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Theorem tgrpset 40250
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 40249 . . . 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 6904 . . 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 6902 . . . . . . 7 (𝑀 = π‘Š β†’ ((LTrnβ€˜πΎ)β€˜π‘€) = ((LTrnβ€˜πΎ)β€˜π‘Š))
65opeq2d 4885 . . . . . 6 (𝑀 = π‘Š β†’ ⟨(Baseβ€˜ndx), ((LTrnβ€˜πΎ)β€˜π‘€)⟩ = ⟨(Baseβ€˜ndx), ((LTrnβ€˜πΎ)β€˜π‘Š)⟩)
7 eqidd 2729 . . . . . . . 8 (𝑀 = π‘Š β†’ (𝑓 ∘ 𝑔) = (𝑓 ∘ 𝑔))
85, 5, 7mpoeq123dv 7501 . . . . . . 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 4885 . . . . . 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 4750 . . . . 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 2728 . . . . 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 5438 . . . . 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 7010 . . . 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 4882 . . . . 5 ⟨(Baseβ€˜ndx), π‘‡βŸ© = ⟨(Baseβ€˜ndx), ((LTrnβ€˜πΎ)β€˜π‘Š)⟩
16 eqid 2728 . . . . . . 7 (𝑓 ∘ 𝑔) = (𝑓 ∘ 𝑔)
1714, 14, 16mpoeq123i 7502 . . . . . 6 (𝑓 ∈ 𝑇, 𝑔 ∈ 𝑇 ↦ (𝑓 ∘ 𝑔)) = (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š), 𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ (𝑓 ∘ 𝑔))
1817opeq2i 4882 . . . . 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 4747 . . . 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 2786 . . 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 2788 . 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 2780 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 394   = wceq 1533   ∈ wcel 2098  {cpr 4634  βŸ¨cop 4638   ↦ cmpt 5235   ∘ ccom 5686  β€˜cfv 6553   ∈ cmpo 7428  ndxcnx 17169  Basecbs 17187  +gcplusg 17240  LHypclh 39489  LTrncltrn 39606  TGrpctgrp 40247
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 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-oprab 7430  df-mpo 7431  df-tgrp 40248
This theorem is referenced by:  tgrpbase  40251  tgrpopr  40252  dvaabl  40529
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