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Mirrors > Home > MPE Home > Th. List > Mathboxes > resvval2 | Structured version Visualization version GIF version |
Description: Value of nontrivial structure restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) |
Ref | Expression |
---|---|
resvsca.r | ⊢ 𝑅 = (𝑊 ↾v 𝐴) |
resvsca.f | ⊢ 𝐹 = (Scalar‘𝑊) |
resvsca.b | ⊢ 𝐵 = (Base‘𝐹) |
Ref | Expression |
---|---|
resvval2 | ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝑅 = (𝑊 sSet 〈(Scalar‘ndx), (𝐹 ↾s 𝐴)〉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resvsca.r | . . . 4 ⊢ 𝑅 = (𝑊 ↾v 𝐴) | |
2 | resvsca.f | . . . 4 ⊢ 𝐹 = (Scalar‘𝑊) | |
3 | resvsca.b | . . . 4 ⊢ 𝐵 = (Base‘𝐹) | |
4 | 1, 2, 3 | resvval 31526 | . . 3 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝑅 = if(𝐵 ⊆ 𝐴, 𝑊, (𝑊 sSet 〈(Scalar‘ndx), (𝐹 ↾s 𝐴)〉))) |
5 | iffalse 4468 | . . 3 ⊢ (¬ 𝐵 ⊆ 𝐴 → if(𝐵 ⊆ 𝐴, 𝑊, (𝑊 sSet 〈(Scalar‘ndx), (𝐹 ↾s 𝐴)〉)) = (𝑊 sSet 〈(Scalar‘ndx), (𝐹 ↾s 𝐴)〉)) | |
6 | 4, 5 | sylan9eqr 2800 | . 2 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ (𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌)) → 𝑅 = (𝑊 sSet 〈(Scalar‘ndx), (𝐹 ↾s 𝐴)〉)) |
7 | 6 | 3impb 1114 | 1 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝑅 = (𝑊 sSet 〈(Scalar‘ndx), (𝐹 ↾s 𝐴)〉)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ⊆ wss 3887 ifcif 4459 〈cop 4567 ‘cfv 6433 (class class class)co 7275 sSet csts 16864 ndxcnx 16894 Basecbs 16912 ↾s cress 16941 Scalarcsca 16965 ↾v cresv 31523 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-sbc 3717 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-iota 6391 df-fun 6435 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-resv 31524 |
This theorem is referenced by: resvsca 31529 resvlem 31530 resvlemOLD 31531 |
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