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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 12122 | . . . 4 ⊢ (dist = Slot (dist‘ndx) ∧ (dist‘ndx) ∈ ℕ) | |
4 | 9re 8810 | . . . . . 6 ⊢ 9 ∈ ℝ | |
5 | 1nn 8734 | . . . . . . 7 ⊢ 1 ∈ ℕ | |
6 | 2nn0 8997 | . . . . . . 7 ⊢ 2 ∈ ℕ0 | |
7 | 9nn0 9004 | . . . . . . 7 ⊢ 9 ∈ ℕ0 | |
8 | 9lt10 9315 | . . . . . . 7 ⊢ 9 < ;10 | |
9 | 5, 6, 7, 8 | declti 9222 | . . . . . 6 ⊢ 9 < ;12 |
10 | 4, 9 | gtneii 7862 | . . . . 5 ⊢ ;12 ≠ 9 |
11 | dsndx 12120 | . . . . . 6 ⊢ (dist‘ndx) = ;12 | |
12 | tsetndx 12110 | . . . . . 6 ⊢ (TopSet‘ndx) = 9 | |
13 | 11, 12 | neeq12i 2325 | . . . . 5 ⊢ ((dist‘ndx) ≠ (TopSet‘ndx) ↔ ;12 ≠ 9) |
14 | 10, 13 | mpbir 145 | . . . 4 ⊢ (dist‘ndx) ≠ (TopSet‘ndx) |
15 | tsetslid 12112 | . . . . 5 ⊢ (TopSet = Slot (TopSet‘ndx) ∧ (TopSet‘ndx) ∈ ℕ) | |
16 | 15 | simpri 112 | . . . 4 ⊢ (TopSet‘ndx) ∈ ℕ |
17 | 3, 14, 16 | setsslnid 12013 | . . 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 5425 | . 2 ⊢ (𝜑 → (dist‘𝐾) = (dist‘(𝑀 sSet 〈(TopSet‘ndx), (MetOpen‘𝐷)〉))) |
21 | 18, 20 | eqtr4d 2175 | 1 ⊢ (𝜑 → (dist‘𝑀) = (dist‘𝐾)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∈ wcel 1480 ≠ wne 2308 〈cop 3530 × cxp 4537 ↾ cres 4541 ‘cfv 5123 (class class class)co 5774 1c1 7624 ℕcn 8723 2c2 8774 9c9 8781 ;cdc 9185 ndxcnx 11959 sSet csts 11960 Slot cslot 11961 Basecbs 11962 TopSetcts 12030 distcds 12033 MetOpencmopn 12157 |
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-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7714 ax-resscn 7715 ax-1cn 7716 ax-1re 7717 ax-icn 7718 ax-addcl 7719 ax-addrcl 7720 ax-mulcl 7721 ax-mulrcl 7722 ax-addcom 7723 ax-mulcom 7724 ax-addass 7725 ax-mulass 7726 ax-distr 7727 ax-i2m1 7728 ax-0lt1 7729 ax-1rid 7730 ax-0id 7731 ax-rnegex 7732 ax-precex 7733 ax-cnre 7734 ax-pre-ltirr 7735 ax-pre-ltwlin 7736 ax-pre-lttrn 7737 ax-pre-ltadd 7739 ax-pre-mulgt0 7740 |
This theorem depends on definitions: df-bi 116 df-3or 963 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-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 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-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-iota 5088 df-fun 5125 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7805 df-mnf 7806 df-xr 7807 df-ltxr 7808 df-le 7809 df-sub 7938 df-neg 7939 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-n0 8981 df-z 9058 df-dec 9186 df-ndx 11965 df-slot 11966 df-sets 11969 df-tset 12043 df-ds 12046 |
This theorem is referenced by: (None) |
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