Theorem dvbdfbdioolem1 40143

Description: Given a function with bounded derivative, on an open interval, here is an absolute bound to the difference of the image of two points in the interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypotheses

Ref	Expression
dvbdfbdioolem1.a	|- ( ph -> A e. RR )
dvbdfbdioolem1.b	|- ( ph -> B e. RR )
dvbdfbdioolem1.f	|- ( ph -> F : ( A (,) B ) --> RR )
dvbdfbdioolem1.dmdv	|- ( ph -> dom ( RR _D F ) = ( A (,) B ) )
dvbdfbdioolem1.k	|- ( ph -> K e. RR )
dvbdfbdioolem1.dvbd	|- ( ph -> A. x e. ( A (,) B ) ( abs ` ( ( RR _D F ) ` x ) ) <_ K )
dvbdfbdioolem1.c	|- ( ph -> C e. ( A (,) B ) )
dvbdfbdioolem1.d	|- ( ph -> D e. ( C (,) B ) )

Assertion

Ref	Expression
dvbdfbdioolem1	|- ( ph -> ( ( abs ` ( ( F ` D ) - ( F ` C ) ) ) <_ ( K x. ( D - C ) ) /\ ( abs ` ( ( F ` D ) - ( F ` C ) ) ) <_ ( K x. ( B - A ) ) ) )

Distinct variable groups:   x, A   x, B   x, C   x, D   x, F   x, K   ph, x

Proof of Theorem dvbdfbdioolem1

Step	Hyp	Ref	Expression
1		ioossre 12235	. . . 4 |- ( A (,) B ) C_ RR
2		dvbdfbdioolem1.c	. . . 4 |- ( ph -> C e. ( A (,) B ) )
3	1, 2	sseldi 3601	. . 3 |- ( ph -> C e. RR )
4		ioossre 12235	. . . 4 |- ( C (,) B ) C_ RR
5		dvbdfbdioolem1.d	. . . 4 |- ( ph -> D e. ( C (,) B ) )
6	4, 5	sseldi 3601	. . 3 |- ( ph -> D e. RR )
7	3	rexrd 10089	. . . 4 |- ( ph -> C e. RR* )
8		dvbdfbdioolem1.b	. . . . 5 |- ( ph -> B e. RR )
9	8	rexrd 10089	. . . 4 |- ( ph -> B e. RR* )
10		ioogtlb 39717	. . . 4 |- ( ( C e. RR* /\ B e. RR* /\ D e. ( C (,) B ) ) -> C < D )
11	7, 9, 5, 10	syl3anc 1326	. . 3 |- ( ph -> C < D )
12		dvbdfbdioolem1.a	. . . . . 6 |- ( ph -> A e. RR )
13	12	rexrd 10089	. . . . 5 |- ( ph -> A e. RR* )
14		ioogtlb 39717	. . . . . 6 |- ( ( A e. RR* /\ B e. RR* /\ C e. ( A (,) B ) ) -> A < C )
15	13, 9, 2, 14	syl3anc 1326	. . . . 5 |- ( ph -> A < C )
16		iooltub 39735	. . . . . 6 |- ( ( C e. RR* /\ B e. RR* /\ D e. ( C (,) B ) ) -> D < B )
17	7, 9, 5, 16	syl3anc 1326	. . . . 5 |- ( ph -> D < B )
18		iccssioo 12242	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( A < C /\ D < B ) ) -> ( C [,] D ) C_ ( A (,) B ) )
19	13, 9, 15, 17, 18	syl22anc 1327	. . . 4 |- ( ph -> ( C [,] D ) C_ ( A (,) B ) )
20		dvbdfbdioolem1.f	. . . . 5 |- ( ph -> F : ( A (,) B ) --> RR )
21		ax-resscn 9993	. . . . . . 7 |- RR C_ CC
22	21	a1i 11	. . . . . 6 |- ( ph -> RR C_ CC )
