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 3999 | . . . 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 3497 | . . . . . . 7 ⊢ 𝑧 ∈ V | |
11 | 10 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 𝑧 ∈ V) |
12 | 11 | resiexd 6979 | . . . . 5 ⊢ (𝜑 → ( I ↾ 𝑧) ∈ V) |
13 | 1, 3, 4, 5, 6, 7, 8, 9, 12 | lindfpropd 30942 | . . . 4 ⊢ (𝜑 → (( I ↾ 𝑧) LIndF 𝐾 ↔ ( I ↾ 𝑧) LIndF 𝐿)) |
14 | 2, 13 | anbi12d 632 | . . 3 ⊢ (𝜑 → ((𝑧 ⊆ (Base‘𝐾) ∧ ( I ↾ 𝑧) LIndF 𝐾) ↔ (𝑧 ⊆ (Base‘𝐿) ∧ ( I ↾ 𝑧) LIndF 𝐿))) |
15 | eqid 2821 | . . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
16 | 15 | islinds 20953 | . . . 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 2821 | . . . . 5 ⊢ (Base‘𝐿) = (Base‘𝐿) | |
19 | 18 | islinds 20953 | . . . 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 313 | . 2 ⊢ (𝜑 → (𝑧 ∈ (LIndS‘𝐾) ↔ 𝑧 ∈ (LIndS‘𝐿))) |
22 | 21 | eqrdv 2819 | 1 ⊢ (𝜑 → (LIndS‘𝐾) = (LIndS‘𝐿)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ⊆ wss 3936 class class class wbr 5066 I cid 5459 ↾ cres 5557 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 +gcplusg 16565 Scalarcsca 16568 ·𝑠 cvsca 16569 0gc0g 16713 LIndF clindf 20948 LIndSclinds 20949 |
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-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 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-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-lss 19704 df-lsp 19744 df-lindf 20950 df-linds 20951 |
This theorem is referenced by: fedgmullem2 31026 |
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