Theorem c1lip2 23761

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 9993	. . . . 5 |- RR C_ CC
5		1nn0 11308	. . . . 5 |- 1 e. NN0
6		elcpn 23697	. . . . 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 708	. . . 4 |- ( F e. ( ( C^n ` RR ) ` 1 ) <-> ( F e. ( CC ^pm RR ) /\ ( ( RR Dn F ) ` 1 ) e. ( dom F -cn-> CC ) ) )
8	7	simplbi 476	. . 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 7877	. . . . . . . . 9 |- ( F e. ( CC ^pm RR ) -> Fun F )
12	9, 11	syl 17	. . . . . . . 8 |- ( ph -> Fun F )
13		funfn 5918	. . . . . . . 8 |- ( Fun F <-> F Fn dom F )
14	12, 13	sylib 208	. . . . . . 7 |- ( ph -> F Fn dom F )
15		c1lip2.rn	. . . . . . 7 |- ( ph -> ran F C_ RR )
16		df-f 5892	. . . . . . 7 |- ( F : dom F --> RR <-> ( F Fn dom F /\ ran F C_ RR ) )
17	14, 15, 16	sylanbrc 698	. . . . . 6 |- ( ph -> F : dom F --> RR )
18		cnex 10017	. . . . . . . . 9 |- CC e. _V
19		reex 10027	. . . . . . . . 9 |- RR e. _V
20	18, 19	elpm2 7889	. . . . . . . 8 |- ( F e. ( CC ^pm RR ) <-> ( F : dom F --> CC /\ dom F C_ RR ) )
21	20	simprbi 480	. . . . . . 7 |- ( F e. ( CC ^pm RR ) -> dom F C_ RR )
22	9, 21	syl 17	. . . . . 6 |- ( ph -> dom F C_ RR )
23		dvfre 23714	. . . . . 6 |- ( ( F : dom F --> RR /\ dom F C_ RR ) -> ( RR _D F ) : dom ( RR _D F ) --> RR )
24	17, 22, 23	syl2anc 693	. . . . 5 |- ( ph -> ( RR _D F ) : dom ( RR _D F ) --> RR )
25		0p1e1 11132	. . . . . . . . . . 11 |- ( 0 + 1 ) = 1
26	25	fveq2i 6194	. . . . . . . . . 10 |- ( ( RR Dn F ) ` ( 0 + 1 ) ) = ( ( RR Dn F ) ` 1 )
27		0nn0 11307	. . . . . . . . . . . 12 |- 0 e. NN0
28		dvnp1 23688	. . . . . . . . . . . 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 1415	. . . . . . . . . . 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 2670	. . . . . . . . 9 |- ( ph -> ( ( RR Dn F ) ` 1 ) = ( RR _D ( ( RR Dn F ) ` 0 ) ) )
32		dvn0 23687	. . . . . . . . . . 11 |- ( ( RR C_ CC /\ F e. ( CC ^pm RR ) ) -> ( ( RR Dn F ) ` 0 ) = F )
33	4, 9, 32	sylancr 695	. . . . . . . . . 10 |- ( ph -> ( ( RR Dn F ) ` 0 ) = F )
34	33	oveq2d 6666	. . . . . . . . 9 |- ( ph -> ( RR _D ( ( RR Dn F ) ` 0 ) ) = ( RR _D F ) )
35	31, 34	eqtrd 2656	. . . . . . . 8 |- ( ph -> ( ( RR Dn F ) ` 1 ) = ( RR _D F ) )
36	7	simprbi 480	. . . . . . . . 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 2702	. . . . . . 7 |- ( ph -> ( RR _D F ) e. ( dom F -cn-> CC ) )
39		cncff 22696	. . . . . . 7 |- ( ( RR _D F ) e. ( dom F -cn-> CC ) -> ( RR _D F ) : dom F --> CC )
40		fdm 6051	. . . . . . 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 6031	. . . . 5 |- ( ph -> ( ( RR _D F ) : dom ( RR _D F ) --> RR <-> ( RR _D F ) : dom F --> RR ) )
43	24, 42	mpbid 222	. . . 4 |- ( ph -> ( RR _D F ) : dom F --> RR )
44		cncffvrn 22701	. . . . 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 695	. . . 4 |- ( ph -> ( ( RR _D F ) e. ( dom F -cn-> RR ) <-> ( RR _D F ) : dom F --> RR ) )
46	43, 45	mpbird 247	. . 3 |- ( ph -> ( RR _D F ) e. ( dom F -cn-> RR ) )
47		rescncf 22700	. . 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 65	. 2 |- ( ph -> ( ( RR _D F ) |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> RR ) )
49	19	prid1 4297	. . . . . . . . 9 |- RR e. { RR , CC }
50		1eluzge0 11732	. . . . . . . . 9 |- 1 e. ( ZZ>= ` 0 )
51		cpnord 23698	. . . . . . . . 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 1424	. . . . . . . 8 |- ( ( C^n ` RR ) ` 1 ) C_ ( ( C^n ` RR ) ` 0 )
53	52, 3	sseldi 3601	. . . . . . 7 |- ( ph -> F e. ( ( C^n ` RR ) ` 0 ) )
54		elcpn 23697	. . . . . . . . 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 708	. . . . . . . 8 |- ( F e. ( ( C^n ` RR ) ` 0 ) <-> ( F e. ( CC ^pm RR ) /\ ( ( RR Dn F ) ` 0 ) e. ( dom F -cn-> CC ) ) )
56	55	simprbi 480	. . . . . . 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 2702	. . . . 5 |- ( ph -> F e. ( dom F -cn-> CC ) )
59		cncffvrn 22701	. . . . 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 695	. . . 4 |- ( ph -> ( F e. ( dom F -cn-> RR ) <-> F : dom F --> RR ) )
61	17, 60	mpbird 247	. . 3 |- ( ph -> F e. ( dom F -cn-> RR ) )
62		rescncf 22700	. . 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 65	. 2 |- ( ph -> ( F |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> RR ) )
64	1, 2, 9, 48, 63	c1lip1 23760	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 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   C_ wss 3574   {cpr 4179   class class class wbr 4653   dom cdm 5114   ran crn 5115   |` cres 5116   Fun wfun 5882   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^pm cpm 7858   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   <_ cle 10075   - cmin 10266   NN0cn0 11292   ZZ>=cuz 11687   [,]cicc 12178   abscabs 13974   -cn->ccncf 22679   _D cdv 23627   Dncdvn 23628   C^nccpn 23629

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-cmp 21190  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-limc 23630  df-dv 23631  df-dvn 23632  df-cpn 23633

This theorem is referenced by:  c1lip3  23762