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Theorem tngval 23243
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 3487 . . 3 (𝐺𝑉𝐺 ∈ V)
3 elex 3487 . . 3 (𝑁𝑊𝑁 ∈ V)
4 simpl 486 . . . . . 6 ((𝑔 = 𝐺𝑓 = 𝑁) → 𝑔 = 𝐺)
5 simpr 488 . . . . . . . . 9 ((𝑔 = 𝐺𝑓 = 𝑁) → 𝑓 = 𝑁)
64fveq2d 6656 . . . . . . . . . 10 ((𝑔 = 𝐺𝑓 = 𝑁) → (-g𝑔) = (-g𝐺))
7 tngval.m . . . . . . . . . 10 = (-g𝐺)
86, 7eqtr4di 2875 . . . . . . . . 9 ((𝑔 = 𝐺𝑓 = 𝑁) → (-g𝑔) = )
95, 8coeq12d 5712 . . . . . . . 8 ((𝑔 = 𝐺𝑓 = 𝑁) → (𝑓 ∘ (-g𝑔)) = (𝑁 ))
10 tngval.d . . . . . . . 8 𝐷 = (𝑁 )
119, 10eqtr4di 2875 . . . . . . 7 ((𝑔 = 𝐺𝑓 = 𝑁) → (𝑓 ∘ (-g𝑔)) = 𝐷)
1211opeq2d 4785 . . . . . 6 ((𝑔 = 𝐺𝑓 = 𝑁) → ⟨(dist‘ndx), (𝑓 ∘ (-g𝑔))⟩ = ⟨(dist‘ndx), 𝐷⟩)
134, 12oveq12d 7158 . . . . 5 ((𝑔 = 𝐺𝑓 = 𝑁) → (𝑔 sSet ⟨(dist‘ndx), (𝑓 ∘ (-g𝑔))⟩) = (𝐺 sSet ⟨(dist‘ndx), 𝐷⟩))
1411fveq2d 6656 . . . . . . 7 ((𝑔 = 𝐺𝑓 = 𝑁) → (MetOpen‘(𝑓 ∘ (-g𝑔))) = (MetOpen‘𝐷))
15 tngval.j . . . . . . 7 𝐽 = (MetOpen‘𝐷)
1614, 15eqtr4di 2875 . . . . . 6 ((𝑔 = 𝐺𝑓 = 𝑁) → (MetOpen‘(𝑓 ∘ (-g𝑔))) = 𝐽)
1716opeq2d 4785 . . . . 5 ((𝑔 = 𝐺𝑓 = 𝑁) → ⟨(TopSet‘ndx), (MetOpen‘(𝑓 ∘ (-g𝑔)))⟩ = ⟨(TopSet‘ndx), 𝐽⟩)
1813, 17oveq12d 7158 . . . 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 23189 . . . 4 toNrmGrp = (𝑔 ∈ V, 𝑓 ∈ V ↦ ((𝑔 sSet ⟨(dist‘ndx), (𝑓 ∘ (-g𝑔))⟩) sSet ⟨(TopSet‘ndx), (MetOpen‘(𝑓 ∘ (-g𝑔)))⟩))
20 ovex 7173 . . . 4 ((𝐺 sSet ⟨(dist‘ndx), 𝐷⟩) sSet ⟨(TopSet‘ndx), 𝐽⟩) ∈ V
2118, 19, 20ovmpoa 7289 . . 3 ((𝐺 ∈ V ∧ 𝑁 ∈ V) → (𝐺 toNrmGrp 𝑁) = ((𝐺 sSet ⟨(dist‘ndx), 𝐷⟩) sSet ⟨(TopSet‘ndx), 𝐽⟩))
222, 3, 21syl2an 598 . 2 ((𝐺𝑉𝑁𝑊) → (𝐺 toNrmGrp 𝑁) = ((𝐺 sSet ⟨(dist‘ndx), 𝐷⟩) sSet ⟨(TopSet‘ndx), 𝐽⟩))
231, 22syl5eq 2869 1 ((𝐺𝑉𝑁𝑊) → 𝑇 = ((𝐺 sSet ⟨(dist‘ndx), 𝐷⟩) sSet ⟨(TopSet‘ndx), 𝐽⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2114  Vcvv 3469  cop 4545  ccom 5536  cfv 6334  (class class class)co 7140  ndxcnx 16471   sSet csts 16472  TopSetcts 16562  distcds 16565  -gcsg 18096  MetOpencmopn 20079   toNrmGrp ctng 23183
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-iota 6293  df-fun 6336  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-tng 23189
This theorem is referenced by:  tnglem  23244  tngds  23252  tngtset  23253
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