Theorem c1lip2 22962

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

Step	Hyp	Ref	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 9601	. . . . 5 |- RR C_ CC
5		1nn0 10892	. . . . 5 |- 1 e. NN0
6		elcpn 22900	. . . . 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 ) ) ) )
7	4, 5, 6	mp2an 679	. . . 4 |- ( F e. ( ( C^n ` RR ) ` 1 ) <-> ( F e. ( CC ^pm RR ) /\ ( ( RR Dn F ) ` 1 ) e. ( dom F -cn-> CC ) ) )
8	7	simplbi 462	. . 3 |- ( F e. ( ( C^n ` RR ) ` 1 ) -> F e. ( CC ^pm RR ) )
9	3, 8	syl 17	. 2 |- ( ph -> F e. ( CC ^pm RR ) )
10		c1lip2.dm	. . 3 |- ( ph -> ( A [,] B ) C_ dom F )
11		pmfun 7496	. . . . . . . . 9 |- ( F e. ( CC ^pm RR ) -> Fun F )
12	9, 11	syl 17	. . . . . . . 8 |- ( ph -> Fun F )
13		funfn 5614	. . . . . . . 8 |- ( Fun F <-> F Fn dom F )
14	12, 13	sylib 200	. . . . . . 7 |- ( ph -> F Fn dom F )
15		c1lip2.rn	. . . . . . 7 |- ( ph -> ran F C_ RR )
16		df-f 5589	. . . . . . 7 |- ( F : dom F --> RR <-> ( F Fn dom F /\ ran F C_ RR ) )
17	14, 15, 16	sylanbrc 671	. . . . . 6 |- ( ph -> F : dom F --> RR )
18		cnex 9625	. . . . . . . . 9 |- CC e. _V
19		reex 9635	. . . . . . . . 9 |- RR e. _V
20	18, 19	elpm2 7508	. . . . . . . 8 |- ( F e. ( CC ^pm RR ) <-> ( F : dom F --> CC /\ dom F C_ RR ) )
21	20	simprbi 466	. . . . . . 7 |- ( F e. ( CC ^pm RR ) -> dom F C_ RR )
22	9, 21	syl 17	. . . . . 6 |- ( ph -> dom F C_ RR )
23		dvfre 22917	. . . . . 6 |- ( ( F : dom F --> RR /\ dom F C_ RR ) -> ( RR _D F ) : dom ( RR _D F ) --> RR )
24	17, 22, 23	syl2anc 667	. . . . 5 |- ( ph -> ( RR _D F ) : dom ( RR _D F ) --> RR )
25		0p1e1 10728	. . . . . . . . . . 11 |- ( 0 + 1 ) = 1
26	25	fveq2i 5873	. . . . . . . . . 10 |- ( ( RR Dn F ) ` ( 0 + 1 ) ) = ( ( RR Dn F ) ` 1 )
27		0nn0 10891	. . . . . . . . . . . 12 |- 0 e. NN0
28		dvnp1 22891	. . . . . . . . . . . 12 |- ( ( RR C_ CC /\ F e. ( CC ^pm RR ) /\ 0 e. NN0 ) -> ( ( RR Dn F ) ` ( 0 + 1 ) ) = ( RR _D ( ( RR Dn F ) ` 0 ) ) )
29	4, 27, 28	mp3an13 1357	. . . . . . . . . . 11 |- ( F e. ( CC ^pm RR ) -> ( ( RR Dn F ) ` ( 0 + 1 ) ) = ( RR _D ( ( RR Dn F ) ` 0 ) ) )
30	9, 29	syl 17	. . . . . . . . . 10 |- ( ph -> ( ( RR Dn F ) ` ( 0 + 1 ) ) = ( RR _D ( ( RR Dn F ) ` 0 ) ) )
31	26, 30	syl5eqr 2501	. . . . . . . . 9 |- ( ph -> ( ( RR Dn F ) ` 1 ) = ( RR _D ( ( RR Dn F ) ` 0 ) ) )
32		dvn0 22890	. . . . . . . . . . 11 |- ( ( RR C_ CC /\ F e. ( CC ^pm RR ) ) -> ( ( RR Dn F ) ` 0 ) = F )
33	4, 9, 32	sylancr 670	. . . . . . . . . 10 |- ( ph -> ( ( RR Dn F ) ` 0 ) = F )
34	33	oveq2d 6311	. . . . . . . . 9 |- ( ph -> ( RR _D ( ( RR Dn F ) ` 0 ) ) = ( RR _D F ) )
35	31, 34	eqtrd 2487	. . . . . . . 8 |- ( ph -> ( ( RR Dn F ) ` 1 ) = ( RR _D F ) )
36	7	simprbi 466	. . . . . . . . 9 |- ( F e. ( ( C^n ` RR ) ` 1 ) -> ( ( RR Dn F ) ` 1 ) e. ( dom F -cn-> CC ) )
37	3, 36	syl 17	. . . . . . . 8 |- ( ph -> ( ( RR Dn F ) ` 1 ) e. ( dom F -cn-> CC ) )
38	35, 37	eqeltrrd 2532	. . . . . . 7 |- ( ph -> ( RR _D F ) e. ( dom F -cn-> CC ) )
39		cncff 21937	. . . . . . 7 |- ( ( RR _D F ) e. ( dom F -cn-> CC ) -> ( RR _D F ) : dom F --> CC )
