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Theorem c1lip2 19843
Description: C1 functions are Lipschitz continuous on closed intervals. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Revised by Stefan O'Rear, 6-May-2015.)
Hypotheses
Ref Expression
c1lip2.a  |-  ( ph  ->  A  e.  RR )
c1lip2.b  |-  ( ph  ->  B  e.  RR )
c1lip2.f  |-  ( ph  ->  F  e.  ( ( C ^n `  RR ) `  1 )
)
c1lip2.rn  |-  ( ph  ->  ran  F  C_  RR )
c1lip2.dm  |-  ( ph  ->  ( A [,] B
)  C_  dom  F )
Assertion
Ref Expression
c1lip2  |-  ( ph  ->  E. k  e.  RR  A. x  e.  ( A [,] B ) A. y  e.  ( A [,] B ) ( abs `  ( ( F `  y )  -  ( F `  x )
) )  <_  (
k  x.  ( abs `  ( y  -  x
) ) ) )
Distinct variable groups:    ph, x, y, k    x, A, y, k    x, B, y, k    x, F, y, k

Proof of Theorem c1lip2
StepHypRef Expression
1 c1lip2.a . 2  |-  ( ph  ->  A  e.  RR )
2 c1lip2.b . 2  |-  ( ph  ->  B  e.  RR )
3 c1lip2.f . . 3  |-  ( ph  ->  F  e.  ( ( C ^n `  RR ) `  1 )
)
4 ax-resscn 9011 . . . . 5  |-  RR  C_  CC
5 1nn0 10201 . . . . 5  |-  1  e.  NN0
6 elcpn 19781 . . . . 5  |-  ( ( RR  C_  CC  /\  1  e.  NN0 )  ->  ( F  e.  ( (
C ^n `  RR ) `  1 )  <->  ( F  e.  ( CC 
^pm  RR )  /\  (
( RR  D n F ) `  1
)  e.  ( dom 
F -cn-> CC ) ) ) )
74, 5, 6mp2an 654 . . . 4  |-  ( F  e.  ( ( C ^n `  RR ) `
 1 )  <->  ( F  e.  ( CC  ^pm  RR )  /\  ( ( RR  D n F ) `
 1 )  e.  ( dom  F -cn-> CC ) ) )
87simplbi 447 . . 3  |-  ( F  e.  ( ( C ^n `  RR ) `
 1 )  ->  F  e.  ( CC  ^pm 
RR ) )
93, 8syl 16 . 2  |-  ( ph  ->  F  e.  ( CC 
^pm  RR ) )
10 c1lip2.dm . . 3  |-  ( ph  ->  ( A [,] B
)  C_  dom  F )
11 pmfun 7003 . . . . . . . . 9  |-  ( F  e.  ( CC  ^pm  RR )  ->  Fun  F )
129, 11syl 16 . . . . . . . 8  |-  ( ph  ->  Fun  F )
13 funfn 5449 . . . . . . . 8  |-  ( Fun 
F  <->  F  Fn  dom  F )
1412, 13sylib 189 . . . . . . 7  |-  ( ph  ->  F  Fn  dom  F
)
15 c1lip2.rn . . . . . . 7  |-  ( ph  ->  ran  F  C_  RR )
16 df-f 5425 . . . . . . 7  |-  ( F : dom  F --> RR  <->  ( F  Fn  dom  F  /\  ran  F 
C_  RR ) )
1714, 15, 16sylanbrc 646 . . . . . 6  |-  ( ph  ->  F : dom  F --> RR )
18 cnex 9035 . . . . . . . . 9  |-  CC  e.  _V
19 reex 9045 . . . . . . . . 9  |-  RR  e.  _V
2018, 19elpm2 7012 . . . . . . . 8  |-  ( F  e.  ( CC  ^pm  RR )  <->  ( F : dom  F --> CC  /\  dom  F 
C_  RR ) )
2120simprbi 451 . . . . . . 7  |-  ( F  e.  ( CC  ^pm  RR )  ->  dom  F  C_  RR )
229, 21syl 16 . . . . . 6  |-  ( ph  ->  dom  F  C_  RR )
23 dvfre 19798 . . . . . 6  |-  ( ( F : dom  F --> RR  /\  dom  F  C_  RR )  ->  ( RR 
