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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 | β’ πΉ = (π₯ β (0[,]1) β¦ if(π₯ = 0, +β, -(logβπ₯))) |
xrge0iifhmeo.k | β’ π½ = ((ordTopβ β€ ) βΎt (0[,]+β)) |
Ref | Expression |
---|---|
xrge0iifmhm | β’ πΉ β (((mulGrpββfld) βΎs (0[,]1)) MndHom (β*π βΎs (0[,]+β))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2724 | . . . . 5 β’ ((mulGrpββfld) βΎs (0[,]1)) = ((mulGrpββfld) βΎs (0[,]1)) | |
2 | 1 | iistmd 33371 | . . . 4 β’ ((mulGrpββfld) βΎs (0[,]1)) β TopMnd |
3 | tmdmnd 23901 | . . . 4 β’ (((mulGrpββfld) βΎs (0[,]1)) β TopMnd β ((mulGrpββfld) βΎs (0[,]1)) β Mnd) | |
4 | 2, 3 | ax-mp 5 | . . 3 β’ ((mulGrpββfld) βΎs (0[,]1)) β Mnd |
5 | xrge0cmn 21271 | . . . 4 β’ (β*π βΎs (0[,]+β)) β CMnd | |
6 | cmnmnd 19707 | . . . 4 β’ ((β*π βΎs (0[,]+β)) β CMnd β (β*π βΎs (0[,]+β)) β Mnd) | |
7 | 5, 6 | ax-mp 5 | . . 3 β’ (β*π βΎs (0[,]+β)) β Mnd |
8 | 4, 7 | pm3.2i 470 | . 2 β’ (((mulGrpββfld) βΎs (0[,]1)) β Mnd β§ (β*π βΎs (0[,]+β)) β Mnd) |
9 | xrge0iifhmeo.1 | . . . . . 6 β’ πΉ = (π₯ β (0[,]1) β¦ if(π₯ = 0, +β, -(logβπ₯))) | |
10 | 9 | xrge0iifcnv 33402 | . . . . 5 β’ (πΉ:(0[,]1)β1-1-ontoβ(0[,]+β) β§ β‘πΉ = (π¦ β (0[,]+β) β¦ if(π¦ = +β, 0, (expβ-π¦)))) |
11 | 10 | simpli 483 | . . . 4 β’ πΉ:(0[,]1)β1-1-ontoβ(0[,]+β) |
12 | f1of 6823 | . . . 4 β’ (πΉ:(0[,]1)β1-1-ontoβ(0[,]+β) β πΉ:(0[,]1)βΆ(0[,]+β)) | |
13 | 11, 12 | ax-mp 5 | . . 3 β’ πΉ:(0[,]1)βΆ(0[,]+β) |
14 | xrge0iifhmeo.k | . . . . 5 β’ π½ = ((ordTopβ β€ ) βΎt (0[,]+β)) | |
15 | 9, 14 | xrge0iifhom 33406 | . . . 4 β’ ((π¦ β (0[,]1) β§ π§ β (0[,]1)) β (πΉβ(π¦ Β· π§)) = ((πΉβπ¦) +π (πΉβπ§))) |
16 | 15 | rgen2 3189 | . . 3 β’ βπ¦ β (0[,]1)βπ§ β (0[,]1)(πΉβ(π¦ Β· π§)) = ((πΉβπ¦) +π (πΉβπ§)) |
17 | 9, 14 | xrge0iif1 33407 | . . 3 β’ (πΉβ1) = 0 |
18 | 13, 16, 17 | 3pm3.2i 1336 | . 2 β’ (πΉ:(0[,]1)βΆ(0[,]+β) β§ βπ¦ β (0[,]1)βπ§ β (0[,]1)(πΉβ(π¦ Β· π§)) = ((πΉβπ¦) +π (πΉβπ§)) β§ (πΉβ1) = 0) |
19 | unitsscn 13474 | . . . 4 β’ (0[,]1) β β | |
20 | eqid 2724 | . . . . . 6 β’ (mulGrpββfld) = (mulGrpββfld) | |
21 | cnfldbas 21232 | . . . . . 6 β’ β = (Baseββfld) | |
22 | 20, 21 | mgpbas 20035 | . . . . 5 β’ β = (Baseβ(mulGrpββfld)) |
23 | 1, 22 | ressbas2 17181 | . . . 4 β’ ((0[,]1) β β β (0[,]1) = (Baseβ((mulGrpββfld) βΎs (0[,]1)))) |
24 | 19, 23 | ax-mp 5 | . . 3 β’ (0[,]1) = (Baseβ((mulGrpββfld) βΎs (0[,]1))) |
25 | xrge0base 32651 | . . 3 β’ (0[,]+β) = (Baseβ(β*π βΎs (0[,]+β))) | |
26 | cnfldex 21231 | . . . . 5 β’ βfld β V | |
27 | ovex 7434 | . . . . 5 β’ (0[,]1) β V | |
28 | eqid 2724 | . . . . . 6 β’ (βfld βΎs (0[,]1)) = (βfld βΎs (0[,]1)) | |
29 | 28, 20 | mgpress 20044 | . . . . 5 β’ ((βfld β V β§ (0[,]1) β V) β ((mulGrpββfld) βΎs (0[,]1)) = (mulGrpβ(βfld βΎs (0[,]1)))) |
30 | 26, 27, 29 | mp2an 689 | . . . 4 β’ ((mulGrpββfld) βΎs (0[,]1)) = (mulGrpβ(βfld βΎs (0[,]1))) |
31 | cnfldmul 21234 | . . . . . 6 β’ Β· = (.rββfld) | |
32 | 28, 31 | ressmulr 17251 | . . . . 5 β’ ((0[,]1) β V β Β· = (.rβ(βfld βΎs (0[,]1)))) |
33 | 27, 32 | ax-mp 5 | . . . 4 β’ Β· = (.rβ(βfld βΎs (0[,]1))) |
34 | 30, 33 | mgpplusg 20033 | . . 3 β’ Β· = (+gβ((mulGrpββfld) βΎs (0[,]1))) |
35 | xrge0plusg 32653 | . . 3 β’ +π = (+gβ(β*π βΎs (0[,]+β))) | |
36 | cnring 21251 | . . . 4 β’ βfld β Ring | |
37 | 1elunit 13444 | . . . 4 β’ 1 β (0[,]1) | |
38 | cnfld1 21254 | . . . . 5 β’ 1 = (1rββfld) | |