40		fdm 5738	. . . . . . 7 |- ( ( RR _D F ) : dom F --> CC -> dom ( RR _D F ) = dom F )
41	38, 39, 40	3syl 18	. . . . . 6 |- ( ph -> dom ( RR _D F ) = dom F )
42	41	feq2d 5720	. . . . 5 |- ( ph -> ( ( RR _D F ) : dom ( RR _D F ) --> RR <-> ( RR _D F ) : dom F --> RR ) )
43	24, 42	mpbid 214	. . . 4 |- ( ph -> ( RR _D F ) : dom F --> RR )
44		cncffvrn 21942	. . . . 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 ) )
45	4, 38, 44	sylancr 670	. . . 4 |- ( ph -> ( ( RR _D F ) e. ( dom F -cn-> RR ) <-> ( RR _D F ) : dom F --> RR ) )
46	43, 45	mpbird 236	. . 3 |- ( ph -> ( RR _D F ) e. ( dom F -cn-> RR ) )
47		rescncf 21941	. . 3 |- ( ( A [,] B ) C_ dom F -> ( ( RR _D F ) e. ( dom F -cn-> RR ) -> ( ( RR _D F ) |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> RR ) ) )
48	10, 46, 47	sylc 62	. 2 |- ( ph -> ( ( RR _D F ) |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> RR ) )
49	19	prid1 4083	. . . . . . . . 9 |- RR e. { RR , CC }
50		1eluzge0 11209	. . . . . . . . 9 |- 1 e. ( ZZ>= ` 0 )
51		cpnord 22901	. . . . . . . . 9 |- ( ( RR e. { RR , CC } /\ 0 e. NN0 /\ 1 e. ( ZZ>= ` 0 ) ) -> ( ( C^n ` RR ) ` 1 ) C_ ( ( C^n ` RR ) ` 0 ) )
52	49, 27, 50, 51	mp3an 1366	. . . . . . . 8 |- ( ( C^n ` RR ) ` 1 ) C_ ( ( C^n ` RR ) ` 0 )
53	52, 3	sseldi 3432	. . . . . . 7 |- ( ph -> F e. ( ( C^n ` RR ) ` 0 ) )
54		elcpn 22900	. . . . . . . . 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 ) ) ) )
55	4, 27, 54	mp2an 679	. . . . . . . 8 |- ( F e. ( ( C^n ` RR ) ` 0 ) <-> ( F e. ( CC ^pm RR ) /\ ( ( RR Dn F ) ` 0 ) e. ( dom F -cn-> CC ) ) )
56	55	simprbi 466	. . . . . . 7 |- ( F e. ( ( C^n ` RR ) ` 0 ) -> ( ( RR Dn F ) ` 0 ) e. ( dom F -cn-> CC ) )
57	53, 56	syl 17	. . . . . 6 |- ( ph -> ( ( RR Dn F ) ` 0 ) e. ( dom F -cn-> CC ) )
58	33, 57	eqeltrrd 2532	. . . . 5 |- ( ph -> F e. ( dom F -cn-> CC ) )
59		cncffvrn 21942	. . . . 5 |- ( ( RR C_ CC /\ F e. ( dom F -cn-> CC ) ) -> ( F e. ( dom F -cn-> RR ) <-> F : dom F --> RR ) )
60	4, 58, 59	sylancr 670	. . . 4 |- ( ph -> ( F e. ( dom F -cn-> RR ) <-> F : dom F --> RR ) )
61	17, 60	mpbird 236	. . 3 |- ( ph -> F e. ( dom F -cn-> RR ) )
62		rescncf 21941	. . 3 |- ( ( A [,] B ) C_ dom F -> ( F e. ( dom F -cn-> RR ) -> ( F |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> RR ) ) )
63	10, 61, 62	sylc 62	. 2 |- ( ph -> ( F |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> RR ) )
64	1, 2, 9, 48, 63	c1lip1 22961	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 ) ) ) )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   = wceq 1446   e. wcel 1889   A.wral 2739   E.wrex 2740   C_ wss 3406   {cpr 3972   class class class wbr 4405   dom cdm 4837   ran crn 4838   |` cres 4839   Fun wfun 5579   Fn wfn 5580   -->wf 5581   ` cfv 5585  (class class class)co 6295   ^pm cpm 7478   CCcc 9542   RRcr 9543   0cc0 9544   1c1 9545   + caddc 9547   x. cmul 9549   <_ cle 9681   - cmin 9865   NN0cn0 10876   ZZ>=cuz 11166   [,]cicc 11645   abscabs 13309   -cn->ccncf 21920   _D cdv 22830   Dncdvn 22831   C^nccpn 22832

