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Theorem resvval2 32438
Description: Value of nontrivial structure restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
Hypotheses
Ref Expression
resvsca.r 𝑅 = (π‘Š β†Ύv 𝐴)
resvsca.f 𝐹 = (Scalarβ€˜π‘Š)
resvsca.b 𝐡 = (Baseβ€˜πΉ)
Assertion
Ref Expression
resvval2 ((Β¬ 𝐡 βŠ† 𝐴 ∧ π‘Š ∈ 𝑋 ∧ 𝐴 ∈ π‘Œ) β†’ 𝑅 = (π‘Š sSet ⟨(Scalarβ€˜ndx), (𝐹 β†Ύs 𝐴)⟩))
Proof of Theorem resvval2
StepHypRef Expression
1 resvsca.r . . . 4 𝑅 = (π‘Š β†Ύv 𝐴)
2 resvsca.f . . . 4 𝐹 = (Scalarβ€˜π‘Š)
3 resvsca.b . . . 4 𝐡 = (Baseβ€˜πΉ)
41, 2, 3resvval 32436 . . 3 ((π‘Š ∈ 𝑋 ∧ 𝐴 ∈ π‘Œ) β†’ 𝑅 = if(𝐡 βŠ† 𝐴, π‘Š, (π‘Š sSet ⟨(Scalarβ€˜ndx), (𝐹 β†Ύs 𝐴)⟩)))
5 iffalse 4537 . . 3 (Β¬ 𝐡 βŠ† 𝐴 β†’ if(𝐡 βŠ† 𝐴, π‘Š, (π‘Š sSet ⟨(Scalarβ€˜ndx), (𝐹 β†Ύs 𝐴)⟩)) = (π‘Š sSet ⟨(Scalarβ€˜ndx), (𝐹 β†Ύs 𝐴)⟩))
64, 5sylan9eqr 2794 . 2 ((Β¬ 𝐡 βŠ† 𝐴 ∧ (π‘Š ∈ 𝑋 ∧ 𝐴 ∈ π‘Œ)) β†’ 𝑅 = (π‘Š sSet ⟨(Scalarβ€˜ndx), (𝐹 β†Ύs 𝐴)⟩))
763impb 1115 1 ((Β¬ 𝐡 βŠ† 𝐴 ∧ π‘Š ∈ 𝑋 ∧ 𝐴 ∈ π‘Œ) β†’ 𝑅 = (π‘Š sSet ⟨(Scalarβ€˜ndx), (𝐹 β†Ύs 𝐴)⟩))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   βŠ† wss 3948  ifcif 4528  βŸ¨cop 4634  β€˜cfv 6543  (class class class)co 7408   sSet csts 17095  ndxcnx 17125  Basecbs 17143   β†Ύs cress 17172  Scalarcsca 17199   β†Ύv cresv 32433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-resv 32434
This theorem is referenced by:  resvsca  32439  resvlem  32440  resvlemOLD  32441
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