Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > xrge0iifmhm | Structured version Visualization version GIF version |
Description: The defined function from the closed unit interval to the extended nonnegative reals is a monoid homomorphism. (Contributed by Thierry Arnoux, 21-Jun-2017.) |
Ref | Expression |
---|---|
xrge0iifhmeo.1 | ⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥))) |
xrge0iifhmeo.k | ⊢ 𝐽 = ((ordTop‘ ≤ ) ↾t (0[,]+∞)) |
Ref | Expression |
---|---|
xrge0iifmhm | ⊢ 𝐹 ∈ (((mulGrp‘ℂfld) ↾s (0[,]1)) MndHom (ℝ*𝑠 ↾s (0[,]+∞))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2818 | . . . . 5 ⊢ ((mulGrp‘ℂfld) ↾s (0[,]1)) = ((mulGrp‘ℂfld) ↾s (0[,]1)) | |
2 | 1 | iistmd 31044 | . . . 4 ⊢ ((mulGrp‘ℂfld) ↾s (0[,]1)) ∈ TopMnd |
3 | tmdmnd 22611 | . . . 4 ⊢ (((mulGrp‘ℂfld) ↾s (0[,]1)) ∈ TopMnd → ((mulGrp‘ℂfld) ↾s (0[,]1)) ∈ Mnd) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ ((mulGrp‘ℂfld) ↾s (0[,]1)) ∈ Mnd |
5 | xrge0cmn 20515 | . . . 4 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd | |
6 | cmnmnd 18851 | . . . 4 ⊢ ((ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd → (ℝ*𝑠 ↾s (0[,]+∞)) ∈ Mnd) | |
7 | 5, 6 | ax-mp 5 | . . 3 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ Mnd |
8 | 4, 7 | pm3.2i 471 | . 2 ⊢ (((mulGrp‘ℂfld) ↾s (0[,]1)) ∈ Mnd ∧ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ Mnd) |
9 | xrge0iifhmeo.1 | . . . . . 6 ⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥))) | |
10 | 9 | xrge0iifcnv 31075 | . . . . 5 ⊢ (𝐹:(0[,]1)–1-1-onto→(0[,]+∞) ∧ ◡𝐹 = (𝑦 ∈ (0[,]+∞) ↦ if(𝑦 = +∞, 0, (exp‘-𝑦)))) |
11 | 10 | simpli 484 | . . . 4 ⊢ 𝐹:(0[,]1)–1-1-onto→(0[,]+∞) |
12 | f1of 6608 | . . . 4 ⊢ (𝐹:(0[,]1)–1-1-onto→(0[,]+∞) → 𝐹:(0[,]1)⟶(0[,]+∞)) | |
13 | 11, 12 | ax-mp 5 | . . 3 ⊢ 𝐹:(0[,]1)⟶(0[,]+∞) |
14 | xrge0iifhmeo.k | . . . . 5 ⊢ 𝐽 = ((ordTop‘ ≤ ) ↾t (0[,]+∞)) | |
15 | 9, 14 | xrge0iifhom 31079 | . . . 4 ⊢ ((𝑦 ∈ (0[,]1) ∧ 𝑧 ∈ (0[,]1)) → (𝐹‘(𝑦 · 𝑧)) = ((𝐹‘𝑦) +𝑒 (𝐹‘𝑧))) |
16 | 15 | rgen2 3200 | . . 3 ⊢ ∀𝑦 ∈ (0[,]1)∀𝑧 ∈ (0[,]1)(𝐹‘(𝑦 · 𝑧)) = ((𝐹‘𝑦) +𝑒 (𝐹‘𝑧)) |
17 | 9, 14 | xrge0iif1 31080 | . . 3 ⊢ (𝐹‘1) = 0 |
18 | 13, 16, 17 | 3pm3.2i 1331 | . 2 ⊢ (𝐹:(0[,]1)⟶(0[,]+∞) ∧ ∀𝑦 ∈ (0[,]1)∀𝑧 ∈ (0[,]1)(𝐹‘(𝑦 · 𝑧)) = ((𝐹‘𝑦) +𝑒 (𝐹‘𝑧)) ∧ (𝐹‘1) = 0) |
19 | unitsscn 31038 | . . . 4 ⊢ (0[,]1) ⊆ ℂ | |
20 | eqid 2818 | . . . . . 6 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld) | |
