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Theorem resvval 29158
 Description: Value of structure restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
Hypotheses
Ref Expression
resvsca.r 𝑅 = (𝑊v 𝐴)
resvsca.f 𝐹 = (Scalar‘𝑊)
resvsca.b 𝐵 = (Base‘𝐹)
Assertion
Ref Expression
resvval ((𝑊𝑋𝐴𝑌) → 𝑅 = if(𝐵𝐴, 𝑊, (𝑊 sSet ⟨(Scalar‘ndx), (𝐹s 𝐴)⟩)))

Proof of Theorem resvval
Dummy variables 𝑥 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 resvsca.r . 2 𝑅 = (𝑊v 𝐴)
2 elex 3185 . . 3 (𝑊𝑋𝑊 ∈ V)
3 elex 3185 . . 3 (𝐴𝑌𝐴 ∈ V)
4 ovex 6577 . . . . . 6 (𝑊 sSet ⟨(Scalar‘ndx), (𝐹s 𝐴)⟩) ∈ V
5 ifcl 4080 . . . . . 6 ((𝑊 ∈ V ∧ (𝑊 sSet ⟨(Scalar‘ndx), (𝐹s 𝐴)⟩) ∈ V) → if(𝐵𝐴, 𝑊, (𝑊 sSet ⟨(Scalar‘ndx), (𝐹s 𝐴)⟩)) ∈ V)
64, 5mpan2 703 . . . . 5 (𝑊 ∈ V → if(𝐵𝐴, 𝑊, (𝑊 sSet ⟨(Scalar‘ndx), (𝐹s 𝐴)⟩)) ∈ V)
76adantr 480 . . . 4 ((𝑊 ∈ V ∧ 𝐴 ∈ V) → if(𝐵𝐴, 𝑊, (𝑊 sSet ⟨(Scalar‘ndx), (𝐹s 𝐴)⟩)) ∈ V)
8 simpl 472 . . . . . . . . . . 11 ((𝑤 = 𝑊𝑥 = 𝐴) → 𝑤 = 𝑊)
98fveq2d 6107 . . . . . . . . . 10 ((𝑤 = 𝑊𝑥 = 𝐴) → (Scalar‘𝑤) = (Scalar‘𝑊))
10 resvsca.f . . . . . . . . . 10 𝐹 = (Scalar‘𝑊)
119, 10syl6eqr 2662 . . . . . . . . 9 ((𝑤 = 𝑊𝑥 = 𝐴) → (Scalar‘𝑤) = 𝐹)
1211fveq2d 6107 . . . . . . . 8 ((𝑤 = 𝑊𝑥 = 𝐴) → (Base‘(Scalar‘𝑤)) = (Base‘𝐹))
13 resvsca.b . . . . . . . 8 𝐵 = (Base‘𝐹)
1412, 13syl6eqr 2662 . . . . . . 7 ((𝑤 = 𝑊𝑥 = 𝐴) → (Base‘(Scalar‘𝑤)) = 𝐵)
15 simpr 476 . . . . . . 7 ((𝑤 = 𝑊𝑥 = 𝐴) → 𝑥 = 𝐴)
1614, 15sseq12d 3597 . . . . . 6 ((𝑤 = 𝑊𝑥 = 𝐴) → ((Base‘(Scalar‘𝑤)) ⊆ 𝑥𝐵𝐴))
1711, 15oveq12d 6567 . . . . . . . 8 ((𝑤 = 𝑊𝑥 = 𝐴) → ((Scalar‘𝑤) ↾s 𝑥) = (𝐹s 𝐴))
1817opeq2d 4347 . . . . . . 7 ((𝑤 = 𝑊𝑥 = 𝐴) → ⟨(Scalar‘ndx), ((Scalar‘𝑤) ↾s 𝑥)⟩ = ⟨(Scalar‘ndx), (𝐹s 𝐴)⟩)
198, 18oveq12d 6567 . . . . . 6 ((𝑤 = 𝑊𝑥 = 𝐴) → (𝑤 sSet ⟨(Scalar‘ndx), ((Scalar‘𝑤) ↾s 𝑥)⟩) = (𝑊 sSet ⟨(Scalar‘ndx), (𝐹s 𝐴)⟩))
2016, 8, 19ifbieq12d 4063 . . . . 5 ((𝑤 = 𝑊𝑥 = 𝐴) → if((Base‘(Scalar‘𝑤)) ⊆ 𝑥, 𝑤, (𝑤 sSet ⟨(Scalar‘ndx), ((Scalar‘𝑤) ↾s 𝑥)⟩)) = if(𝐵𝐴, 𝑊, (𝑊 sSet ⟨(Scalar‘ndx), (𝐹s 𝐴)⟩)))
21 df-resv 29156 . . . . 5 v = (𝑤 ∈ V, 𝑥 ∈ V ↦ if((Base‘(Scalar‘𝑤)) ⊆ 𝑥, 𝑤, (𝑤 sSet ⟨(Scalar‘ndx), ((Scalar‘𝑤) ↾s 𝑥)⟩)))
2220, 21ovmpt2ga 6688 . . . 4 ((𝑊 ∈ V ∧ 𝐴 ∈ V ∧ if(𝐵𝐴, 𝑊, (𝑊 sSet ⟨(Scalar‘ndx), (𝐹s 𝐴)⟩)) ∈ V) → (𝑊v 𝐴) = if(𝐵𝐴, 𝑊, (𝑊 sSet ⟨(Scalar‘ndx), (𝐹s 𝐴)⟩)))
237, 22mpd3an3 1417 . . 3 ((𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊v 𝐴) = if(𝐵𝐴, 𝑊, (𝑊 sSet ⟨(Scalar‘ndx), (𝐹s 𝐴)⟩)))
242, 3, 23syl2an 493 . 2 ((𝑊𝑋𝐴𝑌) → (𝑊v 𝐴) = if(𝐵𝐴, 𝑊, (𝑊 sSet ⟨(Scalar‘ndx), (𝐹s 𝐴)⟩)))
251, 24syl5eq 2656 1 ((𝑊𝑋𝐴𝑌) → 𝑅 = if(𝐵𝐴, 𝑊, (𝑊 sSet ⟨(Scalar‘ndx), (𝐹s 𝐴)⟩)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ⊆ wss 3540  ifcif 4036  ⟨cop 4131  ‘cfv 5804  (class class class)co 6549  ndxcnx 15692   sSet csts 15693  Basecbs 15695   ↾s cress 15696  Scalarcsca 15771   ↾v cresv 29155 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-resv 29156 This theorem is referenced by:  resvid2  29159  resvval2  29160
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