Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  elghomlem1OLD Structured version   Visualization version   GIF version

Theorem elghomlem1OLD 37057
Description: Obsolete as of 15-Mar-2020. Lemma for elghomOLD 37059. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
elghomlem1OLD.1 𝑆 = {𝑓 ∣ (𝑓:ran 𝐺⟢ran 𝐻 ∧ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((π‘“β€˜π‘₯)𝐻(π‘“β€˜π‘¦)) = (π‘“β€˜(π‘₯𝐺𝑦)))}
Assertion
Ref Expression
elghomlem1OLD ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) β†’ (𝐺 GrpOpHom 𝐻) = 𝑆)
Distinct variable groups:   π‘₯,𝑓,𝑦,𝐺   𝑓,𝐻,π‘₯,𝑦
Allowed substitution hints:   𝑆(π‘₯,𝑦,𝑓)

Proof of Theorem elghomlem1OLD
Dummy variables 𝑔 β„Ž are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rnexg 7898 . . 3 (𝐺 ∈ GrpOp β†’ ran 𝐺 ∈ V)
2 rnexg 7898 . . 3 (𝐻 ∈ GrpOp β†’ ran 𝐻 ∈ V)
3 elghomlem1OLD.1 . . . 4 𝑆 = {𝑓 ∣ (𝑓:ran 𝐺⟢ran 𝐻 ∧ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((π‘“β€˜π‘₯)𝐻(π‘“β€˜π‘¦)) = (π‘“β€˜(π‘₯𝐺𝑦)))}
43fabexg 7928 . . 3 ((ran 𝐺 ∈ V ∧ ran 𝐻 ∈ V) β†’ 𝑆 ∈ V)
51, 2, 4syl2an 595 . 2 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) β†’ 𝑆 ∈ V)
6 rneq 5936 . . . . . 6 (𝑔 = 𝐺 β†’ ran 𝑔 = ran 𝐺)
76feq2d 6704 . . . . 5 (𝑔 = 𝐺 β†’ (𝑓:ran π‘”βŸΆran β„Ž ↔ 𝑓:ran 𝐺⟢ran β„Ž))
8 oveq 7418 . . . . . . . . 9 (𝑔 = 𝐺 β†’ (π‘₯𝑔𝑦) = (π‘₯𝐺𝑦))
98fveq2d 6896 . . . . . . . 8 (𝑔 = 𝐺 β†’ (π‘“β€˜(π‘₯𝑔𝑦)) = (π‘“β€˜(π‘₯𝐺𝑦)))
109eqeq2d 2742 . . . . . . 7 (𝑔 = 𝐺 β†’ (((π‘“β€˜π‘₯)β„Ž(π‘“β€˜π‘¦)) = (π‘“β€˜(π‘₯𝑔𝑦)) ↔ ((π‘“β€˜π‘₯)β„Ž(π‘“β€˜π‘¦)) = (π‘“β€˜(π‘₯𝐺𝑦))))
116, 10raleqbidv 3341 . . . . . 6 (𝑔 = 𝐺 β†’ (βˆ€π‘¦ ∈ ran 𝑔((π‘“β€˜π‘₯)β„Ž(π‘“β€˜π‘¦)) = (π‘“β€˜(π‘₯𝑔𝑦)) ↔ βˆ€π‘¦ ∈ ran 𝐺((π‘“β€˜π‘₯)β„Ž(π‘“β€˜π‘¦)) = (π‘“β€˜(π‘₯𝐺𝑦))))
126, 11raleqbidv 3341 . . . . 5 (𝑔 = 𝐺 β†’ (βˆ€π‘₯ ∈ ran π‘”βˆ€π‘¦ ∈ ran 𝑔((π‘“β€˜π‘₯)β„Ž(π‘“β€˜π‘¦)) = (π‘“β€˜(π‘₯𝑔𝑦)) ↔ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((π‘“β€˜π‘₯)β„Ž(π‘“β€˜π‘¦)) = (π‘“β€˜(π‘₯𝐺𝑦))))
137, 12anbi12d 630 . . . 4 (𝑔 = 𝐺 β†’ ((𝑓:ran π‘”βŸΆran β„Ž ∧ βˆ€π‘₯ ∈ ran π‘”βˆ€π‘¦ ∈ ran 𝑔((π‘“β€˜π‘₯)β„Ž(π‘“β€˜π‘¦)) = (π‘“β€˜(π‘₯𝑔𝑦))) ↔ (𝑓:ran 𝐺⟢ran β„Ž ∧ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((π‘“β€˜π‘₯)β„Ž(π‘“β€˜π‘¦)) = (π‘“β€˜(π‘₯𝐺𝑦)))))
1413abbidv 2800 . . 3 (𝑔 = 𝐺 β†’ {𝑓 ∣ (𝑓:ran π‘”βŸΆran β„Ž ∧ βˆ€π‘₯ ∈ ran π‘”βˆ€π‘¦ ∈ ran 𝑔((π‘“β€˜π‘₯)β„Ž(π‘“β€˜π‘¦)) = (π‘“β€˜(π‘₯𝑔𝑦)))} = {𝑓 ∣ (𝑓:ran 𝐺⟢ran β„Ž ∧ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((π‘“β€˜π‘₯)β„Ž(π‘“β€˜π‘¦)) = (π‘“β€˜(π‘₯𝐺𝑦)))})
