Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 2860 | . . . 4 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿)) |
4 | 3 | pweqd 4560 | . . 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 19791 | . . . . 5 ⊢ (𝜑 → (LSubSp‘𝐾) = (LSubSp‘𝐿)) |
12 | rabeq 3485 | . . . . 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 4883 | . . 3 ⊢ (𝜑 → ∩ {𝑡 ∈ (LSubSp‘𝐾) ∣ 𝑠 ⊆ 𝑡} = ∩ {𝑡 ∈ (LSubSp‘𝐿) ∣ 𝑠 ⊆ 𝑡}) |
15 | 4, 14 | mpteq12dv 5153 | . 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 2823 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
18 | eqid 2823 | . . . 4 ⊢ (LSubSp‘𝐾) = (LSubSp‘𝐾) | |
19 | eqid 2823 | . . . 4 ⊢ (LSpan‘𝐾) = (LSpan‘𝐾) | |
20 | 17, 18, 19 | lspfval 19747 | . . 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 2823 | . . . 4 ⊢ (Base‘𝐿) = (Base‘𝐿) | |
24 | eqid 2823 | . . . 4 ⊢ (LSubSp‘𝐿) = (LSubSp‘𝐿) | |
25 | eqid 2823 | . . . 4 ⊢ (LSpan‘𝐿) = (LSpan‘𝐿) | |
26 | 23, 24, 25 | lspfval 19747 | . . 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 2868 | 1 ⊢ (𝜑 → (LSpan‘𝐾) = (LSpan‘𝐿)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 {crab 3144 Vcvv 3496 ⊆ wss 3938 𝒫 cpw 4541 ∩ cint 4878 ↦ cmpt 5148 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 +gcplusg 16567 Scalarcsca 16570 ·𝑠 cvsca 16571 LSubSpclss 19705 LSpanclspn 19745 |
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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-lss 19706 df-lsp 19746 |
This theorem is referenced by: lbspropd 19873 lidlrsppropd 20005 lindfpropd 30944 |
Copyright terms: Public domain | W3C validator |