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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lindspropd | Structured version Visualization version GIF version |
Description: Property deduction for linearly independent sets. (Contributed by Thierry Arnoux, 16-Jul-2023.) |
Ref | Expression |
---|---|
lindfpropd.1 | ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿)) |
lindfpropd.2 | ⊢ (𝜑 → (Base‘(Scalar‘𝐾)) = (Base‘(Scalar‘𝐿))) |
lindfpropd.3 | ⊢ (𝜑 → (0g‘(Scalar‘𝐾)) = (0g‘(Scalar‘𝐿))) |
lindfpropd.4 | ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) |
lindfpropd.5 | ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐾)) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥( ·𝑠 ‘𝐾)𝑦) ∈ (Base‘𝐾)) |
lindfpropd.6 | ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐾)) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥( ·𝑠 ‘𝐾)𝑦) = (𝑥( ·𝑠 ‘𝐿)𝑦)) |
lindfpropd.k | ⊢ (𝜑 → 𝐾 ∈ 𝑉) |
lindfpropd.l | ⊢ (𝜑 → 𝐿 ∈ 𝑊) |
Ref | Expression |
---|---|
lindspropd | ⊢ (𝜑 → (LIndS‘𝐾) = (LIndS‘𝐿)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lindfpropd.1 | . . . . 5 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿)) | |
2 | 1 | sseq2d 4035 | . . . 4 ⊢ (𝜑 → (𝑧 ⊆ (Base‘𝐾) ↔ 𝑧 ⊆ (Base‘𝐿))) |
3 | lindfpropd.2 | . . . . 5 ⊢ (𝜑 → (Base‘(Scalar‘𝐾)) = (Base‘(Scalar‘𝐿))) | |
4 | lindfpropd.3 | . . . . 5 ⊢ (𝜑 → (0g‘(Scalar‘𝐾)) = (0g‘(Scalar‘𝐿))) | |
5 | lindfpropd.4 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) | |
6 | lindfpropd.5 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐾)) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥( ·𝑠 ‘𝐾)𝑦) ∈ (Base‘𝐾)) | |
7 | lindfpropd.6 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐾)) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥( ·𝑠 ‘𝐾)𝑦) = (𝑥( ·𝑠 ‘𝐿)𝑦)) | |
8 | lindfpropd.k | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ 𝑉) | |
9 | lindfpropd.l | . . . . 5 ⊢ (𝜑 → 𝐿 ∈ 𝑊) | |
10 | vex 3486 | . . . . . . 7 ⊢ 𝑧 ∈ V | |
11 | 10 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 𝑧 ∈ V) |
12 | 11 | resiexd 7251 | . . . . 5 ⊢ (𝜑 → ( I ↾ 𝑧) ∈ V) |
13 | 1, 3, 4, 5, 6, 7, 8, 9, 12 | lindfpropd 33367 | . . . 4 ⊢ (𝜑 → (( I ↾ 𝑧) LIndF 𝐾 ↔ ( I ↾ 𝑧) LIndF 𝐿)) |
14 | 2, 13 | anbi12d 631 | . . 3 ⊢ (𝜑 → ((𝑧 ⊆ (Base‘𝐾) ∧ ( I ↾ 𝑧) LIndF 𝐾) ↔ (𝑧 ⊆ (Base‘𝐿) ∧ ( I ↾ 𝑧) LIndF 𝐿))) |
15 | eqid 2734 | . . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
16 | 15 | islinds 21847 | . . . 4 ⊢ (𝐾 ∈ 𝑉 → (𝑧 ∈ (LIndS‘𝐾) ↔ (𝑧 ⊆ (Base‘𝐾) ∧ ( I ↾ 𝑧) LIndF 𝐾))) |
17 | 8, 16 | syl 17 | . . 3 ⊢ (𝜑 → (𝑧 ∈ (LIndS‘𝐾) ↔ (𝑧 ⊆ (Base‘𝐾) ∧ ( I ↾ 𝑧) LIndF 𝐾))) |
18 | eqid 2734 | . . . . 5 ⊢ (Base‘𝐿) = (Base‘𝐿) | |
19 | 18 | islinds 21847 | . . . 4 ⊢ (𝐿 ∈ 𝑊 → (𝑧 ∈ (LIndS‘𝐿) ↔ (𝑧 ⊆ (Base‘𝐿) ∧ ( I ↾ 𝑧) LIndF 𝐿))) |
20 | 9, 19 | syl 17 | . . 3 ⊢ (𝜑 → (𝑧 ∈ (LIndS‘𝐿) ↔ (𝑧 ⊆ (Base‘𝐿) ∧ ( I ↾ 𝑧) LIndF 𝐿))) |
21 | 14, 17, 20 | 3bitr4d 311 | . 2 ⊢ (𝜑 → (𝑧 ∈ (LIndS‘𝐾) ↔ 𝑧 ∈ (LIndS‘𝐿))) |
22 | 21 | eqrdv 2732 | 1 ⊢ (𝜑 → (LIndS‘𝐾) = (LIndS‘𝐿)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2103 Vcvv 3482 ⊆ wss 3970 class class class wbr 5169 I cid 5596 ↾ cres 5701 ‘cfv 6572 (class class class)co 7445 Basecbs 17253 +gcplusg 17306 Scalarcsca 17309 ·𝑠 cvsca 17310 0gc0g 17494 LIndF clindf 21842 LIndSclinds 21843 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-rep 5306 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-ral 3064 df-rex 3073 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4973 df-iun 5021 df-br 5170 df-opab 5232 df-mpt 5253 df-id 5597 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-ov 7448 df-lss 20948 df-lsp 20988 df-lindf 21844 df-linds 21845 |
This theorem is referenced by: fedgmullem2 33635 |
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