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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 31095 | . . 3 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝑅 = if(𝐵 ⊆ 𝐴, 𝑊, (𝑊 sSet 〈(Scalar‘ndx), (𝐹 ↾s 𝐴)〉))) |
5 | iffalse 4420 | . . 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 2795 | . 2 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ (𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌)) → 𝑅 = (𝑊 sSet 〈(Scalar‘ndx), (𝐹 ↾s 𝐴)〉)) |
7 | 6 | 3impb 1116 | 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 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2113 ⊆ wss 3841 ifcif 4411 〈cop 4519 ‘cfv 6333 (class class class)co 7164 ndxcnx 16576 sSet csts 16577 Basecbs 16579 ↾s cress 16580 Scalarcsca 16664 ↾v cresv 31092 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pr 5293 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3399 df-sbc 3680 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-if 4412 df-sn 4514 df-pr 4516 df-op 4520 df-uni 4794 df-br 5028 df-opab 5090 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6291 df-fun 6335 df-fv 6341 df-ov 7167 df-oprab 7168 df-mpo 7169 df-resv 31093 |
This theorem is referenced by: resvsca 31098 resvlem 31099 |
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