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Theorem resmgmhm2b 43899
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 43880 . . . . . 6 (𝐹 ∈ (𝑆 MgmHom 𝑇) → (𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm))
21simpld 495 . . . . 5 (𝐹 ∈ (𝑆 MgmHom 𝑇) → 𝑆 ∈ Mgm)
32adantl 482 . . . 4 (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → 𝑆 ∈ Mgm)
4 resmgmhm2.u . . . . . 6 𝑈 = (𝑇s 𝑋)
54submgmmgm 43894 . . . . 5 (𝑋 ∈ (SubMgm‘𝑇) → 𝑈 ∈ Mgm)
65ad2antrr 722 . . . 4 (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → 𝑈 ∈ Mgm)
73, 6jca 512 . . 3 (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → (𝑆 ∈ Mgm ∧ 𝑈 ∈ Mgm))
8 eqid 2826 . . . . . . . . 9 (Base‘𝑆) = (Base‘𝑆)
9 eqid 2826 . . . . . . . . 9 (Base‘𝑇) = (Base‘𝑇)
108, 9mgmhmf 43883 . . . . . . . 8 (𝐹 ∈ (𝑆 MgmHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
1110adantl 482 . . . . . . 7 (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
1211ffnd 6512 . . . . . 6 (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → 𝐹 Fn (Base‘𝑆))
13 simplr 765 . . . . . 6 (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → ran 𝐹𝑋)
14 df-f 6356 . . . . . 6 (𝐹:(Base‘𝑆)⟶𝑋 ↔ (𝐹 Fn (Base‘𝑆) ∧ ran 𝐹𝑋))
1512, 13, 14sylanbrc 583 . . . . 5 (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → 𝐹:(Base‘𝑆)⟶𝑋)
164submgmbas 43895 . . . . . . 7 (𝑋 ∈ (SubMgm‘𝑇) → 𝑋 = (Base‘𝑈))
1716ad2antrr 722 . . . . . 6 (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → 𝑋 = (Base‘𝑈))
1817feq3d 6498 . . . . 5 (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → (𝐹:(Base‘𝑆)⟶𝑋𝐹:(Base‘𝑆)⟶(Base‘𝑈)))
1915, 18mpbid 233 . . . 4 (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → 𝐹:(Base‘𝑆)⟶(Base‘𝑈))
20 eqid 2826 . . . . . . . . 9 (+g𝑆) = (+g𝑆)
21 eqid 2826 . . . . . . . . 9 (+g𝑇) = (+g𝑇)
228, 20, 21mgmhmlin 43885 . . . . . . . 8 ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
23223expb 1114 . . . . . . 7 ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
2423adantll 710 . . . . . 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 16602 . . . . . . . 8 (𝑋 ∈ (SubMgm‘𝑇) → (+g𝑇) = (+g𝑈))
2625ad3antrrr 726 . . . . . . 7 ((((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (+g𝑇) = (+g𝑈))
2726oveqd 7165 . . . . . 6 ((((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → ((𝐹𝑥)(+g𝑇)(𝐹𝑦)) = ((𝐹𝑥)(+g𝑈)(𝐹𝑦)))
2824, 27eqtrd 2861 . . . . 5 ((((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑈)(𝐹𝑦)))
2928ralrimivva 3196 . . . 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 512 . . 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 2826 . . . 4 (Base‘𝑈) = (Base‘𝑈)
32 eqid 2826 . . . 4 (+g𝑈) = (+g𝑈)
338, 31, 20, 32ismgmhm 43882 . . 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 583 . 2 (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → 𝐹 ∈ (𝑆 MgmHom 𝑈))
354resmgmhm2 43898 . . . 4 ((𝐹 ∈ (𝑆 MgmHom 𝑈) ∧ 𝑋 ∈ (SubMgm‘𝑇)) → 𝐹 ∈ (𝑆 MgmHom 𝑇))
3635ancoms 459 . . 3 ((𝑋 ∈ (SubMgm‘𝑇) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑈)) → 𝐹 ∈ (𝑆 MgmHom 𝑇))
3736adantlr 711 . 2 (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑈)) → 𝐹 ∈ (𝑆 MgmHom 𝑇))
3834, 37impbida 797 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 207  wa 396   = wceq 1530  wcel 2107  wral 3143  wss 3940  ran crn 5555   Fn wfn 6347  wf 6348  cfv 6352  (class class class)co 7148  Basecbs 16473  s cress 16474  +gcplusg 16555  Mgmcmgm 17840   MgmHom cmgmhm 43876  SubMgmcsubmgm 43877
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7569  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-er 8279  df-map 8398  df-en 8499  df-dom 8500  df-sdom 8501  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11628  df-2 11689  df-ndx 16476  df-slot 16477  df-base 16479  df-sets 16480  df-ress 16481  df-plusg 16568  df-mgm 17842  df-mgmhm 43878  df-submgm 43879
This theorem is referenced by: (None)
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