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Theorem c1lip2 21312
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 9327 . . . . 5  |-  RR  C_  CC
5 1nn0 10583 . . . . 5  |-  1  e.  NN0
6 elcpn 21250 . . . . 5  |-  ( ( RR  C_  CC  /\  1  e.  NN0 )  ->  ( F  e.  ( (
C^n `  RR ) `  1 )  <->  ( F  e.  ( CC 
^pm  RR )  /\  (
( RR  Dn
F ) `  1
)  e.  ( dom 
F -cn-> CC ) ) ) )
74, 5, 6mp2an 665 . . . 4  |-  ( F  e.  ( ( C^n `  RR ) `
 1 )  <->  ( F  e.  ( CC  ^pm  RR )  /\  ( ( RR  Dn F ) `
 1 )  e.  ( dom  F -cn-> CC ) ) )
87simplbi 457 . . 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 7220 . . . . . . . . 9  |-  ( F  e.  ( CC  ^pm  RR )  ->  Fun  F )
129, 11syl 16 . . . . . . . 8  |-  ( ph  ->  Fun  F )
13 funfn 5435 . . . . . . . 8  |-  ( Fun 
F  <->  F  Fn  dom  F )
1412, 13sylib 196 . . . . . . 7  |-  ( ph  ->  F  Fn  dom  F
)
15 c1lip2.rn . . . . . . 7  |-  ( ph  ->  ran  F  C_  RR )
16 df-f 5410 . . . . . . 7  |-  ( F : dom  F --> RR  <->  ( F  Fn  dom  F  /\  ran  F 
C_  RR ) )
1714, 15, 16sylanbrc 657 . . . . . 6  |-  ( ph  ->  F : dom  F --> RR )
18 cnex 9351 . . . . . . . . 9  |-  CC  e.  _V
19 reex 9361 . . . . . . . . 9  |-  RR  e.  _V
2018, 19elpm2 7232 . . . . . . . 8  |-  ( F  e.  ( CC  ^pm  RR )  <->  ( F : dom  F --> CC  /\  dom  F 
C_  RR ) )
2120simprbi 461 . . . . . . 7  |-  ( F  e.  ( CC  ^pm  RR )  ->  dom  F  C_  RR )
229, 21syl 16 . . . . . 6  |-  ( ph  ->  dom  F  C_  RR )
23 dvfre 21267 . . . . . 6  |-  ( ( F : dom  F --> RR  /\  dom  F  C_  RR )  ->  ( RR 
_D  F ) : dom  ( RR  _D  F ) --> RR )
2417, 22, 23syl2anc 654 . . . . 5  |-  ( ph  ->  ( RR  _D  F
) : dom  ( RR  _D  F ) --> RR )
25 0p1e1 10421 . . . . . . . . . . 11  |-  ( 0  +  1 )  =  1
2625fveq2i 5682 . . . . . . . . . 10  |-  ( ( RR  Dn F ) `  ( 0  +  1 ) )  =  ( ( RR  Dn F ) `
 1 )
27 0nn0 10582 . . . . . . . . . . . 12  |-  0  e.  NN0
28 dvnp1 21241 . . . . . . . . . . . 12  |-  ( ( RR  C_  CC  /\  F  e.  ( CC  ^pm  RR )  /\  0  e.  NN0 )  ->  ( ( RR  Dn F ) `
 ( 0  +  1 ) )  =  ( RR  _D  (
( RR  Dn
F ) `  0
) ) )
294, 27, 28mp3an13 1298 . . . . . . . . . . 11  |-  ( F  e.  ( CC  ^pm  RR )  ->  ( ( RR  Dn F ) `
 ( 0  +  1 ) )  =  ( RR  _D  (
( RR  Dn
F ) `  0
) ) )
309, 29syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( ( RR  Dn F ) `  ( 0  +  1 ) )  =  ( RR  _D  ( ( RR  Dn F ) `  0 ) ) )
3126, 30syl5eqr 2479 . . . . . . . . 9  |-  ( ph  ->  ( ( RR  Dn F ) ` 
1 )  =  ( RR  _D  ( ( RR  Dn F ) `  0 ) ) )
32 dvn0 21240 . . . . . . . . . . 11  |-  ( ( RR  C_  CC  /\  F  e.  ( CC  ^pm  RR ) )  ->  (
( RR  Dn
F ) `  0
)  =  F )
334, 9, 32sylancr 656 . . . . . . . . . 10  |-  ( ph  ->  ( ( RR  Dn F ) ` 
0 )  =  F )
3433oveq2d 6096 . . . . . . . . 9  |-  ( ph  ->  ( RR  _D  (
( RR  Dn
F ) `  0
) )  =  ( RR  _D  F ) )
3531, 34eqtrd 2465 . . . . . . . 8  |-  ( ph  ->  ( ( RR  Dn F ) ` 
1 )  =  ( RR  _D  F ) )
