Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  resvval2 Structured version   Visualization version   GIF version

Theorem resvval2 31097
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 31095 . . 3 ((𝑊𝑋𝐴𝑌) → 𝑅 = if(𝐵𝐴, 𝑊, (𝑊 sSet ⟨(Scalar‘ndx), (𝐹s 𝐴)⟩)))
5 iffalse 4420 . . 3 𝐵𝐴 → if(𝐵𝐴, 𝑊, (𝑊 sSet ⟨(Scalar‘ndx), (𝐹s 𝐴)⟩)) = (𝑊 sSet ⟨(Scalar‘ndx), (𝐹s 𝐴)⟩))
64, 5sylan9eqr 2795 . 2 ((¬ 𝐵𝐴 ∧ (𝑊𝑋𝐴𝑌)) → 𝑅 = (𝑊 sSet ⟨(Scalar‘ndx), (𝐹s 𝐴)⟩))
763impb 1116 1 ((¬ 𝐵𝐴𝑊𝑋𝐴𝑌) → 𝑅 = (𝑊 sSet ⟨(Scalar‘ndx), (𝐹s 𝐴)⟩))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  w3a 1088   = wceq 1542  wcel 2113  wss 3841  ifcif 4411  cop 4519  cfv 6333  (class class class)co 7164  ndxcnx 16576   sSet csts 16577  Basecbs 16579  s cress 16580  Scalarcsca 16664  v cresv 31092
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-sep 5164  ax-nul 5171  ax-pr 5293
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3399  df-sbc 3680  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-if 4412  df-sn 4514  df-pr 4516  df-op 4520  df-uni 4794  df-br 5028  df-opab 5090  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6291  df-fun 6335  df-fv 6341  df-ov 7167  df-oprab 7168  df-mpo 7169  df-resv 31093
This theorem is referenced by:  resvsca  31098  resvlem  31099
  Copyright terms: Public domain W3C validator