23	20, 22	fssd 6057	. . . . . . 7 |- ( ph -> F : ( A (,) B ) --> CC )
24	1	a1i 11	. . . . . . 7 |- ( ph -> ( A (,) B ) C_ RR )
25		dvbdfbdioolem1.dmdv	. . . . . . 7 |- ( ph -> dom ( RR _D F ) = ( A (,) B ) )
26		dvcn 23684	. . . . . . 7 |- ( ( ( RR C_ CC /\ F : ( A (,) B ) --> CC /\ ( A (,) B ) C_ RR ) /\ dom ( RR _D F ) = ( A (,) B ) ) -> F e. ( ( A (,) B ) -cn-> CC ) )
27	22, 23, 24, 25, 26	syl31anc 1329	. . . . . 6 |- ( ph -> F e. ( ( A (,) B ) -cn-> CC ) )
28		cncffvrn 22701	. . . . . 6 |- ( ( RR C_ CC /\ F e. ( ( A (,) B ) -cn-> CC ) ) -> ( F e. ( ( A (,) B ) -cn-> RR ) <-> F : ( A (,) B ) --> RR ) )
29	22, 27, 28	syl2anc 693	. . . . 5 |- ( ph -> ( F e. ( ( A (,) B ) -cn-> RR ) <-> F : ( A (,) B ) --> RR ) )
30	20, 29	mpbird 247	. . . 4 |- ( ph -> F e. ( ( A (,) B ) -cn-> RR ) )
31		rescncf 22700	. . . 4 |- ( ( C [,] D ) C_ ( A (,) B ) -> ( F e. ( ( A (,) B ) -cn-> RR ) -> ( F |` ( C [,] D ) ) e. ( ( C [,] D ) -cn-> RR ) ) )
32	19, 30, 31	sylc 65	. . 3 |- ( ph -> ( F |` ( C [,] D ) ) e. ( ( C [,] D ) -cn-> RR ) )
33	19, 24	sstrd 3613	. . . . . . 7 |- ( ph -> ( C [,] D ) C_ RR )
34		eqid 2622	. . . . . . . 8 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
35	34	tgioo2 22606	. . . . . . . 8 |- ( topGen ` ran (,) ) = ( ( TopOpen ` ℂfld ) ↾t RR )
36	34, 35	dvres 23675	. . . . . . 7 |- ( ( ( RR C_ CC /\ F : ( A (,) B ) --> CC ) /\ ( ( A (,) B ) C_ RR /\ ( C [,] D ) C_ RR ) ) -> ( RR _D ( F |` ( C [,] D ) ) ) = ( ( RR _D F ) |` ( ( int ` ( topGen ` ran (,) ) ) ` ( C [,] D ) ) ) )
37	22, 23, 24, 33, 36	syl22anc 1327	. . . . . 6 |- ( ph -> ( RR _D ( F |` ( C [,] D ) ) ) = ( ( RR _D F ) |` ( ( int ` ( topGen ` ran (,) ) ) ` ( C [,] D ) ) ) )
38		iccntr 22624	. . . . . . . 8 |- ( ( C e. RR /\ D e. RR ) -> ( ( int ` ( topGen ` ran (,) ) ) ` ( C [,] D ) ) = ( C (,) D ) )
39	3, 6, 38	syl2anc 693	. . . . . . 7 |- ( ph -> ( ( int ` ( topGen ` ran (,) ) ) ` ( C [,] D ) ) = ( C (,) D ) )
40	39	reseq2d 5396	. . . . . 6 |- ( ph -> ( ( RR _D F ) |` ( ( int ` ( topGen ` ran (,) ) ) ` ( C [,] D ) ) ) = ( ( RR _D F ) |` ( C (,) D ) ) )
41	37, 40	eqtrd 2656	. . . . 5 |- ( ph -> ( RR _D ( F |` ( C [,] D ) ) ) = ( ( RR _D F ) |` ( C (,) D ) ) )
42	41	dmeqd 5326	. . . 4 |- ( ph -> dom ( RR _D ( F |` ( C [,] D ) ) ) = dom ( ( RR _D F ) |` ( C (,) D ) ) )
43	12, 3, 15	ltled 10185	. . . . . . 7 |- ( ph -> A <_ C )
44	6, 8, 17	ltled 10185	. . . . . . 7 |- ( ph -> D <_ B )
