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Theorem resvval 32223
Description: Value of 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
resvval ((π‘Š ∈ 𝑋 ∧ 𝐴 ∈ π‘Œ) β†’ 𝑅 = if(𝐡 βŠ† 𝐴, π‘Š, (π‘Š sSet ⟨(Scalarβ€˜ndx), (𝐹 β†Ύs 𝐴)⟩)))
Proof of Theorem resvval
Dummy variables π‘₯ 𝑀 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 resvsca.r . 2 𝑅 = (π‘Š β†Ύv 𝐴)
2 elex 3477 . . 3 (π‘Š ∈ 𝑋 β†’ π‘Š ∈ V)
3 elex 3477 . . 3 (𝐴 ∈ π‘Œ β†’ 𝐴 ∈ V)
4 ovex 7410 . . . . . 6 (π‘Š sSet ⟨(Scalarβ€˜ndx), (𝐹 β†Ύs 𝐴)⟩) ∈ V
5 ifcl 4551 . . . . . 6 ((π‘Š ∈ V ∧ (π‘Š sSet ⟨(Scalarβ€˜ndx), (𝐹 β†Ύs 𝐴)⟩) ∈ V) β†’ if(𝐡 βŠ† 𝐴, π‘Š, (π‘Š sSet ⟨(Scalarβ€˜ndx), (𝐹 β†Ύs 𝐴)⟩)) ∈ V)
64, 5mpan2 689 . . . . 5 (π‘Š ∈ V β†’ if(𝐡 βŠ† 𝐴, π‘Š, (π‘Š sSet ⟨(Scalarβ€˜ndx), (𝐹 β†Ύs 𝐴)⟩)) ∈ V)
76adantr 481 . . . 4 ((π‘Š ∈ V ∧ 𝐴 ∈ V) β†’ if(𝐡 βŠ† 𝐴, π‘Š, (π‘Š sSet ⟨(Scalarβ€˜ndx), (𝐹 β†Ύs 𝐴)⟩)) ∈ V)
8 simpl 483 . . . . . . . . . . 11 ((𝑀 = π‘Š ∧ π‘₯ = 𝐴) β†’ 𝑀 = π‘Š)
98fveq2d 6866 . . . . . . . . . 10 ((𝑀 = π‘Š ∧ π‘₯ = 𝐴) β†’ (Scalarβ€˜π‘€) = (Scalarβ€˜π‘Š))
10 resvsca.f . . . . . . . . . 10 𝐹 = (Scalarβ€˜π‘Š)
119, 10eqtr4di 2789 . . . . . . . . 9 ((𝑀 = π‘Š ∧ π‘₯ = 𝐴) β†’ (Scalarβ€˜π‘€) = 𝐹)
1211fveq2d 6866 . . . . . . . 8 ((𝑀 = π‘Š ∧ π‘₯ = 𝐴) β†’ (Baseβ€˜(Scalarβ€˜π‘€)) = (Baseβ€˜πΉ))
13 resvsca.b . . . . . . . 8 𝐡 = (Baseβ€˜πΉ)
1412, 13eqtr4di 2789 . . . . . . 7 ((𝑀 = π‘Š ∧ π‘₯ = 𝐴) β†’ (Baseβ€˜(Scalarβ€˜π‘€)) = 𝐡)
15 simpr 485 . . . . . . 7 ((𝑀 = π‘Š ∧ π‘₯ = 𝐴) β†’ π‘₯ = 𝐴)
1614, 15sseq12d 3995 . . . . . 6 ((𝑀 = π‘Š ∧ π‘₯ = 𝐴) β†’ ((Baseβ€˜(Scalarβ€˜π‘€)) βŠ† π‘₯ ↔ 𝐡 βŠ† 𝐴))
1711, 15oveq12d 7395 . . . . . . . 8 ((𝑀 = π‘Š ∧ π‘₯ = 𝐴) β†’ ((Scalarβ€˜π‘€) β†Ύs π‘₯) = (𝐹 β†Ύs 𝐴))
1817opeq2d 4857 . . . . . . 7 ((𝑀 = π‘Š ∧ π‘₯ = 𝐴) β†’ ⟨(Scalarβ€˜ndx), ((Scalarβ€˜π‘€) β†Ύs π‘₯)⟩ = ⟨(Scalarβ€˜ndx), (𝐹 β†Ύs 𝐴)⟩)
198, 18oveq12d 7395 . . . . . 6 ((𝑀 = π‘Š ∧ π‘₯ = 𝐴) β†’ (𝑀 sSet ⟨(Scalarβ€˜ndx), ((Scalarβ€˜π‘€) β†Ύs π‘₯)⟩) = (π‘Š sSet ⟨(Scalarβ€˜ndx), (𝐹 β†Ύs 𝐴)⟩))
