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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 2798 | . . . . 5 ⊢ ((mulGrp‘ℂfld) ↾s (0[,]1)) = ((mulGrp‘ℂfld) ↾s (0[,]1)) | |
2 | 1 | iistmd 31255 | . . . 4 ⊢ ((mulGrp‘ℂfld) ↾s (0[,]1)) ∈ TopMnd |
3 | tmdmnd 22680 | . . . 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 20133 | . . . 4 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd | |
6 | cmnmnd 18914 | . . . 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 474 | . 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 31286 | . . . . 5 ⊢ (𝐹:(0[,]1)–1-1-onto→(0[,]+∞) ∧ ◡𝐹 = (𝑦 ∈ (0[,]+∞) ↦ if(𝑦 = +∞, 0, (exp‘-𝑦)))) |
11 | 10 | simpli 487 | . . . 4 ⊢ 𝐹:(0[,]1)–1-1-onto→(0[,]+∞) |
12 | f1of 6590 | . . . 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 31290 | . . . 4 ⊢ ((𝑦 ∈ (0[,]1) ∧ 𝑧 ∈ (0[,]1)) → (𝐹‘(𝑦 · 𝑧)) = ((𝐹‘𝑦) +𝑒 (𝐹‘𝑧))) |
16 | 15 | rgen2 3168 | . . 3 ⊢ ∀𝑦 ∈ (0[,]1)∀𝑧 ∈ (0[,]1)(𝐹‘(𝑦 · 𝑧)) = ((𝐹‘𝑦) +𝑒 (𝐹‘𝑧)) |
17 | 9, 14 | xrge0iif1 31291 | . . 3 ⊢ (𝐹‘1) = 0 |
18 | 13, 16, 17 | 3pm3.2i 1336 | . 2 ⊢ (𝐹:(0[,]1)⟶(0[,]+∞) ∧ ∀𝑦 ∈ (0[,]1)∀𝑧 ∈ (0[,]1)(𝐹‘(𝑦 · 𝑧)) = ((𝐹‘𝑦) +𝑒 (𝐹‘𝑧)) ∧ (𝐹‘1) = 0) |
19 | unitsscn 12878 | . . . 4 ⊢ (0[,]1) ⊆ ℂ | |
20 | eqid 2798 | . . . . . 6 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld) | |
21 | cnfldbas 20095 | . . . . . 6 ⊢ ℂ = (Base‘ℂfld) | |
22 | 20, 21 | mgpbas 19238 | . . . . 5 ⊢ ℂ = (Base‘(mulGrp‘ℂfld)) |
23 | 1, 22 | ressbas2 16547 | . . . 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 30719 | . . 3 ⊢ (0[,]+∞) = (Base‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
26 | cnfldex 20094 | . . . . 5 ⊢ ℂfld ∈ V | |
27 | ovex 7168 | . . . . 5 ⊢ (0[,]1) ∈ V | |
28 | eqid 2798 | . . . . . 6 ⊢ (ℂfld ↾s (0[,]1)) = (ℂfld ↾s (0[,]1)) | |
29 | 28, 20 | mgpress 19243 | . . . . 5 ⊢ ((ℂfld ∈ V ∧ (0[,]1) ∈ V) → ((mulGrp‘ℂfld) ↾s (0[,]1)) = (mulGrp‘(ℂfld ↾s (0[,]1)))) |
30 | 26, 27, 29 | mp2an 691 | . . . 4 ⊢ ((mulGrp‘ℂfld) ↾s (0[,]1)) = (mulGrp‘(ℂfld ↾s (0[,]1))) |
31 | cnfldmul 20097 | . . . . . 6 ⊢ · = (.r‘ℂfld) | |
32 | 28, 31 | ressmulr 16617 | . . . . 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 19236 | . . 3 ⊢ · = (+g‘((mulGrp‘ℂfld) ↾s (0[,]1))) |
35 | xrge0plusg 30721 | . . 3 ⊢ +𝑒 = (+g‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
36 | cnring 20113 | . . . 4 ⊢ ℂfld ∈ Ring | |
37 | 1elunit 12848 | . . . 4 ⊢ 1 ∈ (0[,]1) | |
38 | cnfld1 20116 | . . . . 5 ⊢ 1 = (1r‘ℂfld) | |
39 | 1, 21, 38 | ringidss 19323 | . . . 4 ⊢ ((ℂfld ∈ Ring ∧ (0[,]1) ⊆ ℂ ∧ 1 ∈ (0[,]1)) → 1 = (0g‘((mulGrp‘ℂfld) ↾s (0[,]1)))) |
