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Theorem xrge0iifmhm 26509
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 2452 . . . . 5  |-  ( (mulGrp ` fld )s  ( 0 [,] 1
) )  =  ( (mulGrp ` fld )s  ( 0 [,] 1 ) )
21iistmd 26472 . . . 4  |-  ( (mulGrp ` fld )s  ( 0 [,] 1
) )  e. TopMnd
3 tmdmnd 19773 . . . 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 17975 . . . 4  |-  ( RR*ss  ( 0 [,] +oo ) )  e. CMnd
6 cmnmnd 16408 . . . 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 455 . 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 26503 . . . . 5  |-  ( F : ( 0 [,] 1 ) -1-1-onto-> ( 0 [,] +oo )  /\  `' F  =  ( y  e.  ( 0 [,] +oo )  |->  if ( y  = +oo ,  0 ,  ( exp `  -u y
) ) ) )
1110simpli 458 . . . 4  |-  F :
( 0 [,] 1
)
-1-1-onto-> ( 0 [,] +oo )
12 f1of 5744 . . . 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 26507 . . . 4  |-  ( ( y  e.  ( 0 [,] 1 )  /\  z  e.  ( 0 [,] 1 ) )  ->  ( F `  ( y  x.  z
) )  =  ( ( F `  y
) +e ( F `  z ) ) )
1615rgen2a 2894 . . 3  |-  A. y  e.  ( 0 [,] 1
) A. z  e.  ( 0 [,] 1
) ( F `  ( y  x.  z
) )  =  ( ( F `  y
) +e ( F `  z ) )
179, 14xrge0iif1 26508 . . 3  |-  ( F `
 1 )  =  0
1813, 16, 173pm3.2i 1166 . 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 26466 . . . 4  |-  ( 0 [,] 1 )  C_  CC
20 eqid 2452 . . . . . 6  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
21 cnfldbas 17942 . . . . . 6  |-  CC  =  ( Base ` fld )
2220, 21mgpbas 16714 . . . . 5  |-  CC  =  ( Base `  (mulGrp ` fld ) )
231, 22ressbas2 14343 . . . 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 26286 . . 3  |-  ( 0 [,] +oo )  =  ( Base `  ( RR*ss  ( 0 [,] +oo ) ) )
26 cnfldex 17941 . . . . 5  |-fld  e.  _V
27 ovex 6220 . . . . 5  |-  ( 0 [,] 1 )  e. 
_V
28 eqid 2452 . . . . . 6  |-  (flds  ( 0 [,] 1 ) )  =  (flds  ( 0 [,] 1 ) )
2928, 20mgpress 16719 . . . . 5  |-  ( (fld  e. 
_V  /\  ( 0 [,] 1 )  e. 
_V )  ->  (
(mulGrp ` fld )s  ( 0 [,] 1 ) )  =  (mulGrp `  (flds  ( 0 [,] 1 ) ) ) )
3026, 27, 29mp2an 672 . . . 4  |-  ( (mulGrp ` fld )s  ( 0 [,] 1
) )  =  (mulGrp `  (flds  ( 0 [,] 1 ) ) )
31 cnfldmul 17944 . . . . . 6  |-  x.  =  ( .r ` fld )
3228, 31ressmulr 14405 . . . . 5  |-  ( ( 0 [,] 1 )  e.  _V  ->  x.  =  ( .r `  (flds  (
0 [,] 1 ) ) ) )
3327, 32ax-mp 5 . . . 4  |-  x.  =  ( .r `  (flds  ( 0 [,] 1 ) ) )
3430, 33mgpplusg 16712 . . 3  |-  x.  =  ( +g  `  ( (mulGrp ` fld )s  ( 0 [,] 1
) ) )
35 xrge0plusg 26288 . . 3  |-  +e 
=  ( +g  `  ( RR*ss  ( 0 [,] +oo ) ) )
36 cnrng 17958 . . . 4  |-fld  e.  Ring
37 1elunit 11516 . . . 4  |-  1  e.  ( 0 [,] 1
)
38 cnfld1 17961 . . . . 5  |-  1  =  ( 1r ` fld )
391, 21, 38rngidss 16789 . . . 4  |-  ( (fld  e. 
