Theorem resvval 33630

Description: Value of structure restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)

Hypotheses

Ref	Expression
resvsca.r	⊢ 𝑅 = (𝑊 ↾v 𝐴)
resvsca.f	⊢ 𝐹 = (Scalar‘𝑊)
resvsca.b	⊢ 𝐵 = (Base‘𝐹)

Assertion

Ref	Expression
resvval	⊢ ((𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝑅 = if(𝐵 ⊆ 𝐴, 𝑊, (𝑊 sSet ⟨(Scalar‘ndx), (𝐹 ↾s 𝐴)⟩)))

Proof of Theorem resvval

Dummy variables 𝑥 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		resvsca.r	. 2 ⊢ 𝑅 = (𝑊 ↾v 𝐴)
2		elex 3476	. . 3 ⊢ (𝑊 ∈ 𝑋 → 𝑊 ∈ V)
3		elex 3476	. . 3 ⊢ (𝐴 ∈ 𝑌 → 𝐴 ∈ V)
4		ovex 7445	. . . . . 6 ⊢ (𝑊 sSet ⟨(Scalar‘ndx), (𝐹 ↾s 𝐴)⟩) ∈ V
5		ifcl 4534	. . . . . 6 ⊢ ((𝑊 ∈ V ∧ (𝑊 sSet ⟨(Scalar‘ndx), (𝐹 ↾s 𝐴)⟩) ∈ V) → if(𝐵 ⊆ 𝐴, 𝑊, (𝑊 sSet ⟨(Scalar‘ndx), (𝐹 ↾s 𝐴)⟩)) ∈ V)
6	4, 5	mpan2 703	. . . . 5 ⊢ (𝑊 ∈ V → if(𝐵 ⊆ 𝐴, 𝑊, (𝑊 sSet ⟨(Scalar‘ndx), (𝐹 ↾s 𝐴)⟩)) ∈ V)
7	6	adantr 485	. . . 4 ⊢ ((𝑊 ∈ V ∧ 𝐴 ∈ V) → if(𝐵 ⊆ 𝐴, 𝑊, (𝑊 sSet ⟨(Scalar‘ndx), (𝐹 ↾s 𝐴)⟩)) ∈ V)
8		simpl 487	. . . . . . . . . . 11 ⊢ ((𝑤 = 𝑊 ∧ 𝑥 = 𝐴) → 𝑤 = 𝑊)
9	8	fveq2d 6887	. . . . . . . . . 10 ⊢ ((𝑤 = 𝑊 ∧ 𝑥 = 𝐴) → (Scalar‘𝑤) = (Scalar‘𝑊))
10		resvsca.f	. . . . . . . . . 10 ⊢ 𝐹 = (Scalar‘𝑊)
11	9, 10	eqtr4di 2816	. . . . . . . . 9 ⊢ ((𝑤 = 𝑊 ∧ 𝑥 = 𝐴) → (Scalar‘𝑤) = 𝐹)
12	11	fveq2d 6887	. . . . . . . 8 ⊢ ((𝑤 = 𝑊 ∧ 𝑥 = 𝐴) → (Base‘(Scalar‘𝑤)) = (Base‘𝐹))
13		resvsca.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐹)
14	12, 13	eqtr4di 2816	. . . . . . 7 ⊢ ((𝑤 = 𝑊 ∧ 𝑥 = 𝐴) → (Base‘(Scalar‘𝑤)) = 𝐵)
15		simpr 489	. . . . . . 7 ⊢ ((𝑤 = 𝑊 ∧ 𝑥 = 𝐴) → 𝑥 = 𝐴)
16	14, 15	sseq12d 3971	. . . . . 6 ⊢ ((𝑤 = 𝑊 ∧ 𝑥 = 𝐴) → ((Base‘(Scalar‘𝑤)) ⊆ 𝑥 ↔ 𝐵 ⊆ 𝐴))
17	11, 15	oveq12d 7430	. . . . . . . 8 ⊢ ((𝑤 = 𝑊 ∧ 𝑥 = 𝐴) → ((Scalar‘𝑤) ↾s 𝑥) = (𝐹 ↾s 𝐴))
18	17	opeq2d 4846	. . . . . . 7 ⊢ ((𝑤 = 𝑊 ∧ 𝑥 = 𝐴) → ⟨(Scalar‘ndx), ((Scalar‘𝑤) ↾s 𝑥)⟩ = ⟨(Scalar‘ndx), (𝐹 ↾s 𝐴)⟩)
19	8, 18	oveq12d 7430	. . . . . 6 ⊢ ((𝑤 = 𝑊 ∧ 𝑥 = 𝐴) → (𝑤 sSet ⟨(Scalar‘ndx), ((Scalar‘𝑤) ↾s 𝑥)⟩) = (𝑊 sSet ⟨(Scalar‘ndx), (𝐹 ↾s 𝐴)⟩))
20	16, 8, 19	ifbieq12d 4517	. . . . 5 ⊢ ((𝑤 = 𝑊 ∧ 𝑥 = 𝐴) → if((Base‘(Scalar‘𝑤)) ⊆ 𝑥, 𝑤, (𝑤 sSet ⟨(Scalar‘ndx), ((Scalar‘𝑤) ↾s 𝑥)⟩)) = if(𝐵 ⊆ 𝐴, 𝑊, (𝑊 sSet ⟨(Scalar‘ndx), (𝐹 ↾s 𝐴)⟩)))
21		df-resv 33628	. . . . 5 ⊢ ↾v = (𝑤 ∈ V, 𝑥 ∈ V ↦ if((Base‘(Scalar‘𝑤)) ⊆ 𝑥, 𝑤, (𝑤 sSet ⟨(Scalar‘ndx), ((Scalar‘𝑤) ↾s 𝑥)⟩)))
22	20, 21	ovmpoga 7566	. . . 4 ⊢ ((𝑊 ∈ V ∧ 𝐴 ∈ V ∧ if(𝐵 ⊆ 𝐴, 𝑊, (𝑊 sSet ⟨(Scalar‘ndx), (𝐹 ↾s 𝐴)⟩)) ∈ V) → (𝑊 ↾v 𝐴) = if(𝐵 ⊆ 𝐴, 𝑊, (𝑊 sSet ⟨(Scalar‘ndx), (𝐹 ↾s 𝐴)⟩)))
23	7, 22	mpd3an3 1491	. . 3 ⊢ ((𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊 ↾v 𝐴) = if(𝐵 ⊆ 𝐴, 𝑊, (𝑊 sSet ⟨(Scalar‘ndx), (𝐹 ↾s 𝐴)⟩)))
24	2, 3, 23	syl2an 607	. 2 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → (𝑊 ↾v 𝐴) = if(𝐵 ⊆ 𝐴, 𝑊, (𝑊 sSet ⟨(Scalar‘ndx), (𝐹 ↾s 𝐴)⟩)))
25	1, 24	eqtrid 2810	1 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝑅 = if(𝐵 ⊆ 𝐴, 𝑊, (𝑊 sSet ⟨(Scalar‘ndx), (𝐹 ↾s 𝐴)⟩)))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  Vcvv 3455   ⊆ wss 3906  ifcif 4488  ⟨cop 4596  ‘cfv 6538  (class class class)co 7412   sSet csts 17224  ndxcnx 17254  Basecbs 17270   ↾_s cress 17291  Scalarcsca 17314   ↾_v cresv 33627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  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 3417  df-v 3457  df-sbc 3746  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-iota 6494  df-fun 6540  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417  df-resv 33628

This theorem is referenced by:  resvid2  33631  resvval2  33632