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 2821 | . . . . 5 ⊢ ((mulGrp‘ℂfld) ↾s (0[,]1)) = ((mulGrp‘ℂfld) ↾s (0[,]1)) | |
2 | 1 | iistmd 31145 | . . . 4 ⊢ ((mulGrp‘ℂfld) ↾s (0[,]1)) ∈ TopMnd |
3 | tmdmnd 22683 | . . . 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 20587 | . . . 4 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd | |
6 | cmnmnd 18922 | . . . 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 473 | . 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 31176 | . . . . 5 ⊢ (𝐹:(0[,]1)–1-1-onto→(0[,]+∞) ∧ ◡𝐹 = (𝑦 ∈ (0[,]+∞) ↦ if(𝑦 = +∞, 0, (exp‘-𝑦)))) |
11 | 10 | simpli 486 | . . . 4 ⊢ 𝐹:(0[,]1)–1-1-onto→(0[,]+∞) |
12 | f1of 6615 | . . . 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 31180 | . . . 4 ⊢ ((𝑦 ∈ (0[,]1) ∧ 𝑧 ∈ (0[,]1)) → (𝐹‘(𝑦 · 𝑧)) = ((𝐹‘𝑦) +𝑒 (𝐹‘𝑧))) |
16 | 15 | rgen2 3203 | . . 3 ⊢ ∀𝑦 ∈ (0[,]1)∀𝑧 ∈ (0[,]1)(𝐹‘(𝑦 · 𝑧)) = ((𝐹‘𝑦) +𝑒 (𝐹‘𝑧)) |
17 | 9, 14 | xrge0iif1 31181 | . . 3 ⊢ (𝐹‘1) = 0 |
18 | 13, 16, 17 | 3pm3.2i 1335 | . 2 ⊢ (𝐹:(0[,]1)⟶(0[,]+∞) ∧ ∀𝑦 ∈ (0[,]1)∀𝑧 ∈ (0[,]1)(𝐹‘(𝑦 · 𝑧)) = ((𝐹‘𝑦) +𝑒 (𝐹‘𝑧)) ∧ (𝐹‘1) = 0) |
19 | unitsscn 31139 | . . . 4 ⊢ (0[,]1) ⊆ ℂ | |
20 | eqid 2821 | . . . . . 6 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld) | |
21 | cnfldbas 20549 | . . . . . 6 ⊢ ℂ = (Base‘ℂfld) | |
22 | 20, 21 | mgpbas 19245 | . . . . 5 ⊢ ℂ = (Base‘(mulGrp‘ℂfld)) |
23 | 1, 22 | ressbas2 16555 | . . . 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 30672 | . . 3 ⊢ (0[,]+∞) = (Base‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
26 | cnfldex 20548 | . . . . 5 ⊢ ℂfld ∈ V | |
27 | ovex 7189 | . . . . 5 ⊢ (0[,]1) ∈ V | |
28 | eqid 2821 | . . . . . 6 ⊢ (ℂfld ↾s (0[,]1)) = (ℂfld ↾s (0[,]1)) | |
29 | 28, 20 | mgpress 19250 | . . . . 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 20551 | . . . . . 6 ⊢ · = (.r‘ℂfld) | |
32 | 28, 31 | ressmulr 16625 | . . . . 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 19243 | . . 3 ⊢ · = (+g‘((mulGrp‘ℂfld) ↾s (0[,]1))) |
35 | xrge0plusg 30674 | . . 3 ⊢ +𝑒 = (+g‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
36 | cnring 20567 | . . . 4 ⊢ ℂfld ∈ Ring | |
37 | 1elunit 12857 | . . . 4 ⊢ 1 ∈ (0[,]1) | |
38 | cnfld1 20570 | . . . . 5 ⊢ 1 = (1r‘ℂfld) | |
39 | 1, 21, 38 | ringidss 19327 | . . . 4 ⊢ ((ℂfld ∈ Ring ∧ (0[,]1) ⊆ ℂ ∧ 1 ∈ (0[,]1)) → 1 = (0g‘((mulGrp‘ℂfld) ↾s (0[,]1)))) |
40 | 36, 19, 37, 39 | mp3an 1457 | . . 3 ⊢ 1 = (0g‘((mulGrp‘ℂfld) ↾s (0[,]1))) |
41 | xrge00 30673 | . . 3 ⊢ 0 = (0g‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
42 | 24, 25, 34, 35, 40, 41 | ismhm 17958 | . 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 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∀wral 3138 Vcvv 3494 ⊆ wss 3936 ifcif 4467 ↦ cmpt 5146 ◡ccnv 5554 ⟶wf 6351 –1-1-onto→wf1o 6354 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 0cc0 10537 1c1 10538 · cmul 10542 +∞cpnf 10672 ≤ cle 10676 -cneg 10871 +𝑒 cxad 12506 [,]cicc 12742 expce 15415 Basecbs 16483 ↾s cress 16484 .rcmulr 16566 ↾t crest 16694 0gc0g 16713 ordTopcordt 16772 ℝ*𝑠cxrs 16773 Mndcmnd 17911 MndHom cmhm 17954 CMndccmn 18906 mulGrpcmgp 19239 Ringcrg 19297 ℂfldccnfld 20545 TopMndctmd 22678 logclog 25138 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-addf 10616 ax-mulf 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-fi 8875 df-sup 8906 df-inf 8907 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-q 12350 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-ioo 12743 df-ioc 12744 df-ico 12745 df-icc 12746 df-fz 12894 df-fzo 13035 df-fl 13163 df-mod 13239 df-seq 13371 df-exp 13431 df-fac 13635 df-bc 13664 df-hash 13692 df-shft 14426 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-limsup 14828 df-clim 14845 df-rlim 14846 df-sum 15043 df-ef 15421 df-sin 15423 df-cos 15424 df-pi 15426 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-hom 16589 df-cco 16590 df-rest 16696 df-topn 16697 df-0g 16715 df-gsum 16716 df-topgen 16717 df-pt 16718 df-prds 16721 df-xrs 16775 df-qtop 16780 df-imas 16781 df-xps 16783 df-mre 16857 df-mrc 16858 df-acs 16860 df-plusf 17851 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-mhm 17956 df-submnd 17957 df-grp 18106 df-minusg 18107 df-sbg 18108 df-mulg 18225 df-subg 18276 df-cntz 18447 df-cmn 18908 df-abl 18909 df-mgp 19240 df-ur 19252 df-ring 19299 df-cring 19300 df-subrg 19533 df-abv 19588 df-lmod 19636 df-scaf 19637 df-sra 19944 df-rgmod 19945 df-psmet 20537 df-xmet 20538 df-met 20539 df-bl 20540 df-mopn 20541 df-fbas 20542 df-fg 20543 df-cnfld 20546 df-top 21502 df-topon 21519 df-topsp 21541 df-bases 21554 df-cld 21627 df-ntr 21628 df-cls 21629 df-nei 21706 df-lp 21744 df-perf 21745 df-cn 21835 df-cnp 21836 df-haus 21923 df-tx 22170 df-hmeo 22363 df-fil 22454 df-fm 22546 df-flim 22547 df-flf 22548 df-tmd 22680 df-tgp 22681 df-trg 22768 df-xms 22930 df-ms 22931 df-tms 22932 df-nm 23192 df-ngp 23193 df-nrg 23195 df-nlm 23196 df-cncf 23486 df-limc 24464 df-dv 24465 df-log 25140 |
This theorem is referenced by: xrge0tmd 31188 |
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