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 2738 | . . . . 5 ⊢ ((mulGrp‘ℂfld) ↾s (0[,]1)) = ((mulGrp‘ℂfld) ↾s (0[,]1)) | |
2 | 1 | iistmd 31852 | . . . 4 ⊢ ((mulGrp‘ℂfld) ↾s (0[,]1)) ∈ TopMnd |
3 | tmdmnd 23226 | . . . 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 20640 | . . . 4 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd | |
6 | cmnmnd 19402 | . . . 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 31883 | . . . . 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 6716 | . . . 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 31887 | . . . 4 ⊢ ((𝑦 ∈ (0[,]1) ∧ 𝑧 ∈ (0[,]1)) → (𝐹‘(𝑦 · 𝑧)) = ((𝐹‘𝑦) +𝑒 (𝐹‘𝑧))) |
16 | 15 | rgen2 3120 | . . 3 ⊢ ∀𝑦 ∈ (0[,]1)∀𝑧 ∈ (0[,]1)(𝐹‘(𝑦 · 𝑧)) = ((𝐹‘𝑦) +𝑒 (𝐹‘𝑧)) |
17 | 9, 14 | xrge0iif1 31888 | . . 3 ⊢ (𝐹‘1) = 0 |
18 | 13, 16, 17 | 3pm3.2i 1338 | . 2 ⊢ (𝐹:(0[,]1)⟶(0[,]+∞) ∧ ∀𝑦 ∈ (0[,]1)∀𝑧 ∈ (0[,]1)(𝐹‘(𝑦 · 𝑧)) = ((𝐹‘𝑦) +𝑒 (𝐹‘𝑧)) ∧ (𝐹‘1) = 0) |
19 | unitsscn 13232 | . . . 4 ⊢ (0[,]1) ⊆ ℂ | |
20 | eqid 2738 | . . . . . 6 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld) | |
21 | cnfldbas 20601 | . . . . . 6 ⊢ ℂ = (Base‘ℂfld) | |
22 | 20, 21 | mgpbas 19726 | . . . . 5 ⊢ ℂ = (Base‘(mulGrp‘ℂfld)) |
23 | 1, 22 | ressbas2 16949 | . . . 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 31294 | . . 3 ⊢ (0[,]+∞) = (Base‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
26 | cnfldex 20600 | . . . . 5 ⊢ ℂfld ∈ V | |
27 | ovex 7308 | . . . . 5 ⊢ (0[,]1) ∈ V | |
28 | eqid 2738 | . . . . . 6 ⊢ (ℂfld ↾s (0[,]1)) = (ℂfld ↾s (0[,]1)) | |
29 | 28, 20 | mgpress 19735 | . . . . 5 ⊢ ((ℂfld ∈ V ∧ (0[,]1) ∈ V) → ((mulGrp‘ℂfld) ↾s (0[,]1)) = (mulGrp‘(ℂfld ↾s (0[,]1)))) |
30 | 26, 27, 29 | mp2an 689 | . . . 4 ⊢ ((mulGrp‘ℂfld) ↾s (0[,]1)) = (mulGrp‘(ℂfld ↾s (0[,]1))) |
31 | cnfldmul 20603 | . . . . . 6 ⊢ · = (.r‘ℂfld) | |
32 | 28, 31 | ressmulr 17017 | . . . . 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 19724 | . . 3 ⊢ · = (+g‘((mulGrp‘ℂfld) ↾s (0[,]1))) |
35 | xrge0plusg 31296 | . . 3 ⊢ +𝑒 = (+g‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
36 | cnring 20620 | . . . 4 ⊢ ℂfld ∈ Ring | |
37 | 1elunit 13202 | . . . 4 ⊢ 1 ∈ (0[,]1) | |
38 | cnfld1 20623 | . . . . 5 ⊢ 1 = (1r‘ℂfld) | |
39 | 1, 21, 38 | ringidss 19816 | . . . 4 ⊢ ((ℂfld ∈ Ring ∧ (0[,]1) ⊆ ℂ ∧ 1 ∈ (0[,]1)) → 1 = (0g‘((mulGrp‘ℂfld) ↾s (0[,]1)))) |
40 | 36, 19, 37, 39 | mp3an 1460 | . . 3 ⊢ 1 = (0g‘((mulGrp‘ℂfld) ↾s (0[,]1))) |
41 | xrge00 31295 | . . 3 ⊢ 0 = (0g‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
