Theorem c1lip2 21602

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 9449	. . . . 5 |- RR C_ CC
5		1nn0 10705	. . . . 5 |- 1 e. NN0
6		elcpn 21540	. . . . 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 672	. . . 4 |- ( F e. ( ( C^n ` RR ) ` 1 ) <-> ( F e. ( CC ^pm RR ) /\ ( ( RR Dn F ) ` 1 ) e. ( dom F -cn-> CC ) ) )
8	7	simplbi 460	. . 3 |- ( F e. ( ( C^n ` RR ) ` 1 ) -> F e. ( CC ^pm RR ) )
9	3, 8	syl 16	. 2 |- ( ph -> F e. ( CC ^pm RR ) )
10		c1lip2.dm	. . 3 |- ( ph -> ( A [,] B ) C_ dom F )
11		pmfun 7341	. . . . . . . . 9 |- ( F e. ( CC ^pm RR ) -> Fun F )
12	9, 11	syl 16	. . . . . . . 8 |- ( ph -> Fun F )
13		funfn 5554	. . . . . . . 8 |- ( Fun F <-> F Fn dom F )
14	12, 13	sylib 196	. . . . . . 7 |- ( ph -> F Fn dom F )
15		c1lip2.rn	. . . . . . 7 |- ( ph -> ran F C_ RR )
16		df-f 5529	. . . . . . 7 |- ( F : dom F --> RR <-> ( F Fn dom F /\ ran F C_ RR ) )
17	14, 15, 16	sylanbrc 664	. . . . . 6 |- ( ph -> F : dom F --> RR )
18		cnex 9473	. . . . . . . . 9 |- CC e. _V
19		reex 9483	. . . . . . . . 9 |- RR e. _V
20	18, 19	elpm2 7353	. . . . . . . 8 |- ( F e. ( CC ^pm RR ) <-> ( F : dom F --> CC /\ dom F C_ RR ) )
21	20	simprbi 464	. . . . . . 7 |- ( F e. ( CC ^pm RR ) -> dom F C_ RR )
22	9, 21	syl 16	. . . . . 6 |- ( ph -> dom F C_ RR )
23		dvfre 21557	. . . . . 6 |- ( ( F : dom F --> RR /\ dom F C_ RR ) -> ( RR _D F ) : dom ( RR _D F ) --> RR )
24	17, 22, 23	syl2anc 661	. . . . 5 |- ( ph -> ( RR _D F ) : dom ( RR _D F ) --> RR )
25		0p1e1 10543	. . . . . . . . . . 11 |- ( 0 + 1 ) = 1
26	25	fveq2i 5801	. . . . . . . . . 10 |- ( ( RR Dn F ) ` ( 0 + 1 ) ) = ( ( RR Dn F ) ` 1 )
27		0nn0 10704	. . . . . . . . . . . 12 |- 0 e. NN0
28		dvnp1 21531	. . . . . . . . . . . 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 1306	. . . . . . . . . . 11 |- ( F e. ( CC ^pm RR ) -> ( ( RR Dn F ) ` ( 0 + 1 ) ) = ( RR _D ( ( RR Dn F ) ` 0 ) ) )
30	9, 29	syl 16	. . . . . . . . . 10 |- ( ph -> ( ( RR Dn F ) ` ( 0 + 1 ) ) = ( RR _D ( ( RR Dn F ) ` 0 ) ) )
31	26, 30	syl5eqr 2509	. . . . . . . . 9 |- ( ph -> ( ( RR Dn F ) ` 1 ) = ( RR _D ( ( RR Dn F ) ` 0 ) ) )
32		dvn0 21530	. . . . . . . . . . 11 |- ( ( RR C_ CC /\ F e. ( CC ^pm RR ) ) -> ( ( RR Dn F ) ` 0 ) = F )
33	4, 9, 32	sylancr 663	. . . . . . . . . 10 |- ( ph -> ( ( RR Dn F ) ` 0 ) = F )
34	33	oveq2d 6215	. . . . . . . . 9 |- ( ph -> ( RR _D ( ( RR Dn F ) ` 0 ) ) = ( RR _D F ) )
35	31, 34	eqtrd 2495	. . . . . . . 8 |- ( ph -> ( ( RR Dn F ) ` 1 ) = ( RR _D F ) )
36	7	simprbi 464	. . . . . . . . 9 |- ( F e. ( ( C^n ` RR ) ` 1 ) -> ( ( RR Dn F ) ` 1 ) e. ( dom F -cn-> CC ) )
37	3, 36	syl 16	. . . . . . . 8 |- ( ph -> ( ( RR Dn F ) ` 1 ) e. ( dom F -cn-> CC ) )
