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Theorem tnglem 24668
Description: Lemma for tngbas 24670 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 17242 . . . 4 (𝐸𝐺) = (𝐸‘(𝐺 sSet ⟨(dist‘ndx), (𝑁 ∘ (-g𝐺))⟩))
4 tnglem.t . . . . 5 (𝐸‘ndx) ≠ (TopSet‘ndx)
51, 4setsnid 17242 . . . 4 (𝐸‘(𝐺 sSet ⟨(dist‘ndx), (𝑁 ∘ (-g𝐺))⟩)) = (𝐸‘((𝐺 sSet ⟨(dist‘ndx), (𝑁 ∘ (-g𝐺))⟩) sSet ⟨(TopSet‘ndx), (MetOpen‘(𝑁 ∘ (-g𝐺)))⟩))
63, 5eqtri 2762 . . 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 2734 . . . . 5 (-g𝐺) = (-g𝐺)
9 eqid 2734 . . . . 5 (𝑁 ∘ (-g𝐺)) = (𝑁 ∘ (-g𝐺))
10 eqid 2734 . . . . 5 (MetOpen‘(𝑁 ∘ (-g𝐺))) = (MetOpen‘(𝑁 ∘ (-g𝐺)))
117, 8, 9, 10tngval 24667 . . . 4 ((𝐺 ∈ V ∧ 𝑁𝑉) → 𝑇 = ((𝐺 sSet ⟨(dist‘ndx), (𝑁 ∘ (-g𝐺))⟩) sSet ⟨(TopSet‘ndx), (MetOpen‘(𝑁 ∘ (-g𝐺)))⟩))
1211fveq2d 6910 . . 3 ((𝐺 ∈ V ∧ 𝑁𝑉) → (𝐸𝑇) = (𝐸‘((𝐺 sSet ⟨(dist‘ndx), (𝑁 ∘ (-g𝐺))⟩) sSet ⟨(TopSet‘ndx), (MetOpen‘(𝑁 ∘ (-g𝐺)))⟩)))
136, 12eqtr4id 2793 . 2 ((𝐺 ∈ V ∧ 𝑁𝑉) → (𝐸𝐺) = (𝐸𝑇))
141str0 17222 . . . . 5 ∅ = (𝐸‘∅)
1514eqcomi 2743 . . . 4 (𝐸‘∅) = ∅
16 reldmtng 24666 . . . 4 Rel dom toNrmGrp
1715, 7, 16oveqprc 17225 . . 3 𝐺 ∈ V → (𝐸𝐺) = (𝐸𝑇))
1817adantr 480 . 2 ((¬ 𝐺 ∈ V ∧ 𝑁𝑉) → (𝐸𝐺) = (𝐸𝑇))
1913, 18pm2.61ian 812 1 (𝑁𝑉 → (𝐸𝐺) = (𝐸𝑇))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1536  wcel 2105  wne 2937  Vcvv 3477  c0 4338  cop 4636  ccom 5692  cfv 6562  (class class class)co 7430   sSet csts 17196  Slot cslot 17214  ndxcnx 17226  TopSetcts 17303  distcds 17306  -gcsg 18965  MetOpencmopn 21371   toNrmGrp ctng 24606
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-res 5700  df-iota 6515  df-fun 6564  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-sets 17197  df-slot 17215  df-tng 24612
This theorem is referenced by:  tngbas  24670  tngplusg  24672  tngmulr  24675  tngsca  24677  tngvsca  24679  tngip  24681
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