45		ioossioo 12265	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( A <_ C /\ D <_ B ) ) -> ( C (,) D ) C_ ( A (,) B ) )
46	13, 9, 43, 44, 45	syl22anc 1327	. . . . . 6 |- ( ph -> ( C (,) D ) C_ ( A (,) B ) )
47	46, 25	sseqtr4d 3642	. . . . 5 |- ( ph -> ( C (,) D ) C_ dom ( RR _D F ) )
48		ssdmres 5420	. . . . 5 |- ( ( C (,) D ) C_ dom ( RR _D F ) <-> dom ( ( RR _D F ) |` ( C (,) D ) ) = ( C (,) D ) )
49	47, 48	sylib 208	. . . 4 |- ( ph -> dom ( ( RR _D F ) |` ( C (,) D ) ) = ( C (,) D ) )
50	42, 49	eqtrd 2656	. . 3 |- ( ph -> dom ( RR _D ( F |` ( C [,] D ) ) ) = ( C (,) D ) )
51	3, 6, 11, 32, 50	mvth 23755	. 2 |- ( ph -> E. x e. ( C (,) D ) ( ( RR _D ( F |` ( C [,] D ) ) ) ` x ) = ( ( ( ( F |` ( C [,] D ) ) ` D ) - ( ( F |` ( C [,] D ) ) ` C ) ) / ( D - C ) ) )
52	41	fveq1d 6193	. . . . . . . . 9 |- ( ph -> ( ( RR _D ( F |` ( C [,] D ) ) ) ` x ) = ( ( ( RR _D F ) |` ( C (,) D ) ) ` x ) )
53		fvres 6207	. . . . . . . . 9 |- ( x e. ( C (,) D ) -> ( ( ( RR _D F ) |` ( C (,) D ) ) ` x ) = ( ( RR _D F ) ` x ) )
54	52, 53	sylan9eq 2676	. . . . . . . 8 |- ( ( ph /\ x e. ( C (,) D ) ) -> ( ( RR _D ( F |` ( C [,] D ) ) ) ` x ) = ( ( RR _D F ) ` x ) )
55	54	eqcomd 2628	. . . . . . 7 |- ( ( ph /\ x e. ( C (,) D ) ) -> ( ( RR _D F ) ` x ) = ( ( RR _D ( F |` ( C [,] D ) ) ) ` x ) )
56	55	3adant3 1081	. . . . . 6 |- ( ( ph /\ x e. ( C (,) D ) /\ ( ( RR _D ( F |` ( C [,] D ) ) ) ` x ) = ( ( ( ( F |` ( C [,] D ) ) ` D ) - ( ( F |` ( C [,] D ) ) ` C ) ) / ( D - C ) ) ) -> ( ( RR _D F ) ` x ) = ( ( RR _D ( F |` ( C [,] D ) ) ) ` x ) )
57		simp3 1063	. . . . . 6 |- ( ( ph /\ x e. ( C (,) D ) /\ ( ( RR _D ( F |` ( C [,] D ) ) ) ` x ) = ( ( ( ( F |` ( C [,] D ) ) ` D ) - ( ( F |` ( C [,] D ) ) ` C ) ) / ( D - C ) ) ) -> ( ( RR _D ( F |` ( C [,] D ) ) ) ` x ) = ( ( ( ( F |` ( C [,] D ) ) ` D ) - ( ( F |` ( C [,] D ) ) ` C ) ) / ( D - C ) ) )
58	6	rexrd 10089	. . . . . . . . . . 11 |- ( ph -> D e. RR* )
59	3, 6, 11	ltled 10185	. . . . . . . . . . 11 |- ( ph -> C <_ D )
60		ubicc2 12289	. . . . . . . . . . 11 |- ( ( C e. RR* /\ D e. RR* /\ C <_ D ) -> D e. ( C [,] D ) )
61	7, 58, 59, 60	syl3anc 1326	. . . . . . . . . 10 |- ( ph -> D e. ( C [,] D ) )
62		fvres 6207	. . . . . . . . . 10 |- ( D e. ( C [,] D ) -> ( ( F |` ( C [,] D ) ) ` D ) = ( F ` D ) )
63	61, 62	syl 17	. . . . . . . . 9 |- ( ph -> ( ( F |` ( C [,] D ) ) ` D ) = ( F ` D ) )
