Theorem tngval 24147

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 3492	. . 3 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V)
3		elex 3492	. . 3 ⊢ (𝑁 ∈ 𝑊 → 𝑁 ∈ V)
4		simpl 483	. . . . . 6 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝑁) → 𝑔 = 𝐺)
5		simpr 485	. . . . . . . . 9 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝑁) → 𝑓 = 𝑁)
6	4	fveq2d 6895	. . . . . . . . . 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 5864	. . . . . . . 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 4880	. . . . . 6 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝑁) → ⟨(dist‘ndx), (𝑓 ∘ (-g‘𝑔))⟩ = ⟨(dist‘ndx), 𝐷⟩)
13	4, 12	oveq12d 7426	. . . . 5 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝑁) → (𝑔 sSet ⟨(dist‘ndx), (𝑓 ∘ (-g‘𝑔))⟩) = (𝐺 sSet ⟨(dist‘ndx), 𝐷⟩))
14	11	fveq2d 6895	. . . . . . 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 4880	. . . . 5 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝑁) → ⟨(TopSet‘ndx), (MetOpen‘(𝑓 ∘ (-g‘𝑔)))⟩ = ⟨(TopSet‘ndx), 𝐽⟩)
18	13, 17	oveq12d 7426	. . . 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 24092	. . . 4 ⊢ toNrmGrp = (𝑔 ∈ V, 𝑓 ∈ V ↦ ((𝑔 sSet ⟨(dist‘ndx), (𝑓 ∘ (-g‘𝑔))⟩) sSet ⟨(TopSet‘ndx), (MetOpen‘(𝑓 ∘ (-g‘𝑔)))⟩))
20		ovex 7441	. . . 4 ⊢ ((𝐺 sSet ⟨(dist‘ndx), 𝐷⟩) sSet ⟨(TopSet‘ndx), 𝐽⟩) ∈ V
21	18, 19, 20	ovmpoa 7562	. . 3 ⊢ ((𝐺 ∈ V ∧ 𝑁 ∈ V) → (𝐺 toNrmGrp 𝑁) = ((𝐺 sSet ⟨(dist‘ndx), 𝐷⟩) sSet ⟨(TopSet‘ndx), 𝐽⟩))
22	2, 3, 21	syl2an 596	. 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 396   = wceq 1541   ∈ wcel 2106  Vcvv 3474  ⟨cop 4634   ∘ ccom 5680  ‘cfv 6543  (class class class)co 7408   sSet csts 17095  ndxcnx 17125  TopSetcts 17202  distcds 17205  -_gcsg 18820  MetOpencmopn 20933   toNrmGrp ctng 24086

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-tng 24092

This theorem is referenced by:  tnglem  24148  tnglemOLD  24149  tngds  24163  tngdsOLD  24164  tngtset  24165