Theorem tngval 24617

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 3451	. . 3 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V)
3		elex 3451	. . 3 ⊢ (𝑁 ∈ 𝑊 → 𝑁 ∈ V)
4		simpl 482	. . . . . 6 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝑁) → 𝑔 = 𝐺)
5		simpr 484	. . . . . . . . 9 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝑁) → 𝑓 = 𝑁)
6	4	fveq2d 6839	. . . . . . . . . 10 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝑁) → (-g‘𝑔) = (-g‘𝐺))
7		tngval.m	. . . . . . . . . 10 ⊢ − = (-g‘𝐺)
8	6, 7	eqtr4di 2790	. . . . . . . . 9 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝑁) → (-g‘𝑔) = − )
9	5, 8	coeq12d 5814	. . . . . . . 8 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝑁) → (𝑓 ∘ (-g‘𝑔)) = (𝑁 ∘ − ))
10		tngval.d	. . . . . . . 8 ⊢ 𝐷 = (𝑁 ∘ − )
11	9, 10	eqtr4di 2790	. . . . . . 7 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝑁) → (𝑓 ∘ (-g‘𝑔)) = 𝐷)
12	11	opeq2d 4824	. . . . . 6 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝑁) → ⟨(dist‘ndx), (𝑓 ∘ (-g‘𝑔))⟩ = ⟨(dist‘ndx), 𝐷⟩)
13	4, 12	oveq12d 7379	. . . . 5 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝑁) → (𝑔 sSet ⟨(dist‘ndx), (𝑓 ∘ (-g‘𝑔))⟩) = (𝐺 sSet ⟨(dist‘ndx), 𝐷⟩))
14	11	fveq2d 6839	. . . . . . 7 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝑁) → (MetOpen‘(𝑓 ∘ (-g‘𝑔))) = (MetOpen‘𝐷))
15		tngval.j	. . . . . . 7 ⊢ 𝐽 = (MetOpen‘𝐷)
16	14, 15	eqtr4di 2790	. . . . . 6 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝑁) → (MetOpen‘(𝑓 ∘ (-g‘𝑔))) = 𝐽)
17	16	opeq2d 4824	. . . . 5 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝑁) → ⟨(TopSet‘ndx), (MetOpen‘(𝑓 ∘ (-g‘𝑔)))⟩ = ⟨(TopSet‘ndx), 𝐽⟩)
18	13, 17	oveq12d 7379	. . . 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 24562	. . . 4 ⊢ toNrmGrp = (𝑔 ∈ V, 𝑓 ∈ V ↦ ((𝑔 sSet ⟨(dist‘ndx), (𝑓 ∘ (-g‘𝑔))⟩) sSet ⟨(TopSet‘ndx), (MetOpen‘(𝑓 ∘ (-g‘𝑔)))⟩))
20		ovex 7394	. . . 4 ⊢ ((𝐺 sSet ⟨(dist‘ndx), 𝐷⟩) sSet ⟨(TopSet‘ndx), 𝐽⟩) ∈ V
21	18, 19, 20	ovmpoa 7516	. . 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 2784	1 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑁 ∈ 𝑊) → 𝑇 = ((𝐺 sSet ⟨(dist‘ndx), 𝐷⟩) sSet ⟨(TopSet‘ndx), 𝐽⟩))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3430  ⟨cop 4574   ∘ ccom 5629  ‘cfv 6493  (class class class)co 7361   sSet csts 17127  ndxcnx 17157  TopSetcts 17220  distcds 17223  -_gcsg 18905  MetOpencmopn 21337   toNrmGrp ctng 24556

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-iota 6449  df-fun 6495  df-fv 6501  df-ov 7364  df-oprab 7365  df-mpo 7366  df-tng 24562

This theorem is referenced by:  tnglem  24618  tngds  24626  tngtset  24627