15 rneq 5936 . . . . . . 7 (β„Ž = 𝐻 β†’ ran β„Ž = ran 𝐻)
1615feq3d 6705 . . . . . 6 (β„Ž = 𝐻 β†’ (𝑓:ran 𝐺⟢ran β„Ž ↔ 𝑓:ran 𝐺⟢ran 𝐻))
17 oveq 7418 . . . . . . . 8 (β„Ž = 𝐻 β†’ ((π‘“β€˜π‘₯)β„Ž(π‘“β€˜π‘¦)) = ((π‘“β€˜π‘₯)𝐻(π‘“β€˜π‘¦)))
1817eqeq1d 2733 . . . . . . 7 (β„Ž = 𝐻 β†’ (((π‘“β€˜π‘₯)β„Ž(π‘“β€˜π‘¦)) = (π‘“β€˜(π‘₯𝐺𝑦)) ↔ ((π‘“β€˜π‘₯)𝐻(π‘“β€˜π‘¦)) = (π‘“β€˜(π‘₯𝐺𝑦))))
19182ralbidv 3217 . . . . . 6 (β„Ž = 𝐻 β†’ (βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((π‘“β€˜π‘₯)β„Ž(π‘“β€˜π‘¦)) = (π‘“β€˜(π‘₯𝐺𝑦)) ↔ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((π‘“β€˜π‘₯)𝐻(π‘“β€˜π‘¦)) = (π‘“β€˜(π‘₯𝐺𝑦))))
2016, 19anbi12d 630 . . . . 5 (β„Ž = 𝐻 β†’ ((𝑓:ran 𝐺⟢ran β„Ž ∧ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((π‘“β€˜π‘₯)β„Ž(π‘“β€˜π‘¦)) = (π‘“β€˜(π‘₯𝐺𝑦))) ↔ (𝑓:ran 𝐺⟢ran 𝐻 ∧ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((π‘“β€˜π‘₯)𝐻(π‘“β€˜π‘¦)) = (π‘“β€˜(π‘₯𝐺𝑦)))))
2120abbidv 2800 . . . 4 (β„Ž = 𝐻 β†’ {𝑓 ∣ (𝑓:ran 𝐺⟢ran β„Ž ∧ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((π‘“β€˜π‘₯)β„Ž(π‘“β€˜π‘¦)) = (π‘“β€˜(π‘₯𝐺𝑦)))} = {𝑓 ∣ (𝑓:ran 𝐺⟢ran 𝐻 ∧ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((π‘“β€˜π‘₯)𝐻(π‘“β€˜π‘¦)) = (π‘“β€˜(π‘₯𝐺𝑦)))})
2221, 3eqtr4di 2789 . . 3 (β„Ž = 𝐻 β†’ {𝑓 ∣ (𝑓:ran 𝐺⟢ran β„Ž ∧ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((π‘“β€˜π‘₯)β„Ž(π‘“β€˜π‘¦)) = (π‘“β€˜(π‘₯𝐺𝑦)))} = 𝑆)
23 df-ghomOLD 37056 . . 3 GrpOpHom = (𝑔 ∈ GrpOp, β„Ž ∈ GrpOp ↦ {𝑓 ∣ (𝑓:ran π‘”βŸΆran β„Ž ∧ βˆ€π‘₯ ∈ ran π‘”βˆ€π‘¦ ∈ ran 𝑔((π‘“β€˜π‘₯)β„Ž(π‘“β€˜π‘¦)) = (π‘“β€˜(π‘₯𝑔𝑦)))})
2414, 22, 23ovmpog 7570 . 2 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝑆 ∈ V) β†’ (𝐺 GrpOpHom 𝐻) = 𝑆)
255, 24mpd3an3 1461 1 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) β†’ (𝐺 GrpOpHom 𝐻) = 𝑆)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2105  {cab 2708  βˆ€wral 3060  Vcvv 3473  ran crn 5678  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7412  GrpOpcgr 30006   GrpOpHom cghomOLD 37055
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7728
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-ov 7415  df-oprab 7416  df-mpo 7417  df-ghomOLD 37056
This theorem is referenced by:  elghomlem2OLD  37058
  Copyright terms: Public domain W3C validator