xrge0iifmhm - Mathbox for Thierry Arnoux

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Theorem xrge0iifmhm 33573

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.)

**Hypotheses**
Ref	Expression
xrge0iifhmeo.1	⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥)))
xrge0iifhmeo.k	⊢ 𝐽 = ((ordTop‘ ≤ ) ↾_t (0[,]+∞))

**Assertion**
Ref	Expression
xrge0iifmhm	⊢ 𝐹 ∈ (((mulGrp‘ℂ_fld) ↾_s (0[,]1)) MndHom (ℝ^*_𝑠 ↾_s (0[,]+∞)))

Distinct variable group: 𝑥,𝐹 Allowed substitution hint: 𝐽(𝑥)

Proof of Theorem xrge0iifmhm Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2728	. . . . 5 ⊢ ((mulGrp‘ℂ_fld) ↾_s (0[,]1)) = ((mulGrp‘ℂ_fld) ↾_s (0[,]1))
2	1	iistmd 33536	. . . 4 ⊢ ((mulGrp‘ℂ_fld) ↾_s (0[,]1)) ∈ TopMnd
3		tmdmnd 23999	. . . 4 ⊢ (((mulGrp‘ℂ_fld) ↾_s (0[,]1)) ∈ TopMnd → ((mulGrp‘ℂ_fld) ↾_s (0[,]1)) ∈ Mnd)
4	2, 3	ax-mp 5	. . 3 ⊢ ((mulGrp‘ℂ_fld) ↾_s (0[,]1)) ∈ Mnd
5		xrge0cmn 21348	. . . 4 ⊢ (ℝ^*_𝑠 ↾_s (0[,]+∞)) ∈ CMnd
6		cmnmnd 19759	. . . 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 469	. 2 ⊢ (((mulGrp‘ℂ_fld) ↾_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 33567	. . . . 5 ⊢ (𝐹:(0[,]1)–1-1-onto→(0[,]+∞) ∧ ^◡𝐹 = (𝑦 ∈ (0[,]+∞) ↦ if(𝑦 = +∞, 0, (exp‘-𝑦))))
11	10	simpli 482	. . . 4 ⊢ 𝐹:(0[,]1)–1-1-onto→(0[,]+∞)
12		f1of 6844	. . . 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 33571	. . . 4 ⊢ ((𝑦 ∈ (0[,]1) ∧ 𝑧 ∈ (0[,]1)) → (𝐹‘(𝑦 · 𝑧)) = ((𝐹‘𝑦) +_𝑒 (𝐹‘𝑧)))
16	15	rgen2 3195	. . 3 ⊢ ∀𝑦 ∈ (0[,]1)∀𝑧 ∈ (0[,]1)(𝐹‘(𝑦 · 𝑧)) = ((𝐹‘𝑦) +_𝑒 (𝐹‘𝑧))
17	9, 14	xrge0iif1 33572	. . 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 13517	. . . 4 ⊢ (0[,]1) ⊆ ℂ
20		eqid 2728	. . . . . 6 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
21		cnfldbas 21290	. . . . . 6 ⊢ ℂ = (Base‘ℂ_fld)
22	20, 21	mgpbas 20087	. . . . 5 ⊢ ℂ = (Base‘(mulGrp‘ℂ_fld))
23	1, 22	ressbas2 17225	. . . 4 ⊢ ((0[,]1) ⊆ ℂ → (0[,]1) = (Base‘((mulGrp‘ℂ_fld) ↾_s (0[,]1))))
24	19, 23	ax-mp 5	. . 3 ⊢ (0[,]1) = (Base‘((mulGrp‘ℂ_fld) ↾_s (0[,]1)))
25		xrge0base 32762	. . 3 ⊢ (0[,]+∞) = (Base‘(ℝ^*_𝑠 ↾_s (0[,]+∞)))
26		cnfldex 21289	. . . . 5 ⊢ ℂ_fld ∈ V
27		ovex 7459	. . . . 5 ⊢ (0[,]1) ∈ V
28		eqid 2728	. . . . . 6 ⊢ (ℂ_fld ↾_s (0[,]1)) = (ℂ_fld ↾_s (0[,]1))
29	28, 20	mgpress 20096	. . . . 5 ⊢ ((ℂ_fld ∈ V ∧ (0[,]1) ∈ V) → ((mulGrp‘ℂ_fld) ↾_s (0[,]1)) = (mulGrp‘(ℂ_fld ↾_s (0[,]1))))
30	26, 27, 29	mp2an 690	. . . 4 ⊢ ((mulGrp‘ℂ_fld) ↾_s (0[,]1)) = (mulGrp‘(ℂ_fld ↾_s (0[,]1)))
31		cnfldmul 21294	. . . . . 6 ⊢ · = (._r‘ℂ_fld)
32	28, 31	ressmulr 17295	. . . . 5 ⊢ ((0[,]1) ∈ V → · = (._r‘(ℂ_fld ↾_s (0[,]1))))
33	27, 32	ax-mp 5	. . . 4 ⊢ · = (._r‘(ℂ_fld ↾_s (0[,]1)))
34	30, 33	mgpplusg 20085	. . 3 ⊢ · = (+_g‘((mulGrp‘ℂ_fld) ↾_s (0[,]1)))
35		xrge0plusg 32764	. . 3 ⊢ +_𝑒 = (+_g‘(ℝ^*_𝑠 ↾_s (0[,]+∞)))
36		cnring 21325	. . . 4 ⊢ ℂ_fld ∈ Ring
37		1elunit 13487	. . . 4 ⊢ 1 ∈ (0[,]1)
38		cnfld1 21328	. . . . 5 ⊢ 1 = (1_r‘ℂ_fld)
39	1, 21, 38	ringidss 20220	. . . 4 ⊢ ((ℂ_fld ∈ Ring ∧ (0[,]1) ⊆ ℂ ∧ 1 ∈ (0[,]1)) → 1 = (0_g‘((mulGrp‘ℂ_fld) ↾_s (0[,]1))))
40	36, 19, 37, 39	mp3an 1457	. . 3 ⊢ 1 = (0_g‘((mulGrp‘ℂ_fld) ↾_s (0[,]1)))
41		xrge00 32763	. . 3 ⊢ 0 = (0_g‘(ℝ^*_𝑠 ↾_s (0[,]+∞)))
42	24, 25, 34, 35, 40, 41	ismhm 18749	. 2 ⊢ (𝐹 ∈ (((mulGrp‘ℂ_fld) ↾_s (0[,]1)) MndHom (ℝ^_𝑠 ↾_s (0[,]+∞))) ↔ ((((mulGrp‘ℂ_fld) ↾_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‘ℂ_fld) ↾_s (0[,]1)) MndHom (ℝ^*_𝑠 ↾_s (0[,]+∞)))

