Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  tgrpset Structured version   Visualization version   GIF version

Theorem tgrpset 40128
Description: The translation group for a fiducial co-atom π‘Š. (Contributed by NM, 5-Jun-2013.)
Hypotheses
Ref Expression
tgrpset.h 𝐻 = (LHypβ€˜πΎ)
tgrpset.t 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
tgrpset.g 𝐺 = ((TGrpβ€˜πΎ)β€˜π‘Š)
Assertion
Ref Expression
tgrpset ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ 𝐺 = {⟨(Baseβ€˜ndx), π‘‡βŸ©, ⟨(+gβ€˜ndx), (𝑓 ∈ 𝑇, 𝑔 ∈ 𝑇 ↦ (𝑓 ∘ 𝑔))⟩})
Distinct variable groups:   𝑓,𝑔,𝐾   𝑇,𝑓,𝑔   𝑓,π‘Š,𝑔
Allowed substitution hints:   𝐺(𝑓,𝑔)   𝐻(𝑓,𝑔)   𝑉(𝑓,𝑔)

Proof of Theorem tgrpset
Dummy variable 𝑀 is distinct from all other variables.
StepHypRef Expression
1 tgrpset.g . 2 𝐺 = ((TGrpβ€˜πΎ)β€˜π‘Š)
2 tgrpset.h . . . . 5 𝐻 = (LHypβ€˜πΎ)
32tgrpfset 40127 . . . 4 (𝐾 ∈ 𝑉 β†’ (TGrpβ€˜πΎ) = (𝑀 ∈ 𝐻 ↦ {⟨(Baseβ€˜ndx), ((LTrnβ€˜πΎ)β€˜π‘€)⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€), 𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (𝑓 ∘ 𝑔))⟩}))
43fveq1d 6886 . . 3 (𝐾 ∈ 𝑉 β†’ ((TGrpβ€˜πΎ)β€˜π‘Š) = ((𝑀 ∈ 𝐻 ↦ {⟨(Baseβ€˜ndx), ((LTrnβ€˜πΎ)β€˜π‘€)⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€), 𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (𝑓 ∘ 𝑔))⟩})β€˜π‘Š))
5 fveq2 6884 . . . . . . 7 (𝑀 = π‘Š β†’ ((LTrnβ€˜πΎ)β€˜π‘€) = ((LTrnβ€˜πΎ)β€˜π‘Š))
65opeq2d 4875 . . . . . 6 (𝑀 = π‘Š β†’ ⟨(Baseβ€˜ndx), ((LTrnβ€˜πΎ)β€˜π‘€)⟩ = ⟨(Baseβ€˜ndx), ((LTrnβ€˜πΎ)β€˜π‘Š)⟩)
7 eqidd 2727 . . . . . . . 8 (𝑀 = π‘Š β†’ (𝑓 ∘ 𝑔) = (𝑓 ∘ 𝑔))
85, 5, 7mpoeq123dv 7479 . . . . . . 7 (𝑀 = π‘Š β†’ (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€), 𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (𝑓 ∘ 𝑔)) = (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š), 𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ (𝑓 ∘ 𝑔)))
98opeq2d 4875 . . . . . 6 (𝑀 = π‘Š β†’ ⟨(+gβ€˜ndx), (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€), 𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (𝑓 ∘ 𝑔))⟩ = ⟨(+gβ€˜ndx), (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š), 𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ (𝑓 ∘ 𝑔))⟩)
106, 9preq12d 4740 . . . . 5 (𝑀 = π‘Š β†’ {⟨(Baseβ€˜ndx), ((LTrnβ€˜πΎ)β€˜π‘€)⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€), 𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (𝑓 ∘ 𝑔))⟩} = {⟨(Baseβ€˜ndx), ((LTrnβ€˜πΎ)β€˜π‘Š)⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š), 𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ (𝑓 ∘ 𝑔))⟩})
