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Theorem c1lip2 22565
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 9538 . . . . 5  |-  RR  C_  CC
5 1nn0 10807 . . . . 5  |-  1  e.  NN0
6 elcpn 22503 . . . . 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 670 . . . 4  |-  ( F  e.  ( ( C^n `  RR ) `
 1 )  <->  ( F  e.  ( CC  ^pm  RR )  /\  ( ( RR  Dn F ) `
 1 )  e.  ( dom  F -cn-> CC ) ) )
87simplbi 458 . . 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 7431 . . . . . . . . 9  |-  ( F  e.  ( CC  ^pm  RR )  ->  Fun  F )
129, 11syl 16 . . . . . . . 8  |-  ( ph  ->  Fun  F )
13 funfn 5599 . . . . . . . 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 5574 . . . . . . 7  |-  ( F : dom  F --> RR  <->  ( F  Fn  dom  F  /\  ran  F 
C_  RR ) )
1714, 15, 16sylanbrc 662 . . . . . 6  |-  ( ph  ->  F : dom  F --> RR )
18 cnex 9562 . . . . . . . . 9  |-  CC  e.  _V
19 reex 9572 . . . . . . . . 9  |-  RR  e.  _V
2018, 19elpm2 7443 . . . . . . . 8  |-  ( F  e.  ( CC  ^pm  RR )  <->  ( F : dom  F --> CC  /\  dom  F 
C_  RR ) )
2120simprbi 462 . . . . . . 7  |-  ( F  e.  ( CC  ^pm  RR )  ->  dom  F  C_  RR )
229, 21syl 16 . . . . . 6  |-  ( ph  ->  dom  F  C_  RR )
23 dvfre 22520 . . . . . 6  |-  ( ( F : dom  F --> RR  /\  dom  F  C_  RR )  ->  ( RR 
_D  F ) : dom  ( RR  _D  F ) --> RR )
2417, 22, 23syl2anc 659 . . . . 5  |-  ( ph  ->  ( RR  _D  F
) : dom  ( RR  _D  F ) --> RR )
25 0p1e1 10643 . . . . . . . . . . 11  |-  ( 0  +  1 )  =  1
2625fveq2i 5851 . . . . . . . . . 10  |-  ( ( RR  Dn F ) `  ( 0  +  1 ) )  =  ( ( RR  Dn F ) `
 1 )
27 0nn0 10806 . . . . . . . . . . . 12  |-  0  e.  NN0
28 dvnp1 22494 . . . . . . . . . . . 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 1313 . . . . . . . . . . 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 2509 . . . . . . . . 9  |-  ( ph  ->  ( ( RR  Dn F ) ` 
1 )  =  ( RR  _D  ( ( RR  Dn F ) `  0 ) ) )
32 dvn0 22493 . . . . . . . . . . 11  |-  ( ( RR  C_  CC  /\  F  e.  ( CC  ^pm  RR ) )  ->  (
( RR  Dn
F ) `  0
)  =  F )
334, 9, 32sylancr 661 . . . . . . . . . 10  |-  ( ph  ->  ( ( RR  Dn F ) ` 
0 )  =  F )
3433oveq2d 6286 . . . . . . . . 9  |-  ( ph  ->  ( RR  _D  (
( RR  Dn
F ) `  0
) )  =  ( RR  _D  F ) )
3531, 34eqtrd 2495 . . . . . . . 8  |-  ( ph  ->  ( ( RR  Dn F ) ` 
1 )  =  ( RR  _D  F ) )
367simprbi 462 . . . . . . . . 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 2543 . . . . . . 7  |-  ( ph  ->  ( RR  _D  F
)  e.  ( dom 
F -cn-> CC ) )
39 cncff 21563 . . . . . . 7  |-  ( ( RR  _D  F )  e.  ( dom  F -cn->
CC )  ->  ( RR  _D  F ) : dom  F --> CC )
40 fdm 5717 . . . . . . 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 5700 . . . . 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 21568 . . . . 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 661 . . . 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 21567 . . 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 4124 . . . . . . . . 9  |-  RR  e.  { RR ,  CC }
50 1eluzge0 11125 . . . . . . . . 9  |-  1  e.  ( ZZ>= `  0 )
51 cpnord 22504 . . . . . . . . 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 1322 . . . . . . . 8  |-  ( ( C^n `  RR ) `  1 )  C_  ( ( C^n `
 RR ) ` 
0 )
5352, 3sseldi 3487 . . . . . . 7  |-  ( ph  ->  F  e.  ( ( C^n `  RR ) `  0 )
)
