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Mirrors > Home > ILE Home > Th. List > mspropd | GIF version |
Description: Property deduction for a 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 |
---|---|
mspropd | ⊢ (𝜑 → (𝐾 ∈ MetSp ↔ 𝐿 ∈ MetSp)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xmspropd.1 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐾)) | |
2 | xmspropd.2 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) | |
3 | xmspropd.3 | . . . 4 ⊢ (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ (𝐵 × 𝐵))) | |
4 | xmspropd.4 | . . . 4 ⊢ (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿)) | |
5 | 1, 2, 3, 4 | xmspropd 13839 | . . 3 ⊢ (𝜑 → (𝐾 ∈ ∞MetSp ↔ 𝐿 ∈ ∞MetSp)) |
6 | 1 | sqxpeqd 4651 | . . . . . . 7 ⊢ (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐾) × (Base‘𝐾))) |
7 | 6 | reseq2d 4905 | . . . . . 6 ⊢ (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))) |
8 | 3, 7 | eqtr3d 2212 | . . . . 5 ⊢ (𝜑 → ((dist‘𝐿) ↾ (𝐵 × 𝐵)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))) |
9 | 2 | sqxpeqd 4651 | . . . . . 6 ⊢ (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐿) × (Base‘𝐿))) |
10 | 9 | reseq2d 4905 | . . . . 5 ⊢ (𝜑 → ((dist‘𝐿) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿)))) |
11 | 8, 10 | eqtr3d 2212 | . . . 4 ⊢ (𝜑 → ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿)))) |
12 | 1, 2 | eqtr3d 2212 | . . . . 5 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿)) |
13 | 12 | fveq2d 5517 | . . . 4 ⊢ (𝜑 → (Met‘(Base‘𝐾)) = (Met‘(Base‘𝐿))) |
14 | 11, 13 | eleq12d 2248 | . . 3 ⊢ (𝜑 → (((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (Met‘(Base‘𝐾)) ↔ ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))) ∈ (Met‘(Base‘𝐿)))) |
15 | 5, 14 | anbi12d 473 | . 2 ⊢ (𝜑 → ((𝐾 ∈ ∞MetSp ∧ ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (Met‘(Base‘𝐾))) ↔ (𝐿 ∈ ∞MetSp ∧ ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))) ∈ (Met‘(Base‘𝐿))))) |
16 | eqid 2177 | . . 3 ⊢ (TopOpen‘𝐾) = (TopOpen‘𝐾) | |
17 | eqid 2177 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
18 | eqid 2177 | . . 3 ⊢ ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) | |
19 | 16, 17, 18 | isms 13815 | . 2 ⊢ (𝐾 ∈ MetSp ↔ (𝐾 ∈ ∞MetSp ∧ ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (Met‘(Base‘𝐾)))) |
20 | eqid 2177 | . . 3 ⊢ (TopOpen‘𝐿) = (TopOpen‘𝐿) | |
21 | eqid 2177 | . . 3 ⊢ (Base‘𝐿) = (Base‘𝐿) | |
22 | eqid 2177 | . . 3 ⊢ ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))) | |
23 | 20, 21, 22 | isms 13815 | . 2 ⊢ (𝐿 ∈ MetSp ↔ (𝐿 ∈ ∞MetSp ∧ ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))) ∈ (Met‘(Base‘𝐿)))) |
24 | 15, 19, 23 | 3bitr4g 223 | 1 ⊢ (𝜑 → (𝐾 ∈ MetSp ↔ 𝐿 ∈ MetSp)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1353 ∈ wcel 2148 × cxp 4623 ↾ cres 4627 ‘cfv 5214 Basecbs 12453 distcds 12536 TopOpenctopn 12676 Metcmet 13301 ∞MetSpcxms 13698 MetSpcms 13699 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4117 ax-sep 4120 ax-pow 4173 ax-pr 4208 ax-un 4432 ax-cnex 7898 ax-resscn 7899 ax-1re 7901 ax-addrcl 7904 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4003 df-opab 4064 df-mpt 4065 df-id 4292 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-rn 4636 df-res 4637 df-ima 4638 df-iota 5176 df-fun 5216 df-fn 5217 df-f 5218 df-f1 5219 df-fo 5220 df-f1o 5221 df-fv 5222 df-ov 5874 df-oprab 5875 df-mpo 5876 df-1st 6137 df-2nd 6138 df-inn 8915 df-2 8973 df-3 8974 df-4 8975 df-5 8976 df-6 8977 df-7 8978 df-8 8979 df-9 8980 df-ndx 12456 df-slot 12457 df-base 12459 df-tset 12546 df-rest 12677 df-topn 12678 df-top 13358 df-topon 13371 df-topsp 13391 df-xms 13701 df-ms 13702 |
This theorem is referenced by: (None) |
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