38	35, 37	eqeltrrd 2543	. . . . . . 7 |- ( ph -> ( RR _D F ) e. ( dom F -cn-> CC ) )
39		cncff 20600	. . . . . . 7 |- ( ( RR _D F ) e. ( dom F -cn-> CC ) -> ( RR _D F ) : dom F --> CC )
40		fdm 5670	. . . . . . 7 |- ( ( RR _D F ) : dom F --> CC -> dom ( RR _D F ) = dom F )
41	38, 39, 40	3syl 20	. . . . . 6 |- ( ph -> dom ( RR _D F ) = dom F )
42	41	feq2d 5654	. . . . 5 |- ( ph -> ( ( RR _D F ) : dom ( RR _D F ) --> RR <-> ( RR _D F ) : dom F --> RR ) )
43	24, 42	mpbid 210	. . . 4 |- ( ph -> ( RR _D F ) : dom F --> RR )
44		cncffvrn 20605	. . . . 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 663	. . . 4 |- ( ph -> ( ( RR _D F ) e. ( dom F -cn-> RR ) <-> ( RR _D F ) : dom F --> RR ) )
46	43, 45	mpbird 232	. . 3 |- ( ph -> ( RR _D F ) e. ( dom F -cn-> RR ) )
47		rescncf 20604	. . 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 60	. 2 |- ( ph -> ( ( RR _D F ) |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> RR ) )
49	19	prid1 4090	. . . . . . . . 9 |- RR e. { RR , CC }
50		nn0uz 11005	. . . . . . . . . 10 |- NN0 = ( ZZ>= ` 0 )
51	5, 50	eleqtri 2540	. . . . . . . . 9 |- 1 e. ( ZZ>= ` 0 )
52		cpnord 21541	. . . . . . . . 9 |- ( ( RR e. { RR , CC } /\ 0 e. NN0 /\ 1 e. ( ZZ>= ` 0 ) ) -> ( ( C^n ` RR ) ` 1 ) C_ ( ( C^n ` RR ) ` 0 ) )
53	49, 27, 51, 52	mp3an 1315	. . . . . . . 8 |- ( ( C^n ` RR ) ` 1 ) C_ ( ( C^n ` RR ) ` 0 )
54	53, 3	sseldi 3461	. . . . . . 7 |- ( ph -> F e. ( ( C^n ` RR ) ` 0 ) )
55		elcpn 21540	. . . . . . . . 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 ) ) ) )
56	4, 27, 55	mp2an 672	. . . . . . . 8 |- ( F e. ( ( C^n ` RR ) ` 0 ) <-> ( F e. ( CC ^pm RR ) /\ ( ( RR Dn F ) ` 0 ) e. ( dom F -cn-> CC ) ) )
57	56	simprbi 464	. . . . . . 7 |- ( F e. ( ( C^n ` RR ) ` 0 ) -> ( ( RR Dn F ) ` 0 ) e. ( dom F -cn-> CC ) )
58	54, 57	syl 16	. . . . . 6 |- ( ph -> ( ( RR Dn F ) ` 0 ) e. ( dom F -cn-> CC ) )
59	33, 58	eqeltrrd 2543	. . . . 5 |- ( ph -> F e. ( dom F -cn-> CC ) )
60		cncffvrn 20605	. . . . 5 |- ( ( RR C_ CC /\ F e. ( dom F -cn-> CC ) ) -> ( F e. ( dom F -cn-> RR ) <-> F : dom F --> RR ) )
61	4, 59, 60	sylancr 663	. . . 4 |- ( ph -> ( F e. ( dom F -cn-> RR ) <-> F : dom F --> RR ) )
62	17, 61	mpbird 232	. . 3 |- ( ph -> F e. ( dom F -cn-> RR ) )
63		rescncf 20604	. . 3 |- ( ( A [,] B ) C_ dom F -> ( F e. ( dom F -cn-> RR ) -> ( F |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> RR ) ) )
64	10, 62, 63	sylc 60	. 2 |- ( ph -> ( F |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> RR ) )
