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Theorem tngval 22662
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 3361 . . 3 (𝐺𝑉𝐺 ∈ V)
3 elex 3361 . . 3 (𝑁𝑊𝑁 ∈ V)
4 simpl 468 . . . . . 6 ((𝑔 = 𝐺𝑓 = 𝑁) → 𝑔 = 𝐺)
5 simpr 471 . . . . . . . . 9 ((𝑔 = 𝐺𝑓 = 𝑁) → 𝑓 = 𝑁)
64fveq2d 6336 . . . . . . . . . 10 ((𝑔 = 𝐺𝑓 = 𝑁) → (-g𝑔) = (-g𝐺))
7 tngval.m . . . . . . . . . 10 = (-g𝐺)
86, 7syl6eqr 2822 . . . . . . . . 9 ((𝑔 = 𝐺𝑓 = 𝑁) → (-g𝑔) = )
95, 8coeq12d 5425 . . . . . . . 8 ((𝑔 = 𝐺𝑓 = 𝑁) → (𝑓 ∘ (-g𝑔)) = (𝑁 ))
10 tngval.d . . . . . . . 8 𝐷 = (𝑁 )
119, 10syl6eqr 2822 . . . . . . 7 ((𝑔 = 𝐺𝑓 = 𝑁) → (𝑓 ∘ (-g𝑔)) = 𝐷)
1211opeq2d 4544 . . . . . 6 ((𝑔 = 𝐺𝑓 = 𝑁) → ⟨(dist‘ndx), (𝑓 ∘ (-g𝑔))⟩ = ⟨(dist‘ndx), 𝐷⟩)
134, 12oveq12d 6810 . . . . 5 ((𝑔 = 𝐺𝑓 = 𝑁) → (𝑔 sSet ⟨(dist‘ndx), (𝑓 ∘ (-g𝑔))⟩) = (𝐺 sSet ⟨(dist‘ndx), 𝐷⟩))
1411fveq2d 6336 . . . . . . 7 ((𝑔 = 𝐺𝑓 = 𝑁) → (MetOpen‘(𝑓 ∘ (-g𝑔))) = (MetOpen‘𝐷))
15 tngval.j . . . . . . 7 𝐽 = (MetOpen‘𝐷)
1614, 15syl6eqr 2822 . . . . . 6 ((𝑔 = 𝐺𝑓 = 𝑁) → (MetOpen‘(𝑓 ∘ (-g𝑔))) = 𝐽)
1716opeq2d 4544 . . . . 5 ((𝑔 = 𝐺𝑓 = 𝑁) → ⟨(TopSet‘ndx), (MetOpen‘(𝑓 ∘ (-g𝑔)))⟩ = ⟨(TopSet‘ndx), 𝐽⟩)
1813, 17oveq12d 6810 . . . 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 22608 . . . 4 toNrmGrp = (𝑔 ∈ V, 𝑓 ∈ V ↦ ((𝑔 sSet ⟨(dist‘ndx), (𝑓 ∘ (-g𝑔))⟩) sSet ⟨(TopSet‘ndx), (MetOpen‘(𝑓 ∘ (-g𝑔)))⟩))
20 ovex 6822 . . . 4 ((𝐺 sSet ⟨(dist‘ndx), 𝐷⟩) sSet ⟨(TopSet‘ndx), 𝐽⟩) ∈ V
2118, 19, 20ovmpt2a 6937 . . 3 ((𝐺 ∈ V ∧ 𝑁 ∈ V) → (𝐺 toNrmGrp 𝑁) = ((𝐺 sSet ⟨(dist‘ndx), 𝐷⟩) sSet ⟨(TopSet‘ndx), 𝐽⟩))
222, 3, 21syl2an 575 . 2 ((𝐺𝑉𝑁𝑊) → (𝐺 toNrmGrp 𝑁) = ((𝐺 sSet ⟨(dist‘ndx), 𝐷⟩) sSet ⟨(TopSet‘ndx), 𝐽⟩))
231, 22syl5eq 2816 1 ((𝐺𝑉𝑁𝑊) → 𝑇 = ((𝐺 sSet ⟨(dist‘ndx), 𝐷⟩) sSet ⟨(TopSet‘ndx), 𝐽⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1630  wcel 2144  Vcvv 3349  cop 4320  ccom 5253  cfv 6031  (class class class)co 6792  ndxcnx 16060   sSet csts 16061  TopSetcts 16154  distcds 16157  -gcsg 17631  MetOpencmopn 19950   toNrmGrp ctng 22602
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-tng 22608
This theorem is referenced by:  tnglem  22663  tngds  22671  tngtset  22672
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