21 | cnfldbas 20477 | . . . . . 6 ⊢ ℂ = (Base‘ℂfld) | |
22 | 20, 21 | mgpbas 19174 | . . . . 5 ⊢ ℂ = (Base‘(mulGrp‘ℂfld)) |
23 | 1, 22 | ressbas2 16543 | . . . 4 ⊢ ((0[,]1) ⊆ ℂ → (0[,]1) = (Base‘((mulGrp‘ℂfld) ↾s (0[,]1)))) |
24 | 19, 23 | ax-mp 5 | . . 3 ⊢ (0[,]1) = (Base‘((mulGrp‘ℂfld) ↾s (0[,]1))) |
25 | xrge0base 30599 | . . 3 ⊢ (0[,]+∞) = (Base‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
26 | cnfldex 20476 | . . . . 5 ⊢ ℂfld ∈ V | |
27 | ovex 7178 | . . . . 5 ⊢ (0[,]1) ∈ V | |
28 | eqid 2818 | . . . . . 6 ⊢ (ℂfld ↾s (0[,]1)) = (ℂfld ↾s (0[,]1)) | |
29 | 28, 20 | mgpress 19179 | . . . . 5 ⊢ ((ℂfld ∈ V ∧ (0[,]1) ∈ V) → ((mulGrp‘ℂfld) ↾s (0[,]1)) = (mulGrp‘(ℂfld ↾s (0[,]1)))) |
30 | 26, 27, 29 | mp2an 688 | . . . 4 ⊢ ((mulGrp‘ℂfld) ↾s (0[,]1)) = (mulGrp‘(ℂfld ↾s (0[,]1))) |
31 | cnfldmul 20479 | . . . . . 6 ⊢ · = (.r‘ℂfld) | |
32 | 28, 31 | ressmulr 16613 | . . . . 5 ⊢ ((0[,]1) ∈ V → · = (.r‘(ℂfld ↾s (0[,]1)))) |
33 | 27, 32 | ax-mp 5 | . . . 4 ⊢ · = (.r‘(ℂfld ↾s (0[,]1))) |
34 | 30, 33 | mgpplusg 19172 | . . 3 ⊢ · = (+g‘((mulGrp‘ℂfld) ↾s (0[,]1))) |
35 | xrge0plusg 30601 | . . 3 ⊢ +𝑒 = (+g‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
36 | cnring 20495 | . . . 4 ⊢ ℂfld ∈ Ring | |
37 | 1elunit 12844 | . . . 4 ⊢ 1 ∈ (0[,]1) | |
38 | cnfld1 20498 | . . . . 5 ⊢ 1 = (1r‘ℂfld) | |
39 | 1, 21, 38 | ringidss 19256 | . . . 4 ⊢ ((ℂfld ∈ Ring ∧ (0[,]1) ⊆ ℂ ∧ 1 ∈ (0[,]1)) → 1 = (0g‘((mulGrp‘ℂfld) ↾s (0[,]1)))) |
40 | 36, 19, 37, 39 | mp3an 1452 | . . 3 ⊢ 1 = (0g‘((mulGrp‘ℂfld) ↾s (0[,]1))) |
41 | xrge00 30600 | . . 3 ⊢ 0 = (0g‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
42 | 24, 25, 34, 35, 40, 41 | ismhm 17946 | . 2 ⊢ (𝐹 ∈ (((mulGrp‘ℂfld) ↾s (0[,]1)) MndHom (ℝ*𝑠 ↾s (0[,]+∞))) ↔ ((((mulGrp‘ℂfld) ↾s (0[,]1)) ∈ Mnd ∧ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ Mnd) ∧ (𝐹:(0[,]1)⟶(0[,]+∞) ∧ ∀𝑦 ∈ (0[,]1)∀𝑧 ∈ (0[,]1)(𝐹‘(𝑦 · 𝑧)) = ((𝐹‘𝑦) +𝑒 (𝐹‘𝑧)) ∧ (𝐹‘1) = 0))) |
43 | 8, 18, 42 | mpbir2an 707 | 1 ⊢ 𝐹 ∈ (((mulGrp‘ℂfld) ↾s (0[,]1)) MndHom (ℝ*𝑠 ↾s (0[,]+∞))) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ∀wral 3135 Vcvv 3492 ⊆ wss 3933 ifcif 4463 ↦ cmpt 5137 ◡ccnv 5547 ⟶wf 6344 –1-1-onto→wf1o 6347 ‘cfv 6348 (class class class)co 7145 ℂcc 10523 0cc0 10525 1c1 10526 · cmul 10530 +∞cpnf 10660 ≤ cle 10664 -cneg 10859 +𝑒 cxad 12493 [,]cicc 12729 expce 15403 Basecbs 