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Theorem tgrpset 35051
Description: The translation group for a fiducial co-atom 𝑊. (Contributed by NM, 5-Jun-2013.)
Hypotheses
Ref Expression
tgrpset.h 𝐻 = (LHyp‘𝐾)
tgrpset.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
tgrpset.g 𝐺 = ((TGrp‘𝐾)‘𝑊)
Assertion
Ref Expression
tgrpset ((𝐾𝑉𝑊𝐻) → 𝐺 = {⟨(Base‘ndx), 𝑇⟩, ⟨(+g‘ndx), (𝑓𝑇, 𝑔𝑇 ↦ (𝑓𝑔))⟩})
Distinct variable groups:   𝑓,𝑔,𝐾   𝑇,𝑓,𝑔   𝑓,𝑊,𝑔
Allowed substitution hints:   𝐺(𝑓,𝑔)   𝐻(𝑓,𝑔)   𝑉(𝑓,𝑔)

Proof of Theorem tgrpset
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 tgrpset.g . 2 𝐺 = ((TGrp‘𝐾)‘𝑊)
2 tgrpset.h . . . . 5 𝐻 = (LHyp‘𝐾)
32tgrpfset 35050 . . . 4 (𝐾𝑉 → (TGrp‘𝐾) = (𝑤𝐻 ↦ {⟨(Base‘ndx), ((LTrn‘𝐾)‘𝑤)⟩, ⟨(+g‘ndx), (𝑓 ∈ ((LTrn‘𝐾)‘𝑤), 𝑔 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑓𝑔))⟩}))
43fveq1d 6105 . . 3 (𝐾𝑉 → ((TGrp‘𝐾)‘𝑊) = ((𝑤𝐻 ↦ {⟨(Base‘ndx), ((LTrn‘𝐾)‘𝑤)⟩, ⟨(+g‘ndx), (𝑓 ∈ ((LTrn‘𝐾)‘𝑤), 𝑔 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑓𝑔))⟩})‘𝑊))
5 fveq2 6103 . . . . . . 7 (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = ((LTrn‘𝐾)‘𝑊))
65opeq2d 4347 . . . . . 6 (𝑤 = 𝑊 → ⟨(Base‘ndx), ((LTrn‘𝐾)‘𝑤)⟩ = ⟨(Base‘ndx), ((LTrn‘𝐾)‘𝑊)⟩)
7 eqidd 2611 . . . . . . . 8 (𝑤 = 𝑊 → (𝑓𝑔) = (𝑓𝑔))
85, 5, 7mpt2eq123dv 6615 . . . . . . 7 (𝑤 = 𝑊 → (𝑓 ∈ ((LTrn‘𝐾)‘𝑤), 𝑔 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑓𝑔)) = (𝑓 ∈ ((LTrn‘𝐾)‘𝑊), 𝑔 ∈ ((LTrn‘𝐾)‘𝑊) ↦ (𝑓𝑔)))
98opeq2d 4347 . . . . . 6 (𝑤 = 𝑊 → ⟨(+g‘ndx), (𝑓 ∈ ((LTrn‘𝐾)‘𝑤), 𝑔 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑓𝑔))⟩ = ⟨(+g‘ndx), (𝑓 ∈ ((LTrn‘𝐾)‘𝑊), 𝑔 ∈ ((LTrn‘𝐾)‘𝑊) ↦ (𝑓𝑔))⟩)
106, 9preq12d 4220 . . . . 5 (𝑤 = 𝑊 → {⟨(Base‘ndx), ((LTrn‘𝐾)‘𝑤)⟩, ⟨(+g‘ndx), (𝑓 ∈ ((LTrn‘𝐾)‘𝑤), 𝑔 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑓𝑔))⟩} = {⟨(Base‘ndx), ((LTrn‘𝐾)‘𝑊)⟩, ⟨(+g‘ndx), (𝑓 ∈ ((LTrn‘𝐾)‘𝑊), 𝑔 ∈ ((LTrn‘𝐾)‘𝑊) ↦ (𝑓𝑔))⟩})