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-inf2 8151  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621  ax-pre-sup 9622  ax-addf 9623  ax-mulf 9624

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-iin 4284  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-se 4797  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-isom 5594  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6536  df-om 6698  df-1st 6798  df-2nd 6799  df-supp 6920  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-2o 7188  df-oadd 7191  df-er 7368  df-map 7479  df-pm 7480  df-ixp 7528  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-fsupp 7889  df-fi 7930  df-sup 7961  df-inf 7962  df-oi 8030  df-card 8378  df-cda 8603  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-div 10277  df-nn 10617  df-2 10675  df-3 10676  df-4 10677  df-5 10678  df-6 10679  df-7 10680  df-8 10681  df-9 10682  df-10 10683  df-n0 10877  df-z 10945  df-dec 11059  df-uz 11167  df-q 11272  df-rp 11310  df-xneg 11416  df-xadd 11417  df-xmul 11418  df-ioo 11646  df-ico 11648  df-icc 11649  df-fz 11792  df-fzo 11923  df-seq 12221  df-exp 12280  df-hash 12523  df-cj 13174  df-re 13175  df-im 13176  df-sqrt 13310  df-abs 13311  df-struct 15135  df-ndx 15136  df-slot 15137  df-base 15138  df-sets 15139  df-ress 15140  df-plusg 15215  df-mulr 15216  df-starv 15217  df-sca 15218  df-vsca 15219  df-ip 15220  df-tset 15221  df-ple 15222  df-ds 15224  df-unif 15225  df-hom 15226  df-cco 15227  df-rest 15333  df-topn 15334  df-0g 15352  df-gsum 15353  df-topgen 15354  df-pt 15355  df-prds 15358  df-xrs 15412  df-qtop 15418  df-imas 15419  df-xps 15422  df-mre 15504  df-mrc 15505  df-acs 15507  df-mgm 16500  df-sgrp 16539  df-mnd 16549  df-submnd 16595  df-mulg 16688  df-cntz 16983  df-cmn 17444  df-psmet 18974  df-xmet 18975  df-met 18976  df-bl 18977  df-mopn 18978  df-fbas 18979  df-fg 18980  df-cnfld 18983  df-top 19933  df-bases 19934  df-topon 19935  df-topsp 19936  df-cld 20046  df-ntr 20047  df-cls 20048  df-nei 20126  df-lp 20164  df-perf 20165  df-cn 20255  df-cnp 20256  df-haus 20343  df-cmp 20414  df-tx 20589  df-hmeo 20782  df-fil 20873  df-fm 20965  df-flim 20966  df-flf 20967  df-xms 21347  df-ms 21348  df-tms 21349  df-cncf 21922  df-limc 22833  df-dv 22834  df-dvn 22835  df-cpn 22836

This theorem is referenced by:  c1lip3  22963