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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 30900 | . . 3 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝑅 = if(𝐵 ⊆ 𝐴, 𝑊, (𝑊 sSet 〈(Scalar‘ndx), (𝐹 ↾s 𝐴)〉))) |
5 | iftrue 4473 | . . 3 ⊢ (𝐵 ⊆ 𝐴 → if(𝐵 ⊆ 𝐴, 𝑊, (𝑊 sSet 〈(Scalar‘ndx), (𝐹 ↾s 𝐴)〉)) = 𝑊) | |
6 | 4, 5 | sylan9eqr 2878 | . 2 ⊢ ((𝐵 ⊆ 𝐴 ∧ (𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌)) → 𝑅 = 𝑊) |
7 | 6 | 3impb 1111 | 1 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝑅 = 𝑊) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ⊆ wss 3936 ifcif 4467 〈cop 4573 ‘cfv 6355 (class class class)co 7156 ndxcnx 16480 sSet csts 16481 Basecbs 16483 ↾s cress 16484 Scalarcsca 16568 ↾v cresv 30897 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-iota 6314 df-fun 6357 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-resv 30898 |
This theorem is referenced by: resvsca 30903 resvlem 30904 |
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