xrge0iifhmeo - Mathbox for Thierry Arnoux

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Theorem xrge0iifhmeo 33537

Description: Expose a homeomorphism from the closed unit interval to the extended nonnegative reals. (Contributed by Thierry Arnoux, 1-Apr-2017.)

**Hypotheses**
Ref	Expression
xrge0iifhmeo.1	⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥)))
xrge0iifhmeo.k	⊢ 𝐽 = ((ordTop‘ ≤ ) ↾_t (0[,]+∞))

**Assertion**
Ref	Expression
xrge0iifhmeo	⊢ 𝐹 ∈ (IIHomeo𝐽)

Distinct variable group: 𝑥,𝐹 Allowed substitution hint: 𝐽(𝑥)

Proof of Theorem xrge0iifhmeo Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		letsr 18585	. . . . . 6 ⊢ ≤ ∈ TosetRel
2		tsrps 18579	. . . . . 6 ⊢ ( ≤ ∈ TosetRel → ≤ ∈ PosetRel)
3	1, 2	ax-mp 5	. . . . 5 ⊢ ≤ ∈ PosetRel
4	3	elexi 3491	. . . 4 ⊢ ≤ ∈ V
5	4	inex1 5317	. . 3 ⊢ ( ≤ ∩ ((0[,]1) × (0[,]1))) ∈ V
6		cnvps 18570	. . . . . 6 ⊢ ( ≤ ∈ PosetRel → ^◡ ≤ ∈ PosetRel)
7	3, 6	ax-mp 5	. . . . 5 ⊢ ^◡ ≤ ∈ PosetRel
8	7	elexi 3491	. . . 4 ⊢ ^◡ ≤ ∈ V
9	8	inex1 5317	. . 3 ⊢ (^◡ ≤ ∩ ((0[,]+∞) × (0[,]+∞))) ∈ V
10		xrge0iifhmeo.1	. . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥)))
11	10	xrge0iifiso 33536	. . . . . 6 ⊢ 𝐹 Isom < , ^◡ < ((0[,]1), (0[,]+∞))
12		iccssxr 13440	. . . . . . 7 ⊢ (0[,]1) ⊆ ℝ^*
13		iccssxr 13440	. . . . . . 7 ⊢ (0[,]+∞) ⊆ ℝ^*
14		gtiso 32493	. . . . . . 7 ⊢ (((0[,]1) ⊆ ℝ^* ∧ (0[,]+∞) ⊆ ℝ^*) → (𝐹 Isom < , ^◡ < ((0[,]1), (0[,]+∞)) ↔ 𝐹 Isom ≤ , ^◡ ≤ ((0[,]1), (0[,]+∞))))
15	12, 13, 14	mp2an 691	. . . . . 6 ⊢ (𝐹 Isom < , ^◡ < ((0[,]1), (0[,]+∞)) ↔ 𝐹 Isom ≤ , ^◡ ≤ ((0[,]1), (0[,]+∞)))
16	11, 15	mpbi 229	. . . . 5 ⊢ 𝐹 Isom ≤ , ^◡ ≤ ((0[,]1), (0[,]+∞))
17		isores1 7342	. . . . 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 7341	. . . 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 18582	. . . . . . 7 ⊢ ℝ^* = dom ≤
22	21	psssdm 18574	. . . . . 6 ⊢ (( ≤ ∈ PosetRel ∧ (0[,]1) ⊆ ℝ^*) → dom ( ≤ ∩ ((0[,]1) × (0[,]1))) = (0[,]1))
23	3, 12, 22	mp2an 691	. . . . 5 ⊢ dom ( ≤ ∩ ((0[,]1) × (0[,]1))) = (0[,]1)
24	23	eqcomi 2737	. . . 4 ⊢ (0[,]1) = dom ( ≤ ∩ ((0[,]1) × (0[,]1)))
25		lern 18583	. . . . . . . 8 ⊢ ℝ^* = ran ≤
26		df-rn 5689	. . . . . . . 8 ⊢ ran ≤ = dom ^◡ ≤
27	25, 26	eqtri 2756	. . . . . . 7 ⊢ ℝ^* = dom ^◡ ≤
28	27	psssdm 18574	. . . . . 6 ⊢ ((^◡ ≤ ∈ PosetRel ∧ (0[,]+∞) ⊆ ℝ^*) → dom (^◡ ≤ ∩ ((0[,]+∞) × (0[,]+∞))) = (0[,]+∞))
29	7, 13, 28	mp2an 691	. . . . 5 ⊢ dom (^◡ ≤ ∩ ((0[,]+∞) × (0[,]+∞))) = (0[,]+∞)
30	29	eqcomi 2737	. . . 4 ⊢ (0[,]+∞) = dom (^◡ ≤ ∩ ((0[,]+∞) × (0[,]+∞)))
31	24, 30	ordthmeo 23719	. . 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 1458	. 2 ⊢ 𝐹 ∈ ((ordTop‘( ≤ ∩ ((0[,]1) × (0[,]1))))Homeo(ordTop‘(^◡ ≤ ∩ ((0[,]+∞) × (0[,]+∞)))))
33		dfii5 24818	. . 3 ⊢ II = (ordTop‘( ≤ ∩ ((0[,]1) × (0[,]1))))
34		xrge0iifhmeo.k	. . . 4 ⊢ 𝐽 = ((ordTop‘ ≤ ) ↾_t (0[,]+∞))
35		iccss2 13428	. . . . 5 ⊢ ((𝑥 ∈ (0[,]+∞) ∧ 𝑦 ∈ (0[,]+∞)) → (𝑥[,]𝑦) ⊆ (0[,]+∞))
36	13, 35	cnvordtrestixx 33514	. . . 4 ⊢ ((ordTop‘ ≤ ) ↾_t (0[,]+∞)) = (ordTop‘(^◡ ≤ ∩ ((0[,]+∞) × (0[,]+∞))))
37	34, 36	eqtri 2756	. . 3 ⊢ 𝐽 = (ordTop‘(^◡ ≤ ∩ ((0[,]+∞) × (0[,]+∞))))
38	33, 37	oveq12i 7432	. 2 ⊢ (IIHomeo𝐽) = ((ordTop‘( ≤ ∩ ((0[,]1) × (0[,]1))))Homeo(ordTop‘(^◡ ≤ ∩ ((0[,]+∞) × (0[,]+∞)))))
39	32, 38	eleqtrri 2828	1 ⊢ 𝐹 ∈ (IIHomeo𝐽)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 205 = wceq 1534 ∈ wcel 2099 Vcvv 3471 ∩ cin 3946 ⊆ wss 3947 ifcif 4529 ↦ cmpt 5231 × cxp 5676 ^◡ccnv 5677 dom cdm 5678 ran crn 5679 ‘cfv 6548 Isom wiso 6549 (class class class)co 7420 0cc0 11139 1c1 11140 +∞cpnf 11276 ℝ^*cxr 11278 < clt 11279 ≤ cle 11280 -cneg 11476 [,]cicc 13360 ↾_t crest 17402 ordTopcordt 17481 PosetRelcps 18556 TosetRel ctsr 18557 Homeochmeo 23670 IIcii 24808 logclog 26501