367simprbi 461 . . . . . . . . 9  |-  ( F  e.  ( ( C^n `  RR ) `
 1 )  -> 
( ( RR  Dn F ) ` 
1 )  e.  ( dom  F -cn-> CC ) )
373, 36syl 16 . . . . . . . 8  |-  ( ph  ->  ( ( RR  Dn F ) ` 
1 )  e.  ( dom  F -cn-> CC ) )
3835, 37eqeltrrd 2508 . . . . . . 7  |-  ( ph  ->  ( RR  _D  F
)  e.  ( dom 
F -cn-> CC ) )
39 cncff 20311 . . . . . . 7  |-  ( ( RR  _D  F )  e.  ( dom  F -cn->
CC )  ->  ( RR  _D  F ) : dom  F --> CC )
40 fdm 5551 . . . . . . 7  |-  ( ( RR  _D  F ) : dom  F --> CC  ->  dom  ( RR  _D  F
)  =  dom  F
)
4138, 39, 403syl 20 . . . . . 6  |-  ( ph  ->  dom  ( RR  _D  F )  =  dom  F )
4241feq2d 5535 . . . . 5  |-  ( ph  ->  ( ( RR  _D  F ) : dom  ( RR  _D  F
) --> RR  <->  ( RR  _D  F ) : dom  F --> RR ) )
4324, 42mpbid 210 . . . 4  |-  ( ph  ->  ( RR  _D  F
) : dom  F --> RR )
44 cncffvrn 20316 . . . . 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 656 . . . 4  |-  ( ph  ->  ( ( RR  _D  F )  e.  ( dom  F -cn-> RR )  <-> 
( RR  _D  F
) : dom  F --> RR ) )
4643, 45mpbird 232 . . 3  |-  ( ph  ->  ( RR  _D  F
)  e.  ( dom 
F -cn-> RR ) )
47 rescncf 20315 . . 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 60 . 2  |-  ( ph  ->  ( ( RR  _D  F )  |`  ( A [,] B ) )  e.  ( ( A [,] B ) -cn-> RR ) )
4919prid1 3971 . . . . . . . . 9  |-  RR  e.  { RR ,  CC }
50 nn0uz 10883 . . . . . . . . . 10  |-  NN0  =  ( ZZ>= `  0 )
515, 50eleqtri 2505 . . . . . . . . 9  |-  1  e.  ( ZZ>= `  0 )
52 cpnord 21251 . . . . . . . . 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 1307 . . . . . . . 8  |-  ( ( C^n `  RR ) `  1 )  C_  ( ( C^n `
 RR ) ` 
0 )
5453, 3sseldi 3342 . . . . . . 7  |-  ( ph  ->  F  e.  ( ( C^n `  RR ) `  0 )
)
55 elcpn 21250 . . . . . . . . 9  |-  ( ( RR  C_  CC  /\  0  e.  NN0 )  ->  ( F  e.  ( (
C^n `  RR ) `  0 )  <->  ( F  e.  ( CC 
^pm  RR )  /\  (
( RR  Dn
F ) `  0
)  e.  ( dom 
F -cn-> CC ) ) ) )
564, 27, 55mp2an 665 . . . . . . . 8  |-  ( F  e.  ( ( C^n `  RR ) `
 0 )  <->  ( F  e.  ( CC  ^pm  RR )  /\  ( ( RR  Dn F ) `
 0 )  e.  ( dom  F -cn-> CC ) ) )
5756simprbi 461 . . . . . . 7  |-  ( F  e.  ( ( C^n `  RR ) `
 0 )  -> 
( ( RR  Dn F ) ` 
0 )  e.  ( dom  F -cn-> CC ) )
5854, 57syl 16 . . . . . 6  |-  ( ph  ->  ( ( RR  Dn F ) ` 
0 )  e.  ( dom  F -cn-> CC ) )
5933, 58eqeltrrd 2508 . . . . 5  |-  ( ph  ->  F  e.  ( dom 
F -cn-> CC ) )
60 cncffvrn 20316 . . . . 5  |-  ( ( RR  C_  CC  /\  F  e.  ( dom  F -cn-> CC ) )  ->  ( F  e.  ( dom  F
-cn-> RR )  <->  F : dom  F --> RR ) )
614, 59, 60sylancr 656 . . . 4  |-  ( ph  ->  ( F  e.  ( dom  F -cn-> RR )  <-> 
F : dom  F --> RR ) )
6217, 61mpbird 232 . . 3  |-  ( ph  ->  F  e.  ( dom 
F -cn-> RR ) )
63 rescncf 20315 . . 3  |-  ( ( A [,] B ) 
C_  dom  F  ->  ( F  e.  ( dom 
F -cn-> RR )  ->  ( F  |`  ( A [,] B ) )  e.  ( ( A [,] B ) -cn-> RR ) ) )
6410, 62, 63sylc 60 . 2  |-  ( ph  ->  ( F  |`  ( A [,] B ) )  e.  ( ( A [,] B ) -cn-> RR ) )