Colors of variables: wff setvar class

Syntax hints: ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3058 Vcvv 3473 ⊆ wss 3949 ifcif 4532 ↦ cmpt 5235 ^◡ccnv 5681 ⟶wf 6549 –1-1-onto→wf1o 6552 ‘cfv 6553 (class class class)co 7426 ℂcc 11144 0cc0 11146 1c1 11147 · cmul 11151 +∞cpnf 11283 ≤ cle 11287 -cneg 11483 +_𝑒 cxad 13130 [,]cicc 13367 expce 16045 Basecbs 17187 ↾_s cress 17216 ._rcmulr 17241 ↾_t crest 17409 0_gc0g 17428 ordTopcordt 17488 ℝ^*_𝑠cxrs 17489 Mndcmnd 18701 MndHom cmhm 18745 CMndccmn 19742 mulGrpcmgp 20081 Ringcrg 20180 ℂ_fldccnfld 21286 TopMndctmd 23994 logclog 26508

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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9672 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 ax-addf 11225 ax-mulf 11226

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7691 df-om 7877 df-1st 7999 df-2nd 8000 df-supp 8172 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-2o 8494 df-er 8731 df-map 8853 df-pm 8854 df-ixp 8923 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-fsupp 9394 df-fi 9442 df-sup 9473 df-inf 9474 df-oi 9541 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-q 12971 df-rp 13015 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-ioo 13368 df-ioc 13369 df-ico 13370 df-icc 13371 df-fz 13525 df-fzo 13668 df-fl 13797 df-mod 13875 df-seq 14007 df-exp 14067 df-fac 14273 df-bc 14302 df-hash 14330 df-shft 15054 df-cj 15086 df-re 15087 df-im 15088 df-sqrt 15222 df-abs 15223 df-limsup 15455 df-clim 15472 df-rlim 15473 df-sum 15673 df-ef 16051 df-sin 16053 df-cos 16054 df-pi 16056 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-starv 17255 df-sca 17256 df-vsca 17257 df-ip 17258 df-tset 17259 df-ple 17260 df-ds 17262 df-unif 17263 df-hom 17264 df-cco 17265 df-rest 17411 df-topn 17412 df-0g 17430 df-gsum 17431 df-topgen 17432 df-pt 17433 df-prds 17436 df-xrs 17491 df-qtop 17496 df-imas 17497 df-xps 17499 df-mre 17573 df-mrc 17574 df-acs 17576 df-plusf 18606 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-mhm 18747 df-submnd 18748 df-grp 18900 df-minusg 18901 df-sbg 18902 df-mulg 19031 df-subg 19085 df-cntz 19275 df-cmn 19744 df-abl 19745 df-mgp 20082 df-rng 20100 df-ur 20129 df-ring 20182 df-cring 20183 df-subrng 20490 df-subrg 20515 df-abv 20704 df-lmod 20752 df-scaf 20753 df-sra 21065 df-rgmod 21066 df-psmet 21278 df-xmet 21279 df-met 21280 df-bl 21281 df-mopn 21282 df-fbas 21283 df-fg 21284 df-cnfld 21287 df-top 22816 df-topon 22833 df-topsp 22855 df-bases 22869 df-cld 22943 df-ntr 22944 df-cls 22945 df-nei 23022 df-lp 23060 df-perf 23061 df-cn 23151 df-cnp 23152 df-haus 23239 df-tx 23486 df-hmeo 23679 df-fil 23770 df-fm 23862 df-flim 23863 df-flf 23864 df-tmd 23996 df-tgp 23997 df-trg 24084 df-xms 24246 df-ms 24247 df-tms 24248 df-nm 24511 df-ngp 24512 df-nrg 24514 df-nlm 24515 df-cncf 24818 df-limc 25815 df-dv 25816 df-log 26510

This theorem is referenced by: xrge0tmd 33579

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