40 | 36, 19, 37, 39 | mp3an 1458 | . . 3 ⊢ 1 = (0g‘((mulGrp‘ℂfld) ↾s (0[,]1))) |
41 | xrge00 30720 | . . 3 ⊢ 0 = (0g‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
42 | 24, 25, 34, 35, 40, 41 | ismhm 17950 | . 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 710 | 1 ⊢ 𝐹 ∈ (((mulGrp‘ℂfld) ↾s (0[,]1)) MndHom (ℝ*𝑠 ↾s (0[,]+∞))) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ∀wral 3106 Vcvv 3441 ⊆ wss 3881 ifcif 4425 ↦ cmpt 5110 ◡ccnv 5518 ⟶wf 6320 –1-1-onto→wf1o 6323 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 0cc0 10526 1c1 10527 · cmul 10531 +∞cpnf 10661 ≤ cle 10665 -cneg 10860 +𝑒 cxad 12493 [,]cicc 12729 expce 15407 Basecbs 16475 ↾s cress 16476 .rcmulr 16558 ↾t crest 16686 0gc0g 16705 ordTopcordt 16764 ℝ*𝑠cxrs 16765 Mndcmnd 17903 MndHom cmhm 17946 CMndccmn 18898 mulGrpcmgp 19232 Ringcrg 19290 ℂfldccnfld 20091 TopMndctmd 22675 logclog 25146 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-fi 8859 df-sup 8890 df-inf 8891 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 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 12886 df-fzo 13029 df-fl 13157 df-mod 13233 df-seq 13365 df-exp 13426 df-fac 13630 df-bc 13659 df-hash 13687 df-shft 14418 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-limsup 14820 df-clim 14837 df-rlim 14838 df-sum 15035 df-ef 15413 df-sin 15415 df-cos 15416 df-pi 15418 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-rest 16688 df-topn 16689 df-0g 16707 df-gsum 16708 df-topgen 16709 df-pt 16710 df-prds 16713 df-xrs 16767 df-qtop 16772 df-imas 16773 df-xps 16775 df-mre 16849 df-mrc 16850 df-acs 16852 df-plusf 17843 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mhm 17948 df-submnd 17949 df-grp 18098 df-minusg 18099 df-sbg 18100 df-mulg 18217 df-subg 18268 df-cntz 18439 df-cmn 18900 df-abl 18901 df-mgp 19233 df-ur 19245 df-ring 19292 df-cring 19293 df-subrg 19526 df-abv 19581 df-lmod 19629 df-scaf 19630 df-sra 19937 df-rgmod 19938 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-fbas 20088 df-fg 20089 df-cnfld 20092 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 df-cld 21624 df-ntr 21625 df-cls 21626 df-nei 21703 df-lp 21741 df-perf 21742 df-cn 21832 df-cnp 21833 df-haus 21920 df-tx 22167 df-hmeo 22360 df-fil 22451 df-fm 22543 df-flim 22544 df-flf 22545 df-tmd 22677 df-tgp 22678 df-trg 22765 df-xms 22927 df-ms 22928 df-tms 22929 df-nm 23189 df-ngp 23190 df-nrg 23192 df-nlm 23193 df-cncf 23483 df-limc 24469 df-dv 24470 df-log 25148 |
This theorem is referenced by: xrge0tmd 31298 |
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