65	1, 2, 9, 48, 64	c1lip1 21601	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 184   /\ wa 369   = wceq 1370   e. wcel 1758   A.wral 2798   E.wrex 2799   C_ wss 3435   {cpr 3986   class class class wbr 4399   dom cdm 4947   ran crn 4948   |` cres 4949   Fun wfun 5519   Fn wfn 5520   -->wf 5521   ` cfv 5525  (class class class)co 6199   ^pm cpm 7324   CCcc 9390   RRcr 9391   0cc0 9392   1c1 9393   + caddc 9395   x. cmul 9397   <_ cle 9529   - cmin 9705   NN0cn0 10689   ZZ>=cuz 10971   [,]cicc 11413   abscabs 12840   -cn->ccncf 20583   _D cdv 21470   Dncdvn 21471   C^nccpn 21472

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-inf2 7957  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469  ax-pre-sup 9470  ax-addf 9471  ax-mulf 9472

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-int 4236  df-iun 4280  df-iin 4281  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-se 4787  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-isom 5534  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-of 6429  df-om 6586  df-1st 6686  df-2nd 6687  df-supp 6800  df-recs 6941  df-rdg 6975  df-1o 7029  df-2o 7030  df-oadd 7033  df-er 7210  df-map 7325  df-pm 7326  df-ixp 7373  df-en 7420  df-dom 7421  df-sdom 7422  df-fin 7423  df-fsupp 7731  df-fi 7771  df-sup 7801  df-oi 7834  df-card 8219  df-cda 8447  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-div 10104  df-nn 10433  df-2 10490  df-3 10491  df-4 10492  df-5 10493  df-6 10494  df-7 10495  df-8 10496  df-9 10497  df-10 10498  df-n0 10690  df-z 10757  df-dec 10866  df-uz 10972  df-q 11064  df-rp 11102  df-xneg 11199  df-xadd 11200  df-xmul 11201  df-ioo 11414  df-ico 11416  df-icc 11417  df-fz 11554  df-fzo 11665  df-seq 11923  df-exp 11982  df-hash 12220  df-cj 12705  df-re 12706  df-im 12707  df-sqr 12841  df-abs 12842  df-struct 14293  df-ndx 14294  df-slot 14295  df-base 14296  df-sets 14297  df-ress 14298  df-plusg 14369  df-mulr 14370  df-starv 14371  df-sca 14372  df-vsca 14373  df-ip 14374  df-tset 14375  df-ple 14376  df-ds 14378  df-unif 14379  df-hom 14380  df-cco 14381  df-rest 14479  df-topn 14480  df-0g 14498  df-gsum 14499  df-topgen 14500  df-pt 14501  df-prds 14504  df-xrs 14558  df-qtop 14563  df-imas 14564  df-xps 14566  df-mre 14642  df-mrc 14643  df-acs 14645  df-mnd 15533  df-submnd 15583  df-mulg 15666  df-cntz 15953  df-cmn 16399  df-psmet 17933  df-xmet 17934  df-met 17935  df-bl 17936  df-mopn 17937  df-fbas 17938  df-fg 17939  df-cnfld 17943  df-top 18634  df-bases 18636  df-topon 18637  df-topsp 18638  df-cld 18754  df-ntr 18755  df-cls 18756  df-nei 18833  df-lp 18871  df-perf 18872  df-cn 18962  df-cnp 18963  df-haus 19050  df-cmp 19121  df-tx 19266  df-hmeo 19459  df-fil 19550  df-fm 19642  df-flim 19643  df-flf 19644  df-xms 20026  df-ms 20027  df-tms 20028  df-cncf 20585  df-limc 21473  df-dv 21474  df-dvn 21475  df-cpn 21476

This theorem is referenced by:  c1lip3  21603