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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 2732 | . . . . 5 ⊢ ((mulGrp‘ℂfld) ↾s (0[,]1)) = ((mulGrp‘ℂfld) ↾s (0[,]1)) | |
2 | 1 | iistmd 32870 | . . . 4 ⊢ ((mulGrp‘ℂfld) ↾s (0[,]1)) ∈ TopMnd |
3 | tmdmnd 23570 | . . . 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 20979 | . . . 4 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd | |
6 | cmnmnd 19659 | . . . 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 32901 | . . . . 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 6830 | . . . 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 32905 | . . . 4 ⊢ ((𝑦 ∈ (0[,]1) ∧ 𝑧 ∈ (0[,]1)) → (𝐹‘(𝑦 · 𝑧)) = ((𝐹‘𝑦) +𝑒 (𝐹‘𝑧))) |
16 | 15 | rgen2 3197 | . . 3 ⊢ ∀𝑦 ∈ (0[,]1)∀𝑧 ∈ (0[,]1)(𝐹‘(𝑦 · 𝑧)) = ((𝐹‘𝑦) +𝑒 (𝐹‘𝑧)) |
17 | 9, 14 | xrge0iif1 32906 | . . 3 ⊢ (𝐹‘1) = 0 |
18 | 13, 16, 17 | 3pm3.2i 1339 | . 2 ⊢ (𝐹:(0[,]1)⟶(0[,]+∞) ∧ ∀𝑦 ∈ (0[,]1)∀𝑧 ∈ (0[,]1)(𝐹‘(𝑦 · 𝑧)) = ((𝐹‘𝑦) +𝑒 (𝐹‘𝑧)) ∧ (𝐹‘1) = 0) |
19 | unitsscn 13473 | . . . 4 ⊢ (0[,]1) ⊆ ℂ | |
20 | eqid 2732 | . . . . . 6 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld) | |
21 | cnfldbas 20940 | . . . . . 6 ⊢ ℂ = (Base‘ℂfld) | |
22 | 20, 21 | mgpbas 19987 | . . . . 5 ⊢ ℂ = (Base‘(mulGrp‘ℂfld)) |
23 | 1, 22 | ressbas2 17178 | . . . 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 32173 | . . 3 ⊢ (0[,]+∞) = (Base‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
26 | cnfldex 20939 | . . . . 5 ⊢ ℂfld ∈ V | |
27 | ovex 7438 | . . . . 5 ⊢ (0[,]1) ∈ V | |
28 | eqid 2732 | . . . . . 6 ⊢ (ℂfld ↾s (0[,]1)) = (ℂfld ↾s (0[,]1)) | |
29 | 28, 20 | mgpress 19996 | . . . . 5 ⊢ ((ℂfld ∈ V ∧ (0[,]1) ∈ V) → ((mulGrp‘ℂfld) ↾s (0[,]1)) = (mulGrp‘(ℂfld ↾s (0[,]1)))) |
30 | 26, 27, 29 | mp2an 690 | . . . 4 ⊢ ((mulGrp‘ℂfld) ↾s (0[,]1)) = (mulGrp‘(ℂfld ↾s (0[,]1))) |
31 | cnfldmul 20942 | . . . . . 6 ⊢ · = (.r‘ℂfld) | |
32 | 28, 31 | ressmulr 17248 | . . . . 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 19985 | . . 3 ⊢ · = (+g‘((mulGrp‘ℂfld) ↾s (0[,]1))) |
35 | xrge0plusg 32175 | . . 3 ⊢ +𝑒 = (+g‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
36 | cnring 20959 | . . . 4 ⊢ ℂfld ∈ Ring | |
37 | 1elunit 13443 | . . . 4 ⊢ 1 ∈ (0[,]1) | |
38 | cnfld1 20962 | . . . . 5 ⊢ 1 = (1r‘ℂfld) | |
39 | 1, 21, 38 | ringidss 20087 | . . . 4 ⊢ ((ℂfld ∈ Ring ∧ (0[,]1) ⊆ ℂ ∧ 1 ∈ (0[,]1)) → 1 = (0g‘((mulGrp‘ℂfld) ↾s (0[,]1)))) |
40 | 36, 19, 37, 39 | mp3an 1461 | . . 3 ⊢ 1 = (0g‘((mulGrp‘ℂfld) ↾s (0[,]1))) |
41 | xrge00 32174 | . . 3 ⊢ 0 = (0g‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
42 | 24, 25, 34, 35, 40, 41 | ismhm 18669 | . 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 709 | 1 ⊢ 𝐹 ∈ (((mulGrp‘ℂfld) ↾s (0[,]1)) MndHom (ℝ*𝑠 ↾s (0[,]+∞))) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3061 Vcvv 3474 ⊆ wss 3947 ifcif 4527 ↦ cmpt 5230 ◡ccnv 5674 ⟶wf 6536 –1-1-onto→wf1o 6539 ‘cfv 6540 (class class class)co 7405 ℂcc 11104 0cc0 11106 1c1 11107 · cmul 11111 +∞cpnf 11241 ≤ cle 11245 -cneg 11441 +𝑒 cxad 13086 [,]cicc 13323 expce 16001 Basecbs 17140 ↾s cress 17169 .rcmulr 17194 ↾t crest 17362 0gc0g 17381 ordTopcordt 17441 ℝ*𝑠cxrs 17442 Mndcmnd 18621 MndHom cmhm 18665 CMndccmn 19642 mulGrpcmgp 19981 Ringcrg 20049 ℂfldccnfld 20936 TopMndctmd 23565 logclog 26054 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 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 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 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 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ioo 13324 df-ioc 13325 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-fl 13753 df-mod 13831 df-seq 13963 df-exp 14024 df-fac 14230 df-bc 14259 df-hash 14287 df-shft 15010 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-limsup 15411 df-clim 15428 df-rlim 15429 df-sum 15629 df-ef 16007 df-sin 16009 df-cos 16010 df-pi 16012 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-hom 17217 df-cco 17218 df-rest 17364 df-topn 17365 df-0g 17383 df-gsum 17384 df-topgen 17385 df-pt 17386 df-prds 17389 df-xrs 17444 df-qtop 17449 df-imas 17450 df-xps 17452 df-mre 17526 df-mrc 17527 df-acs 17529 df-plusf 18556 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-mhm 18667 df-submnd 18668 df-grp 18818 df-minusg 18819 df-sbg 18820 df-mulg 18945 df-subg 18997 df-cntz 19175 df-cmn 19644 df-abl 19645 df-mgp 19982 df-ur 19999 df-ring 20051 df-cring 20052 df-subrg 20353 df-abv 20417 df-lmod 20465 df-scaf 20466 df-sra 20777 df-rgmod 20778 df-psmet 20928 df-xmet 20929 df-met 20930 df-bl 20931 df-mopn 20932 df-fbas 20933 df-fg 20934 df-cnfld 20937 df-top 22387 df-topon 22404 df-topsp 22426 df-bases 22440 df-cld 22514 df-ntr 22515 df-cls 22516 df-nei 22593 df-lp 22631 df-perf 22632 df-cn 22722 df-cnp 22723 df-haus 22810 df-tx 23057 df-hmeo 23250 df-fil 23341 df-fm 23433 df-flim 23434 df-flf 23435 df-tmd 23567 df-tgp 23568 df-trg 23655 df-xms 23817 df-ms 23818 df-tms 23819 df-nm 24082 df-ngp 24083 df-nrg 24085 df-nlm 24086 df-cncf 24385 df-limc 25374 df-dv 25375 df-log 26056 |
This theorem is referenced by: xrge0tmd 32913 |
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