64		lbicc2 12288	. . . . . . . . . . 11 |- ( ( C e. RR* /\ D e. RR* /\ C <_ D ) -> C e. ( C [,] D ) )
65	7, 58, 59, 64	syl3anc 1326	. . . . . . . . . 10 |- ( ph -> C e. ( C [,] D ) )
66		fvres 6207	. . . . . . . . . 10 |- ( C e. ( C [,] D ) -> ( ( F |` ( C [,] D ) ) ` C ) = ( F ` C ) )
67	65, 66	syl 17	. . . . . . . . 9 |- ( ph -> ( ( F |` ( C [,] D ) ) ` C ) = ( F ` C ) )
68	63, 67	oveq12d 6668	. . . . . . . 8 |- ( ph -> ( ( ( F |` ( C [,] D ) ) ` D ) - ( ( F |` ( C [,] D ) ) ` C ) ) = ( ( F ` D ) - ( F ` C ) ) )
69	68	oveq1d 6665	. . . . . . 7 |- ( ph -> ( ( ( ( F |` ( C [,] D ) ) ` D ) - ( ( F |` ( C [,] D ) ) ` C ) ) / ( D - C ) ) = ( ( ( F ` D ) - ( F ` C ) ) / ( D - C ) ) )
70	69	3ad2ant1 1082	. . . . . 6 |- ( ( ph /\ x e. ( C (,) D ) /\ ( ( RR _D ( F |` ( C [,] D ) ) ) ` x ) = ( ( ( ( F |` ( C [,] D ) ) ` D ) - ( ( F |` ( C [,] D ) ) ` C ) ) / ( D - C ) ) ) -> ( ( ( ( F |` ( C [,] D ) ) ` D ) - ( ( F |` ( C [,] D ) ) ` C ) ) / ( D - C ) ) = ( ( ( F ` D ) - ( F ` C ) ) / ( D - C ) ) )
71	56, 57, 70	3eqtrd 2660	. . . . 5 |- ( ( ph /\ x e. ( C (,) D ) /\ ( ( RR _D ( F |` ( C [,] D ) ) ) ` x ) = ( ( ( ( F |` ( C [,] D ) ) ` D ) - ( ( F |` ( C [,] D ) ) ` C ) ) / ( D - C ) ) ) -> ( ( RR _D F ) ` x ) = ( ( ( F ` D ) - ( F ` C ) ) / ( D - C ) ) )
72		simp3 1063	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( C (,) D ) /\ ( ( RR _D F ) ` x ) = ( ( ( F ` D ) - ( F ` C ) ) / ( D - C ) ) ) -> ( ( RR _D F ) ` x ) = ( ( ( F ` D ) - ( F ` C ) ) / ( D - C ) ) )
73	72	eqcomd 2628	. . . . . . . . . 10 |- ( ( ph /\ x e. ( C (,) D ) /\ ( ( RR _D F ) ` x ) = ( ( ( F ` D ) - ( F ` C ) ) / ( D - C ) ) ) -> ( ( ( F ` D ) - ( F ` C ) ) / ( D - C ) ) = ( ( RR _D F ) ` x ) )
74	19, 61	sseldd 3604	. . . . . . . . . . . . . . 15 |- ( ph -> D e. ( A (,) B ) )
75	20, 74	ffvelrnd 6360	. . . . . . . . . . . . . 14 |- ( ph -> ( F ` D ) e. RR )
76	20, 2	ffvelrnd 6360	. . . . . . . . . . . . . 14 |- ( ph -> ( F ` C ) e. RR )
77	75, 76	resubcld 10458	. . . . . . . . . . . . 13 |- ( ph -> ( ( F ` D ) - ( F ` C ) ) e. RR )
78	77	recnd 10068	. . . . . . . . . . . 12 |- ( ph -> ( ( F ` D ) - ( F ` C ) ) e. CC )
79	78	3ad2ant1 1082	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( C (,) D ) /\ ( ( RR _D F ) ` x ) = ( ( ( F ` D ) - ( F ` C ) ) / ( D - C ) ) ) -> ( ( F ` D ) - ( F ` C ) ) e. CC )
80		dvfre 23714	. . . . . . . . . . . . . . . . 17 |- ( ( F : ( A (,) B ) --> RR /\ ( A (,) B ) C_ RR ) -> ( RR _D F ) : dom ( RR _D F ) --> RR )