_D  F ) : dom  ( RR  _D  F ) --> RR )
2417, 22, 23syl2anc 643 . . . . 5  |-  ( ph  ->  ( RR  _D  F
) : dom  ( RR  _D  F ) --> RR )
25 0p1e1 10057 . . . . . . . . . . 11  |-  ( 0  +  1 )  =  1
2625fveq2i 5698 . . . . . . . . . 10  |-  ( ( RR  D n F ) `  ( 0  +  1 ) )  =  ( ( RR  D n F ) `
 1 )
27 0nn0 10200 . . . . . . . . . . . 12  |-  0  e.  NN0
28 dvnp1 19772 . . . . . . . . . . . 12  |-  ( ( RR  C_  CC  /\  F  e.  ( CC  ^pm  RR )  /\  0  e.  NN0 )  ->  ( ( RR  D n F ) `
 ( 0  +  1 ) )  =  ( RR  _D  (
( RR  D n F ) `  0
) ) )
294, 27, 28mp3an13 1270 . . . . . . . . . . 11  |-  ( F  e.  ( CC  ^pm  RR )  ->  ( ( RR  D n F ) `
 ( 0  +  1 ) )  =  ( RR  _D  (
( RR  D n F ) `  0
) ) )
309, 29syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( ( RR  D n F ) `  (
0  +  1 ) )  =  ( RR 
_D  ( ( RR  D n F ) `
 0 ) ) )
3126, 30syl5eqr 2458 . . . . . . . . 9  |-  ( ph  ->  ( ( RR  D n F ) `  1
)  =  ( RR 
_D  ( ( RR  D n F ) `
 0 ) ) )
32 dvn0 19771 . . . . . . . . . . 11  |-  ( ( RR  C_  CC  /\  F  e.  ( CC  ^pm  RR ) )  ->  (
( RR  D n F ) `  0
)  =  F )
334, 9, 32sylancr 645 . . . . . . . . . 10  |-  ( ph  ->  ( ( RR  D n F ) `  0
)  =  F )
3433oveq2d 6064 . . . . . . . . 9  |-  ( ph  ->  ( RR  _D  (
( RR  D n F ) `  0
) )  =  ( RR  _D  F ) )
3531, 34eqtrd 2444 . . . . . . . 8  |-  ( ph  ->  ( ( RR  D n F ) `  1
)  =  ( RR 
_D  F ) )
367simprbi 451 . . . . . . . . 9  |-  ( F  e.  ( ( C ^n `  RR ) `
 1 )  -> 
( ( RR  D n F ) `  1
)  e.  ( dom 
F -cn-> CC ) )
373, 36syl 16 . . . . . . . 8  |-  ( ph  ->  ( ( RR  D n F ) `  1
)  e.  ( dom 
F -cn-> CC ) )
3835, 37eqeltrrd 2487 . . . . . . 7  |-  ( ph  ->  ( RR  _D  F
)  e.  ( dom 
F -cn-> CC ) )
39 cncff 18884 . . . . . . 7  |-  ( ( RR  _D  F )  e.  ( dom  F -cn->
CC )  ->  ( RR  _D  F ) : dom  F --> CC )
40 fdm 5562 . . . . . . 7  |-  ( ( RR  _D  F ) : dom  F --> CC  ->  dom  ( RR  _D  F
)  =  dom  F
)
4138, 39, 403syl 19 . . . . . 6  |-  ( ph  ->  dom  ( RR  _D  F )  =  dom  F )
4241feq2d 5548 . . . . 5  |-  ( ph  ->  ( ( RR  _D  F ) : dom  ( RR  _D  F
) --> RR  <->  ( RR  _D  F ) : dom  F --> RR ) )
4324, 42mpbid 202 . . . 4  |-  ( ph  ->  ( RR  _D  F
) : dom  F --> RR )
44 cncffvrn 18889 . . . . 5  |-  ( ( RR  C_  CC  /\  ( RR  _D  F )  e.  ( dom  F -cn-> CC ) )  ->  (
( RR  _D  F
)  e.  ( dom 
F -cn-> RR )  <->  ( RR  _D  F ) : dom  F --> RR ) )
454, 38, 44sylancr 645 . . . 4  |-  ( ph  ->  ( ( RR  _D  F )  e.  ( dom  F -cn-> RR )  <-> 
( RR  _D  F
) : dom  F --> RR ) )
4643, 45mpbird 224 . . 3  |-  ( ph  ->  ( RR  _D  F
