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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 22623 | . . 3 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ) | |
2 | unitssre 12357 | . . . 4 ⊢ (0[,]1) ⊆ ℝ | |
3 | ax-resscn 10031 | . . . 4 ⊢ ℝ ⊆ ℂ | |
4 | 2, 3 | sstri 3645 | . . 3 ⊢ (0[,]1) ⊆ ℂ |
5 | xmetres2 22213 | . . 3 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ (0[,]1) ⊆ ℂ) → ((abs ∘ − ) ↾ ((0[,]1) × (0[,]1))) ∈ (∞Met‘(0[,]1))) | |
6 | 1, 4, 5 | mp2an 708 | . 2 ⊢ ((abs ∘ − ) ↾ ((0[,]1) × (0[,]1))) ∈ (∞Met‘(0[,]1)) |
7 | df-ii 22727 | . . 3 ⊢ II = (MetOpen‘((abs ∘ − ) ↾ ((0[,]1) × (0[,]1)))) | |
8 | 7 | mopntopon 22291 | . 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 2030 ⊆ wss 3607 × cxp 5141 ↾ cres 5145 ∘ ccom 5147 ‘cfv 5926 (class class class)co 6690 ℂcc 9972 ℝcr 9973 0cc0 9974 1c1 9975 − cmin 10304 [,]cicc 12216 abscabs 14018 ∞Metcxmt 19779 TopOnctopon 20763 IIcii 22725 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-pre-sup 10052 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-er 7787 df-map 7901 df-en 7998 df-dom 7999 df-sdom 8000 df-sup 8389 df-inf 8390 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 df-nn 11059 df-2 11117 df-3 11118 df-n0 11331 df-z 11416 df-uz 11726 df-q 11827 df-rp 11871 df-xneg 11984 df-xadd 11985 df-xmul 11986 df-icc 12220 df-seq 12842 df-exp 12901 df-cj 13883 df-re 13884 df-im 13885 df-sqrt 14019 df-abs 14020 df-topgen 16151 df-psmet 19786 df-xmet 19787 df-met 19788 df-bl 19789 df-mopn 19790 df-top 20747 df-topon 20764 df-bases 20798 df-ii 22727 |
This theorem is referenced by: iitop 22730 iiuni 22731 icchmeo 22787 htpycom 22822 htpyid 22823 htpyco1 22824 htpyco2 22825 htpycc 22826 phtpycn 22829 phtpy01 22831 isphtpy2d 22833 phtpycom 22834 phtpyid 22835 phtpyco2 22836 phtpycc 22837 reparphti 22843 pcocn 22863 pcohtpylem 22865 pcoptcl 22867 pcopt 22868 pcopt2 22869 pcoass 22870 pcorevcl 22871 pcorevlem 22872 pi1xfrf 22899 pi1xfr 22901 pi1xfrcnvlem 22902 pi1xfrcnv 22903 pi1cof 22905 pi1coghm 22907 xrge0pluscn 30114 ptpconn 31341 indispconn 31342 connpconn 31343 txsconnlem 31348 txsconn 31349 cvxsconn 31351 cvmliftlem8 31400 cvmlift2lem2 31412 cvmlift2lem3 31413 cvmlift2lem6 31416 cvmlift2lem9 31419 cvmlift2lem11 31421 cvmlift2lem12 31422 cvmliftphtlem 31425 cvmlift3lem6 31432 cvmlift3lem9 31435 |
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