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Theorem resvval 32711
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 3491 . . 3 (π‘Š ∈ 𝑋 β†’ π‘Š ∈ V)
3 elex 3491 . . 3 (𝐴 ∈ π‘Œ β†’ 𝐴 ∈ V)
4 ovex 7444 . . . . . 6 (π‘Š sSet ⟨(Scalarβ€˜ndx), (𝐹 β†Ύs 𝐴)⟩) ∈ V
5 ifcl 4572 . . . . . 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 687 . . . . 5 (π‘Š ∈ V β†’ if(𝐡 βŠ† 𝐴, π‘Š, (π‘Š sSet ⟨(Scalarβ€˜ndx), (𝐹 β†Ύs 𝐴)⟩)) ∈ V)
76adantr 479 . . . 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 481 . . . . . . . . . . 11 ((𝑀 = π‘Š ∧ π‘₯ = 𝐴) β†’ 𝑀 = π‘Š)
98fveq2d 6894 . . . . . . . . . 10 ((𝑀 = π‘Š ∧ π‘₯ = 𝐴) β†’ (Scalarβ€˜π‘€) = (Scalarβ€˜π‘Š))
10 resvsca.f . . . . . . . . . 10 𝐹 = (Scalarβ€˜π‘Š)
119, 10eqtr4di 2788 . . . . . . . . 9 ((𝑀 = π‘Š ∧ π‘₯ = 𝐴) β†’ (Scalarβ€˜π‘€) = 𝐹)
1211fveq2d 6894 . . . . . . . 8 ((𝑀 = π‘Š ∧ π‘₯ = 𝐴) β†’ (Baseβ€˜(Scalarβ€˜π‘€)) = (Baseβ€˜πΉ))
13 resvsca.b . . . . . . . 8 𝐡 = (Baseβ€˜πΉ)
1412, 13eqtr4di 2788 . . . . . . 7 ((𝑀 = π‘Š ∧ π‘₯ = 𝐴) β†’ (Baseβ€˜(Scalarβ€˜π‘€)) = 𝐡)
15 simpr 483 . . . . . . 7 ((𝑀 = π‘Š ∧ π‘₯ = 𝐴) β†’ π‘₯ = 𝐴)
1614, 15sseq12d 4014 . . . . . 6 ((𝑀 = π‘Š ∧ π‘₯ = 𝐴) β†’ ((Baseβ€˜(Scalarβ€˜π‘€)) βŠ† π‘₯ ↔ 𝐡 βŠ† 𝐴))
1711, 15oveq12d 7429 . . . . . . . 8 ((𝑀 = π‘Š ∧ π‘₯ = 𝐴) β†’ ((Scalarβ€˜π‘€) β†Ύs π‘₯) = (𝐹 β†Ύs 𝐴))
1817opeq2d 4879 . . . . . . 7 ((𝑀 = π‘Š ∧ π‘₯ = 𝐴) β†’ ⟨(Scalarβ€˜ndx), ((Scalarβ€˜π‘€) β†Ύs π‘₯)⟩ = ⟨(Scalarβ€˜ndx), (𝐹 β†Ύs 𝐴)⟩)
198, 18oveq12d 7429 . . . . . 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 4555 . . . . 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 32709 . . . . 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 7564 . . . 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 1460 . . 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 594 . 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 2782 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 394   = wceq 1539   ∈ wcel 2104  Vcvv 3472   βŠ† wss 3947  ifcif 4527  βŸ¨cop 4633  β€˜cfv 6542  (class class class)co 7411   sSet csts 17100  ndxcnx 17130  Basecbs 17148   β†Ύs cress 17177  Scalarcsca 17204   β†Ύv cresv 32708
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6494  df-fun 6544  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-resv 32709
This theorem is referenced by:  resvid2  32712  resvval2  32713
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