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Mirrors > Home > ILE Home > Th. List > xmspropd | GIF version |
Description: Property deduction for an extended metric space. (Contributed by Mario Carneiro, 4-Oct-2015.) |
Ref | Expression |
---|---|
xmspropd.1 | ⊢ (𝜑 → 𝐵 = (Base‘𝐾)) |
xmspropd.2 | ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) |
xmspropd.3 | ⊢ (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ (𝐵 × 𝐵))) |
xmspropd.4 | ⊢ (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿)) |
Ref | Expression |
---|---|
xmspropd | ⊢ (𝜑 → (𝐾 ∈ ∞MetSp ↔ 𝐿 ∈ ∞MetSp)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xmspropd.1 | . . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘𝐾)) | |
2 | xmspropd.2 | . . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) | |
3 | 1, 2 | eqtr3d 2174 | . . . 4 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿)) |
4 | xmspropd.4 | . . . 4 ⊢ (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿)) | |
5 | 3, 4 | tpspropd 12206 | . . 3 ⊢ (𝜑 → (𝐾 ∈ TopSp ↔ 𝐿 ∈ TopSp)) |
6 | xmspropd.3 | . . . . . . 7 ⊢ (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ (𝐵 × 𝐵))) | |
7 | 1 | sqxpeqd 4565 | . . . . . . . 8 ⊢ (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐾) × (Base‘𝐾))) |
8 | 7 | reseq2d 4819 | . . . . . . 7 ⊢ (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))) |
9 | 6, 8 | eqtr3d 2174 | . . . . . 6 ⊢ (𝜑 → ((dist‘𝐿) ↾ (𝐵 × 𝐵)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))) |
10 | 2 | sqxpeqd 4565 | . . . . . . 7 ⊢ (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐿) × (Base‘𝐿))) |
11 | 10 | reseq2d 4819 | . . . . . 6 ⊢ (𝜑 → ((dist‘𝐿) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿)))) |
12 | 9, 11 | eqtr3d 2174 | . . . . 5 ⊢ (𝜑 → ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿)))) |
13 | 12 | fveq2d 5425 | . . . 4 ⊢ (𝜑 → (MetOpen‘((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))) = (MetOpen‘((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))))) |
14 | 4, 13 | eqeq12d 2154 | . . 3 ⊢ (𝜑 → ((TopOpen‘𝐾) = (MetOpen‘((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))) ↔ (TopOpen‘𝐿) = (MetOpen‘((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿)))))) |
15 | 5, 14 | anbi12d 464 | . 2 ⊢ (𝜑 → ((𝐾 ∈ TopSp ∧ (TopOpen‘𝐾) = (MetOpen‘((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))) ↔ (𝐿 ∈ TopSp ∧ (TopOpen‘𝐿) = (MetOpen‘((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))))))) |
16 | eqid 2139 | . . 3 ⊢ (TopOpen‘𝐾) = (TopOpen‘𝐾) | |
17 | eqid 2139 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
18 | eqid 2139 | . . 3 ⊢ ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) | |
19 | 16, 17, 18 | isxms 12623 | . 2 ⊢ (𝐾 ∈ ∞MetSp ↔ (𝐾 ∈ TopSp ∧ (TopOpen‘𝐾) = (MetOpen‘((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))))) |
20 | eqid 2139 | . . 3 ⊢ (TopOpen‘𝐿) = (TopOpen‘𝐿) | |
21 | eqid 2139 | . . 3 ⊢ (Base‘𝐿) = (Base‘𝐿) | |
22 | eqid 2139 | . . 3 ⊢ ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))) | |
23 | 20, 21, 22 | isxms 12623 | . 2 ⊢ (𝐿 ∈ ∞MetSp ↔ (𝐿 ∈ TopSp ∧ (TopOpen‘𝐿) = (MetOpen‘((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿)))))) |
24 | 15, 19, 23 | 3bitr4g 222 | 1 ⊢ (𝜑 → (𝐾 ∈ ∞MetSp ↔ 𝐿 ∈ ∞MetSp)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1331 ∈ wcel 1480 × cxp 4537 ↾ cres 4541 ‘cfv 5123 Basecbs 11962 distcds 12033 TopOpenctopn 12124 MetOpencmopn 12157 TopSpctps 12200 ∞MetSpcxms 12508 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-cnex 7714 ax-resscn 7715 ax-1re 7717 ax-addrcl 7720 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-inn 8724 df-2 8782 df-3 8783 df-4 8784 df-5 8785 df-6 8786 df-7 8787 df-8 8788 df-9 8789 df-ndx 11965 df-slot 11966 df-base 11968 df-tset 12043 df-rest 12125 df-topn 12126 df-top 12168 df-topon 12181 df-topsp 12201 df-xms 12511 |
This theorem is referenced by: mspropd 12650 |
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