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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > xrge0iifhmeo | Structured version Visualization version GIF version |
Description: Expose a homeomorphism from the closed unit interval to the extended nonnegative reals. (Contributed by Thierry Arnoux, 1-Apr-2017.) |
Ref | Expression |
---|---|
xrge0iifhmeo.1 | ⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥))) |
xrge0iifhmeo.k | ⊢ 𝐽 = ((ordTop‘ ≤ ) ↾t (0[,]+∞)) |
Ref | Expression |
---|---|
xrge0iifhmeo | ⊢ 𝐹 ∈ (IIHomeo𝐽) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | letsr 18442 | . . . . . 6 ⊢ ≤ ∈ TosetRel | |
2 | tsrps 18436 | . . . . . 6 ⊢ ( ≤ ∈ TosetRel → ≤ ∈ PosetRel) | |
3 | 1, 2 | ax-mp 5 | . . . . 5 ⊢ ≤ ∈ PosetRel |
4 | 3 | elexi 3462 | . . . 4 ⊢ ≤ ∈ V |
5 | 4 | inex1 5272 | . . 3 ⊢ ( ≤ ∩ ((0[,]1) × (0[,]1))) ∈ V |
6 | cnvps 18427 | . . . . . 6 ⊢ ( ≤ ∈ PosetRel → ◡ ≤ ∈ PosetRel) | |
7 | 3, 6 | ax-mp 5 | . . . . 5 ⊢ ◡ ≤ ∈ PosetRel |
8 | 7 | elexi 3462 | . . . 4 ⊢ ◡ ≤ ∈ V |
9 | 8 | inex1 5272 | . . 3 ⊢ (◡ ≤ ∩ ((0[,]+∞) × (0[,]+∞))) ∈ V |
10 | xrge0iifhmeo.1 | . . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥))) | |
11 | 10 | xrge0iifiso 32328 | . . . . . 6 ⊢ 𝐹 Isom < , ◡ < ((0[,]1), (0[,]+∞)) |
12 | iccssxr 13301 | . . . . . . 7 ⊢ (0[,]1) ⊆ ℝ* | |
13 | iccssxr 13301 | . . . . . . 7 ⊢ (0[,]+∞) ⊆ ℝ* | |
14 | gtiso 31441 | . . . . . . 7 ⊢ (((0[,]1) ⊆ ℝ* ∧ (0[,]+∞) ⊆ ℝ*) → (𝐹 Isom < , ◡ < ((0[,]1), (0[,]+∞)) ↔ 𝐹 Isom ≤ , ◡ ≤ ((0[,]1), (0[,]+∞)))) | |
15 | 12, 13, 14 | mp2an 690 | . . . . . 6 ⊢ (𝐹 Isom < , ◡ < ((0[,]1), (0[,]+∞)) ↔ 𝐹 Isom ≤ , ◡ ≤ ((0[,]1), (0[,]+∞))) |
16 | 11, 15 | mpbi 229 | . . . . 5 ⊢ 𝐹 Isom ≤ , ◡ ≤ ((0[,]1), (0[,]+∞)) |
17 | isores1 7275 | . . . . 5 ⊢ (𝐹 Isom ≤ , ◡ ≤ ((0[,]1), (0[,]+∞)) ↔ 𝐹 Isom ( ≤ ∩ ((0[,]1) × (0[,]1))), ◡ ≤ ((0[,]1), (0[,]+∞))) | |
18 | 16, 17 | mpbi 229 | . . . 4 ⊢ 𝐹 Isom ( ≤ ∩ ((0[,]1) × (0[,]1))), ◡ ≤ ((0[,]1), (0[,]+∞)) |
19 | isores2 7274 | . . . 4 ⊢ (𝐹 Isom ( ≤ ∩ ((0[,]1) × (0[,]1))), ◡ ≤ ((0[,]1), (0[,]+∞)) ↔ 𝐹 Isom ( ≤ ∩ ((0[,]1) × (0[,]1))), (◡ ≤ ∩ ((0[,]+∞) × (0[,]+∞)))((0[,]1), (0[,]+∞))) | |
20 | 18, 19 | mpbi 229 | . . 3 ⊢ 𝐹 Isom ( ≤ ∩ ((0[,]1) × (0[,]1))), (◡ ≤ ∩ ((0[,]+∞) × (0[,]+∞)))((0[,]1), (0[,]+∞)) |
21 | ledm 18439 | . . . . . . 7 ⊢ ℝ* = dom ≤ | |
22 | 21 | psssdm 18431 | . . . . . 6 ⊢ (( ≤ ∈ PosetRel ∧ (0[,]1) ⊆ ℝ*) → dom ( ≤ ∩ ((0[,]1) × (0[,]1))) = (0[,]1)) |
