Theorem c1lip2 21312

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 9327	. . . . 5 |- RR C_ CC
5		1nn0 10583	. . . . 5 |- 1 e. NN0
6		elcpn 21250	. . . . 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 665	. . . 4 |- ( F e. ( ( C^n ` RR ) ` 1 ) <-> ( F e. ( CC ^pm RR ) /\ ( ( RR Dn F ) ` 1 ) e. ( dom F -cn-> CC ) ) )
8	7	simplbi 457	. . 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 7220	. . . . . . . . 9 |- ( F e. ( CC ^pm RR ) -> Fun F )
12	9, 11	syl 16	. . . . . . . 8 |- ( ph -> Fun F )
13		funfn 5435	. . . . . . . 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 5410	. . . . . . 7 |- ( F : dom F --> RR <-> ( F Fn dom F /\ ran F C_ RR ) )
17	14, 15, 16	sylanbrc 657	. . . . . 6 |- ( ph -> F : dom F --> RR )
18		cnex 9351	. . . . . . . . 9 |- CC e. _V
19		reex 9361	. . . . . . . . 9 |- RR e. _V
20	18, 19	elpm2 7232	. . . . . . . 8 |- ( F e. ( CC ^pm RR ) <-> ( F : dom F --> CC /\ dom F C_ RR ) )
21	20	simprbi 461	. . . . . . 7 |- ( F e. ( CC ^pm RR ) -> dom F C_ RR )
22	9, 21	syl 16	. . . . . 6 |- ( ph -> dom F C_ RR )
23		dvfre 21267	. . . . . 6 |- ( ( F : dom F --> RR /\ dom F C_ RR ) -> ( RR _D F ) : dom ( RR _D F ) --> RR )
24	17, 22, 23	syl2anc 654	. . . . 5 |- ( ph -> ( RR _D F ) : dom ( RR _D F ) --> RR )
25		0p1e1 10421	. . . . . . . . . . 11 |- ( 0 + 1 ) = 1
26	25	fveq2i 5682	. . . . . . . . . 10 |- ( ( RR Dn F ) ` ( 0 + 1 ) ) = ( ( RR Dn F ) ` 1 )
27		0nn0 10582	. . . . . . . . . . . 12 |- 0 e. NN0
28		dvnp1 21241	. . . . . . . . . . . 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 1298	. . . . . . . . . . 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 2479	. . . . . . . . 9 |- ( ph -> ( ( RR Dn F ) ` 1 ) = ( RR _D ( ( RR Dn F ) ` 0 ) ) )
32		dvn0 21240	. . . . . . . . . . 11 |- ( ( RR C_ CC /\ F e. ( CC ^pm RR ) ) -> ( ( RR Dn F ) ` 0 ) = F )
33	4, 9, 32	sylancr 656	. . . . . . . . . 10 |- ( ph -> ( ( RR Dn F ) ` 0 ) = F )
34	33	oveq2d 6096	. . . . . . . . 9 |- ( ph -> ( RR _D ( ( RR Dn F ) ` 0 ) ) = ( RR _D F ) )
35	31, 34	eqtrd 2465	. . . . . . . 8 |- ( ph -> ( ( RR Dn F ) ` 1 ) = ( RR _D F ) )
36	7	simprbi 461	. . . . . . . . 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 2508	. . . . . . 7 |- ( ph -> ( RR _D F ) e. ( dom F -cn-> CC ) )
39		cncff 20311	. . . . . . 7 |- ( ( RR _D F ) e. ( dom F -cn-> CC ) -> ( RR _D F ) : dom F --> CC )
40		fdm 5551	. . . . . . 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 5535	. . . . 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 20316	. . . . 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 656	. . . 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 20315	. . 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 3971	. . . . . . . . 9 |- RR e. { RR , CC }
50		nn0uz 10883	. . . . . . . . . 10 |- NN0 = ( ZZ>= ` 0 )
51	5, 50	eleqtri 2505	. . . . . . . . 9 |- 1 e. ( ZZ>= ` 0 )
52		cpnord 21251	. . . . . . . . 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 1307	. . . . . . . 8 |- ( ( C^n ` RR ) ` 1 ) C_ ( ( C^n ` RR ) ` 0 )
54	53, 3	sseldi 3342	. . . . . . 7 |- ( ph -> F e. ( ( C^n ` RR ) ` 0 ) )
55		elcpn 21250	. . . . . . . . 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 665	. . . . . . . 8 |- ( F e. ( ( C^n ` RR ) ` 0 ) <-> ( F e. ( CC ^pm RR ) /\ ( ( RR Dn F ) ` 0 ) e. ( dom F -cn-> CC ) ) )
57	56	simprbi 461	. . . . . . 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 2508	. . . . 5 |- ( ph -> F e. ( dom F -cn-> CC ) )
60		cncffvrn 20316	. . . . 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 656	. . . 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 20315	. . 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 21311	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 1362   e. wcel 1755   A.wral 2705   E.wrex 2706   C_ wss 3316   {cpr 3867   class class class wbr 4280   dom cdm 4827   ran crn 4828   |` cres 4829   Fun wfun 5400   Fn wfn 5401   -->wf 5402   ` cfv 5406  (class class class)co 6080   ^pm cpm 7203   CCcc 9268   RRcr 9269   0cc0 9270   1c1 9271   + caddc 9273   x. cmul 9275   <_ cle 9407   - cmin 9583   NN0cn0 10567   ZZ>=cuz 10849   [,]cicc 11291   abscabs 12707   -cn->ccncf 20294   _D cdv 21180   Dncdvn 21181   C^nccpn 21182

