Theorem resvval 33345

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 3480	. . 3 ⊢ (𝑊 ∈ 𝑋 → 𝑊 ∈ V)
3		elex 3480	. . 3 ⊢ (𝐴 ∈ 𝑌 → 𝐴 ∈ V)
4		ovex 7438	. . . . . 6 ⊢ (𝑊 sSet ⟨(Scalar‘ndx), (𝐹 ↾s 𝐴)⟩) ∈ V
5		ifcl 4546	. . . . . 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 691	. . . . 5 ⊢ (𝑊 ∈ V → if(𝐵 ⊆ 𝐴, 𝑊, (𝑊 sSet ⟨(Scalar‘ndx), (𝐹 ↾s 𝐴)⟩)) ∈ V)
7	6	adantr 480	. . . 4 ⊢ ((𝑊 ∈ V ∧ 𝐴 ∈ V) → if(𝐵 ⊆ 𝐴, 𝑊, (𝑊 sSet ⟨(Scalar‘ndx), (𝐹 ↾s 𝐴)⟩)) ∈ V)
8		simpl 482	. . . . . . . . . . 11 ⊢ ((𝑤 = 𝑊 ∧ 𝑥 = 𝐴) → 𝑤 = 𝑊)
9	8	fveq2d 6880	. . . . . . . . . 10 ⊢ ((𝑤 = 𝑊 ∧ 𝑥 = 𝐴) → (Scalar‘𝑤) = (Scalar‘𝑊))
10		resvsca.f	. . . . . . . . . 10 ⊢ 𝐹 = (Scalar‘𝑊)
11	9, 10	eqtr4di 2788	. . . . . . . . 9 ⊢ ((𝑤 = 𝑊 ∧ 𝑥 = 𝐴) → (Scalar‘𝑤) = 𝐹)
12	11	fveq2d 6880	. . . . . . . 8 ⊢ ((𝑤 = 𝑊 ∧ 𝑥 = 𝐴) → (Base‘(Scalar‘𝑤)) = (Base‘𝐹))
13		resvsca.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐹)
14	12, 13	eqtr4di 2788	. . . . . . 7 ⊢ ((𝑤 = 𝑊 ∧ 𝑥 = 𝐴) → (Base‘(Scalar‘𝑤)) = 𝐵)
15		simpr 484	. . . . . . 7 ⊢ ((𝑤 = 𝑊 ∧ 𝑥 = 𝐴) → 𝑥 = 𝐴)
16	14, 15	sseq12d 3992	. . . . . 6 ⊢ ((𝑤 = 𝑊 ∧ 𝑥 = 𝐴) → ((Base‘(Scalar‘𝑤)) ⊆ 𝑥 ↔ 𝐵 ⊆ 𝐴))
17	11, 15	oveq12d 7423	. . . . . . . 8 ⊢ ((𝑤 = 𝑊 ∧ 𝑥 = 𝐴) → ((Scalar‘𝑤) ↾s 𝑥) = (𝐹 ↾s 𝐴))
18	17	opeq2d 4856	. . . . . . 7 ⊢ ((𝑤 = 𝑊 ∧ 𝑥 = 𝐴) → ⟨(Scalar‘ndx), ((Scalar‘𝑤) ↾s 𝑥)⟩ = ⟨(Scalar‘ndx), (𝐹 ↾s 𝐴)⟩)
19	8, 18	oveq12d 7423	. . . . . 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 4529	. . . . 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 33343	. . . . 5 ⊢ ↾v = (𝑤 ∈ V, 𝑥 ∈ V ↦ if((Base‘(Scalar‘𝑤)) ⊆ 𝑥, 𝑤, (𝑤 sSet ⟨(Scalar‘ndx), ((Scalar‘𝑤) ↾s 𝑥)⟩)))
22	20, 21	ovmpoga 7561	. . . 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 1464	. . 3 ⊢ ((𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊 ↾v 𝐴) = if(𝐵 ⊆ 𝐴, 𝑊, (𝑊 sSet ⟨(Scalar‘ndx), (𝐹 ↾s 𝐴)⟩)))
24	2, 3, 23	syl2an 596	. 2 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → (𝑊 ↾v 𝐴) = if(𝐵 ⊆ 𝐴, 𝑊, (𝑊 sSet ⟨(Scalar‘ndx), (𝐹 ↾s 𝐴)⟩)))
25	1, 24	eqtrid 2782	1 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝑅 = if(𝐵 ⊆ 𝐴, 𝑊, (𝑊 sSet ⟨(Scalar‘ndx), (𝐹 ↾s 𝐴)⟩)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  Vcvv 3459   ⊆ wss 3926  ifcif 4500  ⟨cop 4607  ‘cfv 6531  (class class class)co 7405   sSet csts 17182  ndxcnx 17212  Basecbs 17228   ↾_s cress 17251  Scalarcsca 17274   ↾_v cresv 33342

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-iota 6484  df-fun 6533  df-fv 6539  df-ov 7408  df-oprab 7409  df-mpo 7410  df-resv 33343

This theorem is referenced by:  resvid2  33346  resvval2  33347