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Theorem tngval 22353
Description: Value of the function which augments a given structure 𝐺 with a norm 𝑁. (Contributed by Mario Carneiro, 2-Oct-2015.)
Hypotheses
Ref Expression
tngval.t 𝑇 = (𝐺 toNrmGrp 𝑁)
tngval.m = (-g𝐺)
tngval.d 𝐷 = (𝑁 )
tngval.j 𝐽 = (MetOpen‘𝐷)
Assertion
Ref Expression
tngval ((𝐺𝑉𝑁𝑊) → 𝑇 = ((𝐺 sSet ⟨(dist‘ndx), 𝐷⟩) sSet ⟨(TopSet‘ndx), 𝐽⟩))

Proof of Theorem tngval
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tngval.t . 2 𝑇 = (𝐺 toNrmGrp 𝑁)
2 elex 3198 . . 3 (𝐺𝑉𝐺 ∈ V)
3 elex 3198 . . 3 (𝑁𝑊𝑁 ∈ V)
4 simpl 473 . . . . . 6 ((𝑔 = 𝐺𝑓 = 𝑁) → 𝑔 = 𝐺)
5 simpr 477 . . . . . . . . 9 ((𝑔 = 𝐺𝑓 = 𝑁) → 𝑓 = 𝑁)
64fveq2d 6152 . . . . . . . . . 10 ((𝑔 = 𝐺𝑓 = 𝑁) → (-g𝑔) = (-g𝐺))
7 tngval.m . . . . . . . . . 10 = (-g𝐺)
86, 7syl6eqr 2673 . . . . . . . . 9 ((𝑔 = 𝐺𝑓 = 𝑁) → (-g𝑔) = )
95, 8coeq12d 5246 . . . . . . . 8 ((𝑔 = 𝐺𝑓 = 𝑁) → (𝑓 ∘ (-g𝑔)) = (𝑁 ))
10 tngval.d . . . . . . . 8 𝐷 = (𝑁 )
119, 10syl6eqr 2673 . . . . . . 7 ((𝑔 = 𝐺𝑓 = 𝑁) → (𝑓 ∘ (-g𝑔)) = 𝐷)
1211opeq2d 4377 . . . . . 6 ((𝑔 = 𝐺𝑓 = 𝑁) → ⟨(dist‘ndx), (𝑓 ∘ (-g𝑔))⟩ = ⟨(dist‘ndx), 𝐷⟩)
134, 12oveq12d 6622 . . . . 5 ((𝑔 = 𝐺𝑓 = 𝑁) → (𝑔 sSet ⟨(dist‘ndx), (𝑓 ∘ (-g𝑔))⟩) = (𝐺 sSet ⟨(dist‘ndx), 𝐷⟩))
1411fveq2d 6152 . . . . . . 7 ((𝑔 = 𝐺𝑓 = 𝑁) → (MetOpen‘(𝑓 ∘ (-g𝑔))) = (MetOpen‘𝐷))
15 tngval.j . . . . . . 7 𝐽 = (MetOpen‘𝐷)
1614, 15syl6eqr 2673 . . . . . 6 ((𝑔 = 𝐺𝑓 = 𝑁) → (MetOpen‘(𝑓 ∘ (-g𝑔))) = 𝐽)
1716opeq2d 4377 . . . . 5 ((𝑔 = 𝐺𝑓 = 𝑁) → ⟨(TopSet‘ndx), (MetOpen‘(𝑓 ∘ (-g𝑔)))⟩ = ⟨(TopSet‘ndx), 𝐽⟩)
1813, 17oveq12d 6622 . . . 4 ((𝑔 = 𝐺𝑓 = 𝑁) → ((𝑔 sSet ⟨(dist‘ndx), (𝑓 ∘ (-g𝑔))⟩) sSet ⟨(TopSet‘ndx), (MetOpen‘(𝑓 ∘ (-g𝑔)))⟩) = ((𝐺 sSet ⟨(dist‘ndx), 𝐷⟩) sSet ⟨(TopSet‘ndx), 𝐽⟩))
19 df-tng 22299 . . . 4 toNrmGrp = (𝑔 ∈ V, 𝑓 ∈ V ↦ ((𝑔 sSet ⟨(dist‘ndx), (𝑓 ∘ (-g𝑔))⟩) sSet ⟨(TopSet‘ndx), (MetOpen‘(𝑓 ∘ (-g𝑔)))⟩))
20 ovex 6632 . . . 4 ((𝐺 sSet ⟨(dist‘ndx), 𝐷⟩) sSet ⟨(TopSet‘ndx), 𝐽⟩) ∈ V
2118, 19, 20ovmpt2a 6744 . . 3 ((𝐺 ∈ V ∧ 𝑁 ∈ V) → (𝐺 toNrmGrp 𝑁) = ((𝐺 sSet ⟨(dist‘ndx), 𝐷⟩) sSet ⟨(TopSet‘ndx), 𝐽⟩))
222, 3, 21syl2an 494 . 2 ((𝐺𝑉𝑁𝑊) → (𝐺 toNrmGrp 𝑁) = ((𝐺 sSet ⟨(dist‘ndx), 𝐷⟩) sSet ⟨(TopSet‘ndx), 𝐽⟩))
231, 22syl5eq 2667 1 ((𝐺𝑉𝑁𝑊) → 𝑇 = ((𝐺 sSet ⟨(dist‘ndx), 𝐷⟩) sSet ⟨(TopSet‘ndx), 𝐽⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  Vcvv 3186  cop 4154  ccom 5078  cfv 5847  (class class class)co 6604  ndxcnx 15778   sSet csts 15779  TopSetcts 15868  distcds 15871  -gcsg 17345  MetOpencmopn 19655   toNrmGrp ctng 22293
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-iota 5810  df-fun 5849  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-tng 22299
This theorem is referenced by:  tnglem  22354  tngds  22362  tngtset  22363
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