Theorem resvval 33974

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 3475	. . 3 ⊢ (𝑊 ∈ 𝑋 → 𝑊 ∈ V)
3		elex 3475	. . 3 ⊢ (𝐴 ∈ 𝑌 → 𝐴 ∈ V)
4		ovex 7479	. . . . . 6 ⊢ (𝑊 sSet ⟨(Scalar‘ndx), (𝐹 ↾s 𝐴)⟩) ∈ V
5		ifcl 4531	. . . . . 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 704	. . . . 5 ⊢ (𝑊 ∈ V → if(𝐵 ⊆ 𝐴, 𝑊, (𝑊 sSet ⟨(Scalar‘ndx), (𝐹 ↾s 𝐴)⟩)) ∈ V)
7	6	adantr 486	. . . 4 ⊢ ((𝑊 ∈ V ∧ 𝐴 ∈ V) → if(𝐵 ⊆ 𝐴, 𝑊, (𝑊 sSet ⟨(Scalar‘ndx), (𝐹 ↾s 𝐴)⟩)) ∈ V)
8		simpl 488	. . . . . . . . . . 11 ⊢ ((𝑤 = 𝑊 ∧ 𝑥 = 𝐴) → 𝑤 = 𝑊)
9	8	fveq2d 6915	. . . . . . . . . 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 6915	. . . . . . . 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 490	. . . . . . 7 ⊢ ((𝑤 = 𝑊 ∧ 𝑥 = 𝐴) → 𝑥 = 𝐴)
16	14, 15	sseq12d 3946	. . . . . 6 ⊢ ((𝑤 = 𝑊 ∧ 𝑥 = 𝐴) → ((Base‘(Scalar‘𝑤)) ⊆ 𝑥 ↔ 𝐵 ⊆ 𝐴))
17	11, 15	oveq12d 7464	. . . . . . . 8 ⊢ ((𝑤 = 𝑊 ∧ 𝑥 = 𝐴) → ((Scalar‘𝑤) ↾s 𝑥) = (𝐹 ↾s 𝐴))
18	17	opeq2d 4850	. . . . . . 7 ⊢ ((𝑤 = 𝑊 ∧ 𝑥 = 𝐴) → ⟨(Scalar‘ndx), ((Scalar‘𝑤) ↾s 𝑥)⟩ = ⟨(Scalar‘ndx), (𝐹 ↾s 𝐴)⟩)
19	8, 18	oveq12d 7464	. . . . . 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 4514	. . . . 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 33972	. . . . 5 ⊢ ↾v = (𝑤 ∈ V, 𝑥 ∈ V ↦ if((Base‘(Scalar‘𝑤)) ⊆ 𝑥, 𝑤, (𝑤 sSet ⟨(Scalar‘ndx), ((Scalar‘𝑤) ↾s 𝑥)⟩)))
22	20, 21	ovmpoga 7601	. . . 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 1492	. . 3 ⊢ ((𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊 ↾v 𝐴) = if(𝐵 ⊆ 𝐴, 𝑊, (𝑊 sSet ⟨(Scalar‘ndx), (𝐹 ↾s 𝐴)⟩)))
24	2, 3, 23	syl2an 608	. 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)⟩)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  Vcvv 3454   ⊆ wss 3900  ifcif 4485  ⟨cop 4838  ‘cfv 6561  (class class class)co 7446   sSet csts 17373  ndxcnx 17403  Basecbs 17419   ↾_s cress 17440  Scalarcsca 17463   ↾_v cresv 33971

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5172  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-iota 6515  df-fun 6563  df-fv 6569  df-ov 7449  df-oprab 7450  df-mpo 7451  df-resv 33972

This theorem is used by:  resvid2  33975  resvval2  33976