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Mirrors > Home > MPE Home > Th. List > iitopon | Structured version Visualization version GIF version |
Description: The unit interval is a topological space. (Contributed by Mario Carneiro, 3-Sep-2015.) |
Ref | Expression |
---|---|
iitopon | ⊢ II ∈ (TopOn‘(0[,]1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnxmet 24082 | . . 3 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ) | |
2 | unitssre 13370 | . . . 4 ⊢ (0[,]1) ⊆ ℝ | |
3 | ax-resscn 11066 | . . . 4 ⊢ ℝ ⊆ ℂ | |
4 | 2, 3 | sstri 3951 | . . 3 ⊢ (0[,]1) ⊆ ℂ |
5 | xmetres2 23660 | . . 3 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ (0[,]1) ⊆ ℂ) → ((abs ∘ − ) ↾ ((0[,]1) × (0[,]1))) ∈ (∞Met‘(0[,]1))) | |
6 | 1, 4, 5 | mp2an 690 | . 2 ⊢ ((abs ∘ − ) ↾ ((0[,]1) × (0[,]1))) ∈ (∞Met‘(0[,]1)) |
7 | df-ii 24186 | . . 3 ⊢ II = (MetOpen‘((abs ∘ − ) ↾ ((0[,]1) × (0[,]1)))) | |
8 | 7 | mopntopon 23738 | . 2 ⊢ (((abs ∘ − ) ↾ ((0[,]1) × (0[,]1))) ∈ (∞Met‘(0[,]1)) → II ∈ (TopOn‘(0[,]1))) |
9 | 6, 8 | ax-mp 5 | 1 ⊢ II ∈ (TopOn‘(0[,]1)) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2106 ⊆ wss 3908 × cxp 5629 ↾ cres 5633 ∘ ccom 5635 ‘cfv 6493 (class class class)co 7351 ℂcc 11007 ℝcr 11008 0cc0 11009 1c1 11010 − cmin 11343 [,]cicc 13221 abscabs 15073 ∞Metcxmet 20728 TopOnctopon 22205 IIcii 24184 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-om 7795 df-1st 7913 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-er 8606 df-map 8725 df-en 8842 df-dom 8843 df-sdom 8844 df-sup 9336 df-inf 9337 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-div 11771 df-nn 12112 df-2 12174 df-3 12175 df-n0 12372 df-z 12458 df-uz 12722 df-q 12828 df-rp 12870 df-xneg 12987 df-xadd 12988 df-xmul 12989 df-icc 13225 df-seq 13861 df-exp 13922 df-cj 14938 df-re 14939 df-im 14940 df-sqrt 15074 df-abs 15075 df-topgen 17279 df-psmet 20735 df-xmet 20736 df-met 20737 df-bl 20738 df-mopn 20739 df-top 22189 df-topon 22206 df-bases 22242 df-ii 24186 |
This theorem is referenced by: iitop 24189 iiuni 24190 icchmeo 24250 htpycom 24285 htpyid 24286 htpyco1 24287 htpyco2 24288 htpycc 24289 phtpycn 24292 phtpy01 24294 isphtpy2d 24296 phtpycom 24297 phtpyid 24298 phtpyco2 24299 phtpycc 24300 reparphti 24306 pcocn 24326 pcohtpylem 24328 pcoptcl 24330 pcopt 24331 pcopt2 24332 pcoass 24333 pcorevcl 24334 pcorevlem 24335 pi1xfrf 24362 pi1xfr 24364 pi1xfrcnvlem 24365 pi1xfrcnv 24366 pi1cof 24368 pi1coghm 24370 xrge0pluscn 32349 ptpconn 33655 indispconn 33656 connpconn 33657 txsconnlem 33662 txsconn 33663 cvxsconn 33665 cvmliftlem8 33714 cvmlift2lem2 33726 cvmlift2lem3 33727 cvmlift2lem6 33730 cvmlift2lem9 33733 cvmlift2lem11 33735 cvmlift2lem12 33736 cvmliftphtlem 33739 cvmlift3lem6 33746 cvmlift3lem9 33749 |
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