42 | 24, 25, 34, 35, 40, 41 | ismhm 18432 | . 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 708 | 1 ⊢ 𝐹 ∈ (((mulGrp‘ℂfld) ↾s (0[,]1)) MndHom (ℝ*𝑠 ↾s (0[,]+∞))) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ∀wral 3064 Vcvv 3432 ⊆ wss 3887 ifcif 4459 ↦ cmpt 5157 ◡ccnv 5588 ⟶wf 6429 –1-1-onto→wf1o 6432 ‘cfv 6433 (class class class)co 7275 ℂcc 10869 0cc0 10871 1c1 10872 · cmul 10876 +∞cpnf 11006 ≤ cle 11010 -cneg 11206 +𝑒 cxad 12846 [,]cicc 13082 expce 15771 Basecbs 16912 ↾s cress 16941 .rcmulr 16963 ↾t crest 17131 0gc0g 17150 ordTopcordt 17210 ℝ*𝑠cxrs 17211 Mndcmnd 18385 MndHom cmhm 18428 CMndccmn 19386 mulGrpcmgp 19720 Ringcrg 19783 ℂfldccnfld 20597 TopMndctmd 23221 logclog 25710 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 ax-addf 10950 ax-mulf 10951 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-er 8498 df-map 8617 df-pm 8618 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-fi 9170 df-sup 9201 df-inf 9202 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-q 12689 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-ioo 13083 df-ioc 13084 df-ico 13085 df-icc 13086 df-fz 13240 df-fzo 13383 df-fl 13512 df-mod 13590 df-seq 13722 df-exp 13783 df-fac 13988 df-bc 14017 df-hash 14045 df-shft 14778 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-limsup 15180 df-clim 15197 df-rlim 15198 df-sum 15398 df-ef 15777 df-sin 15779 df-cos 15780 df-pi 15782 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-hom 16986 df-cco 16987 df-rest 17133 df-topn 17134 df-0g 17152 df-gsum 17153 df-topgen 17154 df-pt 17155 df-prds 17158 df-xrs 17213 df-qtop 17218 df-imas 17219 df-xps 17221 df-mre 17295 df-mrc 17296 df-acs 17298 df-plusf 18325 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-mhm 18430 df-submnd 18431 df-grp 18580 df-minusg 18581 df-sbg 18582 df-mulg 18701 df-subg 18752 df-cntz 18923 df-cmn 19388 df-abl 19389 df-mgp 19721 df-ur 19738 df-ring 19785 df-cring 19786 df-subrg 20022 df-abv 20077 df-lmod 20125 df-scaf 20126 df-sra 20434 df-rgmod 20435 df-psmet 20589 df-xmet 20590 df-met 20591 df-bl 20592 df-mopn 20593 df-fbas 20594 df-fg 20595 df-cnfld 20598 df-top 22043 df-topon 22060 df-topsp 22082 df-bases 22096 df-cld 22170 df-ntr 22171 df-cls 22172 df-nei 22249 df-lp 22287 df-perf 22288 df-cn 22378 df-cnp 22379 df-haus 22466 df-tx 22713 df-hmeo 22906 df-fil 22997 df-fm 23089 df-flim 23090 df-flf 23091 df-tmd 23223 df-tgp 23224 df-trg 23311 df-xms 23473 df-ms 23474 df-tms 23475 df-nm 23738 df-ngp 23739 df-nrg 23741 df-nlm 23742 df-cncf 24041 df-limc 25030 df-dv 25031 df-log 25712 |
This theorem is referenced by: xrge0tmd 31895 |
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