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Theorem tnglem 24148
Description: Lemma for tngbas 24150 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 17141 . . . 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 17141 . . . 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 2760 . . 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 2732 . . . . 5 (-gβ€˜πΊ) = (-gβ€˜πΊ)
9 eqid 2732 . . . . 5 (𝑁 ∘ (-gβ€˜πΊ)) = (𝑁 ∘ (-gβ€˜πΊ))
10 eqid 2732 . . . . 5 (MetOpenβ€˜(𝑁 ∘ (-gβ€˜πΊ))) = (MetOpenβ€˜(𝑁 ∘ (-gβ€˜πΊ)))
117, 8, 9, 10tngval 24147 . . . 4 ((𝐺 ∈ V ∧ 𝑁 ∈ 𝑉) β†’ 𝑇 = ((𝐺 sSet ⟨(distβ€˜ndx), (𝑁 ∘ (-gβ€˜πΊ))⟩) sSet ⟨(TopSetβ€˜ndx), (MetOpenβ€˜(𝑁 ∘ (-gβ€˜πΊ)))⟩))
1211fveq2d 6895 . . 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 2791 . 2 ((𝐺 ∈ V ∧ 𝑁 ∈ 𝑉) β†’ (πΈβ€˜πΊ) = (πΈβ€˜π‘‡))
141str0 17121 . . . . 5 βˆ… = (πΈβ€˜βˆ…)
1514eqcomi 2741 . . . 4 (πΈβ€˜βˆ…) = βˆ…
16 reldmtng 24146 . . . 4 Rel dom toNrmGrp
1715, 7, 16oveqprc 17124 . . 3 (Β¬ 𝐺 ∈ V β†’ (πΈβ€˜πΊ) = (πΈβ€˜π‘‡))
1817adantr 481 . 2 ((Β¬ 𝐺 ∈ V ∧ 𝑁 ∈ 𝑉) β†’ (πΈβ€˜πΊ) = (πΈβ€˜π‘‡))
1913, 18pm2.61ian 810 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 396   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  Vcvv 3474  βˆ…c0 4322  βŸ¨cop 4634   ∘ ccom 5680  β€˜cfv 6543  (class class class)co 7408   sSet csts 17095  Slot cslot 17113  ndxcnx 17125  TopSetcts 17202  distcds 17205  -gcsg 18820  MetOpencmopn 20933   toNrmGrp ctng 24086
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 5299  ax-nul 5306  ax-pr 5427  ax-un 7724
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 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-res 5688  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-sets 17096  df-slot 17114  df-tng 24092
This theorem is referenced by:  tngbas  24150  tngplusg  24152  tngmulr  24155  tngsca  24157  tngvsca  24159  tngip  24161
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