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-inf2 9665 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-pre-sup 11217 ax-addf 11218

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7685 df-om 7871 df-1st 7993 df-2nd 7994 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-er 8725 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9387 df-fi 9435 df-sup 9466 df-inf 9467 df-oi 9534 df-card 9963 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-dec 12709 df-uz 12854 df-q 12964 df-rp 13008 df-xneg 13125 df-xadd 13126 df-xmul 13127 df-ioo 13361 df-ioc 13362 df-ico 13363 df-icc 13364 df-fz 13518 df-fzo 13661 df-fl 13790 df-mod 13868 df-seq 14000 df-exp 14060 df-fac 14266 df-bc 14295 df-hash 14323 df-shft 15047 df-cj 15079 df-re 15080 df-im 15081 df-sqrt 15215 df-abs 15216 df-limsup 15448 df-clim 15465 df-rlim 15466 df-sum 15666 df-ef 16044 df-sin 16046 df-cos 16047 df-pi 16049 df-struct 17116 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-ress 17210 df-plusg 17246 df-mulr 17247 df-starv 17248 df-sca 17249 df-vsca 17250 df-ip 17251 df-tset 17252 df-ple 17253 df-ds 17255 df-unif 17256 df-hom 17257 df-cco 17258 df-rest 17404 df-topn 17405 df-0g 17423 df-gsum 17424 df-topgen 17425 df-pt 17426 df-prds 17429 df-ordt 17483 df-xrs 17484 df-qtop 17489 df-imas 17490 df-xps 17492 df-mre 17566 df-mrc 17567 df-acs 17569 df-ps 18558 df-tsr 18559 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-submnd 18741 df-mulg 19024 df-cntz 19268 df-cmn 19737 df-psmet 21271 df-xmet 21272 df-met 21273 df-bl 21274 df-mopn 21275 df-fbas 21276 df-fg 21277 df-cnfld 21280 df-top 22809 df-topon 22826 df-topsp 22848 df-bases 22862 df-cld 22936 df-ntr 22937 df-cls 22938 df-nei 23015 df-lp 23053 df-perf 23054 df-cn 23144 df-cnp 23145 df-haus 23232 df-tx 23479 df-hmeo 23672 df-fil 23763 df-fm 23855 df-flim 23856 df-flf 23857 df-xms 24239 df-ms 24240 df-tms 24241 df-ii 24810 df-cncf 24811 df-limc 25808 df-dv 25809 df-log 26503

This theorem is referenced by: xrge0pluscn 33541 xrge0tmd 33546

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