Theorem tngval 24148

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	⊢ − = (-g‘𝐺)
tngval.d	⊢ 𝐷 = (𝑁 ∘ − )
tngval.j	⊢ 𝐽 = (MetOpen‘𝐷)

Assertion

Ref	Expression
tngval	⊢ ((𝐺 ∈ 𝑉 ∧ 𝑁 ∈ 𝑊) → 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.

Step	Hyp	Ref	Expression
1		tngval.t	. 2 ⊢ 𝑇 = (𝐺 toNrmGrp 𝑁)
2		elex 3493	. . 3 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V)
3		elex 3493	. . 3 ⊢ (𝑁 ∈ 𝑊 → 𝑁 ∈ V)
4		simpl 484	. . . . . 6 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝑁) → 𝑔 = 𝐺)
5		simpr 486	. . . . . . . . 9 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝑁) → 𝑓 = 𝑁)
6	4	fveq2d 6896	. . . . . . . . . 10 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝑁) → (-g‘𝑔) = (-g‘𝐺))
7		tngval.m	. . . . . . . . . 10 ⊢ − = (-g‘𝐺)
8	6, 7	eqtr4di 2791	. . . . . . . . 9 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝑁) → (-g‘𝑔) = − )
9	5, 8	coeq12d 5865	. . . . . . . 8 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝑁) → (𝑓 ∘ (-g‘𝑔)) = (𝑁 ∘ − ))
10		tngval.d	. . . . . . . 8 ⊢ 𝐷 = (𝑁 ∘ − )
11	9, 10	eqtr4di 2791	. . . . . . 7 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝑁) → (𝑓 ∘ (-g‘𝑔)) = 𝐷)
12	11	opeq2d 4881	. . . . . 6 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝑁) → ⟨(dist‘ndx), (𝑓 ∘ (-g‘𝑔))⟩ = ⟨(dist‘ndx), 𝐷⟩)
13	4, 12	oveq12d 7427	. . . . 5 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝑁) → (𝑔 sSet ⟨(dist‘ndx), (𝑓 ∘ (-g‘𝑔))⟩) = (𝐺 sSet ⟨(dist‘ndx), 𝐷⟩))
14	11	fveq2d 6896	. . . . . . 7 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝑁) → (MetOpen‘(𝑓 ∘ (-g‘𝑔))) = (MetOpen‘𝐷))
15		tngval.j	. . . . . . 7 ⊢ 𝐽 = (MetOpen‘𝐷)
16	14, 15	eqtr4di 2791	. . . . . 6 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝑁) → (MetOpen‘(𝑓 ∘ (-g‘𝑔))) = 𝐽)
17	16	opeq2d 4881	. . . . 5 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝑁) → ⟨(TopSet‘ndx), (MetOpen‘(𝑓 ∘ (-g‘𝑔)))⟩ = ⟨(TopSet‘ndx), 𝐽⟩)
18	13, 17	oveq12d 7427	. . . 4 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝑁) → ((𝑔 sSet ⟨(dist‘ndx), (𝑓 ∘ (-g‘𝑔))⟩) sSet ⟨(TopSet‘ndx), (MetOpen‘(𝑓 ∘ (-g‘𝑔)))⟩) = ((𝐺 sSet ⟨(dist‘ndx), 𝐷⟩) sSet ⟨(TopSet‘ndx), 𝐽⟩))
19		df-tng 24093	. . . 4 ⊢ toNrmGrp = (𝑔 ∈ V, 𝑓 ∈ V ↦ ((𝑔 sSet ⟨(dist‘ndx), (𝑓 ∘ (-g‘𝑔))⟩) sSet ⟨(TopSet‘ndx), (MetOpen‘(𝑓 ∘ (-g‘𝑔)))⟩))
20		ovex 7442	. . . 4 ⊢ ((𝐺 sSet ⟨(dist‘ndx), 𝐷⟩) sSet ⟨(TopSet‘ndx), 𝐽⟩) ∈ V
21	18, 19, 20	ovmpoa 7563	. . 3 ⊢ ((𝐺 ∈ V ∧ 𝑁 ∈ V) → (𝐺 toNrmGrp 𝑁) = ((𝐺 sSet ⟨(dist‘ndx), 𝐷⟩) sSet ⟨(TopSet‘ndx), 𝐽⟩))
22	2, 3, 21	syl2an 597	. 2 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑁 ∈ 𝑊) → (𝐺 toNrmGrp 𝑁) = ((𝐺 sSet ⟨(dist‘ndx), 𝐷⟩) sSet ⟨(TopSet‘ndx), 𝐽⟩))
23	1, 22	eqtrid 2785	1 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑁 ∈ 𝑊) → 𝑇 = ((𝐺 sSet ⟨(dist‘ndx), 𝐷⟩) sSet ⟨(TopSet‘ndx), 𝐽⟩))

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  Vcvv 3475  ⟨cop 4635   ∘ ccom 5681  ‘cfv 6544  (class class class)co 7409   sSet csts 17096  ndxcnx 17126  TopSetcts 17203  distcds 17206  -_gcsg 18821  MetOpencmopn 20934   toNrmGrp ctng 24087

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-tng 24093

This theorem is referenced by:  tnglem  24149  tnglemOLD  24150  tngds  24164  tngdsOLD  24165  tngtset  24166