Ring  /\  ( 0 [,] 1 )  C_  CC  /\  1  e.  ( 0 [,] 1 ) )  ->  1  =  ( 0g `  ( (mulGrp ` fld )s  ( 0 [,] 1
) ) ) )
4036, 19, 37, 39mp3an 1315 . . 3  |-  1  =  ( 0g `  ( (mulGrp ` fld )s  ( 0 [,] 1 ) ) )
41 xrge00 26287 . . 3  |-  0  =  ( 0g `  ( RR*ss  ( 0 [,] +oo ) ) )
4224, 25, 34, 35, 40, 41ismhm 15580 . 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 911 1  |-  F  e.  ( ( (mulGrp ` fld )s  (
0 [,] 1 ) ) MndHom  ( RR*ss  (
0 [,] +oo )
) )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2796   _Vcvv 3072    C_ wss 3431   ifcif 3894    |-> cmpt 4453   `'ccnv 4942   -->wf 5517   -1-1-onto->wf1o 5520   ` cfv 5521  (class class class)co 6195   CCcc 9386   0cc0 9388   1c1 9389    x. cmul 9393   +oocpnf 9521    <_ cle 9525   -ucneg 9702   +ecxad 11193   [,]cicc 11409   expce 13460   Basecbs 14287   ↾s cress 14288   .rcmulr 14353   ↾t crest 14473   0gc0g 14492  ordTopcordt 14551   RR*scxrs 14552   Mndcmnd 15523   MndHom cmhm 15576  CMndccmn 16393  mulGrpcmgp 16708   Ringcrg 16763  ℂfldccnfld 17938  TopMndctmd 19768   logclog 22134
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-inf2 7953  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465  ax-pre-sup 9466  ax-addf 9467  ax-mulf 9468
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-int 4232  df-iun 4276  df-iin 4277  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-isom 5530  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-of 6425  df-om 6582  df-1st 6682  df-2nd 6683  df-supp 6796  df-recs 6937  df-rdg 6971  df-1o 7025  df-2o 7026  df-oadd 7029  df-er 7206  df-map 7321  df-pm 7322  df-ixp 7369  df-en 7416  df-dom 7417  df-sdom 7418  df-fin 7419  df-fsupp 7727  df-fi 7767  df-sup 7797  df-oi 7830  df-card 8215  df-cda 8443  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-div 10100  df-nn 10429  df-2 10486  df-3 10487  df-4 10488  df-5 10489  df-6 10490  df-7 10491  df-8 10492  df-9 10493  df-10 10494  df-n0 10686  df-z 10753  df-dec 10862  df-uz 10968  df-q 11060  df-rp 11098  df-xneg 11195  df-xadd 11196  df-xmul 11197  df-ioo 11410  df-ioc 11411  df-ico 11412  df-icc 11413  df-fz 11550  df-fzo 11661  df-fl 11754  df-mod 11821  df-seq 11919  df-exp 11978  df-fac 12164  df-bc 12191  df-hash 12216  df-shft 12669  df-cj 12701  df-re 12702  df-im 12703  df-sqr 12837  df-abs 12838  df-limsup 13062  df-clim 13079  df-rlim 13080  df-sum 13277  df-ef 13466  df-sin 13468  df-cos 13469  df-pi 13471  df-struct 14289  df-ndx 14290  df-slot 14291  df-base 14292  df-sets 14293  df-ress 14294  df-plusg 14365  df-mulr 14366  df-starv 14367  df-sca 14368  df-vsca 14369  df-ip 14370  df-tset 14371  df-ple 14372  df-ds 14374  df-unif 14375  df-hom 14376  df-cco 14377  df-rest 14475  df-topn 14476  df-0g 14494  df-gsum 14495  df-topgen 14496  df-pt 14497  df-prds 14500  df-xrs 14554  df-qtop 14559  df-imas 14560  df-xps 14562  df-mre 14638  df-mrc 14639  df-acs 14641  df-mnd 15529  df-plusf 15530  df-mhm 15578  df-submnd 15579  df-grp 15659  df-minusg 15660  df-sbg 15661  df-mulg 15662  df-subg 15792  df-cntz 15949  df-cmn 16395  df-abl 16396  df-mgp 16709  df-ur 16721  df-rng 16765  df-cring 16766  df-subrg 16981  df-abv 17020  df-lmod 17068  df-scaf 17069  df-sra 17371  df-rgmod 17372  df-psmet 17929  df-xmet 17930  df-met 17931  df-bl 17932  df-mopn 17933  df-fbas 17934  df-fg 17935  df-cnfld 17939  df-top 18630  df-bases 18632  df-topon 18633  df-topsp 18634  df-cld 18750  df-ntr 18751  df-cls 18752  df-nei 18829  df-lp 18867  df-perf 18868  df-cn 18958  df-cnp 18959  df-haus 19046  df-tx 19262  df-hmeo 19455  df-fil 19546  df-fm 19638  df-flim 19639  df-flf 19640  df-tmd 19770  df-tgp 19771  df-trg 19861  df-xms 20022  df-ms 20023  df-tms 20024  df-nm 20302  df-ngp 20303  df-nrg 20305  df-nlm 20306  df-cncf 20581  df-limc 21469  df-dv 21470  df-log 22136
This theorem is referenced by:  xrge0tmd  26516
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