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Theorem xrge0iifmhm 26305
Description: The defined function from the closed unit interval and the extended nonnegative reals is a monoid homomorphism. (Contributed by Thierry Arnoux, 21-Jun-2017.)
Hypotheses
Ref Expression
xrge0iifhmeo.1  |-  F  =  ( x  e.  ( 0 [,] 1 ) 
|->  if ( x  =  0 , +oo ,  -u ( log `  x
) ) )
xrge0iifhmeo.k  |-  J  =  ( (ordTop `  <_  )t  ( 0 [,] +oo )
)
Assertion
Ref Expression
xrge0iifmhm  |-  F  e.  ( ( (mulGrp ` fld )s  (
0 [,] 1 ) ) MndHom  ( RR*ss  (
0 [,] +oo )
) )
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.
StepHypRef Expression
1 eqid 2441 . . . . 5  |-  ( (mulGrp ` fld )s  ( 0 [,] 1
) )  =  ( (mulGrp ` fld )s  ( 0 [,] 1 ) )
21iistmd 26268 . . . 4  |-  ( (mulGrp ` fld )s  ( 0 [,] 1
) )  e. TopMnd
3 tmdmnd 19605 . . . 4  |-  ( ( (mulGrp ` fld )s  ( 0 [,] 1 ) )  e. TopMnd  ->  ( (mulGrp ` fld )s  ( 0 [,] 1 ) )  e. 
Mnd )
42, 3ax-mp 5 . . 3  |-  ( (mulGrp ` fld )s  ( 0 [,] 1
) )  e.  Mnd
5 xrge0cmn 17814 . . . 4  |-  ( RR*ss  ( 0 [,] +oo ) )  e. CMnd
6 cmnmnd 16285 . . . 4  |-  ( (
RR*ss  ( 0 [,] +oo ) )  e. CMnd  ->  (
RR*ss  ( 0 [,] +oo ) )  e.  Mnd )
75, 6ax-mp 5 . . 3  |-  ( RR*ss  ( 0 [,] +oo ) )  e.  Mnd
84, 7pm3.2i 452 . 2  |-  ( ( (mulGrp ` fld )s  ( 0 [,] 1 ) )  e. 
Mnd  /\  ( RR*ss  ( 0 [,] +oo ) )  e.  Mnd )
9 xrge0iifhmeo.1 . . . . . 6  |-  F  =  ( x  e.  ( 0 [,] 1 ) 
|->  if ( x  =  0 , +oo ,  -u ( log `  x
) ) )
109xrge0iifcnv 26299 . . . . 5  |-  ( F : ( 0 [,] 1 ) -1-1-onto-> ( 0 [,] +oo )  /\  `' F  =  ( y  e.  ( 0 [,] +oo )  |->  if ( y  = +oo ,  0 ,  ( exp `  -u y
) ) ) )
1110simpli 455 . . . 4  |-  F :
( 0 [,] 1
)
-1-1-onto-> ( 0 [,] +oo )
12 f1of 5638 . . . 4  |-  ( F : ( 0 [,] 1 ) -1-1-onto-> ( 0 [,] +oo )  ->  F : ( 0 [,] 1 ) --> ( 0 [,] +oo ) )
1311, 12ax-mp 5 . . 3  |-  F :
( 0 [,] 1
) --> ( 0 [,] +oo )
14 xrge0iifhmeo.k . . . . 5  |-  J  =  ( (ordTop `  <_  )t  ( 0 [,] +oo )
)
159, 14xrge0iifhom 26303 . . . 4  |-  ( ( y  e.  ( 0 [,] 1 )  /\  z  e.  ( 0 [,] 1 ) )  ->  ( F `  ( y  x.  z
) )  =  ( ( F `  y
) +e ( F `  z ) ) )
1615rgen2a 2780 . . 3  |-  A. y  e.  ( 0 [,] 1
) A. z  e.  ( 0 [,] 1
) ( F `  ( y  x.  z
) )  =  ( ( F `  y
) +e ( F `  z ) )
179, 14xrge0iif1 26304 . . 3  |-  ( F `
