Theorem resvval 33476

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 3474	. . 3 ⊢ (𝑊 ∈ 𝑋 → 𝑊 ∈ V)
3		elex 3474	. . 3 ⊢ (𝐴 ∈ 𝑌 → 𝐴 ∈ V)
4		ovex 7425	. . . . . 6 ⊢ (𝑊 sSet ⟨(Scalar‘ndx), (𝐹 ↾s 𝐴)⟩) ∈ V
5		ifcl 4525	. . . . . 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 701	. . . . 5 ⊢ (𝑊 ∈ V → if(𝐵 ⊆ 𝐴, 𝑊, (𝑊 sSet ⟨(Scalar‘ndx), (𝐹 ↾s 𝐴)⟩)) ∈ V)
7	6	adantr 484	. . . 4 ⊢ ((𝑊 ∈ V ∧ 𝐴 ∈ V) → if(𝐵 ⊆ 𝐴, 𝑊, (𝑊 sSet ⟨(Scalar‘ndx), (𝐹 ↾s 𝐴)⟩)) ∈ V)
8		simpl 486	. . . . . . . . . . 11 ⊢ ((𝑤 = 𝑊 ∧ 𝑥 = 𝐴) → 𝑤 = 𝑊)
9	8	fveq2d 6867	. . . . . . . . . 10 ⊢ ((𝑤 = 𝑊 ∧ 𝑥 = 𝐴) → (Scalar‘𝑤) = (Scalar‘𝑊))
10		resvsca.f	. . . . . . . . . 10 ⊢ 𝐹 = (Scalar‘𝑊)
11	9, 10	eqtr4di 2814	. . . . . . . . 9 ⊢ ((𝑤 = 𝑊 ∧ 𝑥 = 𝐴) → (Scalar‘𝑤) = 𝐹)
12	11	fveq2d 6867	. . . . . . . 8 ⊢ ((𝑤 = 𝑊 ∧ 𝑥 = 𝐴) → (Base‘(Scalar‘𝑤)) = (Base‘𝐹))
13		resvsca.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐹)
14	12, 13	eqtr4di 2814	. . . . . . 7 ⊢ ((𝑤 = 𝑊 ∧ 𝑥 = 𝐴) → (Base‘(Scalar‘𝑤)) = 𝐵)
15		simpr 488	. . . . . . 7 ⊢ ((𝑤 = 𝑊 ∧ 𝑥 = 𝐴) → 𝑥 = 𝐴)
16	14, 15	sseq12d 3969	. . . . . 6 ⊢ ((𝑤 = 𝑊 ∧ 𝑥 = 𝐴) → ((Base‘(Scalar‘𝑤)) ⊆ 𝑥 ↔ 𝐵 ⊆ 𝐴))
17	11, 15	oveq12d 7410	. . . . . . . 8 ⊢ ((𝑤 = 𝑊 ∧ 𝑥 = 𝐴) → ((Scalar‘𝑤) ↾s 𝑥) = (𝐹 ↾s 𝐴))
18	17	opeq2d 4837	. . . . . . 7 ⊢ ((𝑤 = 𝑊 ∧ 𝑥 = 𝐴) → ⟨(Scalar‘ndx), ((Scalar‘𝑤) ↾s 𝑥)⟩ = ⟨(Scalar‘ndx), (𝐹 ↾s 𝐴)⟩)
19	8, 18	oveq12d 7410	. . . . . 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 4508	. . . . 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 33474	. . . . 5 ⊢ ↾v = (𝑤 ∈ V, 𝑥 ∈ V ↦ if((Base‘(Scalar‘𝑤)) ⊆ 𝑥, 𝑤, (𝑤 sSet ⟨(Scalar‘ndx), ((Scalar‘𝑤) ↾s 𝑥)⟩)))
22	20, 21	ovmpoga 7546	. . . 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 1482	. . 3 ⊢ ((𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊 ↾v 𝐴) = if(𝐵 ⊆ 𝐴, 𝑊, (𝑊 sSet ⟨(Scalar‘ndx), (𝐹 ↾s 𝐴)⟩)))
24	2, 3, 23	syl2an 605	. 2 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → (𝑊 ↾v 𝐴) = if(𝐵 ⊆ 𝐴, 𝑊, (𝑊 sSet ⟨(Scalar‘ndx), (𝐹 ↾s 𝐴)⟩)))
25	1, 24	eqtrid 2808	1 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝑅 = if(𝐵 ⊆ 𝐴, 𝑊, (𝑊 sSet ⟨(Scalar‘ndx), (𝐹 ↾s 𝐴)⟩)))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559   ∈ wcel 2141  Vcvv 3453   ⊆ wss 3904  ifcif 4479  ⟨cop 4587  ‘cfv 6517  (class class class)co 7392   sSet csts 17182  ndxcnx 17212  Basecbs 17228   ↾_s cress 17249  Scalarcsca 17272   ↾_v cresv 33473

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3745  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-iota 6473  df-fun 6519  df-fv 6525  df-ov 7395  df-oprab 7396  df-mpo 7397  df-resv 33474

This theorem is referenced by:  resvid2  33477  resvval2  33478