39 | 1, 21, 38 | ringidss 20166 | . . . 4 β’ ((βfld β Ring β§ (0[,]1) β β β§ 1 β (0[,]1)) β 1 = (0gβ((mulGrpββfld) βΎs (0[,]1)))) |
40 | 36, 19, 37, 39 | mp3an 1457 | . . 3 β’ 1 = (0gβ((mulGrpββfld) βΎs (0[,]1))) |
41 | xrge00 32652 | . . 3 β’ 0 = (0gβ(β*π βΎs (0[,]+β))) | |
42 | 24, 25, 34, 35, 40, 41 | ismhm 18705 | . 2 β’ (πΉ β (((mulGrpββfld) βΎs (0[,]1)) MndHom (β*π βΎs (0[,]+β))) β ((((mulGrpββfld) βΎs (0[,]1)) β Mnd β§ (β*π βΎs (0[,]+β)) β Mnd) β§ (πΉ:(0[,]1)βΆ(0[,]+β) β§ βπ¦ β (0[,]1)βπ§ β (0[,]1)(πΉβ(π¦ Β· π§)) = ((πΉβπ¦) +π (πΉβπ§)) β§ (πΉβ1) = 0))) |
43 | 8, 18, 42 | mpbir2an 708 | 1 β’ πΉ β (((mulGrpββfld) βΎs (0[,]1)) MndHom (β*π βΎs (0[,]+β))) |
Colors of variables: wff setvar class |
Syntax hints: β§ wa 395 β§ w3a 1084 = wceq 1533 β wcel 2098 βwral 3053 Vcvv 3466 β wss 3940 ifcif 4520 β¦ cmpt 5221 β‘ccnv 5665 βΆwf 6529 β1-1-ontoβwf1o 6532 βcfv 6533 (class class class)co 7401 βcc 11104 0cc0 11106 1c1 11107 Β· cmul 11111 +βcpnf 11242 β€ cle 11246 -cneg 11442 +π cxad 13087 [,]cicc 13324 expce 16002 Basecbs 17143 βΎs cress 17172 .rcmulr 17197 βΎt crest 17365 0gc0g 17384 ordTopcordt 17444 β*π cxrs 17445 Mndcmnd 18657 MndHom cmhm 18701 CMndccmn 19690 mulGrpcmgp 20029 Ringcrg 20128 βfldccnfld 21228 TopMndctmd 23896 logclog 26405 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-iin 4990 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-se 5622 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-of 7663 df-om 7849 df-1st 7968 df-2nd 7969 df-supp 8141 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-2o 8462 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-fi 9402 df-sup 9433 df-inf 9434 df-oi 9501 df-card 9930 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-z 12556 df-dec 12675 df-uz 12820 df-q 12930 df-rp 12972 df-xneg 13089 df-xadd 13090 df-xmul 13091 df-ioo 13325 df-ioc 13326 df-ico 13327 df-icc 13328 df-fz 13482 df-fzo 13625 df-fl 13754 df-mod 13832 df-seq 13964 df-exp 14025 df-fac 14231 df-bc 14260 df-hash 14288 df-shft 15011 df-cj 15043 df-re 15044 df-im 15045 df-sqrt 15179 df-abs 15180 df-limsup 15412 df-clim 15429 df-rlim 15430 df-sum 15630 df-ef 16008 df-sin 16010 df-cos 16011 df-pi 16013 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-rest 17367 df-topn 17368 df-0g 17386 df-gsum 17387 df-topgen 17388 df-pt 17389 df-prds 17392 df-xrs 17447 df-qtop 17452 df-imas 17453 df-xps 17455 df-mre 17529 df-mrc 17530 df-acs 17532 df-plusf 18562 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-mhm 18703 df-submnd 18704 df-grp 18856 df-minusg 18857 df-sbg 18858 df-mulg 18986 df-subg 19040 df-cntz 19223 df-cmn 19692 df-abl 19693 df-mgp 20030 df-rng 20048 df-ur 20077 df-ring 20130 df-cring 20131 df-subrng 20436 df-subrg 20461 df-abv 20650 df-lmod 20698 df-scaf 20699 df-sra 21011 df-rgmod 21012 df-psmet 21220 df-xmet 21221 df-met 21222 df-bl 21223 df-mopn 21224 df-fbas 21225 df-fg 21226 df-cnfld 21229 df-top 22718 df-topon 22735 df-topsp 22757 df-bases 22771 df-cld 22845 df-ntr 22846 df-cls 22847 df-nei 22924 df-lp 22962 df-perf 22963 df-cn 23053 df-cnp 23054 df-haus 23141 df-tx 23388 df-hmeo 23581 df-fil 23672 df-fm 23764 df-flim 23765 df-flf 23766 df-tmd 23898 df-tgp 23899 df-trg 23986 df-xms 24148 df-ms 24149 df-tms 24150 df-nm 24413 df-ngp 24414 df-nrg 24416 df-nlm 24417 df-cncf 24720 df-limc 25717 df-dv 25718 df-log 26407 |
This theorem is referenced by: xrge0tmd 33414 |
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