 1 )  =  0
1813, 16, 173pm3.2i 1161 . 2  |-  ( F : ( 0 [,] 1 ) --> ( 0 [,] +oo )  /\  A. y  e.  ( 0 [,] 1 ) A. z  e.  ( 0 [,] 1 ) ( F `  ( y  x.  z ) )  =  ( ( F `
 y ) +e ( F `  z ) )  /\  ( F `  1 )  =  0 )
19 unitsscn 26262 . . . 4  |-  ( 0 [,] 1 )  C_  CC
20 eqid 2441 . . . . . 6  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
21 cnfldbas 17781 . . . . . 6  |-  CC  =  ( Base ` fld )
2220, 21mgpbas 16587 . . . . 5  |-  CC  =  ( Base `  (mulGrp ` fld ) )
231, 22ressbas2 14225 . . . 4  |-  ( ( 0 [,] 1 ) 
C_  CC  ->  ( 0 [,] 1 )  =  ( Base `  (
(mulGrp ` fld )s  ( 0 [,] 1 ) ) ) )
2419, 23ax-mp 5 . . 3  |-  ( 0 [,] 1 )  =  ( Base `  (
(mulGrp ` fld )s  ( 0 [,] 1 ) ) )
25 xrge0base 26079 . . 3  |-  ( 0 [,] +oo )  =  ( Base `  ( RR*ss  ( 0 [,] +oo ) ) )
26 cnfldex 17780 . . . . 5  |-fld  e.  _V
27 ovex 6115 . . . . 5  |-  ( 0 [,] 1 )  e. 
_V
28 eqid 2441 . . . . . 6  |-  (flds  ( 0 [,] 1 ) )  =  (flds  ( 0 [,] 1 ) )
2928, 20mgpress 16592 . . . . 5  |-  ( (fld  e. 
_V  /\  ( 0 [,] 1 )  e. 
_V )  ->  (
(mulGrp ` fld )s  ( 0 [,] 1 ) )  =  (mulGrp `  (flds  ( 0 [,] 1 ) ) ) )
3026, 27, 29mp2an 667 . . . 4  |-  ( (mulGrp ` fld )s  ( 0 [,] 1
) )  =  (mulGrp `  (flds  ( 0 [,] 1 ) ) )
31 cnfldmul 17783 . . . . . 6  |-  x.  =  ( .r ` fld )
3228, 31ressmulr 14287 . . . . 5  |-  ( ( 0 [,] 1 )  e.  _V  ->  x.  =  ( .r `  (flds  (
0 [,] 1 ) ) ) )
3327, 32ax-mp 5 . . . 4  |-  x.  =  ( .r `  (flds  ( 0 [,] 1 ) ) )
3430, 33mgpplusg 16585 . . 3  |-  x.  =  ( +g  `  ( (mulGrp ` fld )s  ( 0 [,] 1
) ) )
35 xrge0plusg 26081 . . 3  |-  +e 
=  ( +g  `  ( RR*ss  ( 0 [,] +oo ) ) )
36 cnrng 17797 . . . 4  |-fld  e.  Ring
37 1elunit 11400 . . . 4  |-  1  e.  ( 0 [,] 1
)
38 cnfld1 17800 . . . . 5  |-  1  =  ( 1r ` fld )
391, 21, 38rngidss 16661 . . . 4  |-  ( (fld  e. 
Ring  /\  ( 0 [,] 1 )  C_  CC  /\  1  e.  ( 0 [,] 1 ) )  ->  1  =  ( 0g `  ( (mulGrp ` fld )s  ( 0 [,] 1
) ) ) )
4036, 19, 37, 39mp3an 1309 . . 3  |-  1  =  ( 0g `  ( (mulGrp ` fld )s  ( 0 [,] 1 ) ) )
41 xrge00 26080 . . 3  |-  0  =  ( 0g `  ( RR*ss  ( 0 [,] +oo ) ) )
4224, 25, 34, 35, 40, 41ismhm 15462 . 2  |-  ( F  e.  ( ( (mulGrp ` fld )s  ( 0 [,] 1
) ) MndHom  ( RR*ss  ( 0 [,] +oo ) ) )  <->  ( (
( (mulGrp ` fld )s  ( 0 [,] 1 ) )  e. 