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-inf2 7835  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347  ax-pre-sup 9348  ax-addf 9349  ax-mulf 9350

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-iin 4162  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-se 4667  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-of 6309  df-om 6466  df-1st 6566  df-2nd 6567  df-supp 6680  df-recs 6818  df-rdg 6852  df-1o 6908  df-2o 6909  df-oadd 6912  df-er 7089  df-map 7204  df-pm 7205  df-ixp 7252  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-fsupp 7609  df-fi 7649  df-sup 7679  df-oi 7712  df-card 8097  df-cda 8325  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-div 9982  df-nn 10311  df-2 10368  df-3 10369  df-4 10370  df-5 10371  df-6 10372  df-7 10373  df-8 10374  df-9 10375  df-10 10376  df-n0 10568  df-z 10635  df-dec 10744  df-uz 10850  df-q 10942  df-rp 10980  df-xneg 11077  df-xadd 11078  df-xmul 11079  df-ioo 11292  df-ico 11294  df-icc 11295  df-fz 11425  df-fzo 11533  df-seq 11791  df-exp 11850  df-hash 12088  df-cj 12572  df-re 12573  df-im 12574  df-sqr 12708  df-abs 12709  df-struct 14159  df-ndx 14160  df-slot 14161  df-base 14162  df-sets 14163  df-ress 14164  df-plusg 14234  df-mulr 14235  df-starv 14236  df-sca 14237  df-vsca 14238  df-ip 14239  df-tset 14240  df-ple 14241  df-ds 14243  df-unif 14244  df-hom 14245  df-cco 14246  df-rest 14344  df-topn 14345  df-0g 14363  df-gsum 14364  df-topgen 14365  df-pt 14366  df-prds 14369  df-xrs 14423  df-qtop 14428  df-imas 14429  df-xps 14431  df-mre 14507  df-mrc 14508  df-acs 14510  df-mnd 15398  df-submnd 15448  df-mulg 15528  df-cntz 15815  df-cmn 16259  df-psmet 17653  df-xmet 17654  df-met 17655  df-bl 17656  df-mopn 17657  df-fbas 17658  df-fg 17659  df-cnfld 17663  df-top 18345  df-bases 18347  df-topon 18348  df-topsp 18349  df-cld 18465  df-ntr 18466  df-cls 18467  df-nei 18544  df-lp 18582  df-perf 18583  df-cn 18673  df-cnp 18674  df-haus 18761  df-cmp 18832  df-tx 18977  df-hmeo 19170  df-fil 19261  df-fm 19353  df-flim 19354  df-flf 19355  df-xms 19737  df-ms 19738  df-tms 19739  df-cncf 20296  df-limc 21183  df-dv 21184  df-dvn 21185  df-cpn 21186

This theorem is referenced by:  c1lip3  21313