Mathbox for Thierry Arnoux |
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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 19607 | . . 3 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd | |
2 | cmnmnd 18031 | . . 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 29316 | . 2 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ TopSp | |
5 | eqeq1 2614 | . . . . 5 ⊢ (𝑦 = 𝑥 → (𝑦 = 0 ↔ 𝑥 = 0)) | |
6 | fveq2 6103 | . . . . . 6 ⊢ (𝑦 = 𝑥 → (log‘𝑦) = (log‘𝑥)) | |
7 | 6 | negeqd 10154 | . . . . 5 ⊢ (𝑦 = 𝑥 → -(log‘𝑦) = -(log‘𝑥)) |
8 | 5, 7 | ifbieq2d 4061 | . . . 4 ⊢ (𝑦 = 𝑥 → if(𝑦 = 0, +∞, -(log‘𝑦)) = if(𝑥 = 0, +∞, -(log‘𝑥))) |
9 | 8 | cbvmptv 4678 | . . 3 ⊢ (𝑦 ∈ (0[,]1) ↦ if(𝑦 = 0, +∞, -(log‘𝑦))) = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥))) |
10 | eqid 2610 | . . 3 ⊢ ((ordTop‘ ≤ ) ↾t (0[,]+∞)) = ((ordTop‘ ≤ ) ↾t (0[,]+∞)) | |
11 | eqid 2610 | . . 3 ⊢ ( +𝑒 ↾ ((0[,]+∞) × (0[,]+∞))) = ( +𝑒 ↾ ((0[,]+∞) × (0[,]+∞))) | |
12 | 9, 10, 11 | xrge0pluscn 29314 | . 2 ⊢ ( +𝑒 ↾ ((0[,]+∞) × (0[,]+∞))) ∈ ((((ordTop‘ ≤ ) ↾t (0[,]+∞)) ×t ((ordTop‘ ≤ ) ↾t (0[,]+∞))) Cn ((ordTop‘ ≤ ) ↾t (0[,]+∞))) |
13 | xrsbas 19581 | . . . . 5 ⊢ ℝ* = (Base‘ℝ*𝑠) | |
14 | eqid 2610 | . . . . 5 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) = (ℝ*𝑠 ↾s (0[,]+∞)) | |
15 | xrsadd 19582 | . . . . 5 ⊢ +𝑒 = (+g‘ℝ*𝑠) | |
16 | xaddf 11929 | . . . . . 6 ⊢ +𝑒 :(ℝ* × ℝ*)⟶ℝ* | |
17 | ffn 5958 | . . . . . 6 ⊢ ( +𝑒 :(ℝ* × ℝ*)⟶ℝ* → +𝑒 Fn (ℝ* × ℝ*)) | |
18 | 16, 17 | ax-mp 5 | . . . . 5 ⊢ +𝑒 Fn (ℝ* × ℝ*) |
19 | iccssxr 12127 | . . . . 5 ⊢ (0[,]+∞) ⊆ ℝ* | |
20 | 13, 14, 15, 18, 19 | ressplusf 28981 | . . . 4 ⊢ (+𝑓‘(ℝ*𝑠 ↾s (0[,]+∞))) = ( +𝑒 ↾ ((0[,]+∞) × (0[,]+∞))) |
21 | 20 | eqcomi 2619 | . . 3 ⊢ ( +𝑒 ↾ ((0[,]+∞) × (0[,]+∞))) = (+𝑓‘(ℝ*𝑠 ↾s (0[,]+∞))) |
22 | xrge0base 29016 | . . . 4 ⊢ (0[,]+∞) = (Base‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
23 | ovex 6577 | . . . . 5 ⊢ (0[,]+∞) ∈ V | |
24 | xrstset 19584 | . . . . . 6 ⊢ (ordTop‘ ≤ ) = (TopSet‘ℝ*𝑠) | |
25 | 14, 24 | resstset 15869 | . . . . 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 15918 | . . 3 ⊢ ((ordTop‘ ≤ ) ↾t (0[,]+∞)) = (TopOpen‘(ℝ*𝑠 ↾s (0[,]+∞))) |
28 | 21, 27 | istmd 21688 | . 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 1237 | 1 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ TopMnd |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∈ wcel 1977 Vcvv 3173 ifcif 4036 ↦ cmpt 4643 × cxp 5036 ↾ cres 5040 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 0cc0 9815 1c1 9816 +∞cpnf 9950 ℝ*cxr 9952 ≤ cle 9954 -cneg 10146 +𝑒 cxad 11820 [,]cicc 12049 ↾s cress 15696 TopSetcts 15774 ↾t crest 15904 ordTopcordt 15982 ℝ*𝑠cxrs 15983 +𝑓cplusf 17062 Mndcmnd 17117 CMndccmn 18016 TopSpctps 20519 Cn ccn 20838 ×t ctx 21173 TopMndctmd 21684 logclog 24105 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-ioc 12051 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-fac 12923 df-bc 12952 df-hash 12980 df-shft 13655 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-limsup 14050 df-clim 14067 df-rlim 14068 df-sum 14265 df-ef 14637 df-sin 14639 df-cos 14640 df-pi 14642 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-rest 15906 df-topn 15907 df-0g 15925 df-gsum 15926 df-topgen 15927 df-pt 15928 df-prds 15931 df-ordt 15984 df-xrs 15985 df-qtop 15990 df-imas 15991 df-xps 15993 df-mre 16069 df-mrc 16070 df-acs 16072 df-ps 17023 df-tsr 17024 df-plusf 17064 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-grp 17248 df-minusg 17249 df-sbg 17250 df-mulg 17364 df-subg 17414 df-cntz 17573 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-ring 18372 df-cring 18373 df-subrg 18601 df-abv 18640 df-lmod 18688 df-scaf 18689 df-sra 18993 df-rgmod 18994 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-fbas 19564 df-fg 19565 df-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cld 20633 df-ntr 20634 df-cls 20635 df-nei 20712 df-lp 20750 df-perf 20751 df-cn 20841 df-cnp 20842 df-haus 20929 df-tx 21175 df-hmeo 21368 df-fil 21460 df-fm 21552 df-flim 21553 df-flf 21554 df-tmd 21686 df-tgp 21687 df-trg 21773 df-xms 21935 df-ms 21936 df-tms 21937 df-nm 22197 df-ngp 22198 df-nrg 22200 df-nlm 22201 df-ii 22488 df-cncf 22489 df-limc 23436 df-dv 23437 df-log 24107 |
This theorem is referenced by: (None) |
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