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Theorem resvval2 29957
Description: Value of nontrivial 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
resvval2 ((¬ 𝐵𝐴𝑊𝑋𝐴𝑌) → 𝑅 = (𝑊 sSet ⟨(Scalar‘ndx), (𝐹s 𝐴)⟩))

Proof of Theorem resvval2
StepHypRef Expression
1 resvsca.r . . . 4 𝑅 = (𝑊v 𝐴)
2 resvsca.f . . . 4 𝐹 = (Scalar‘𝑊)
3 resvsca.b . . . 4 𝐵 = (Base‘𝐹)
41, 2, 3resvval 29955 . . 3 ((𝑊𝑋𝐴𝑌) → 𝑅 = if(𝐵𝐴, 𝑊, (𝑊 sSet ⟨(Scalar‘ndx), (𝐹s 𝐴)⟩)))
5 iffalse 4128 . . 3 𝐵𝐴 → if(𝐵𝐴, 𝑊, (𝑊 sSet ⟨(Scalar‘ndx), (𝐹s 𝐴)⟩)) = (𝑊 sSet ⟨(Scalar‘ndx), (𝐹s 𝐴)⟩))
64, 5sylan9eqr 2707 . 2 ((¬ 𝐵𝐴 ∧ (𝑊𝑋𝐴𝑌)) → 𝑅 = (𝑊 sSet ⟨(Scalar‘ndx), (𝐹s 𝐴)⟩))
763impb 1279 1 ((¬ 𝐵𝐴𝑊𝑋𝐴𝑌) → 𝑅 = (𝑊 sSet ⟨(Scalar‘ndx), (𝐹s 𝐴)⟩))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  wss 3607  ifcif 4119  cop 4216  cfv 5926  (class class class)co 6690  ndxcnx 15901   sSet csts 15902  Basecbs 15904  s cress 15905  Scalarcsca 15991  v cresv 29952
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-resv 29953
This theorem is referenced by:  resvsca  29958  resvlem  29959
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