23 | 3, 12, 22 | mp2an 690 | . . . . 5 ⊢ dom ( ≤ ∩ ((0[,]1) × (0[,]1))) = (0[,]1) |
24 | 23 | eqcomi 2746 | . . . 4 ⊢ (0[,]1) = dom ( ≤ ∩ ((0[,]1) × (0[,]1))) |
25 | lern 18440 | . . . . . . . 8 ⊢ ℝ* = ran ≤ | |
26 | df-rn 5642 | . . . . . . . 8 ⊢ ran ≤ = dom ◡ ≤ | |
27 | 25, 26 | eqtri 2765 | . . . . . . 7 ⊢ ℝ* = dom ◡ ≤ |
28 | 27 | psssdm 18431 | . . . . . 6 ⊢ ((◡ ≤ ∈ PosetRel ∧ (0[,]+∞) ⊆ ℝ*) → dom (◡ ≤ ∩ ((0[,]+∞) × (0[,]+∞))) = (0[,]+∞)) |
29 | 7, 13, 28 | mp2an 690 | . . . . 5 ⊢ dom (◡ ≤ ∩ ((0[,]+∞) × (0[,]+∞))) = (0[,]+∞) |
30 | 29 | eqcomi 2746 | . . . 4 ⊢ (0[,]+∞) = dom (◡ ≤ ∩ ((0[,]+∞) × (0[,]+∞))) |
31 | 24, 30 | ordthmeo 23105 | . . 3 ⊢ ((( ≤ ∩ ((0[,]1) × (0[,]1))) ∈ V ∧ (◡ ≤ ∩ ((0[,]+∞) × (0[,]+∞))) ∈ V ∧ 𝐹 Isom ( ≤ ∩ ((0[,]1) × (0[,]1))), (◡ ≤ ∩ ((0[,]+∞) × (0[,]+∞)))((0[,]1), (0[,]+∞))) → 𝐹 ∈ ((ordTop‘( ≤ ∩ ((0[,]1) × (0[,]1))))Homeo(ordTop‘(◡ ≤ ∩ ((0[,]+∞) × (0[,]+∞)))))) |
32 | 5, 9, 20, 31 | mp3an 1461 | . 2 ⊢ 𝐹 ∈ ((ordTop‘( ≤ ∩ ((0[,]1) × (0[,]1))))Homeo(ordTop‘(◡ ≤ ∩ ((0[,]+∞) × (0[,]+∞))))) |
33 | dfii5 24200 | . . 3 ⊢ II = (ordTop‘( ≤ ∩ ((0[,]1) × (0[,]1)))) | |
34 | xrge0iifhmeo.k | . . . 4 ⊢ 𝐽 = ((ordTop‘ ≤ ) ↾t (0[,]+∞)) | |
35 | iccss2 13289 | . . . . 5 ⊢ ((𝑥 ∈ (0[,]+∞) ∧ 𝑦 ∈ (0[,]+∞)) → (𝑥[,]𝑦) ⊆ (0[,]+∞)) | |
36 | 13, 35 | cnvordtrestixx 32306 | . . . 4 ⊢ ((ordTop‘ ≤ ) ↾t (0[,]+∞)) = (ordTop‘(◡ ≤ ∩ ((0[,]+∞) × (0[,]+∞)))) |
37 | 34, 36 | eqtri 2765 | . . 3 ⊢ 𝐽 = (ordTop‘(◡ ≤ ∩ ((0[,]+∞) × (0[,]+∞)))) |
38 | 33, 37 | oveq12i 7363 | . 2 ⊢ (IIHomeo𝐽) = ((ordTop‘( ≤ ∩ ((0[,]1) × (0[,]1))))Homeo(ordTop‘(◡ ≤ ∩ ((0[,]+∞) × (0[,]+∞))))) |
39 | 32, 38 | eleqtrri 2837 | 1 ⊢ 𝐹 ∈ (IIHomeo𝐽) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1541 ∈ wcel 2106 Vcvv 3443 ∩ cin 3907 ⊆ wss 3908 ifcif 4484 ↦ cmpt 5186 × cxp 5629 ◡ccnv 5630 dom cdm 5631 ran crn 5632 ‘cfv 6493 Isom wiso 6494 (class class class)co 7351 0cc0 11009 1c1 11010 +∞cpnf 11144 ℝ*cxr 11146 < clt 11147 ≤ cle 11148 -cneg 11344 [,]cicc 13221 ↾t crest 17262 ordTopcordt 17341 PosetRelcps 18413 TosetRel ctsr 18414 Homeochmeo 23056 IIcii 24190 logclog 25862 |
