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Theorem tnglem 24653
Description: Lemma for tngbas 24655 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 17245 . . . 4 (𝐸𝐺) = (𝐸‘(𝐺 sSet ⟨(dist‘ndx), (𝑁 ∘ (-g𝐺))⟩))
4 tnglem.t . . . . 5 (𝐸‘ndx) ≠ (TopSet‘ndx)
51, 4setsnid 17245 . . . 4 (𝐸‘(𝐺 sSet ⟨(dist‘ndx), (𝑁 ∘ (-g𝐺))⟩)) = (𝐸‘((𝐺 sSet ⟨(dist‘ndx), (𝑁 ∘ (-g𝐺))⟩) sSet ⟨(TopSet‘ndx), (MetOpen‘(𝑁 ∘ (-g𝐺)))⟩))
63, 5eqtri 2765 . . 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 2737 . . . . 5 (-g𝐺) = (-g𝐺)
9 eqid 2737 . . . . 5 (𝑁 ∘ (-g𝐺)) = (𝑁 ∘ (-g𝐺))
10 eqid 2737 . . . . 5 (MetOpen‘(𝑁 ∘ (-g𝐺))) = (MetOpen‘(𝑁 ∘ (-g𝐺)))
117, 8, 9, 10tngval 24652 . . . 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 2796 . 2 ((𝐺 ∈ V ∧ 𝑁𝑉) → (𝐸𝐺) = (𝐸𝑇))
141str0 17226 . . . . 5 ∅ = (𝐸‘∅)
1514eqcomi 2746 . . . 4 (𝐸‘∅) = ∅
16 reldmtng 24651 . . . 4 Rel dom toNrmGrp
1715, 7, 16oveqprc 17229 . . 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 1540  wcel 2108  wne 2940  Vcvv 3480  c0 4333  cop 4632  ccom 5689  cfv 6561  (class class class)co 7431   sSet csts 17200  Slot cslot 17218  ndxcnx 17230  TopSetcts 17303  distcds 17306  -gcsg 18953  MetOpencmopn 21354   toNrmGrp ctng 24591
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-res 5697  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-sets 17201  df-slot 17219  df-tng 24597
This theorem is referenced by:  tngbas  24655  tngplusg  24657  tngmulr  24660  tngsca  24662  tngvsca  24664  tngip  24666
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