Theorem c1lip2 23029

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 9614	. . . . 5 |- RR C_ CC
5		1nn0 10909	. . . . 5 |- 1 e. NN0
6		elcpn 22967	. . . . 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 686	. . . 4 |- ( F e. ( ( C^n ` RR ) ` 1 ) <-> ( F e. ( CC ^pm RR ) /\ ( ( RR Dn F ) ` 1 ) e. ( dom F -cn-> CC ) ) )
8	7	simplbi 467	. . 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 7509	. . . . . . . . 9 |- ( F e. ( CC ^pm RR ) -> Fun F )
12	9, 11	syl 17	. . . . . . . 8 |- ( ph -> Fun F )
13		funfn 5618	. . . . . . . 8 |- ( Fun F <-> F Fn dom F )
14	12, 13	sylib 201	. . . . . . 7 |- ( ph -> F Fn dom F )
15		c1lip2.rn	. . . . . . 7 |- ( ph -> ran F C_ RR )
16		df-f 5593	. . . . . . 7 |- ( F : dom F --> RR <-> ( F Fn dom F /\ ran F C_ RR ) )
17	14, 15, 16	sylanbrc 677	. . . . . 6 |- ( ph -> F : dom F --> RR )
18		cnex 9638	. . . . . . . . 9 |- CC e. _V
19		reex 9648	. . . . . . . . 9 |- RR e. _V
20	18, 19	elpm2 7521	. . . . . . . 8 |- ( F e. ( CC ^pm RR ) <-> ( F : dom F --> CC /\ dom F C_ RR ) )
21	20	simprbi 471	. . . . . . 7 |- ( F e. ( CC ^pm RR ) -> dom F C_ RR )
22	9, 21	syl 17	. . . . . 6 |- ( ph -> dom F C_ RR )
23		dvfre 22984	. . . . . 6 |- ( ( F : dom F --> RR /\ dom F C_ RR ) -> ( RR _D F ) : dom ( RR _D F ) --> RR )
24	17, 22, 23	syl2anc 673	. . . . 5 |- ( ph -> ( RR _D F ) : dom ( RR _D F ) --> RR )
25		0p1e1 10743	. . . . . . . . . . 11 |- ( 0 + 1 ) = 1
26	25	fveq2i 5882	. . . . . . . . . 10 |- ( ( RR Dn F ) ` ( 0 + 1 ) ) = ( ( RR Dn F ) ` 1 )
27		0nn0 10908	. . . . . . . . . . . 12 |- 0 e. NN0
28		dvnp1 22958	. . . . . . . . . . . 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 1381	. . . . . . . . . . 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 2519	. . . . . . . . 9 |- ( ph -> ( ( RR Dn F ) ` 1 ) = ( RR _D ( ( RR Dn F ) ` 0 ) ) )
32		dvn0 22957	. . . . . . . . . . 11 |- ( ( RR C_ CC /\ F e. ( CC ^pm RR ) ) -> ( ( RR Dn F ) ` 0 ) = F )
33	4, 9, 32	sylancr 676	. . . . . . . . . 10 |- ( ph -> ( ( RR Dn F ) ` 0 ) = F )
34	33	oveq2d 6324	. . . . . . . . 9 |- ( ph -> ( RR _D ( ( RR Dn F ) ` 0 ) ) = ( RR _D F ) )
35	31, 34	eqtrd 2505	. . . . . . . 8 |- ( ph -> ( ( RR Dn F ) ` 1 ) = ( RR _D F ) )
36	7	simprbi 471	. . . . . . . . 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 2550	. . . . . . 7 |- ( ph -> ( RR _D F ) e. ( dom F -cn-> CC ) )
39		cncff 22003	. . . . . . 7 |- ( ( RR _D F ) e. ( dom F -cn-> CC ) -> ( RR _D F ) : dom F --> CC )
40		fdm 5745	. . . . . . 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 5725	. . . . 5 |- ( ph -> ( ( RR _D F ) : dom ( RR _D F ) --> RR <-> ( RR _D F ) : dom F --> RR ) )
43	24, 42	mpbid 215	. . . 4 |- ( ph -> ( RR _D F ) : dom F --> RR )
44		cncffvrn 22008	. . . . 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 676	. . . 4 |- ( ph -> ( ( RR _D F ) e. ( dom F -cn-> RR ) <-> ( RR _D F ) : dom F --> RR ) )
46	43, 45	mpbird 240	. . 3 |- ( ph -> ( RR _D F ) e. ( dom F -cn-> RR ) )