)  e.  ( dom 
F -cn-> RR ) )
47 rescncf 18888 . . 3  |-  ( ( A [,] B ) 
C_  dom  F  ->  ( ( RR  _D  F
)  e.  ( dom 
F -cn-> RR )  ->  (
( RR  _D  F
)  |`  ( A [,] B ) )  e.  ( ( A [,] B ) -cn-> RR ) ) )
4810, 46, 47sylc 58 . 2  |-  ( ph  ->  ( ( RR  _D  F )  |`  ( A [,] B ) )  e.  ( ( A [,] B ) -cn-> RR ) )
4919prid1 3880 . . . . . . . . 9  |-  RR  e.  { RR ,  CC }
50 nn0uz 10484 . . . . . . . . . 10  |-  NN0  =  ( ZZ>= `  0 )
515, 50eleqtri 2484 . . . . . . . . 9  |-  1  e.  ( ZZ>= `  0 )
52 cpnord 19782 . . . . . . . . 9  |-  ( ( RR  e.  { RR ,  CC }  /\  0  e.  NN0  /\  1  e.  ( ZZ>= `  0 )
)  ->  ( (
C ^n `  RR ) `  1 )  C_  ( ( C ^n
`  RR ) ` 
0 ) )
5349, 27, 51, 52mp3an 1279 . . . . . . . 8  |-  ( ( C ^n `  RR ) `  1 )  C_  ( ( C ^n
`  RR ) ` 
0 )
5453, 3sseldi 3314 . . . . . . 7  |-  ( ph  ->  F  e.  ( ( C ^n `  RR ) `  0 )
)
55 elcpn 19781 . . . . . . . . 9  |-  ( ( RR  C_  CC  /\  0  e.  NN0 )  ->  ( F  e.  ( (
C ^n `  RR ) `  0 )  <->  ( F  e.  ( CC 
^pm  RR )  /\  (
( RR  D n F ) `  0
)  e.  ( dom 
F -cn-> CC ) ) ) )
564, 27, 55mp2an 654 . . . . . . . 8  |-  ( F  e.  ( ( C ^n `  RR ) `
 0 )  <->  ( F  e.  ( CC  ^pm  RR )  /\  ( ( RR  D n F ) `
 0 )  e.  ( dom  F -cn-> CC ) ) )
5756simprbi 451 . . . . . . 7  |-  ( F  e.  ( ( C ^n `  RR ) `
 0 )  -> 
( ( RR  D n F ) `  0
)  e.  ( dom 
F -cn-> CC ) )
5854, 57syl 16 . . . . . 6  |-  ( ph  ->  ( ( RR  D n F ) `  0
)  e.  ( dom 
F -cn-> CC ) )
5933, 58eqeltrrd 2487 . . . . 5  |-  ( ph  ->  F  e.  ( dom 
F -cn-> CC ) )
60 cncffvrn 18889 . . . . 5  |-  ( ( RR  C_  CC  /\  F  e.  ( dom  F -cn-> CC ) )  ->  ( F  e.  ( dom  F
-cn-> RR )  <->  F : dom  F --> RR ) )
614, 59, 60sylancr 645 . . . 4  |-  ( ph  ->  ( F  e.  ( dom  F -cn-> RR )  <-> 
F : dom  F --> RR ) )
6217, 61mpbird 224 . . 3  |-  ( ph  ->  F  e.  ( dom 
F -cn-> RR ) )
63 rescncf 18888 . . 3  |-  ( ( A [,] B ) 
C_  dom  F  ->  ( F  e.  ( dom 
F -cn-> RR )  ->  ( F  |`  ( A [,] B ) )  e.  ( ( A [,] B ) -cn-> RR ) ) )
6410, 62, 63sylc 58 . 2  |-  ( ph  ->  ( F  |`  ( A [,] B ) )  e.  ( ( A [,] B ) -cn-> RR ) )
651, 2, 9, 48, 64c1lip1 19842 1  |-  ( ph  ->  E. k  e.  RR  A. x  e.  ( A [,] B ) A. y  e.  ( A [,] B ) ( abs `  ( ( F `  y )  -  ( F `  x )
) )  <_  (
k  x.  ( abs `  ( y  -  x
