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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.) |
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))) |
lsspropd.v1 | ⊢ (𝜑 → 𝐾 ∈ V) |
lsspropd.v2 | ⊢ (𝜑 → 𝐿 ∈ V) |
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 2646 | . . . 4 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿)) |
4 | 3 | pweqd 4113 | . . 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 18838 | . . . . 5 ⊢ (𝜑 → (LSubSp‘𝐾) = (LSubSp‘𝐿)) |
12 | rabeq 3166 | . . . . 5 ⊢ ((LSubSp‘𝐾) = (LSubSp‘𝐿) → {𝑡 ∈ (LSubSp‘𝐾) ∣ 𝑠 ⊆ 𝑡} = {𝑡 ∈ (LSubSp‘𝐿) ∣ 𝑠 ⊆ 𝑡}) | |
13 | 11, 12 | syl 17 | . . . 4 ⊢ (𝜑 → {𝑡 ∈ (LSubSp‘𝐾) ∣ 𝑠 ⊆ 𝑡} = {𝑡 ∈ (LSubSp‘𝐿) ∣ 𝑠 ⊆ 𝑡}) |
14 | 13 | inteqd 4415 | . . 3 ⊢ (𝜑 → ∩ {𝑡 ∈ (LSubSp‘𝐾) ∣ 𝑠 ⊆ 𝑡} = ∩ {𝑡 ∈ (LSubSp‘𝐿) ∣ 𝑠 ⊆ 𝑡}) |
15 | 4, 14 | mpteq12dv 4663 | . 2 ⊢ (𝜑 → (𝑠 ∈ 𝒫 (Base‘𝐾) ↦ ∩ {𝑡 ∈ (LSubSp‘𝐾) ∣ 𝑠 ⊆ 𝑡}) = (𝑠 ∈ 𝒫 (Base‘𝐿) ↦ ∩ {𝑡 ∈ (LSubSp‘𝐿) ∣ 𝑠 ⊆ 𝑡})) |
16 | lsspropd.v1 | . . 3 ⊢ (𝜑 → 𝐾 ∈ V) | |
17 | eqid 2610 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
18 | eqid 2610 | . . . 4 ⊢ (LSubSp‘𝐾) = (LSubSp‘𝐾) | |
19 | eqid 2610 | . . . 4 ⊢ (LSpan‘𝐾) = (LSpan‘𝐾) | |
20 | 17, 18, 19 | lspfval 18794 | . . 3 ⊢ (𝐾 ∈ V → (LSpan‘𝐾) = (𝑠 ∈ 𝒫 (Base‘𝐾) ↦ ∩ {𝑡 ∈ (LSubSp‘𝐾) ∣ 𝑠 ⊆ 𝑡})) |
21 | 16, 20 | syl 17 | . 2 ⊢ (𝜑 → (LSpan‘𝐾) = (𝑠 ∈ 𝒫 (Base‘𝐾) ↦ ∩ {𝑡 ∈ (LSubSp‘𝐾) ∣ 𝑠 ⊆ 𝑡})) |
22 | lsspropd.v2 | . . 3 ⊢ (𝜑 → 𝐿 ∈ V) | |
23 | eqid 2610 | . . . 4 ⊢ (Base‘𝐿) = (Base‘𝐿) | |
24 | eqid 2610 | . . . 4 ⊢ (LSubSp‘𝐿) = (LSubSp‘𝐿) | |
25 | eqid 2610 | . . . 4 ⊢ (LSpan‘𝐿) = (LSpan‘𝐿) | |
26 | 23, 24, 25 | lspfval 18794 | . . 3 ⊢ (𝐿 ∈ V → (LSpan‘𝐿) = (𝑠 ∈ 𝒫 (Base‘𝐿) ↦ ∩ {𝑡 ∈ (LSubSp‘𝐿) ∣ 𝑠 ⊆ 𝑡})) |
27 | 22, 26 | syl 17 | . 2 ⊢ (𝜑 → (LSpan‘𝐿) = (𝑠 ∈ 𝒫 (Base‘𝐿) ↦ ∩ {𝑡 ∈ (LSubSp‘𝐿) ∣ 𝑠 ⊆ 𝑡})) |
28 | 15, 21, 27 | 3eqtr4d 2654 | 1 ⊢ (𝜑 → (LSpan‘𝐾) = (LSpan‘𝐿)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {crab 2900 Vcvv 3173 ⊆ wss 3540 𝒫 cpw 4108 ∩ cint 4410 ↦ cmpt 4643 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 +gcplusg 15768 Scalarcsca 15771 ·𝑠 cvsca 15772 LSubSpclss 18753 LSpanclspn 18792 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-lss 18754 df-lsp 18793 |
This theorem is referenced by: lbspropd 18920 lidlrsppropd 19051 |
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