47		rescncf 22007	. . 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 61	. 2 |- ( ph -> ( ( RR _D F ) |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> RR ) )
49	19	prid1 4071	. . . . . . . . 9 |- RR e. { RR , CC }
50		1eluzge0 11226	. . . . . . . . 9 |- 1 e. ( ZZ>= ` 0 )
51		cpnord 22968	. . . . . . . . 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 1390	. . . . . . . 8 |- ( ( C^n ` RR ) ` 1 ) C_ ( ( C^n ` RR ) ` 0 )
53	52, 3	sseldi 3416	. . . . . . 7 |- ( ph -> F e. ( ( C^n ` RR ) ` 0 ) )
54		elcpn 22967	. . . . . . . . 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 686	. . . . . . . 8 |- ( F e. ( ( C^n ` RR ) ` 0 ) <-> ( F e. ( CC ^pm RR ) /\ ( ( RR Dn F ) ` 0 ) e. ( dom F -cn-> CC ) ) )
56	55	simprbi 471	. . . . . . 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 2550	. . . . 5 |- ( ph -> F e. ( dom F -cn-> CC ) )
59		cncffvrn 22008	. . . . 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 676	. . . 4 |- ( ph -> ( F e. ( dom F -cn-> RR ) <-> F : dom F --> RR ) )
61	17, 60	mpbird 240	. . 3 |- ( ph -> F e. ( dom F -cn-> RR ) )
62		rescncf 22007	. . 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 61	. 2 |- ( ph -> ( F |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> RR ) )
64	1, 2, 9, 48, 63	c1lip1 23028	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 189   /\ wa 376   = wceq 1452   e. wcel 1904   A.wral 2756   E.wrex 2757   C_ wss 3390   {cpr 3961   class class class wbr 4395   dom cdm 4839   ran crn 4840   |` cres 4841   Fun wfun 5583   Fn wfn 5584   -->wf 5585   ` cfv 5589  (class class class)co 6308   ^pm cpm 7491   CCcc 9555   RRcr 9556   0cc0 9557   1c1 9558   + caddc 9560   x. cmul 9562   <_ cle 9694   - cmin 9880   NN0cn0 10893   ZZ>=cuz 11182   [,]cicc 11663   abscabs 13374   -cn->ccncf 21986   _D cdv 22897   Dncdvn 22898   C^nccpn 22899

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636  ax-mulf 9637

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-fi 7943  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-ioo 11664  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-hom 15292  df-cco 15293  df-rest 15399  df-topn 15400  df-0g 15418  df-gsum 15419  df-topgen 15420  df-pt 15421  df-prds 15424  df-xrs 15478  df-qtop 15484  df-imas 15485  df-xps 15488  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-mulg 16754  df-cntz 17049  df-cmn 17510  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-fbas 19044  df-fg 19045  df-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cld 20111  df-ntr 20112  df-cls 20113  df-nei 20191  df-lp 20229  df-perf 20230  df-cn 20320  df-cnp 20321  df-haus 20408  df-cmp 20479  df-tx 20654  df-hmeo 20847  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033  df-xms 21413  df-ms 21414  df-tms 21415  df-cncf 21988  df-limc 22900  df-dv 22901  df-dvn 22902  df-cpn 22903

This theorem is referenced by:  c1lip3  23030