Theorem tngval 23793

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 3449	. . 3 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V)
3		elex 3449	. . 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 6775	. . . . . . . . . 10 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝑁) → (-g‘𝑔) = (-g‘𝐺))
7		tngval.m	. . . . . . . . . 10 ⊢ − = (-g‘𝐺)
8	6, 7	eqtr4di 2798	. . . . . . . . 9 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝑁) → (-g‘𝑔) = − )
9	5, 8	coeq12d 5772	. . . . . . . 8 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝑁) → (𝑓 ∘ (-g‘𝑔)) = (𝑁 ∘ − ))
10		tngval.d	. . . . . . . 8 ⊢ 𝐷 = (𝑁 ∘ − )
11	9, 10	eqtr4di 2798	. . . . . . 7 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝑁) → (𝑓 ∘ (-g‘𝑔)) = 𝐷)
12	11	opeq2d 4817	. . . . . 6 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝑁) → ⟨(dist‘ndx), (𝑓 ∘ (-g‘𝑔))⟩ = ⟨(dist‘ndx), 𝐷⟩)
13	4, 12	oveq12d 7289	. . . . 5 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝑁) → (𝑔 sSet ⟨(dist‘ndx), (𝑓 ∘ (-g‘𝑔))⟩) = (𝐺 sSet ⟨(dist‘ndx), 𝐷⟩))
14	11	fveq2d 6775	. . . . . . 7 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝑁) → (MetOpen‘(𝑓 ∘ (-g‘𝑔))) = (MetOpen‘𝐷))
15		tngval.j	. . . . . . 7 ⊢ 𝐽 = (MetOpen‘𝐷)
16	14, 15	eqtr4di 2798	. . . . . 6 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝑁) → (MetOpen‘(𝑓 ∘ (-g‘𝑔))) = 𝐽)
17	16	opeq2d 4817	. . . . 5 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝑁) → ⟨(TopSet‘ndx), (MetOpen‘(𝑓 ∘ (-g‘𝑔)))⟩ = ⟨(TopSet‘ndx), 𝐽⟩)
18	13, 17	oveq12d 7289	. . . 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 23738	. . . 4 ⊢ toNrmGrp = (𝑔 ∈ V, 𝑓 ∈ V ↦ ((𝑔 sSet ⟨(dist‘ndx), (𝑓 ∘ (-g‘𝑔))⟩) sSet ⟨(TopSet‘ndx), (MetOpen‘(𝑓 ∘ (-g‘𝑔)))⟩))
20		ovex 7304	. . . 4 ⊢ ((𝐺 sSet ⟨(dist‘ndx), 𝐷⟩) sSet ⟨(TopSet‘ndx), 𝐽⟩) ∈ V
21	18, 19, 20	ovmpoa 7422	. . 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 2792	1 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑁 ∈ 𝑊) → 𝑇 = ((𝐺 sSet ⟨(dist‘ndx), 𝐷⟩) sSet ⟨(TopSet‘ndx), 𝐽⟩))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1542   ∈ wcel 2110  Vcvv 3431  ⟨cop 4573   ∘ ccom 5594  ‘cfv 6432  (class class class)co 7271   sSet csts 16862  ndxcnx 16892  TopSetcts 16966  distcds 16969  -_gcsg 18577  MetOpencmopn 20585   toNrmGrp ctng 23732

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-sbc 3721  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-br 5080  df-opab 5142  df-id 5490  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-iota 6390  df-fun 6434  df-fv 6440  df-ov 7274  df-oprab 7275  df-mpo 7276  df-tng 23738

This theorem is referenced by:  tnglem  23794  tnglemOLD  23795  tngds  23809  tngdsOLD  23810  tngtset  23811