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Theorem c1lip2 23029
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 9614 . . . . 5  |-  RR  C_  CC
5 1nn0 10909 . . . . 5  |-  1  e.  NN0
6 elcpn 22967 . . . . 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 686 . . . 4  |-  ( F  e.  ( ( C^n `  RR ) `
 1 )  <->  ( F  e.  ( CC  ^pm  RR )  /\  ( ( RR  Dn F ) `
 1 )  e.  ( dom  F -cn-> CC ) ) )
87simplbi 467 . . 3  |-  ( F  e.  ( ( C^n `  RR ) `
 1 )  ->  F  e.  ( CC  ^pm 
RR ) )
93, 8syl 17 . 2  |-  ( ph  ->  F  e.  ( CC 
^pm  RR ) )
10 c1lip2.dm . . 3  |-  ( ph  ->  ( A [,] B
)  C_  dom  F )
11 pmfun 7509 . . . . . . . . 9  |-  ( F  e.  ( CC  ^pm  RR )  ->  Fun  F )
129, 11syl 17 . . . . . . . 8  |-  ( ph  ->  Fun  F )
13 funfn 5618 . . . . . . . 8  |-  ( Fun 
F  <->  F  Fn  dom  F )
1412, 13sylib 201 . . . . . . 7  |-  ( ph  ->  F  Fn  dom  F
)
15 c1lip2.rn . . . . . . 7  |-  ( ph  ->  ran  F  C_  RR )
16 df-f 5593 . . . . . . 7  |-  ( F : dom  F --> RR  <->  ( F  Fn  dom  F  /\  ran  F 
C_  RR ) )
1714, 15, 16sylanbrc 677 . . . . . 6  |-  ( ph  ->  F : dom  F --> RR )
18 cnex 9638 . . . . . . . . 9  |-  CC  e.  _V
19 reex 9648 . . . . . . . . 9  |-  RR  e.  _V
2018, 19elpm2 7521 . . . . . . . 8  |-  ( F  e.  ( CC  ^pm  RR )  <->  ( F : dom  F --> CC  /\  dom  F 
C_  RR ) )
2120simprbi 471 . . . . . . 7  |-  ( F  e.  ( CC  ^pm  RR )  ->  dom  F  C_  RR )
229, 21syl 17 . . . . . 6  |-  ( ph  ->  dom  F  C_  RR )
23 dvfre 22984 . . . . . 6  |-  ( ( F : dom  F --> RR  /\  dom  F  C_  RR )  ->  ( RR 
_D  F ) : dom  ( RR  _D  F ) --> RR )
2417, 22, 23syl2anc 673 . . . . 5  |-  ( ph  ->  ( RR  _D  F
) : dom  ( RR  _D  F ) --> RR )
25 0p1e1 10743 . . . . . . . . . . 11  |-  ( 0  +  1 )  =  1
2625fveq2i 5882 . . . . . . . . . 10  |-  ( ( RR  Dn F ) `  ( 0  +  1 ) )  =  ( ( RR  Dn F ) `
 1 )
27 0nn0 10908 . . . . . . . . . . . 12  |-  0  e.  NN0
28 dvnp1 22958 . . . . . . . . . . . 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 1381 . . . . . . . . . . 11  |-  ( F  e.  ( CC  ^pm  RR )  ->  ( ( RR  Dn F ) `
 ( 0  +  1 ) )  =  ( RR  _D  (
( RR  Dn
F ) `  0
) ) )
309, 29syl 17 . . . . . . . . . 10  |-  ( ph  ->  ( ( RR  Dn F ) `  ( 0  +  1 ) )  =  ( RR  _D  ( ( RR  Dn F ) `  0 ) ) )
3126, 30syl5eqr 2519 . . . . . . . . 9  |-  ( ph  ->  ( ( RR  Dn F ) ` 
1 )  =  ( RR  _D  ( ( RR  Dn F ) `  0 ) ) )
32 dvn0 22957 . . . . . . . . . . 11  |-  ( ( RR  C_  CC  /\  F  e.  ( CC  ^pm  RR ) )  ->  (
( RR  Dn
F ) `  0
)  =  F )
334, 9, 32sylancr 676 . . . . . . . . . 10  |-  ( ph  ->  ( ( RR  Dn F ) ` 
0 )  =  F )
3433oveq2d 6324 . . . . . . . . 9  |-  ( ph  ->  ( RR  _D  (
( RR  Dn
F ) `  0
) )  =  ( RR  _D  F ) )
3531, 34eqtrd 2505 . . . . . . . 8  |-  ( ph  ->  ( ( RR  Dn F ) ` 
1 )  =  ( RR  _D  F ) )
367simprbi 471 . . . . . . . . 9  |-  ( F  e.  ( ( C^n `  RR ) `
 1 )  -> 
( ( RR  Dn F ) ` 
1 )  e.  ( dom  F -cn-> CC ) )
373, 36syl 17 . . . . . . . 8  |-  ( ph  ->  ( ( RR  Dn F ) ` 
1 )  e.  ( dom  F -cn-> CC ) )
3835, 37eqeltrrd 2550 . . . . . . 7  |-  ( ph  ->  ( RR  _D  F
)  e.  ( dom 
F -cn-> CC ) )
39 cncff 22003 . . . . . . 7  |-  ( ( RR  _D  F )  e.  ( dom  F -cn->
CC )  ->  ( RR  _D  F ) : dom  F --> CC )
40 fdm 5745 . . . . . . 7  |-  ( ( RR  _D  F ) : dom  F --> CC  ->  dom  ( RR  _D  F
)  =  dom  F
)
4138, 39, 403syl 18 . . . . . 6  |-  ( ph  ->  dom  ( RR  _D  F )  =  dom  F )
4241feq2d 5725 . . . . 5  |-  ( ph  ->  ( ( RR  _D  F ) : dom  ( RR  _D  F
) --> RR  <->  ( RR  _D  F ) : dom  F --> RR ) )
4324, 42mpbid 215 . . . 4  |-  ( ph  ->  ( RR  _D  F
) : dom  F --> RR )
44 cncffvrn 22008 . . . . 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 676 . . . 4  |-  ( ph  ->  ( ( RR  _D  F )  e.  ( dom  F -cn-> RR )  <-> 
( RR  _D  F
) : dom  F --> RR ) )
4643, 45mpbird 240 . . 3  |-  ( ph  ->  ( RR  _D  F
)  e.  ( dom 
F -cn-> RR ) )
47 rescncf 22007 . . 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 61 . 2  |-  ( ph  ->  ( ( RR  _D  F )  |`  ( A [,] B ) )  e.  ( ( A [,] B ) -cn-> RR ) )
4919prid1 4071 . . . . . . . . 9  |-  RR  e.  { RR ,  CC }
50 1eluzge0 11226 . . . . . . . . 9  |-  1  e.  ( ZZ>= `  0 )
51 cpnord 22968 . . . . . . . . 9  |-  ( ( RR  e.  { RR ,  CC }  /\  0  e.  NN0  /\  1  e.  ( ZZ>= `  0 )
)  ->  ( (
C^n `  RR ) `  1 )  C_  ( ( C^n `
 RR ) ` 
0 ) )
5249, 27, 50, 51mp3an 1390 . . . . . . . 8  |-  ( ( C^n `  RR ) `  1 )  C_  ( ( C^n `
 RR ) ` 
0 )
5352, 3sseldi 3416 . . . . . . 7  |-  ( ph  ->  F  e.  ( ( C^n `  RR ) `  0 )
)
54 elcpn 22967 . . . . . . . . 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 ) ) ) )
554, 27, 54mp2an 686 . . . . . . . 8  |-  ( F  e.  ( ( C^n `  RR ) `
 0 )  <->  ( F  e.  ( CC  ^pm  RR )  /\  ( ( RR  Dn F ) `
 0 )  e.  ( dom  F -cn-> CC ) ) )
5655simprbi 471 . . . . . . 7  |-  ( F  e.  ( ( C^n `  RR ) `
 0 )  -> 
( ( RR  Dn F ) ` 
0 )  e.  ( dom  F -cn-> CC ) )
5753, 56syl 17 . . . . . 6  |-  ( ph  ->  ( ( RR  Dn F ) ` 
0 )  e.  ( dom  F -cn-> CC ) )
5833, 57eqeltrrd 2550 . . . . 5  |-  ( ph  ->  F  e.  ( dom 
F -cn-> CC ) )
59 cncffvrn 22008 . . . . 5  |-  ( ( RR  C_  CC  /\  F  e.  ( dom  F -cn-> CC ) )  ->  ( F  e.  ( dom  F
-cn-> RR )  <->  F : dom  F --> RR ) )
604, 58, 59sylancr 676 . . . 4  |-  ( ph  ->  ( F  e.  ( dom  F -cn-> RR )  <-> 
F : dom  F --> RR ) )
6117, 60mpbird 240 . . 3  |-  ( ph  ->  F  e.  ( dom 
F -cn-> RR ) )
62 rescncf 22007 . . 3  |-  ( ( A [,] B ) 
C_  dom  F  ->  ( F  e.  ( dom 
F -cn-> RR )  ->  ( F  |`  ( A [,] B ) )  e.  ( ( A [,] B ) -cn-> RR ) ) )
6310, 61, 62sylc 61 . 2  |-  ( ph  ->  ( F  |`  ( A [,] B ) )  e.  ( ( A [,] B ) -cn-> RR ) )
641, 2, 9, 48, 63c1lip1 23028 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 189    /\ wa 376    = wceq 1452    e. wcel 1904   A.wral 2756   E.wrex 2757    C_ wss 3390   {cpr 3961   class class class wbr 4395   dom cdm 4839   ran crn 4840    |` cres 4841   Fun wfun 5583    Fn wfn 5584   -->wf 5585   ` cfv 5589  (class class class)co 6308    ^pm cpm 7491   CCcc 9555   RRcr 9556   0cc0 9557   1c1 9558    + caddc 9560    x. cmul 9562    <_ cle 9694    - cmin 9880   NN0cn0 10893   ZZ>=cuz 11182   [,]cicc 11663   abscabs 13374   -cn->ccncf 21986    _D cdv 22897    Dncdvn 22898   C^nccpn 22899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636  ax-mulf 9637
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-fi 7943  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-ioo 11664  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-hom 15292  df-cco 15293  df-rest 15399  df-topn 15400  df-0g 15418  df-gsum 15419  df-topgen 15420  df-pt 15421  df-prds 15424  df-xrs 15478  df-qtop 15484  df-imas 15485  df-xps 15488  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-mulg 16754  df-cntz 17049  df-cmn 17510  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-fbas 19044  df-fg 19045  df-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cld 20111  df-ntr 20112  df-cls 20113  df-nei 20191  df-lp 20229  df-perf 20230  df-cn 20320  df-cnp 20321  df-haus 20408  df-cmp 20479  df-tx 20654  df-hmeo 20847  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033  df-xms 21413  df-ms 21414  df-tms 21415  df-cncf 21988  df-limc 22900  df-dv 22901  df-dvn 22902  df-cpn 22903
This theorem is referenced by:  c1lip3  23030
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