Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > dfii3 | Structured version Visualization version GIF version |
Description: Alternate definition of the unit interval. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 3-Sep-2015.) |
Ref | Expression |
---|---|
dfii3.1 | ⊢ 𝐽 = (TopOpen‘ℂfld) |
Ref | Expression |
---|---|
dfii3 | ⊢ II = (𝐽 ↾t (0[,]1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnxmet 23934 | . . 3 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ) | |
2 | unitssre 13230 | . . . 4 ⊢ (0[,]1) ⊆ ℝ | |
3 | ax-resscn 10929 | . . . 4 ⊢ ℝ ⊆ ℂ | |
4 | 2, 3 | sstri 3935 | . . 3 ⊢ (0[,]1) ⊆ ℂ |
5 | eqid 2740 | . . . 4 ⊢ ((abs ∘ − ) ↾ ((0[,]1) × (0[,]1))) = ((abs ∘ − ) ↾ ((0[,]1) × (0[,]1))) | |
6 | dfii3.1 | . . . . 5 ⊢ 𝐽 = (TopOpen‘ℂfld) | |
7 | 6 | cnfldtopn 23943 | . . . 4 ⊢ 𝐽 = (MetOpen‘(abs ∘ − )) |
8 | df-ii 24038 | . . . 4 ⊢ II = (MetOpen‘((abs ∘ − ) ↾ ((0[,]1) × (0[,]1)))) | |
9 | 5, 7, 8 | metrest 23678 | . . 3 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ (0[,]1) ⊆ ℂ) → (𝐽 ↾t (0[,]1)) = II) |
10 | 1, 4, 9 | mp2an 689 | . 2 ⊢ (𝐽 ↾t (0[,]1)) = II |
11 | 10 | eqcomi 2749 | 1 ⊢ II = (𝐽 ↾t (0[,]1)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2110 ⊆ wss 3892 × cxp 5588 ↾ cres 5592 ∘ ccom 5594 ‘cfv 6432 (class class class)co 7271 ℂcc 10870 ℝcr 10871 0cc0 10872 1c1 10873 − cmin 11205 [,]cicc 13081 abscabs 14943 ↾t crest 17129 TopOpenctopn 17130 ∞Metcxmet 20580 ℂfldccnfld 20595 IIcii 24036 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 ax-pre-sup 10950 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-om 7707 df-1st 7824 df-2nd 7825 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-1o 8288 df-er 8481 df-map 8600 df-en 8717 df-dom 8718 df-sdom 8719 df-fin 8720 df-sup 9179 df-inf 9180 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12437 df-uz 12582 df-q 12688 df-rp 12730 df-xneg 12847 df-xadd 12848 df-xmul 12849 df-icc 13085 df-fz 13239 df-seq 13720 df-exp 13781 df-cj 14808 df-re 14809 df-im 14810 df-sqrt 14944 df-abs 14945 df-struct 16846 df-slot 16881 df-ndx 16893 df-base 16911 df-plusg 16973 df-mulr 16974 df-starv 16975 df-tset 16979 df-ple 16980 df-ds 16982 df-unif 16983 df-rest 17131 df-topn 17132 df-topgen 17152 df-psmet 20587 df-xmet 20588 df-met 20589 df-bl 20590 df-mopn 20591 df-cnfld 20596 df-top 22041 df-topon 22058 df-bases 22094 df-ii 24038 |
This theorem is referenced by: dfii4 24045 iimulcn 24099 icchmeo 24102 reparphti 24158 cvxpconn 33200 cvxsconn 33201 |
Copyright terms: Public domain | W3C validator |