Mnd  /\  ( RR*ss  ( 0 [,] +oo ) )  e.  Mnd )  /\  ( F :
( 0 [,] 1
) --> ( 0 [,] +oo )  /\  A. y  e.  ( 0 [,] 1
) A. z  e.  ( 0 [,] 1
) ( F `  ( y  x.  z
) )  =  ( ( F `  y
) +e ( F `  z ) )  /\  ( F `
 1 )  =  0 ) ) )
438, 18, 42mpbir2an 906 1  |-  F  e.  ( ( (mulGrp ` fld )s  (
0 [,] 1 ) ) MndHom  ( RR*ss  (
0 [,] +oo )
) )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761   A.wral 2713   _Vcvv 2970    C_ wss 3325   ifcif 3788    e. cmpt 4347   `'ccnv 4835   -->wf 5411   -1-1-onto->wf1o 5414   ` cfv 5415  (class class class)co 6090   CCcc 9276   0cc0 9278   1c1 9279    x. cmul 9283   +oocpnf 9411    <_ cle 9415   -ucneg 9592   +ecxad 11083   [,]cicc 11299   expce 13343   Basecbs 14170   ↾s cress 14171   .rcmulr 14235   ↾t crest 14355   0gc0g 14374  ordTopcordt 14433   RR*scxrs 14434   Mndcmnd 15405   MndHom cmhm 15458  CMndccmn 16270  mulGrpcmgp 16581   Ringcrg 16635  ℂfldccnfld 17777  TopMndctmd 19600   logclog 21965
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356  ax-addf 9357  ax-mulf 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-map 7212  df-pm 7213  df-ixp 7260  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fsupp 7617  df-fi 7657  df-sup 7687  df-oi 7720  df-card 8105  df-cda 8333  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-q 10950  df-rp 10988  df-xneg 11085  df-xadd 11086  df-xmul 11087  df-ioo 11300  df-ioc 11301  df-ico 11302  df-icc 11303  df-fz 11434  df-fzo 11545  df-fl 11638  df-mod 11705  df-seq 11803  df-exp 11862  df-fac 12048  df-bc 12075  df-hash 12100  df-shft 12552  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-limsup 12945  df-clim 12962  df-rlim 12963  df-sum 13160  df-ef 13349  df-sin 13351  df-cos 13352  df-pi 13354  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-starv 14249  df-sca 14250  df-vsca 14251  df-ip 14252  df-tset 14253  df-ple 14254  df-ds 14256  df-unif 14257  df-hom 14258  df-cco 14259  df-rest 14357  df-topn 14358  df-0g 14376  df-gsum 14377  df-topgen 14378  df-pt 14379  df-prds 14382  df-xrs 14436  df-qtop 14441  df-imas 14442  df-xps 14444  df-mre 14520  df-mrc 14521  df-acs 14523  df-mnd 15411  df-plusf 15412  df-mhm 15460  df-submnd 15461  df-grp 15538  df-minusg 15539  df-sbg 15540  df-mulg 15541  df-subg 15671  df-cntz 15828  df-cmn 16272  df-abl 16273  df-mgp 16582  df-ur 16594  df-rng 16637  df-cring 16638  df-subrg 16843  df-abv 16882  df-lmod 16930  df-scaf 16931  df-sra 17231  df-rgmod 17232  df-psmet 17768  df-xmet 17769  df-met 17770  df-bl 17771  df-mopn 17772  df-fbas 17773  df-fg 17774  df-cnfld 17778  df-top 18462  df-bases 18464  df-topon 18465  df-topsp 18466  df-cld 18582  df-ntr 18583  df-cls 18584  df-nei 18661  df-lp 18699  df-perf 18700  df-cn 18790  df-cnp 18791  df-haus 18878  df-tx 19094  df-hmeo 19287  df-fil 19378  df-fm 19470  df-flim 19471  df-flf 19472  df-tmd 19602  df-tgp 19603  df-trg 19693  df-xms 19854  df-ms 19855  df-tms 19856  df-nm 20134  df-ngp 20135  df-nrg 20137  df-nlm 20138  df-cncf 20413  df-limc 21300  df-dv 21301  df-log 21967
This theorem is referenced by:  xrge0tmd  26312
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