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2674   E.wrex 2675    C_ wss 3288   {cpr 3783   class class class wbr 4180   dom cdm 4845   ran crn 4846    |` cres 4847   Fun wfun 5415    Fn wfn 5416   -->wf 5417   ` cfv 5421  (class class class)co 6048    ^pm cpm 6986   CCcc 8952   RRcr 8953   0cc0 8954   1c1 8955    + caddc 8957    x. cmul 8959    <_ cle 9085    - cmin 9255   NN0cn0 10185   ZZ>=cuz 10452   [,]cicc 10883   abscabs 12002   -cn->ccncf 18867    _D cdv 19711    D ncdvn 19712   C ^nccpn 19713
This theorem is referenced by:  c1lip3  19844
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-inf2 7560  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031  ax-pre-sup 9032  ax-addf 9033  ax-mulf 9034
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-iin 4064  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-se 4510  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-of 6272  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-2o 6692  df-oadd 6695  df-er 6872  df-map 6987  df-pm 6988  df-ixp 7031  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-fi 7382  df-sup 7412  df-oi 7443  df-card 7790  df-cda 8012  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642  df-nn 9965  df-2 10022  df-3 10023  df-4 10024  df-5 10025  df-6 10026  df-7 10027  df-8 10028  df-9 10029  df-10 10030  df-n0 10186  df-z 10247  df-dec 10347  df-uz 10453  df-q 10539  df-rp 10577  df-xneg 10674  df-xadd 10675  df-xmul 10676  df-ioo 10884  df-ico 10886  df-icc 10887  df-fz 11008  df-fzo 11099  df-seq 11287  df-exp 11346  df-hash 11582  df-cj 11867  df-re 11868  df-im 11869  df-sqr 12003  df-abs 12004  df-struct 13434  df-ndx 13435  df-slot 13436  df-base 13437  df-sets 13438  df-ress 13439  df-plusg 13505  df-mulr 13506  df-starv 13507  df-sca 13508  df-vsca 13509  df-tset 13511  df-ple 13512  df-ds 13514  df-unif 13515  df-hom 13516  df-cco 13517  df-rest 13613  df-topn 13614  df-topgen 13630  df-pt 13631  df-prds 13634  df-xrs 13689  df-0g 13690  df-gsum 13691  df-qtop 13696  df-imas 13697  df-xps 13699  df-mre 13774  df-mrc 13775  df-acs 13777  df-mnd 14653  df-submnd 14702  df-mulg 14778  df-cntz 15079  df-cmn 15377  df-psmet 16657  df-xmet 16658  df-met 16659  df-bl 16660  df-mopn 16661  df-fbas 16662  df-fg 16663  df-cnfld 16667  df-top 16926  df-bases 16928  df-topon 16929  df-topsp 16930  df-cld 17046  df-ntr 17047  df-cls 17048  df-nei 17125  df-lp 17163  df-perf 17164  df-cn 17253  df-cnp 17254  df-haus 17341  df-cmp 17412  df-tx 17555  df-hmeo 17748  df-fil 17839  df-fm 17931  df-flim 17932  df-flf 17933  df-xms 18311  df-ms 18312  df-tms 18313  df-cncf 18869  df-limc 19714  df-dv 19715  df-dvn 19716  df-cpn 19717
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