651, 2, 9, 48, 64c1lip1 21311 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1362    e. wcel 1755   A.wral 2705   E.wrex 2706    C_ wss 3316   {cpr 3867   class class class wbr 4280   dom cdm 4827   ran crn 4828    |` cres 4829   Fun wfun 5400    Fn wfn 5401   -->wf 5402   ` cfv 5406  (class class class)co 6080    ^pm cpm 7203   CCcc 9268   RRcr 9269   0cc0 9270   1c1 9271    + caddc 9273    x. cmul 9275    <_ cle 9407    - cmin 9583   NN0cn0 10567   ZZ>=cuz 10849   [,]cicc 11291   abscabs 12707   -cn->ccncf 20294    _D cdv 21180    Dncdvn 21181   C^nccpn 21182
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-inf2 7835  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347  ax-pre-sup 9348  ax-addf 9349  ax-mulf 9350
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-iin 4162  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-se 4667  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-of 6309  df-om 6466  df-1st 6566  df-2nd 6567  df-supp 6680  df-recs 6818  df-rdg 6852  df-1o 6908  df-2o 6909  df-oadd 6912  df-er 7089  df-map 7204  df-pm 7205  df-ixp 7252  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-fsupp 7609  df-fi 7649  df-sup 7679  df-oi 7712  df-card 8097  df-cda 8325  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-div 9982  df-nn 10311  df-2 10368  df-3 10369  df-4 10370  df-5 10371  df-6 10372  df-7 10373  df-8 10374  df-9 10375  df-10 10376  df-n0 10568  df-z 10635  df-dec 10744  df-uz 10850  df-q 10942  df-rp 10980  df-xneg 11077  df-xadd 11078  df-xmul 11079  df-ioo 11292  df-ico 11294  df-icc 11295  df-fz 11425  df-fzo 11533  df-seq 11791  df-exp 11850  df-hash 12088  df-cj 12572  df-re 12573  df-im 12574  df-sqr 12708  df-abs 12709  df-struct 14159  df-ndx 14160  df-slot 14161  df-base 14162  df-sets 14163  df-ress 14164  df-plusg 14234  df-mulr 14235  df-starv 14236  df-sca 14237  df-vsca 14238  df-ip 14239  df-tset 14240  df-ple 14241  df-ds 14243  df-unif 14244  df-hom 14245  df-cco 14246  df-rest 14344  df-topn 14345  df-0g 14363  df-gsum 14364  df-topgen 14365  df-pt 14366  df-prds 14369  df-xrs 14423  df-qtop 14428  df-imas 14429  df-xps 14431  df-mre 14507  df-mrc 14508  df-acs 14510  df-mnd 15398  df-submnd 15448  df-mulg 15528  df-cntz 15815  df-cmn 16259  df-psmet 17653  df-xmet 17654  df-met 17655  df-bl 17656  df-mopn 17657  df-fbas 17658  df-fg 17659  df-cnfld 17663  df-top 18345  df-bases 18347  df-topon 18348  df-topsp 18349  df-cld 18465  df-ntr 18466  df-cls 18467  df-nei 18544  df-lp 18582  df-perf 18583  df-cn 18673  df-cnp 18674  df-haus 18761  df-cmp 18832  df-tx 18977  df-hmeo 19170  df-fil 19261  df-fm 19353  df-flim 19354  df-flf 19355  df-xms 19737  df-ms 19738  df-tms 19739  df-cncf 20296  df-limc 21183  df-dv 21184  df-dvn 21185  df-cpn 21186
This theorem is referenced by:  c1lip3  21313
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