Theorem tngval 24970

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 3474	. . 3 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V)
3		elex 3474	. . 3 ⊢ (𝑁 ∈ 𝑊 → 𝑁 ∈ V)
4		simpl 488	. . . . . 6 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝑁) → 𝑔 = 𝐺)
5		simpr 490	. . . . . . . . 9 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝑁) → 𝑓 = 𝑁)
6	4	fveq2d 6897	. . . . . . . . . 10 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝑁) → (-g‘𝑔) = (-g‘𝐺))
7		tngval.m	. . . . . . . . . 10 ⊢ − = (-g‘𝐺)
8	6, 7	eqtr4di 2816	. . . . . . . . 9 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝑁) → (-g‘𝑔) = − )
9	5, 8	coeq12d 5846	. . . . . . . 8 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝑁) → (𝑓 ∘ (-g‘𝑔)) = (𝑁 ∘ − ))
10		tngval.d	. . . . . . . 8 ⊢ 𝐷 = (𝑁 ∘ − )
11	9, 10	eqtr4di 2816	. . . . . . 7 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝑁) → (𝑓 ∘ (-g‘𝑔)) = 𝐷)
12	11	opeq2d 4849	. . . . . 6 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝑁) → ⟨(dist‘ndx), (𝑓 ∘ (-g‘𝑔))⟩ = ⟨(dist‘ndx), 𝐷⟩)
13	4, 12	oveq12d 7445	. . . . 5 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝑁) → (𝑔 sSet ⟨(dist‘ndx), (𝑓 ∘ (-g‘𝑔))⟩) = (𝐺 sSet ⟨(dist‘ndx), 𝐷⟩))
14	11	fveq2d 6897	. . . . . . 7 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝑁) → (MetOpen‘(𝑓 ∘ (-g‘𝑔))) = (MetOpen‘𝐷))
15		tngval.j	. . . . . . 7 ⊢ 𝐽 = (MetOpen‘𝐷)
16	14, 15	eqtr4di 2816	. . . . . 6 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝑁) → (MetOpen‘(𝑓 ∘ (-g‘𝑔))) = 𝐽)
17	16	opeq2d 4849	. . . . 5 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝑁) → ⟨(TopSet‘ndx), (MetOpen‘(𝑓 ∘ (-g‘𝑔)))⟩ = ⟨(TopSet‘ndx), 𝐽⟩)
18	13, 17	oveq12d 7445	. . . 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 24915	. . . 4 ⊢ toNrmGrp = (𝑔 ∈ V, 𝑓 ∈ V ↦ ((𝑔 sSet ⟨(dist‘ndx), (𝑓 ∘ (-g‘𝑔))⟩) sSet ⟨(TopSet‘ndx), (MetOpen‘(𝑓 ∘ (-g‘𝑔)))⟩))
20		ovex 7460	. . . 4 ⊢ ((𝐺 sSet ⟨(dist‘ndx), 𝐷⟩) sSet ⟨(TopSet‘ndx), 𝐽⟩) ∈ V
21	18, 19, 20	ovmpoa 7582	. . 3 ⊢ ((𝐺 ∈ V ∧ 𝑁 ∈ V) → (𝐺 toNrmGrp 𝑁) = ((𝐺 sSet ⟨(dist‘ndx), 𝐷⟩) sSet ⟨(TopSet‘ndx), 𝐽⟩))
22	2, 3, 21	syl2an 608	. 2 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑁 ∈ 𝑊) → (𝐺 toNrmGrp 𝑁) = ((𝐺 sSet ⟨(dist‘ndx), 𝐷⟩) sSet ⟨(TopSet‘ndx), 𝐽⟩))
23	1, 22	eqtrid 2810	1 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑁 ∈ 𝑊) → 𝑇 = ((𝐺 sSet ⟨(dist‘ndx), 𝐷⟩) sSet ⟨(TopSet‘ndx), 𝐽⟩))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2146  Vcvv 3453  ⟨cop 4837   ∘ ccom 5658  ‘cfv 6546  (class class class)co 7427   sSet csts 17353  ndxcnx 17383  TopSetcts 17446  distcds 17449  -_gcsg 19158  MetOpencmopn 21680   toNrmGrp ctng 24909

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5251  ax-nul 5262  ax-pr 5394

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3416  df-v 3455  df-sbc 3742  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-br 5106  df-opab 5170  df-id 5549  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-iota 6500  df-fun 6548  df-fv 6554  df-ov 7430  df-oprab 7431  df-mpo 7432  df-tng 24915

This theorem is used by:  tnglem  24971  tngds  24979  tngtset  24980