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Mirrors > Home > ILE Home > Th. List > setsmsdsg | GIF version |
Description: The distance function of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.) |
Ref | Expression |
---|---|
setsms.x | ⊢ (𝜑 → 𝑋 = (Base‘𝑀)) |
setsms.d | ⊢ (𝜑 → 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋))) |
setsms.k | ⊢ (𝜑 → 𝐾 = (𝑀 sSet 〈(TopSet‘ndx), (MetOpen‘𝐷)〉)) |
setsmsbasg.m | ⊢ (𝜑 → 𝑀 ∈ 𝑉) |
setsmsbasg.d | ⊢ (𝜑 → (MetOpen‘𝐷) ∈ 𝑊) |
Ref | Expression |
---|---|
setsmsdsg | ⊢ (𝜑 → (dist‘𝑀) = (dist‘𝐾)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | setsmsbasg.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ 𝑉) | |
2 | setsmsbasg.d | . . 3 ⊢ (𝜑 → (MetOpen‘𝐷) ∈ 𝑊) | |
3 | dsslid 12046 | . . . 4 ⊢ (dist = Slot (dist‘ndx) ∧ (dist‘ndx) ∈ ℕ) | |
4 | 9re 8775 | . . . . . 6 ⊢ 9 ∈ ℝ | |
5 | 1nn 8699 | . . . . . . 7 ⊢ 1 ∈ ℕ | |
6 | 2nn0 8962 | . . . . . . 7 ⊢ 2 ∈ ℕ0 | |
7 | 9nn0 8969 | . . . . . . 7 ⊢ 9 ∈ ℕ0 | |
8 | 9lt10 9280 | . . . . . . 7 ⊢ 9 < ;10 | |
9 | 5, 6, 7, 8 | declti 9187 | . . . . . 6 ⊢ 9 < ;12 |
10 | 4, 9 | gtneii 7827 | . . . . 5 ⊢ ;12 ≠ 9 |
11 | dsndx 12044 | . . . . . 6 ⊢ (dist‘ndx) = ;12 | |
12 | tsetndx 12034 | . . . . . 6 ⊢ (TopSet‘ndx) = 9 | |
13 | 11, 12 | neeq12i 2302 | . . . . 5 ⊢ ((dist‘ndx) ≠ (TopSet‘ndx) ↔ ;12 ≠ 9) |
14 | 10, 13 | mpbir 145 | . . . 4 ⊢ (dist‘ndx) ≠ (TopSet‘ndx) |
15 | tsetslid 12036 | . . . . 5 ⊢ (TopSet = Slot (TopSet‘ndx) ∧ (TopSet‘ndx) ∈ ℕ) | |
16 | 15 | simpri 112 | . . . 4 ⊢ (TopSet‘ndx) ∈ ℕ |
17 | 3, 14, 16 | setsslnid 11937 | . . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ (MetOpen‘𝐷) ∈ 𝑊) → (dist‘𝑀) = (dist‘(𝑀 sSet 〈(TopSet‘ndx), (MetOpen‘𝐷)〉))) |
18 | 1, 2, 17 | syl2anc 408 | . 2 ⊢ (𝜑 → (dist‘𝑀) = (dist‘(𝑀 sSet 〈(TopSet‘ndx), (MetOpen‘𝐷)〉))) |
19 | setsms.k | . . 3 ⊢ (𝜑 → 𝐾 = (𝑀 sSet 〈(TopSet‘ndx), (MetOpen‘𝐷)〉)) | |
20 | 19 | fveq2d 5393 | . 2 ⊢ (𝜑 → (dist‘𝐾) = (dist‘(𝑀 sSet 〈(TopSet‘ndx), (MetOpen‘𝐷)〉))) |
21 | 18, 20 | eqtr4d 2153 | 1 ⊢ (𝜑 → (dist‘𝑀) = (dist‘𝐾)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1316 ∈ wcel 1465 ≠ wne 2285 〈cop 3500 × cxp 4507 ↾ cres 4511 ‘cfv 5093 (class class class)co 5742 1c1 7589 ℕcn 8688 2c2 8739 9c9 8746 ;cdc 9150 ndxcnx 11883 sSet csts 11884 Slot cslot 11885 Basecbs 11886 TopSetcts 11954 distcds 11957 MetOpencmopn 12081 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-mulrcl 7687 ax-addcom 7688 ax-mulcom 7689 ax-addass 7690 ax-mulass 7691 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-1rid 7695 ax-0id 7696 ax-rnegex 7697 ax-precex 7698 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-ltadd 7704 ax-pre-mulgt0 7705 |
This theorem depends on definitions: df-bi 116 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-iota 5058 df-fun 5095 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-inn 8689 df-2 8747 df-3 8748 df-4 8749 df-5 8750 df-6 8751 df-7 8752 df-8 8753 df-9 8754 df-n0 8946 df-z 9023 df-dec 9151 df-ndx 11889 df-slot 11890 df-sets 11893 df-tset 11967 df-ds 11970 |
This theorem is referenced by: (None) |
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