54 elcpn 22503 . . . . . . . . 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 670 . . . . . . . 8  |-  ( F  e.  ( ( C^n `  RR ) `
 0 )  <->  ( F  e.  ( CC  ^pm  RR )  /\  ( ( RR  Dn F ) `
 0 )  e.  ( dom  F -cn-> CC ) ) )
5655simprbi 462 . . . . . . 7  |-  ( F  e.  ( ( C^n `  RR ) `
 0 )  -> 
( ( RR  Dn F ) ` 
0 )  e.  ( dom  F -cn-> CC ) )
5753, 56syl 16 . . . . . 6  |-  ( ph  ->  ( ( RR  Dn F ) ` 
0 )  e.  ( dom  F -cn-> CC ) )
5833, 57eqeltrrd 2543 . . . . 5  |-  ( ph  ->  F  e.  ( dom 
F -cn-> CC ) )
59 cncffvrn 21568 . . . . 5  |-  ( ( RR  C_  CC  /\  F  e.  ( dom  F -cn-> CC ) )  ->  ( F  e.  ( dom  F
-cn-> RR )  <->  F : dom  F --> RR ) )
604, 58, 59sylancr 661 . . . 4  |-  ( ph  ->  ( F  e.  ( dom  F -cn-> RR )  <-> 
F : dom  F --> RR ) )
6117, 60mpbird 232 . . 3  |-  ( ph  ->  F  e.  ( dom 
F -cn-> RR ) )
62 rescncf 21567 . . 3  |-  ( ( A [,] B ) 
C_  dom  F  ->  ( F  e.  ( dom 
F -cn-> RR )  ->  ( F  |`  ( A [,] B ) )  e.  ( ( A [,] B ) -cn-> RR ) ) )
6310, 61, 62sylc 60 . 2  |-  ( ph  ->  ( F  |`  ( A [,] B ) )  e.  ( ( A [,] B ) -cn-> RR ) )
641, 2, 9, 48, 63c1lip1 22564 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 367    = wceq 1398    e. wcel 1823   A.wral 2804   E.wrex 2805    C_ wss 3461   {cpr 4018   class class class wbr 4439   dom cdm 4988   ran crn 4989    |` cres 4990   Fun wfun 5564    Fn wfn 5565   -->wf 5566   ` cfv 5570  (class class class)co 6270    ^pm cpm 7413   CCcc 9479   RRcr 9480   0cc0 9481   1c1 9482    + caddc 9484    x. cmul 9486    <_ cle 9618    - cmin 9796   NN0cn0 10791   ZZ>=cuz 11082   [,]cicc 11535   abscabs 13149   -cn->ccncf 21546    _D cdv 22433    Dncdvn 22434   C^nccpn 22435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560  ax-mulf 9561
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-ixp 7463  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-fi 7863  df-sup 7893  df-oi 7927  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10977  df-uz 11083  df-q 11184  df-rp 11222  df-xneg 11321  df-xadd 11322  df-xmul 11323  df-ioo 11536  df-ico 11538  df-icc 11539  df-fz 11676  df-fzo 11800  df-seq 12090  df-exp 12149  df-hash 12388  df-cj 13014  df-re 13015  df-im 13016  df-sqrt 13150  df-abs 13151  df-struct 14718  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ress 14723  df-plusg 14797  df-mulr 14798  df-starv 14799  df-sca 14800  df-vsca 14801  df-ip 14802  df-tset 14803  df-ple 14804  df-ds 14806  df-unif 14807  df-hom 14808  df-cco 14809  df-rest 14912  df-topn 14913  df-0g 14931  df-gsum 14932  df-topgen 14933  df-pt 14934  df-prds 14937  df-xrs 14991  df-qtop 14996  df-imas 14997  df-xps 14999  df-mre 15075  df-mrc 15076  df-acs 15078  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-submnd 16166  df-mulg 16259  df-cntz 16554  df-cmn 16999  df-psmet 18606  df-xmet 18607  df-met 18608  df-bl 18609  df-mopn 18610  df-fbas 18611  df-fg 18612  df-cnfld 18616  df-top 19566  df-bases 19568  df-topon 19569  df-topsp 19570  df-cld 19687  df-ntr 19688  df-cls 19689  df-nei 19766  df-lp 19804  df-perf 19805  df-cn 19895  df-cnp 19896  df-haus 19983  df-cmp 20054  df-tx 20229  df-hmeo 20422  df-fil 20513  df-fm 20605  df-flim 20606  df-flf 20607  df-xms 20989  df-ms 20990  df-tms 20991  df-cncf 21548  df-limc 22436  df-dv 22437  df-dvn 22438  df-cpn 22439
This theorem is referenced by:  c1lip3  22566
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