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Theorem tgrpfset 39145
Description: The translation group maps for a lattice 𝐾. (Contributed by NM, 5-Jun-2013.)
Hypothesis
Ref Expression
tgrpset.h 𝐻 = (LHypβ€˜πΎ)
Assertion
Ref Expression
tgrpfset (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))⟩}))
Distinct variable groups:   𝑀,𝐻   𝑓,𝑔,𝑀,𝐾
Allowed substitution hints:   𝐻(𝑓,𝑔)   𝑉(𝑀,𝑓,𝑔)

Proof of Theorem tgrpfset
Dummy variable π‘˜ is distinct from all other variables.
StepHypRef Expression
1 elex 3461 . 2 (𝐾 ∈ 𝑉 β†’ 𝐾 ∈ V)
2 fveq2 6839 . . . . 5 (π‘˜ = 𝐾 β†’ (LHypβ€˜π‘˜) = (LHypβ€˜πΎ))
3 tgrpset.h . . . . 5 𝐻 = (LHypβ€˜πΎ)
42, 3eqtr4di 2795 . . . 4 (π‘˜ = 𝐾 β†’ (LHypβ€˜π‘˜) = 𝐻)
5 fveq2 6839 . . . . . . 7 (π‘˜ = 𝐾 β†’ (LTrnβ€˜π‘˜) = (LTrnβ€˜πΎ))
65fveq1d 6841 . . . . . 6 (π‘˜ = 𝐾 β†’ ((LTrnβ€˜π‘˜)β€˜π‘€) = ((LTrnβ€˜πΎ)β€˜π‘€))
76opeq2d 4835 . . . . 5 (π‘˜ = 𝐾 β†’ ⟨(Baseβ€˜ndx), ((LTrnβ€˜π‘˜)β€˜π‘€)⟩ = ⟨(Baseβ€˜ndx), ((LTrnβ€˜πΎ)β€˜π‘€)⟩)
8 eqidd 2738 . . . . . . 7 (π‘˜ = 𝐾 β†’ (𝑓 ∘ 𝑔) = (𝑓 ∘ 𝑔))
96, 6, 8mpoeq123dv 7426 . . . . . 6 (π‘˜ = 𝐾 β†’ (𝑓 ∈ ((LTrnβ€˜π‘˜)β€˜π‘€), 𝑔 ∈ ((LTrnβ€˜π‘˜)β€˜π‘€) ↦ (𝑓 ∘ 𝑔)) = (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€), 𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (𝑓 ∘ 𝑔)))
109opeq2d 4835 . . . . 5 (π‘˜ = 𝐾 β†’ ⟨(+gβ€˜ndx), (𝑓 ∈ ((LTrnβ€˜π‘˜)β€˜π‘€), 𝑔 ∈ ((LTrnβ€˜π‘˜)β€˜π‘€) ↦ (𝑓 ∘ 𝑔))⟩ = ⟨(+gβ€˜ndx), (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€), 𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (𝑓 ∘ 𝑔))⟩)
117, 10preq12d 4700 . . . 4 (π‘˜ = 𝐾 β†’ {⟨(Baseβ€˜ndx), ((LTrnβ€˜π‘˜)β€˜π‘€)⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ ((LTrnβ€˜π‘˜)β€˜π‘€), 𝑔 ∈ ((LTrnβ€˜π‘˜)β€˜π‘€) ↦ (𝑓 ∘ 𝑔))⟩} = {⟨(Baseβ€˜ndx), ((LTrnβ€˜πΎ)β€˜π‘€)⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€), 𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (𝑓 ∘ 𝑔))⟩})
124, 11mpteq12dv 5194 . . 3 (π‘˜ = 𝐾 β†’ (𝑀 ∈ (LHypβ€˜π‘˜) ↦ {⟨(Baseβ€˜ndx), ((LTrnβ€˜π‘˜)β€˜π‘€)⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ ((LTrnβ€˜π‘˜)β€˜π‘€), 𝑔 ∈ ((LTrnβ€˜π‘˜)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))⟩}))
13 df-tgrp 39144 . . 3 TGrp = (π‘˜ ∈ V ↦ (𝑀 ∈ (LHypβ€˜π‘˜) ↦ {⟨(Baseβ€˜ndx), ((LTrnβ€˜π‘˜)β€˜π‘€)⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ ((LTrnβ€˜π‘˜)β€˜π‘€), 𝑔 ∈ ((LTrnβ€˜π‘˜)β€˜π‘€) ↦ (𝑓 ∘ 𝑔))⟩}))
1412, 13, 3mptfvmpt 7174 . 2 (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))⟩}))
151, 14syl 17 1 (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))⟩}))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1541   ∈ wcel 2106  Vcvv 3443  {cpr 4586  βŸ¨cop 4590   ↦ cmpt 5186   ∘ ccom 5635  β€˜cfv 6493   ∈ cmpo 7353  ndxcnx 17025  Basecbs 17043  +gcplusg 17093  LHypclh 38385  LTrncltrn 38502  TGrpctgrp 39143
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-oprab 7355  df-mpo 7356  df-tgrp 39144
This theorem is referenced by:  tgrpset  39146
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