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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 2775 | . . . 4 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿)) |
4 | 3 | pweqd 4615 | . . 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 20605 | . . . . 5 ⊢ (𝜑 → (LSubSp‘𝐾) = (LSubSp‘𝐿)) |
12 | 11 | rabeqdv 3448 | . . . 4 ⊢ (𝜑 → {𝑡 ∈ (LSubSp‘𝐾) ∣ 𝑠 ⊆ 𝑡} = {𝑡 ∈ (LSubSp‘𝐿) ∣ 𝑠 ⊆ 𝑡}) |
13 | 12 | inteqd 4951 | . . 3 ⊢ (𝜑 → ∩ {𝑡 ∈ (LSubSp‘𝐾) ∣ 𝑠 ⊆ 𝑡} = ∩ {𝑡 ∈ (LSubSp‘𝐿) ∣ 𝑠 ⊆ 𝑡}) |
14 | 4, 13 | mpteq12dv 5235 | . 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 2733 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
17 | eqid 2733 | . . . 4 ⊢ (LSubSp‘𝐾) = (LSubSp‘𝐾) | |
18 | eqid 2733 | . . . 4 ⊢ (LSpan‘𝐾) = (LSpan‘𝐾) | |
19 | 16, 17, 18 | lspfval 20561 | . . 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 2733 | . . . 4 ⊢ (Base‘𝐿) = (Base‘𝐿) | |
23 | eqid 2733 | . . . 4 ⊢ (LSubSp‘𝐿) = (LSubSp‘𝐿) | |
24 | eqid 2733 | . . . 4 ⊢ (LSpan‘𝐿) = (LSpan‘𝐿) | |
25 | 22, 23, 24 | lspfval 20561 | . . 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 2783 | 1 ⊢ (𝜑 → (LSpan‘𝐾) = (LSpan‘𝐿)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 {crab 3433 ⊆ wss 3946 𝒫 cpw 4598 ∩ cint 4946 ↦ cmpt 5227 ‘cfv 6535 (class class class)co 7396 Basecbs 17131 +gcplusg 17184 Scalarcsca 17187 ·𝑠 cvsca 17188 LSubSpclss 20519 LSpanclspn 20559 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-int 4947 df-iun 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-ov 7399 df-lss 20520 df-lsp 20560 |
This theorem is referenced by: lbspropd 20687 lidlrsppropd 20831 lindfpropd 32456 |
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