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Mirrors > Home > MPE Home > Th. List > lsppropd | Structured version Visualization version GIF version |
Description: If two structures have the same components (properties), they have the same span function. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 24-Apr-2024.) |
Ref | Expression |
---|---|
lsspropd.b1 | ⊢ (𝜑 → 𝐵 = (Base‘𝐾)) |
lsspropd.b2 | ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) |
lsspropd.w | ⊢ (𝜑 → 𝐵 ⊆ 𝑊) |
lsspropd.p | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) |
lsspropd.s1 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝐾)𝑦) ∈ 𝑊) |
lsspropd.s2 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝐾)𝑦) = (𝑥( ·𝑠 ‘𝐿)𝑦)) |
lsspropd.p1 | ⊢ (𝜑 → 𝑃 = (Base‘(Scalar‘𝐾))) |
lsspropd.p2 | ⊢ (𝜑 → 𝑃 = (Base‘(Scalar‘𝐿))) |
lsppropd.v1 | ⊢ (𝜑 → 𝐾 ∈ 𝑋) |
lsppropd.v2 | ⊢ (𝜑 → 𝐿 ∈ 𝑌) |
Ref | Expression |
---|---|
lsppropd | ⊢ (𝜑 → (LSpan‘𝐾) = (LSpan‘𝐿)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lsspropd.b1 | . . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘𝐾)) | |
2 | lsspropd.b2 | . . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) | |
3 | 1, 2 | eqtr3d 2835 | . . . 4 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿)) |
4 | 3 | pweqd 4516 | . . 3 ⊢ (𝜑 → 𝒫 (Base‘𝐾) = 𝒫 (Base‘𝐿)) |
5 | lsspropd.w | . . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ 𝑊) | |
6 | lsspropd.p | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) | |
7 | lsspropd.s1 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝐾)𝑦) ∈ 𝑊) | |
8 | lsspropd.s2 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝐾)𝑦) = (𝑥( ·𝑠 ‘𝐿)𝑦)) | |
9 | lsspropd.p1 | . . . . . 6 ⊢ (𝜑 → 𝑃 = (Base‘(Scalar‘𝐾))) | |
10 | lsspropd.p2 | . . . . . 6 ⊢ (𝜑 → 𝑃 = (Base‘(Scalar‘𝐿))) | |
11 | 1, 2, 5, 6, 7, 8, 9, 10 | lsspropd 19782 | . . . . 5 ⊢ (𝜑 → (LSubSp‘𝐾) = (LSubSp‘𝐿)) |
12 | 11 | rabeqdv 3432 | . . . 4 ⊢ (𝜑 → {𝑡 ∈ (LSubSp‘𝐾) ∣ 𝑠 ⊆ 𝑡} = {𝑡 ∈ (LSubSp‘𝐿) ∣ 𝑠 ⊆ 𝑡}) |
13 | 12 | inteqd 4843 | . . 3 ⊢ (𝜑 → ∩ {𝑡 ∈ (LSubSp‘𝐾) ∣ 𝑠 ⊆ 𝑡} = ∩ {𝑡 ∈ (LSubSp‘𝐿) ∣ 𝑠 ⊆ 𝑡}) |
14 | 4, 13 | mpteq12dv 5115 | . 2 ⊢ (𝜑 → (𝑠 ∈ 𝒫 (Base‘𝐾) ↦ ∩ {𝑡 ∈ (LSubSp‘𝐾) ∣ 𝑠 ⊆ 𝑡}) = (𝑠 ∈ 𝒫 (Base‘𝐿) ↦ ∩ {𝑡 ∈ (LSubSp‘𝐿) ∣ 𝑠 ⊆ 𝑡})) |
15 | lsppropd.v1 | . . 3 ⊢ (𝜑 → 𝐾 ∈ 𝑋) | |
16 | eqid 2798 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
17 | eqid 2798 | . . . 4 ⊢ (LSubSp‘𝐾) = (LSubSp‘𝐾) | |
18 | eqid 2798 | . . . 4 ⊢ (LSpan‘𝐾) = (LSpan‘𝐾) | |
19 | 16, 17, 18 | lspfval 19738 | . . 3 ⊢ (𝐾 ∈ 𝑋 → (LSpan‘𝐾) = (𝑠 ∈ 𝒫 (Base‘𝐾) ↦ ∩ {𝑡 ∈ (LSubSp‘𝐾) ∣ 𝑠 ⊆ 𝑡})) |
20 | 15, 19 | syl 17 | . 2 ⊢ (𝜑 → (LSpan‘𝐾) = (𝑠 ∈ 𝒫 (Base‘𝐾) ↦ ∩ {𝑡 ∈ (LSubSp‘𝐾) ∣ 𝑠 ⊆ 𝑡})) |
21 | lsppropd.v2 | . . 3 ⊢ (𝜑 → 𝐿 ∈ 𝑌) | |
22 | eqid 2798 | . . . 4 ⊢ (Base‘𝐿) = (Base‘𝐿) | |
23 | eqid 2798 | . . . 4 ⊢ (LSubSp‘𝐿) = (LSubSp‘𝐿) | |
24 | eqid 2798 | . . . 4 ⊢ (LSpan‘𝐿) = (LSpan‘𝐿) | |
25 | 22, 23, 24 | lspfval 19738 | . . 3 ⊢ (𝐿 ∈ 𝑌 → (LSpan‘𝐿) = (𝑠 ∈ 𝒫 (Base‘𝐿) ↦ ∩ {𝑡 ∈ (LSubSp‘𝐿) ∣ 𝑠 ⊆ 𝑡})) |
26 | 21, 25 | syl 17 | . 2 ⊢ (𝜑 → (LSpan‘𝐿) = (𝑠 ∈ 𝒫 (Base‘𝐿) ↦ ∩ {𝑡 ∈ (LSubSp‘𝐿) ∣ 𝑠 ⊆ 𝑡})) |
27 | 14, 20, 26 | 3eqtr4d 2843 | 1 ⊢ (𝜑 → (LSpan‘𝐾) = (LSpan‘𝐿)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 {crab 3110 ⊆ wss 3881 𝒫 cpw 4497 ∩ cint 4838 ↦ cmpt 5110 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 +gcplusg 16557 Scalarcsca 16560 ·𝑠 cvsca 16561 LSubSpclss 19696 LSpanclspn 19736 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-lss 19697 df-lsp 19737 |
This theorem is referenced by: lbspropd 19864 lidlrsppropd 19996 lindfpropd 30996 |
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