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Mirrors > Home > MPE Home > Th. List > setsmsds | Structured version Visualization version 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)〉)) |
Ref | Expression |
---|---|
setsmsds | ⊢ (𝜑 → (dist‘𝑀) = (dist‘𝐾)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | setsms.k | . . 3 ⊢ (𝜑 → 𝐾 = (𝑀 sSet 〈(TopSet‘ndx), (MetOpen‘𝐷)〉)) | |
2 | 1 | fveq2d 6676 | . 2 ⊢ (𝜑 → (dist‘𝐾) = (dist‘(𝑀 sSet 〈(TopSet‘ndx), (MetOpen‘𝐷)〉))) |
3 | dsid 16678 | . . 3 ⊢ dist = Slot (dist‘ndx) | |
4 | 9re 11739 | . . . . 5 ⊢ 9 ∈ ℝ | |
5 | 1nn 11651 | . . . . . 6 ⊢ 1 ∈ ℕ | |
6 | 2nn0 11917 | . . . . . 6 ⊢ 2 ∈ ℕ0 | |
7 | 9nn0 11924 | . . . . . 6 ⊢ 9 ∈ ℕ0 | |
8 | 9lt10 12232 | . . . . . 6 ⊢ 9 < ;10 | |
9 | 5, 6, 7, 8 | declti 12139 | . . . . 5 ⊢ 9 < ;12 |
10 | 4, 9 | gtneii 10754 | . . . 4 ⊢ ;12 ≠ 9 |
11 | dsndx 16677 | . . . . 5 ⊢ (dist‘ndx) = ;12 | |
12 | tsetndx 16661 | . . . . 5 ⊢ (TopSet‘ndx) = 9 | |
13 | 11, 12 | neeq12i 3084 | . . . 4 ⊢ ((dist‘ndx) ≠ (TopSet‘ndx) ↔ ;12 ≠ 9) |
14 | 10, 13 | mpbir 233 | . . 3 ⊢ (dist‘ndx) ≠ (TopSet‘ndx) |
15 | 3, 14 | setsnid 16541 | . 2 ⊢ (dist‘𝑀) = (dist‘(𝑀 sSet 〈(TopSet‘ndx), (MetOpen‘𝐷)〉)) |
16 | 2, 15 | syl6reqr 2877 | 1 ⊢ (𝜑 → (dist‘𝑀) = (dist‘𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ≠ wne 3018 〈cop 4575 × cxp 5555 ↾ cres 5559 ‘cfv 6357 (class class class)co 7158 1c1 10540 2c2 11695 9c9 11702 ;cdc 12101 ndxcnx 16482 sSet csts 16483 Basecbs 16485 TopSetcts 16573 distcds 16576 MetOpencmopn 20537 |
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-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 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-nel 3126 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-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 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-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 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-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-ndx 16488 df-slot 16489 df-sets 16492 df-tset 16586 df-ds 16589 |
This theorem is referenced by: setsxms 23091 setsms 23092 tmslem 23094 |
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