11 eqid 2726 . . . . 5 (𝑀 ∈ 𝐻 ↦ {⟨(Baseβ€˜ndx), ((LTrnβ€˜πΎ)β€˜π‘€)⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€), 𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (𝑓 ∘ 𝑔))⟩}) = (𝑀 ∈ 𝐻 ↦ {⟨(Baseβ€˜ndx), ((LTrnβ€˜πΎ)β€˜π‘€)⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€), 𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (𝑓 ∘ 𝑔))⟩})
12 prex 5425 . . . . 5 {⟨(Baseβ€˜ndx), ((LTrnβ€˜πΎ)β€˜π‘Š)⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š), 𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ (𝑓 ∘ 𝑔))⟩} ∈ V
1310, 11, 12fvmpt 6991 . . . 4 (π‘Š ∈ 𝐻 β†’ ((𝑀 ∈ 𝐻 ↦ {⟨(Baseβ€˜ndx), ((LTrnβ€˜πΎ)β€˜π‘€)⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€), 𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (𝑓 ∘ 𝑔))⟩})β€˜π‘Š) = {⟨(Baseβ€˜ndx), ((LTrnβ€˜πΎ)β€˜π‘Š)⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š), 𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ (𝑓 ∘ 𝑔))⟩})
14 tgrpset.t . . . . . 6 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
1514opeq2i 4872 . . . . 5 ⟨(Baseβ€˜ndx), π‘‡βŸ© = ⟨(Baseβ€˜ndx), ((LTrnβ€˜πΎ)β€˜π‘Š)⟩
16 eqid 2726 . . . . . . 7 (𝑓 ∘ 𝑔) = (𝑓 ∘ 𝑔)
1714, 14, 16mpoeq123i 7480 . . . . . 6 (𝑓 ∈ 𝑇, 𝑔 ∈ 𝑇 ↦ (𝑓 ∘ 𝑔)) = (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š), 𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ (𝑓 ∘ 𝑔))
1817opeq2i 4872 . . . . 5 ⟨(+gβ€˜ndx), (𝑓 ∈ 𝑇, 𝑔 ∈ 𝑇 ↦ (𝑓 ∘ 𝑔))⟩ = ⟨(+gβ€˜ndx), (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š), 𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ (𝑓 ∘ 𝑔))⟩
1915, 18preq12i 4737 . . . 4 {⟨(Baseβ€˜ndx), π‘‡βŸ©, ⟨(+gβ€˜ndx), (𝑓 ∈ 𝑇, 𝑔 ∈ 𝑇 ↦ (𝑓 ∘ 𝑔))⟩} = {⟨(Baseβ€˜ndx), ((LTrnβ€˜πΎ)β€˜π‘Š)⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š), 𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ (𝑓 ∘ 𝑔))⟩}
2013, 19eqtr4di 2784 . . 3 (π‘Š ∈ 𝐻 β†’ ((𝑀 ∈ 𝐻 ↦ {⟨(Baseβ€˜ndx), ((LTrnβ€˜πΎ)β€˜π‘€)⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€), 𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (𝑓 ∘ 𝑔))⟩})β€˜π‘Š) = {⟨(Baseβ€˜ndx), π‘‡βŸ©, ⟨(+gβ€˜ndx), (𝑓 ∈ 𝑇, 𝑔 ∈ 𝑇 ↦ (𝑓 ∘ 𝑔))⟩})
214, 20sylan9eq 2786 . 2 ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ ((TGrpβ€˜πΎ)β€˜π‘Š) = {⟨(Baseβ€˜ndx), π‘‡βŸ©, ⟨(+gβ€˜ndx), (𝑓 ∈ 𝑇, 𝑔 ∈ 𝑇 ↦ (𝑓 ∘ 𝑔))⟩})
221, 21eqtrid 2778 1 ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ 𝐺 = {⟨(Baseβ€˜ndx), π‘‡βŸ©, ⟨(+gβ€˜ndx), (𝑓 ∈ 𝑇, 𝑔 ∈ 𝑇 ↦ (𝑓 ∘ 𝑔))⟩})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  {cpr 4625  βŸ¨cop 4629   ↦ cmpt 5224   ∘ ccom 5673  β€˜cfv 6536   ∈ cmpo 7406  ndxcnx 17132  Basecbs 17150  +gcplusg 17203  LHypclh 39367  LTrncltrn 39484  TGrpctgrp 40125
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-oprab 7408  df-mpo 7409  df-tgrp 40126
This theorem is referenced by:  tgrpbase  40129  tgrpopr  40130  dvaabl  40407
  Copyright terms: Public domain W3C validator