11 eqid 2610 . . . . 5 (𝑤𝐻 ↦ {⟨(Base‘ndx), ((LTrn‘𝐾)‘𝑤)⟩, ⟨(+g‘ndx), (𝑓 ∈ ((LTrn‘𝐾)‘𝑤), 𝑔 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑓𝑔))⟩}) = (𝑤𝐻 ↦ {⟨(Base‘ndx), ((LTrn‘𝐾)‘𝑤)⟩, ⟨(+g‘ndx), (𝑓 ∈ ((LTrn‘𝐾)‘𝑤), 𝑔 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑓𝑔))⟩})
12 prex 4836 . . . . 5 {⟨(Base‘ndx), ((LTrn‘𝐾)‘𝑊)⟩, ⟨(+g‘ndx), (𝑓 ∈ ((LTrn‘𝐾)‘𝑊), 𝑔 ∈ ((LTrn‘𝐾)‘𝑊) ↦ (𝑓𝑔))⟩} ∈ V
1310, 11, 12fvmpt 6191 . . . 4 (𝑊𝐻 → ((𝑤𝐻 ↦ {⟨(Base‘ndx), ((LTrn‘𝐾)‘𝑤)⟩, ⟨(+g‘ndx), (𝑓 ∈ ((LTrn‘𝐾)‘𝑤), 𝑔 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑓𝑔))⟩})‘𝑊) = {⟨(Base‘ndx), ((LTrn‘𝐾)‘𝑊)⟩, ⟨(+g‘ndx), (𝑓 ∈ ((LTrn‘𝐾)‘𝑊), 𝑔 ∈ ((LTrn‘𝐾)‘𝑊) ↦ (𝑓𝑔))⟩})
14 tgrpset.t . . . . . 6 𝑇 = ((LTrn‘𝐾)‘𝑊)
1514opeq2i 4344 . . . . 5 ⟨(Base‘ndx), 𝑇⟩ = ⟨(Base‘ndx), ((LTrn‘𝐾)‘𝑊)⟩
16 eqid 2610 . . . . . . 7 (𝑓𝑔) = (𝑓𝑔)
1714, 14, 16mpt2eq123i 6616 . . . . . 6 (𝑓𝑇, 𝑔𝑇 ↦ (𝑓𝑔)) = (𝑓 ∈ ((LTrn‘𝐾)‘𝑊), 𝑔 ∈ ((LTrn‘𝐾)‘𝑊) ↦ (𝑓𝑔))
1817opeq2i 4344 . . . . 5 ⟨(+g‘ndx), (𝑓𝑇, 𝑔𝑇 ↦ (𝑓𝑔))⟩ = ⟨(+g‘ndx), (𝑓 ∈ ((LTrn‘𝐾)‘𝑊), 𝑔 ∈ ((LTrn‘𝐾)‘𝑊) ↦ (𝑓𝑔))⟩
1915, 18preq12i 4217 . . . 4 {⟨(Base‘ndx), 𝑇⟩, ⟨(+g‘ndx), (𝑓𝑇, 𝑔𝑇 ↦ (𝑓𝑔))⟩} = {⟨(Base‘ndx), ((LTrn‘𝐾)‘𝑊)⟩, ⟨(+g‘ndx), (𝑓 ∈ ((LTrn‘𝐾)‘𝑊), 𝑔 ∈ ((LTrn‘𝐾)‘𝑊) ↦ (𝑓𝑔))⟩}
2013, 19syl6eqr 2662 . . 3 (𝑊𝐻 → ((𝑤𝐻 ↦ {⟨(Base‘ndx), ((LTrn‘𝐾)‘𝑤)⟩, ⟨(+g‘ndx), (𝑓 ∈ ((LTrn‘𝐾)‘𝑤), 𝑔 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑓𝑔))⟩})‘𝑊) = {⟨(Base‘ndx), 𝑇⟩, ⟨(+g‘ndx), (𝑓𝑇, 𝑔𝑇 ↦ (𝑓𝑔))⟩})
214, 20sylan9eq 2664 . 2 ((𝐾𝑉𝑊𝐻) → ((TGrp‘𝐾)‘𝑊) = {⟨(Base‘ndx), 𝑇⟩, ⟨(+g‘ndx), (𝑓𝑇, 𝑔𝑇 ↦ (𝑓𝑔))⟩})
221, 21syl5eq 2656 1 ((𝐾𝑉𝑊𝐻) → 𝐺 = {⟨(Base‘ndx), 𝑇⟩, ⟨(+g‘ndx), (𝑓𝑇, 𝑔𝑇 ↦ (𝑓𝑔))⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  {cpr 4127  cop 4131  cmpt 4643  ccom 5042  cfv 5804  cmpt2 6551  ndxcnx 15692  Basecbs 15695  +gcplusg 15768  LHypclh 34288  LTrncltrn 34405  TGrpctgrp 35048
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-oprab 6553  df-mpt2 6554  df-tgrp 35049
This theorem is referenced by:  tgrpbase  35052  tgrpopr  35053  dvaabl  35331
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