16471 ↾s cress 16472 .rcmulr 16554 ↾t crest 16682 0gc0g 16701 ordTopcordt 16760 ℝ*𝑠cxrs 16761 Mndcmnd 17899 MndHom cmhm 17942 CMndccmn 18835 mulGrpcmgp 19168 Ringcrg 19226 ℂfldccnfld 20473 TopMndctmd 22606 logclog 25065 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 ax-addf 10604 ax-mulf 10605 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7570 df-1st 7678 df-2nd 7679 df-supp 7820 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-er 8278 df-map 8397 df-pm 8398 df-ixp 8450 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fsupp 8822 df-fi 8863 df-sup 8894 df-inf 8895 df-oi 8962 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-ioc 12731 df-ico 12732 df-icc 12733 df-fz 12881 df-fzo 13022 df-fl 13150 df-mod 13226 df-seq 13358 df-exp 13418 df-fac 13622 df-bc 13651 df-hash 13679 df-shft 14414 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-limsup 14816 df-clim 14833 df-rlim 14834 df-sum 15031 df-ef 15409 df-sin 15411 df-cos 15412 df-pi 15414 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-starv 16568 df-sca 16569 df-vsca 16570 df-ip 16571 df-tset 16572 df-ple 16573 df-ds 16575 df-unif 16576 df-hom 16577 df-cco 16578 df-rest 16684 df-topn 16685 df-0g 16703 df-gsum 16704 df-topgen 16705 df-pt 16706 df-prds 16709 df-xrs 16763 df-qtop 16768 df-imas 16769 df-xps 16771 df-mre 16845 df-mrc 16846 df-acs 16848 df-plusf 17839 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-mhm 17944 df-submnd 17945 df-grp 18044 df-minusg 18045 df-sbg 18046 df-mulg 18163 df-subg 18214 df-cntz 18385 df-cmn 18837 df-abl 18838 df-mgp 19169 df-ur 19181 df-ring 19228 df-cring 19229 df-subrg 19462 df-abv 19517 df-lmod 19565 df-scaf 19566 df-sra 19873 df-rgmod 19874 df-psmet 20465 df-xmet 20466 df-met 20467 df-bl 20468 df-mopn 20469 df-fbas 20470 df-fg 20471 df-cnfld 20474 df-top 21430 df-topon 21447 df-topsp 21469 df-bases 21482 df-cld 21555 df-ntr 21556 df-cls 21557 df-nei 21634 df-lp 21672 df-perf 21673 df-cn 21763 df-cnp 21764 df-haus 21851 df-tx 22098 df-hmeo 22291 df-fil 22382 df-fm 22474 df-flim 22475 df-flf 22476 df-tmd 22608 df-tgp 22609 df-trg 22695 df-xms 22857 df-ms 22858 df-tms 22859 df-nm 23119 df-ngp 23120 df-nrg 23122 df-nlm 23123 df-cncf 23413 df-limc 24391 df-dv 24392 df-log 25067 |
This theorem is referenced by: xrge0tmd 31087 |
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