2016, 8, 19ifbieq12d 4534 . . . . 5 ((𝑀 = π‘Š ∧ π‘₯ = 𝐴) β†’ if((Baseβ€˜(Scalarβ€˜π‘€)) βŠ† π‘₯, 𝑀, (𝑀 sSet ⟨(Scalarβ€˜ndx), ((Scalarβ€˜π‘€) β†Ύs π‘₯)⟩)) = if(𝐡 βŠ† 𝐴, π‘Š, (π‘Š sSet ⟨(Scalarβ€˜ndx), (𝐹 β†Ύs 𝐴)⟩)))
21 df-resv 32221 . . . . 5 β†Ύv = (𝑀 ∈ V, π‘₯ ∈ V ↦ if((Baseβ€˜(Scalarβ€˜π‘€)) βŠ† π‘₯, 𝑀, (𝑀 sSet ⟨(Scalarβ€˜ndx), ((Scalarβ€˜π‘€) β†Ύs π‘₯)⟩)))
2220, 21ovmpoga 7529 . . . 4 ((π‘Š ∈ V ∧ 𝐴 ∈ V ∧ if(𝐡 βŠ† 𝐴, π‘Š, (π‘Š sSet ⟨(Scalarβ€˜ndx), (𝐹 β†Ύs 𝐴)⟩)) ∈ V) β†’ (π‘Š β†Ύv 𝐴) = if(𝐡 βŠ† 𝐴, π‘Š, (π‘Š sSet ⟨(Scalarβ€˜ndx), (𝐹 β†Ύs 𝐴)⟩)))
237, 22mpd3an3 1462 . . 3 ((π‘Š ∈ V ∧ 𝐴 ∈ V) β†’ (π‘Š β†Ύv 𝐴) = if(𝐡 βŠ† 𝐴, π‘Š, (π‘Š sSet ⟨(Scalarβ€˜ndx), (𝐹 β†Ύs 𝐴)⟩)))
242, 3, 23syl2an 596 . 2 ((π‘Š ∈ 𝑋 ∧ 𝐴 ∈ π‘Œ) β†’ (π‘Š β†Ύv 𝐴) = if(𝐡 βŠ† 𝐴, π‘Š, (π‘Š sSet ⟨(Scalarβ€˜ndx), (𝐹 β†Ύs 𝐴)⟩)))
251, 24eqtrid 2783 1 ((π‘Š ∈ 𝑋 ∧ 𝐴 ∈ π‘Œ) β†’ 𝑅 = if(𝐡 βŠ† 𝐴, π‘Š, (π‘Š sSet ⟨(Scalarβ€˜ndx), (𝐹 β†Ύs 𝐴)⟩)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  Vcvv 3459   βŠ† wss 3928  ifcif 4506  βŸ¨cop 4612  β€˜cfv 6516  (class class class)co 7377   sSet csts 17061  ndxcnx 17091  Basecbs 17109   β†Ύs cress 17138  Scalarcsca 17165   β†Ύv cresv 32220
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 2702  ax-sep 5276  ax-nul 5283  ax-pr 5404
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 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3419  df-v 3461  df-sbc 3758  df-dif 3931  df-un 3933  df-in 3935  df-ss 3945  df-nul 4303  df-if 4507  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4886  df-br 5126  df-opab 5188  df-id 5551  df-xp 5659  df-rel 5660  df-cnv 5661  df-co 5662  df-dm 5663  df-iota 6468  df-fun 6518  df-fv 6524  df-ov 7380  df-oprab 7381  df-mpo 7382  df-resv 32221
This theorem is referenced by:  resvid2  32224  resvval2  32225
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