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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 2777 | . . . 4 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿)) |
4 | 3 | pweqd 4622 | . . 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 21034 | . . . . 5 ⊢ (𝜑 → (LSubSp‘𝐾) = (LSubSp‘𝐿)) |
12 | 11 | rabeqdv 3449 | . . . 4 ⊢ (𝜑 → {𝑡 ∈ (LSubSp‘𝐾) ∣ 𝑠 ⊆ 𝑡} = {𝑡 ∈ (LSubSp‘𝐿) ∣ 𝑠 ⊆ 𝑡}) |
13 | 12 | inteqd 4956 | . . 3 ⊢ (𝜑 → ∩ {𝑡 ∈ (LSubSp‘𝐾) ∣ 𝑠 ⊆ 𝑡} = ∩ {𝑡 ∈ (LSubSp‘𝐿) ∣ 𝑠 ⊆ 𝑡}) |
14 | 4, 13 | mpteq12dv 5239 | . 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 2735 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
17 | eqid 2735 | . . . 4 ⊢ (LSubSp‘𝐾) = (LSubSp‘𝐾) | |
18 | eqid 2735 | . . . 4 ⊢ (LSpan‘𝐾) = (LSpan‘𝐾) | |
19 | 16, 17, 18 | lspfval 20989 | . . 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 2735 | . . . 4 ⊢ (Base‘𝐿) = (Base‘𝐿) | |
23 | eqid 2735 | . . . 4 ⊢ (LSubSp‘𝐿) = (LSubSp‘𝐿) | |
24 | eqid 2735 | . . . 4 ⊢ (LSpan‘𝐿) = (LSpan‘𝐿) | |
25 | 22, 23, 24 | lspfval 20989 | . . 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 2785 | 1 ⊢ (𝜑 → (LSpan‘𝐾) = (LSpan‘𝐿)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 {crab 3433 ⊆ wss 3963 𝒫 cpw 4605 ∩ cint 4951 ↦ cmpt 5231 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 +gcplusg 17298 Scalarcsca 17301 ·𝑠 cvsca 17302 LSubSpclss 20947 LSpanclspn 20987 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-lss 20948 df-lsp 20988 |
This theorem is referenced by: lbspropd 21116 lidlrsppropd 21272 lindfpropd 33390 |
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