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Theorem tnglem 24149
Description: Lemma for tngbas 24151 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 (πΈβ€˜ndx)
tnglem.t (πΈβ€˜ndx) β‰  (TopSetβ€˜ndx)
tnglem.d (πΈβ€˜ndx) β‰  (distβ€˜ndx)
Assertion
Ref Expression
tnglem (𝑁 ∈ 𝑉 β†’ (πΈβ€˜πΊ) = (πΈβ€˜π‘‡))

Proof of Theorem tnglem
StepHypRef Expression
1 tnglem.e . . . . 5 𝐸 = Slot (πΈβ€˜ndx)
2 tnglem.d . . . . 5 (πΈβ€˜ndx) β‰  (distβ€˜ndx)
31, 2setsnid 17142 . . . 4 (πΈβ€˜πΊ) = (πΈβ€˜(𝐺 sSet ⟨(distβ€˜ndx), (𝑁 ∘ (-gβ€˜πΊ))⟩))
4 tnglem.t . . . . 5 (πΈβ€˜ndx) β‰  (TopSetβ€˜ndx)
51, 4setsnid 17142 . . . 4 (πΈβ€˜(𝐺 sSet ⟨(distβ€˜ndx), (𝑁 ∘ (-gβ€˜πΊ))⟩)) = (πΈβ€˜((𝐺 sSet ⟨(distβ€˜ndx), (𝑁 ∘ (-gβ€˜πΊ))⟩) sSet ⟨(TopSetβ€˜ndx), (MetOpenβ€˜(𝑁 ∘ (-gβ€˜πΊ)))⟩))
63, 5eqtri 2761 . . 3 (πΈβ€˜πΊ) = (πΈβ€˜((𝐺 sSet ⟨(distβ€˜ndx), (𝑁 ∘ (-gβ€˜πΊ))⟩) sSet ⟨(TopSetβ€˜ndx), (MetOpenβ€˜(𝑁 ∘ (-gβ€˜πΊ)))⟩))
7 tngbas.t . . . . 5 𝑇 = (𝐺 toNrmGrp 𝑁)
8 eqid 2733 . . . . 5 (-gβ€˜πΊ) = (-gβ€˜πΊ)
9 eqid 2733 . . . . 5 (𝑁 ∘ (-gβ€˜πΊ)) = (𝑁 ∘ (-gβ€˜πΊ))
10 eqid 2733 . . . . 5 (MetOpenβ€˜(𝑁 ∘ (-gβ€˜πΊ))) = (MetOpenβ€˜(𝑁 ∘ (-gβ€˜πΊ)))
117, 8, 9, 10tngval 24148 . . . 4 ((𝐺 ∈ V ∧ 𝑁 ∈ 𝑉) β†’ 𝑇 = ((𝐺 sSet ⟨(distβ€˜ndx), (𝑁 ∘ (-gβ€˜πΊ))⟩) sSet ⟨(TopSetβ€˜ndx), (MetOpenβ€˜(𝑁 ∘ (-gβ€˜πΊ)))⟩))
1211fveq2d 6896 . . 3 ((𝐺 ∈ V ∧ 𝑁 ∈ 𝑉) β†’ (πΈβ€˜π‘‡) = (πΈβ€˜((𝐺 sSet ⟨(distβ€˜ndx), (𝑁 ∘ (-gβ€˜πΊ))⟩) sSet ⟨(TopSetβ€˜ndx), (MetOpenβ€˜(𝑁 ∘ (-gβ€˜πΊ)))⟩)))
136, 12eqtr4id 2792 . 2 ((𝐺 ∈ V ∧ 𝑁 ∈ 𝑉) β†’ (πΈβ€˜πΊ) = (πΈβ€˜π‘‡))
141str0 17122 . . . . 5 βˆ… = (πΈβ€˜βˆ…)
1514eqcomi 2742 . . . 4 (πΈβ€˜βˆ…) = βˆ…
16 reldmtng 24147 . . . 4 Rel dom toNrmGrp
1715, 7, 16oveqprc 17125 . . 3 (Β¬ 𝐺 ∈ V β†’ (πΈβ€˜πΊ) = (πΈβ€˜π‘‡))
1817adantr 482 . 2 ((Β¬ 𝐺 ∈ V ∧ 𝑁 ∈ 𝑉) β†’ (πΈβ€˜πΊ) = (πΈβ€˜π‘‡))
1913, 18pm2.61ian 811 1 (𝑁 ∈ 𝑉 β†’ (πΈβ€˜πΊ) = (πΈβ€˜π‘‡))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  Vcvv 3475  βˆ…c0 4323  βŸ¨cop 4635   ∘ ccom 5681  β€˜cfv 6544  (class class class)co 7409   sSet csts 17096  Slot cslot 17114  ndxcnx 17126  TopSetcts 17203  distcds 17206  -gcsg 18821  MetOpencmopn 20934   toNrmGrp ctng 24087
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-res 5689  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-sets 17097  df-slot 17115  df-tng 24093
This theorem is referenced by:  tngbas  24151  tngplusg  24153  tngmulr  24156  tngsca  24158  tngvsca  24160  tngip  24162
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