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 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-inf2 9535 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 ax-addf 11088 ax-mulf 11089 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 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-tp 4589 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-iin 4955 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-se 5587 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-isom 6502 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-of 7609 df-om 7795 df-1st 7913 df-2nd 7914 df-supp 8085 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-1o 8404 df-2o 8405 df-er 8606 df-map 8725 df-pm 8726 df-ixp 8794 df-en 8842 df-dom 8843 df-sdom 8844 df-fin 8845 df-fsupp 9264 df-fi 9305 df-sup 9336 df-inf 9337 df-oi 9404 df-card 9833 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-4 12176 df-5 12177 df-6 12178 df-7 12179 df-8 12180 df-9 12181 df-n0 12372 df-z 12458 df-dec 12577 df-uz 12722 df-q 12828 df-rp 12870 df-xneg 12987 df-xadd 12988 df-xmul 12989 df-ioo 13222 df-ioc 13223 df-ico 13224 df-icc 13225 df-fz 13379 df-fzo 13522 df-fl 13651 df-mod 13729 df-seq 13861 df-exp 13922 df-fac 14128 df-bc 14157 df-hash 14185 df-shft 14912 df-cj 14944 df-re 14945 df-im 14946 df-sqrt 15080 df-abs 15081 df-limsup 15313 df-clim 15330 df-rlim 15331 df-sum 15531 df-ef 15910 df-sin 15912 df-cos 15913 df-pi 15915 df-struct 16979 df-sets 16996 df-slot 17014 df-ndx 17026 df-base 17044 df-ress 17073 df-plusg 17106 df-mulr 17107 df-starv 17108 df-sca 17109 df-vsca 17110 df-ip 17111 df-tset 17112 df-ple 17113 df-ds 17115 df-unif 17116 df-hom 17117 df-cco 17118 df-rest 17264 df-topn 17265 df-0g 17283 df-gsum 17284 df-topgen 17285 df-pt 17286 df-prds 17289 df-ordt 17343 df-xrs 17344 df-qtop 17349 df-imas 17350 df-xps 17352 df-mre 17426 df-mrc 17427 df-acs 17429 df-ps 18415 df-tsr 18416 df-mgm 18457 df-sgrp 18506 df-mnd 18517 df-submnd 18562 df-mulg 18832 df-cntz 19056 df-cmn 19523 df-psmet 20741 df-xmet 20742 df-met 20743 df-bl 20744 df-mopn 20745 df-fbas 20746 df-fg 20747 df-cnfld 20750 df-top 22195 df-topon 22212 df-topsp 22234 df-bases 22248 df-cld 22322 df-ntr 22323 df-cls 22324 df-nei 22401 df-lp 22439 df-perf 22440 df-cn 22530 df-cnp 22531 df-haus 22618 df-tx 22865 df-hmeo 23058 df-fil 23149 df-fm 23241 df-flim 23242 df-flf 23243 df-xms 23625 df-ms 23626 df-tms 23627 df-ii 24192 df-cncf 24193 df-limc 25182 df-dv 25183 df-log 25864 |
This theorem is referenced by: xrge0pluscn 32333 xrge0tmd 32338 |
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