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Theorem tngval 24668
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 3499 . . 3 (𝐺𝑉𝐺 ∈ V)
3 elex 3499 . . 3 (𝑁𝑊𝑁 ∈ V)
4 simpl 482 . . . . . 6 ((𝑔 = 𝐺𝑓 = 𝑁) → 𝑔 = 𝐺)
5 simpr 484 . . . . . . . . 9 ((𝑔 = 𝐺𝑓 = 𝑁) → 𝑓 = 𝑁)
64fveq2d 6911 . . . . . . . . . 10 ((𝑔 = 𝐺𝑓 = 𝑁) → (-g𝑔) = (-g𝐺))
7 tngval.m . . . . . . . . . 10 = (-g𝐺)
86, 7eqtr4di 2793 . . . . . . . . 9 ((𝑔 = 𝐺𝑓 = 𝑁) → (-g𝑔) = )
95, 8coeq12d 5878 . . . . . . . 8 ((𝑔 = 𝐺𝑓 = 𝑁) → (𝑓 ∘ (-g𝑔)) = (𝑁 ))
10 tngval.d . . . . . . . 8 𝐷 = (𝑁 )
119, 10eqtr4di 2793 . . . . . . 7 ((𝑔 = 𝐺𝑓 = 𝑁) → (𝑓 ∘ (-g𝑔)) = 𝐷)
1211opeq2d 4885 . . . . . 6 ((𝑔 = 𝐺𝑓 = 𝑁) → ⟨(dist‘ndx), (𝑓 ∘ (-g𝑔))⟩ = ⟨(dist‘ndx), 𝐷⟩)
134, 12oveq12d 7449 . . . . 5 ((𝑔 = 𝐺𝑓 = 𝑁) → (𝑔 sSet ⟨(dist‘ndx), (𝑓 ∘ (-g𝑔))⟩) = (𝐺 sSet ⟨(dist‘ndx), 𝐷⟩))
1411fveq2d 6911 . . . . . . 7 ((𝑔 = 𝐺𝑓 = 𝑁) → (MetOpen‘(𝑓 ∘ (-g𝑔))) = (MetOpen‘𝐷))
15 tngval.j . . . . . . 7 𝐽 = (MetOpen‘𝐷)
1614, 15eqtr4di 2793 . . . . . 6 ((𝑔 = 𝐺𝑓 = 𝑁) → (MetOpen‘(𝑓 ∘ (-g𝑔))) = 𝐽)
1716opeq2d 4885 . . . . 5 ((𝑔 = 𝐺𝑓 = 𝑁) → ⟨(TopSet‘ndx), (MetOpen‘(𝑓 ∘ (-g𝑔)))⟩ = ⟨(TopSet‘ndx), 𝐽⟩)
1813, 17oveq12d 7449 . . . 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 24613 . . . 4 toNrmGrp = (𝑔 ∈ V, 𝑓 ∈ V ↦ ((𝑔 sSet ⟨(dist‘ndx), (𝑓 ∘ (-g𝑔))⟩) sSet ⟨(TopSet‘ndx), (MetOpen‘(𝑓 ∘ (-g𝑔)))⟩))
20 ovex 7464 . . . 4 ((𝐺 sSet ⟨(dist‘ndx), 𝐷⟩) sSet ⟨(TopSet‘ndx), 𝐽⟩) ∈ V
2118, 19, 20ovmpoa 7588 . . 3 ((𝐺 ∈ V ∧ 𝑁 ∈ V) → (𝐺 toNrmGrp 𝑁) = ((𝐺 sSet ⟨(dist‘ndx), 𝐷⟩) sSet ⟨(TopSet‘ndx), 𝐽⟩))
222, 3, 21syl2an 596 . 2 ((𝐺𝑉𝑁𝑊) → (𝐺 toNrmGrp 𝑁) = ((𝐺 sSet ⟨(dist‘ndx), 𝐷⟩) sSet ⟨(TopSet‘ndx), 𝐽⟩))
231, 22eqtrid 2787 1 ((𝐺𝑉𝑁𝑊) → 𝑇 = ((𝐺 sSet ⟨(dist‘ndx), 𝐷⟩) sSet ⟨(TopSet‘ndx), 𝐽⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  Vcvv 3478  cop 4637  ccom 5693  cfv 6563  (class class class)co 7431   sSet csts 17197  ndxcnx 17227  TopSetcts 17304  distcds 17307  -gcsg 18966  MetOpencmopn 21372   toNrmGrp ctng 24607
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-tng 24613
This theorem is referenced by:  tnglem  24669  tnglemOLD  24670  tngds  24684  tngdsOLD  24685  tngtset  24686
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