) → 𝐹 ∈ (𝑆 MgmHom 𝑇))
3837adantlr 746 . 2 (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑈)) → 𝐹 ∈ (𝑆 MgmHom 𝑇))
3935, 38impbida 872 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 194  wa 382   = wceq 1474  wcel 1976  wral 2895  wss 3539  ran crn 5029   Fn wfn 5785  wf 5786  cfv 5790  (class class class)co 6527  Basecbs 15641  s cress 15642  +gcplusg 15714  Mgmcmgm 17009   MgmHom cmgmhm 41562  SubMgmcsubmgm 41563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-er 7606  df-map 7723  df-en 7819  df-dom 7820  df-sdom 7821  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-nn 10868  df-2 10926  df-ndx 15644  df-slot 15645  df-base 15646  df-sets 15647  df-ress 15648  df-plusg 15727  df-mgm 17011  df-mgmhm 41564  df-submgm 41565
This theorem is referenced by: (None)
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