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Theorem tnglem 24140
Description: Lemma for tngbas 24142 and similar theorems. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 31-Oct-2024.)
Hypotheses
Ref Expression
tngbas.t 𝑇 = (𝐺 toNrmGrp 𝑁)
tnglem.e 𝐸 = Slot (𝐸‘ndx)
tnglem.t (𝐸‘ndx) ≠ (TopSet‘ndx)
tnglem.d (𝐸‘ndx) ≠ (dist‘ndx)
Assertion
Ref Expression
tnglem (𝑁𝑉 → (𝐸𝐺) = (𝐸𝑇))

Proof of Theorem tnglem
StepHypRef Expression
1 tnglem.e . . . . 5 𝐸 = Slot (𝐸‘ndx)
2 tnglem.d . . . . 5 (𝐸‘ndx) ≠ (dist‘ndx)
31, 2setsnid 17138 . . . 4 (𝐸𝐺) = (𝐸‘(𝐺 sSet ⟨(dist‘ndx), (𝑁 ∘ (-g𝐺))⟩))
4 tnglem.t . . . . 5 (𝐸‘ndx) ≠ (TopSet‘ndx)
51, 4setsnid 17138 . . . 4 (𝐸‘(𝐺 sSet ⟨(dist‘ndx), (𝑁 ∘ (-g𝐺))⟩)) = (𝐸‘((𝐺 sSet ⟨(dist‘ndx), (𝑁 ∘ (-g𝐺))⟩) sSet ⟨(TopSet‘ndx), (MetOpen‘(𝑁 ∘ (-g𝐺)))⟩))
63, 5eqtri 2760 . . 3 (𝐸𝐺) = (𝐸‘((𝐺 sSet ⟨(dist‘ndx), (𝑁 ∘ (-g𝐺))⟩) sSet ⟨(TopSet‘ndx), (MetOpen‘(𝑁 ∘ (-g𝐺)))⟩))
7 tngbas.t . . . . 5 𝑇 = (𝐺 toNrmGrp 𝑁)
8 eqid 2732 . . . . 5 (-g𝐺) = (-g𝐺)
9 eqid 2732 . . . . 5 (𝑁 ∘ (-g𝐺)) = (𝑁 ∘ (-g𝐺))
10 eqid 2732 . . . . 5 (MetOpen‘(𝑁 ∘ (-g𝐺))) = (MetOpen‘(𝑁 ∘ (-g𝐺)))
117, 8, 9, 10tngval 24139 . . . 4 ((𝐺 ∈ V ∧ 𝑁𝑉) → 𝑇 = ((𝐺 sSet ⟨(dist‘ndx), (𝑁 ∘ (-g𝐺))⟩) sSet ⟨(TopSet‘ndx), (MetOpen‘(𝑁 ∘ (-g𝐺)))⟩))
1211fveq2d 6892 . . 3 ((𝐺 ∈ V ∧ 𝑁𝑉) → (𝐸𝑇) = (𝐸‘((𝐺 sSet ⟨(dist‘ndx), (𝑁 ∘ (-g𝐺))⟩) sSet ⟨(TopSet‘ndx), (MetOpen‘(𝑁 ∘ (-g𝐺)))⟩)))
136, 12eqtr4id 2791 . 2 ((𝐺 ∈ V ∧ 𝑁𝑉) → (𝐸𝐺) = (𝐸𝑇))
141str0 17118 . . . . 5 ∅ = (𝐸‘∅)
1514eqcomi 2741 . . . 4 (𝐸‘∅) = ∅
16 reldmtng 24138 . . . 4 Rel dom toNrmGrp
1715, 7, 16oveqprc 17121 . . 3 𝐺 ∈ V → (𝐸𝐺) = (𝐸𝑇))
1817adantr 481 . 2 ((¬ 𝐺 ∈ V ∧ 𝑁𝑉) → (𝐸𝐺) = (𝐸𝑇))
1913, 18pm2.61ian 810 1 (𝑁𝑉 → (𝐸𝐺) = (𝐸𝑇))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1541  wcel 2106  wne 2940  Vcvv 3474  c0 4321  cop 4633  ccom 5679  cfv 6540  (class class class)co 7405   sSet csts 17092  Slot cslot 17110  ndxcnx 17122  TopSetcts 17199  distcds 17202  -gcsg 18817  MetOpencmopn 20926   toNrmGrp ctng 24078
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 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7721
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-res 5687  df-iota 6492  df-fun 6542  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-sets 17093  df-slot 17111  df-tng 24084
This theorem is referenced by:  tngbas  24142  tngplusg  24144  tngmulr  24147  tngsca  24149  tngvsca  24151  tngip  24153
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