Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  resmgmhm2b Structured version   Visualization version   GIF version

Theorem resmgmhm2b 46570
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 46551 . . . . . 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 46565 . . . . 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 46554 . . . . . . . 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 6719 . . . . . 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 6548 . . . . . 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 46566 . . . . . . 7 (𝑋 ∈ (SubMgm‘𝑇) → 𝑋 = (Base‘𝑈))
1716ad2antrr 725 . . . . . 6 (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → 𝑋 = (Base‘𝑈))
1817feq3d 6705 . . . . 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 46556 . . . . . . . 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 17235 . . . . . . . 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 7426 . . . . . 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 3201 . . . 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 46553 . . 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 46569 . . . 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 3062  wss 3949  ran crn 5678   Fn wfn 6539  wf 6540  cfv 6544  (class class class)co 7409  Basecbs 17144  s cress 17173  +gcplusg 17197  Mgmcmgm 18559   MgmHom cmgmhm 46547  SubMgmcsubmgm 46548
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 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
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 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-er 8703  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-2 12275  df-sets 17097  df-slot 17115  df-ndx 17127  df-base 17145  df-ress 17174  df-plusg 17210  df-mgm 18561  df-mgmhm 46549  df-submgm 46550
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator