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Theorem resvval2 32167
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 32165 . . 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 4496 . . 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 2795 . 2 ((Β¬ 𝐡 βŠ† 𝐴 ∧ (π‘Š ∈ 𝑋 ∧ 𝐴 ∈ π‘Œ)) β†’ 𝑅 = (π‘Š sSet ⟨(Scalarβ€˜ndx), (𝐹 β†Ύs 𝐴)⟩))
763impb 1116 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   βŠ† wss 3911  ifcif 4487  βŸ¨cop 4593  β€˜cfv 6497  (class class class)co 7358   sSet csts 17040  ndxcnx 17070  Basecbs 17088   β†Ύs cress 17117  Scalarcsca 17141   β†Ύv cresv 32162
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6449  df-fun 6499  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-resv 32163
This theorem is referenced by:  resvsca  32168  resvlem  32169  resvlemOLD  32170
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