Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > xrge0tmdOLD | Structured version Visualization version GIF version |
Description: The extended nonnegative real numbers monoid is a topological monoid. (Contributed by Thierry Arnoux, 26-Mar-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
xrge0tmdOLD | ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ TopMnd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrge0cmn 19788 | . . 3 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd | |
2 | cmnmnd 18208 | . . 3 ⊢ ((ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd → (ℝ*𝑠 ↾s (0[,]+∞)) ∈ Mnd) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ Mnd |
4 | xrge0tps 29988 | . 2 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ TopSp | |
5 | eqeq1 2626 | . . . . 5 ⊢ (𝑦 = 𝑥 → (𝑦 = 0 ↔ 𝑥 = 0)) | |
6 | fveq2 6191 | . . . . . 6 ⊢ (𝑦 = 𝑥 → (log‘𝑦) = (log‘𝑥)) | |
7 | 6 | negeqd 10275 | . . . . 5 ⊢ (𝑦 = 𝑥 → -(log‘𝑦) = -(log‘𝑥)) |
8 | 5, 7 | ifbieq2d 4111 | . . . 4 ⊢ (𝑦 = 𝑥 → if(𝑦 = 0, +∞, -(log‘𝑦)) = if(𝑥 = 0, +∞, -(log‘𝑥))) |
9 | 8 | cbvmptv 4750 | . . 3 ⊢ (𝑦 ∈ (0[,]1) ↦ if(𝑦 = 0, +∞, -(log‘𝑦))) = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥))) |
10 | eqid 2622 | . . 3 ⊢ ((ordTop‘ ≤ ) ↾t (0[,]+∞)) = ((ordTop‘ ≤ ) ↾t (0[,]+∞)) | |
11 | eqid 2622 | . . 3 ⊢ ( +𝑒 ↾ ((0[,]+∞) × (0[,]+∞))) = ( +𝑒 ↾ ((0[,]+∞) × (0[,]+∞))) | |
12 | 9, 10, 11 | xrge0pluscn 29986 | . 2 ⊢ ( +𝑒 ↾ ((0[,]+∞) × (0[,]+∞))) ∈ ((((ordTop‘ ≤ ) ↾t (0[,]+∞)) ×t ((ordTop‘ ≤ ) ↾t (0[,]+∞))) Cn ((ordTop‘ ≤ ) ↾t (0[,]+∞))) |
13 | xrsbas 19762 | . . . . 5 ⊢ ℝ* = (Base‘ℝ*𝑠) | |
14 | eqid 2622 | . . . . 5 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) = (ℝ*𝑠 ↾s (0[,]+∞)) | |
15 | xrsadd 19763 | . . . . 5 ⊢ +𝑒 = (+g‘ℝ*𝑠) | |
16 | xaddf 12055 | . . . . . 6 ⊢ +𝑒 :(ℝ* × ℝ*)⟶ℝ* | |
17 | ffn 6045 | . . . . . 6 ⊢ ( +𝑒 :(ℝ* × ℝ*)⟶ℝ* → +𝑒 Fn (ℝ* × ℝ*)) | |
18 | 16, 17 | ax-mp 5 | . . . . 5 ⊢ +𝑒 Fn (ℝ* × ℝ*) |
19 | iccssxr 12256 | . . . . 5 ⊢ (0[,]+∞) ⊆ ℝ* | |
20 | 13, 14, 15, 18, 19 | ressplusf 29650 | . . . 4 ⊢ (+𝑓‘(ℝ*𝑠 ↾s (0[,]+∞))) = ( +𝑒 ↾ ((0[,]+∞) × (0[,]+∞))) |
21 | 20 | eqcomi 2631 | . . 3 ⊢ ( +𝑒 ↾ ((0[,]+∞) × (0[,]+∞))) = (+𝑓‘(ℝ*𝑠 ↾s (0[,]+∞))) |
22 | xrge0base 29685 | . . . 4 ⊢ (0[,]+∞) = (Base‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
23 | ovex 6678 | . . . . 5 ⊢ (0[,]+∞) ∈ V | |
24 | xrstset 19765 | . . . . . 6 ⊢ (ordTop‘ ≤ ) = (TopSet‘ℝ*𝑠) | |
25 | 14, 24 | resstset 16046 | . . . . 5 ⊢ ((0[,]+∞) ∈ V → (ordTop‘ ≤ ) = (TopSet‘(ℝ*𝑠 ↾s (0[,]+∞)))) |
26 | 23, 25 | ax-mp 5 | . . . 4 ⊢ (ordTop‘ ≤ ) = (TopSet‘(ℝ*𝑠 ↾s (0[,]+∞))) |
27 | 22, 26 | topnval 16095 | . . 3 ⊢ ((ordTop‘ ≤ ) ↾t (0[,]+∞)) = (TopOpen‘(ℝ*𝑠 ↾s (0[,]+∞))) |
28 | 21, 27 | istmd 21878 | . 2 ⊢ ((ℝ*𝑠 ↾s (0[,]+∞)) ∈ TopMnd ↔ ((ℝ*𝑠 ↾s (0[,]+∞)) ∈ Mnd ∧ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ TopSp ∧ ( +𝑒 ↾ ((0[,]+∞) × (0[,]+∞))) ∈ ((((ordTop‘ ≤ ) ↾t (0[,]+∞)) ×t ((ordTop‘ ≤ ) ↾t (0[,]+∞))) Cn ((ordTop‘ ≤ ) ↾t (0[,]+∞))))) |
29 | 3, 4, 12, 28 | mpbir3an 1244 | 1 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ TopMnd |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 ∈ wcel 1990 Vcvv 3200 ifcif 4086 ↦ cmpt 4729 × cxp 5112 ↾ cres 5116 Fn wfn 5883 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 0cc0 9936 1c1 9937 +∞cpnf 10071 ℝ*cxr 10073 ≤ cle 10075 -cneg 10267 +𝑒 cxad 11944 [,]cicc 12178 ↾s cress 15858 TopSetcts 15947 ↾t crest 16081 ordTopcordt 16159 ℝ*𝑠cxrs 16160 +𝑓cplusf 17239 Mndcmnd 17294 CMndccmn 18193 TopSpctps 20736 Cn ccn 21028 ×t ctx 21363 TopMndctmd 21874 logclog 24301 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ioc 12180 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-fac 13061 df-bc 13090 df-hash 13118 df-shft 13807 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-limsup 14202 df-clim 14219 df-rlim 14220 df-sum 14417 df-ef 14798 df-sin 14800 df-cos 14801 df-pi 14803 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-ordt 16161 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-ps 17200 df-tsr 17201 df-plusf 17241 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-mulg 17541 df-subg 17591 df-cntz 17750 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-cring 18550 df-subrg 18778 df-abv 18817 df-lmod 18865 df-scaf 18866 df-sra 19172 df-rgmod 19173 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-fbas 19743 df-fg 19744 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-lp 20940 df-perf 20941 df-cn 21031 df-cnp 21032 df-haus 21119 df-tx 21365 df-hmeo 21558 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-tmd 21876 df-tgp 21877 df-trg 21963 df-xms 22125 df-ms 22126 df-tms 22127 df-nm 22387 df-ngp 22388 df-nrg 22390 df-nlm 22391 df-ii 22680 df-cncf 22681 df-limc 23630 df-dv 23631 df-log 24303 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |