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Theorem c1lip2 19887
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 9052 . . . . 5  |-  RR  C_  CC
5 1nn0 10242 . . . . 5  |-  1  e.  NN0
6 elcpn 19825 . . . . 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 655 . . . 4  |-  ( F  e.  ( ( C ^n `  RR ) `
 1 )  <->  ( F  e.  ( CC  ^pm  RR )  /\  ( ( RR  D n F ) `
 1 )  e.  ( dom  F -cn-> CC ) ) )
87simplbi 448 . . 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 7039 . . . . . . . . 9  |-  ( F  e.  ( CC  ^pm  RR )  ->  Fun  F )
129, 11syl 16 . . . . . . . 8  |-  ( ph  ->  Fun  F )
13 funfn 5485 . . . . . . . 8  |-  ( Fun 
F  <->  F  Fn  dom  F )
1412, 13sylib 190 . . . . . . 7  |-  ( ph  ->  F  Fn  dom  F
)
15 c1lip2.rn . . . . . . 7  |-  ( ph  ->  ran  F  C_  RR )
16 df-f 5461 . . . . . . 7  |-  ( F : dom  F --> RR  <->  ( F  Fn  dom  F  /\  ran  F 
C_  RR ) )
1714, 15, 16sylanbrc 647 . . . . . 6  |-  ( ph  ->  F : dom  F --> RR )
18 cnex 9076 . . . . . . . . 9  |-  CC  e.  _V
19 reex 9086 . . . . . . . . 9  |-  RR  e.  _V
2018, 19elpm2 7048 . . . . . . . 8  |-  ( F  e.  ( CC  ^pm  RR )  <->  ( F : dom  F --> CC  /\  dom  F 
C_  RR ) )
2120simprbi 452 . . . . . . 7  |-  ( F  e.  ( CC  ^pm  RR )  ->  dom  F  C_  RR )
229, 21syl 16 . . . . . 6  |-  ( ph  ->  dom  F  C_  RR )
23 dvfre 19842 . . . . . 6  |-  ( ( F : dom  F --> RR  /\  dom  F  C_  RR )  ->  ( RR 
_D  F ) : dom  ( RR  _D  F ) --> RR )
2417, 22, 23syl2anc 644 . . . . 5  |-  ( ph  ->  ( RR  _D  F
) : dom  ( RR  _D  F ) --> RR )
25 0p1e1 10098 . . . . . . . . . . 11  |-  ( 0  +  1 )  =  1
2625fveq2i 5734 . . . . . . . . . 10  |-  ( ( RR  D n F ) `  ( 0  +  1 ) )  =  ( ( RR  D n F ) `
 1 )
27 0nn0 10241 . . . . . . . . . . . 12  |-  0  e.  NN0
28 dvnp1 19816 . . . . . . . . . . . 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 1271 . . . . . . . . . . 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 2484 . . . . . . . . 9  |-  ( ph  ->  ( ( RR  D n F ) `  1
)  =  ( RR 
_D  ( ( RR  D n F ) `
 0 ) ) )
32 dvn0 19815 . . . . . . . . . . 11  |-  ( ( RR  C_  CC  /\  F  e.  ( CC  ^pm  RR ) )  ->  (
( RR  D n F ) `  0
)  =  F )
334, 9, 32sylancr 646 . . . . . . . . . 10  |-  ( ph  ->  ( ( RR  D n F ) `  0
)  =  F )
3433oveq2d 6100 . . . . . . . . 9  |-  ( ph  ->  ( RR  _D  (
( RR  D n F ) `  0
) )  =  ( RR  _D  F ) )
3531, 34eqtrd 2470 . . . . . . . 8  |-  ( ph  ->  ( ( RR  D n F ) `  1
)  =  ( RR 
_D  F ) )
367simprbi 452 . . . . . . . . 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 2513 . . . . . . 7  |-  ( ph  ->  ( RR  _D  F
)  e.  ( dom 
F -cn-> CC ) )
39 cncff 18928 . . . . . . 7  |-  ( ( RR  _D  F )  e.  ( dom  F -cn->
CC )  ->  ( RR  _D  F ) : dom  F --> CC )
40 fdm 5598 . . . . . . 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 5584 . . . . 5  |-  ( ph  ->  ( ( RR  _D  F ) : dom  ( RR  _D  F
) --> RR  <->  ( RR  _D  F ) : dom  F --> RR ) )
4324, 42mpbid 203 . . . 4  |-  ( ph  ->  ( RR  _D  F
) : dom  F --> RR )
44 cncffvrn 18933 . . . . 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 646 . . . 4  |-  ( ph  ->  ( ( RR  _D  F )  e.  ( dom  F -cn-> RR )  <-> 
( RR  _D  F
) : dom  F --> RR ) )
4643, 45mpbird 225 . . 3  |-  ( ph  ->  ( RR  _D  F
)  e.  ( dom 
F -cn-> RR ) )
47 rescncf 18932 . . 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 59 . 2  |-  ( ph  ->  ( ( RR  _D  F )  |`  ( A [,] B ) )  e.  ( ( A [,] B ) -cn-> RR ) )
4919prid1 3914 . . . . . . . . 9  |-  RR  e.  { RR ,  CC }
50 nn0uz 10525 . . . . . . . . . 10  |-  NN0  =  ( ZZ>= `  0 )
515, 50eleqtri 2510 . . . . . . . . 9  |-  1  e.  ( ZZ>= `  0 )
52 cpnord 19826 . . . . . . . . 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 1280 . . . . . . . 8  |-  ( ( C ^n `  RR ) `  1 )  C_  ( ( C ^n
`  RR ) ` 
0 )
5453, 3sseldi 3348 . . . . . . 7  |-  ( ph  ->  F  e.  ( ( C ^n `  RR ) `  0 )
)
55 elcpn 19825 . . . . . . . . 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 655 . . . . . . . 8  |-  ( F  e.  ( ( C ^n `  RR ) `
 0 )  <->  ( F  e.  ( CC  ^pm  RR )  /\  ( ( RR  D n F ) `
 0 )  e.  ( dom  F -cn-> CC ) ) )
5756simprbi 452 . . . . . . 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 2513 . . . . 5  |-  ( ph  ->  F  e.  ( dom 
F -cn-> CC ) )
60 cncffvrn 18933 . . . . 5  |-  ( ( RR  C_  CC  /\  F  e.  ( dom  F -cn-> CC ) )  ->  ( F  e.  ( dom  F
-cn-> RR )  <->  F : dom  F --> RR ) )
614, 59, 60sylancr 646 . . . 4  |-  ( ph  ->  ( F  e.  ( dom  F -cn-> RR )  <-> 
F : dom  F --> RR ) )
6217, 61mpbird 225 . . 3  |-  ( ph  ->  F  e.  ( dom 
F -cn-> RR ) )
63 rescncf 18932 . . 3  |-  ( ( A [,] B ) 
C_  dom  F  ->  ( F  e.  ( dom 
F -cn-> RR )  ->  ( F  |`  ( A [,] B ) )  e.  ( ( A [,] B ) -cn-> RR ) ) )
6410, 62, 63sylc 59 . 2  |-  ( ph  ->  ( F  |`  ( A [,] B ) )  e.  ( ( A [,] B ) -cn-> RR ) )
651, 2, 9, 48, 64c1lip1 19886 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 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   E.wrex 2708    C_ wss 3322   {cpr 3817   class class class wbr 4215   dom cdm 4881   ran crn 4882    |` cres 4883   Fun wfun 5451    Fn wfn 5452   -->wf 5453   ` cfv 5457  (class class class)co 6084    ^pm cpm 7022   CCcc 8993   RRcr 8994   0cc0 8995   1c1 8996    + caddc 8998    x. cmul 9000    <_ cle 9126    - cmin 9296   NN0cn0 10226   ZZ>=cuz 10493   [,]cicc 10924   abscabs 12044   -cn->ccncf 18911    _D cdv 19755    D ncdvn 19756   C ^nccpn 19757
This theorem is referenced by:  c1lip3  19888
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073  ax-addf 9074  ax-mulf 9075
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-of 6308  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-2o 6728  df-oadd 6731  df-er 6908  df-map 7023  df-pm 7024  df-ixp 7067  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-fi 7419  df-sup 7449  df-oi 7482  df-card 7831  df-cda 8053  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-3 10064  df-4 10065  df-5 10066  df-6 10067  df-7 10068  df-8 10069  df-9 10070  df-10 10071  df-n0 10227  df-z 10288  df-dec 10388  df-uz 10494  df-q 10580  df-rp 10618  df-xneg 10715  df-xadd 10716  df-xmul 10717  df-ioo 10925  df-ico 10927  df-icc 10928  df-fz 11049  df-fzo 11141  df-seq 11329  df-exp 11388  df-hash 11624  df-cj 11909  df-re 11910  df-im 11911  df-sqr 12045  df-abs 12046  df-struct 13476  df-ndx 13477  df-slot 13478  df-base 13479  df-sets 13480  df-ress 13481  df-plusg 13547  df-mulr 13548  df-starv 13549  df-sca 13550  df-vsca 13551  df-tset 13553  df-ple 13554  df-ds 13556  df-unif 13557  df-hom 13558  df-cco 13559  df-rest 13655  df-topn 13656  df-topgen 13672  df-pt 13673  df-prds 13676  df-xrs 13731  df-0g 13732  df-gsum 13733  df-qtop 13738  df-imas 13739  df-xps 13741  df-mre 13816  df-mrc 13817  df-acs 13819  df-mnd 14695  df-submnd 14744  df-mulg 14820  df-cntz 15121  df-cmn 15419  df-psmet 16699  df-xmet 16700  df-met 16701  df-bl 16702  df-mopn 16703  df-fbas 16704  df-fg 16705  df-cnfld 16709  df-top 16968  df-bases 16970  df-topon 16971  df-topsp 16972  df-cld 17088  df-ntr 17089  df-cls 17090  df-nei 17167  df-lp 17205  df-perf 17206  df-cn 17296  df-cnp 17297  df-haus 17384  df-cmp 17455  df-tx 17599  df-hmeo 17792  df-fil 17883  df-fm 17975  df-flim 17976  df-flf 17977  df-xms 18355  df-ms 18356  df-tms 18357  df-cncf 18913  df-limc 19758  df-dv 19759  df-dvn 19760  df-cpn 19761
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