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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > resvid2 | Structured version Visualization version GIF version |
Description: General behavior of trivial 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 |
---|---|
resvid2 | ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝑅 = 𝑊) |
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 33333 | . . 3 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝑅 = if(𝐵 ⊆ 𝐴, 𝑊, (𝑊 sSet 〈(Scalar‘ndx), (𝐹 ↾s 𝐴)〉))) |
5 | iftrue 4537 | . . 3 ⊢ (𝐵 ⊆ 𝐴 → if(𝐵 ⊆ 𝐴, 𝑊, (𝑊 sSet 〈(Scalar‘ndx), (𝐹 ↾s 𝐴)〉)) = 𝑊) | |
6 | 4, 5 | sylan9eqr 2797 | . 2 ⊢ ((𝐵 ⊆ 𝐴 ∧ (𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌)) → 𝑅 = 𝑊) |
7 | 6 | 3impb 1114 | 1 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝑅 = 𝑊) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ⊆ wss 3963 ifcif 4531 〈cop 4637 ‘cfv 6563 (class class class)co 7431 sSet csts 17197 ndxcnx 17227 Basecbs 17245 ↾s cress 17274 Scalarcsca 17301 ↾v cresv 33330 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-resv 33331 |
This theorem is referenced by: resvsca 33336 resvlem 33337 resvlemOLD 33338 |
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