81	20, 24, 80	syl2anc 693	. . . . . . . . . . . . . . . 16 |- ( ph -> ( RR _D F ) : dom ( RR _D F ) --> RR )
82	25	feq2d 6031	. . . . . . . . . . . . . . . 16 |- ( ph -> ( ( RR _D F ) : dom ( RR _D F ) --> RR <-> ( RR _D F ) : ( A (,) B ) --> RR ) )
83	81, 82	mpbid 222	. . . . . . . . . . . . . . 15 |- ( ph -> ( RR _D F ) : ( A (,) B ) --> RR )
84	83	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. ( C (,) D ) ) -> ( RR _D F ) : ( A (,) B ) --> RR )
85	46	sselda 3603	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. ( C (,) D ) ) -> x e. ( A (,) B ) )
86	84, 85	ffvelrnd 6360	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. ( C (,) D ) ) -> ( ( RR _D F ) ` x ) e. RR )
87	86	recnd 10068	. . . . . . . . . . . 12 |- ( ( ph /\ x e. ( C (,) D ) ) -> ( ( RR _D F ) ` x ) e. CC )
88	87	3adant3 1081	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( C (,) D ) /\ ( ( RR _D F ) ` x ) = ( ( ( F ` D ) - ( F ` C ) ) / ( D - C ) ) ) -> ( ( RR _D F ) ` x ) e. CC )
89	6, 3	resubcld 10458	. . . . . . . . . . . . 13 |- ( ph -> ( D - C ) e. RR )
90	89	recnd 10068	. . . . . . . . . . . 12 |- ( ph -> ( D - C ) e. CC )
91	90	3ad2ant1 1082	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( C (,) D ) /\ ( ( RR _D F ) ` x ) = ( ( ( F ` D ) - ( F ` C ) ) / ( D - C ) ) ) -> ( D - C ) e. CC )
92	3, 6	posdifd 10614	. . . . . . . . . . . . . 14 |- ( ph -> ( C < D <-> 0 < ( D - C ) ) )
93	11, 92	mpbid 222	. . . . . . . . . . . . 13 |- ( ph -> 0 < ( D - C ) )
94	93	gt0ne0d 10592	. . . . . . . . . . . 12 |- ( ph -> ( D - C ) =/= 0 )
95	94	3ad2ant1 1082	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( C (,) D ) /\ ( ( RR _D F ) ` x ) = ( ( ( F ` D ) - ( F ` C ) ) / ( D - C ) ) ) -> ( D - C ) =/= 0 )
96	79, 88, 91, 95	divmul3d 10835	. . . . . . . . . 10 |- ( ( ph /\ x e. ( C (,) D ) /\ ( ( RR _D F ) ` x ) = ( ( ( F ` D ) - ( F ` C ) ) / ( D - C ) ) ) -> ( ( ( ( F ` D ) - ( F ` C ) ) / ( D - C ) ) = ( ( RR _D F ) ` x ) <-> ( ( F ` D ) - ( F ` C ) ) = ( ( ( RR _D F ) ` x ) x. ( D - C ) ) ) )
97	73, 96	mpbid 222	. . . . . . . . 9 |- ( ( ph /\ x e. ( C (,) D ) /\ ( ( RR _D F ) ` x ) = ( ( ( F ` D ) - ( F ` C ) ) / ( D - C ) ) ) -> ( ( F ` D ) - ( F ` C ) ) = ( ( ( RR _D F ) ` x ) x. ( D - C ) ) )
98	97	fveq2d 6195	. . . . . . . 8 |- ( ( ph /\ x e. ( C (,) D ) /\ ( ( RR _D F ) ` x ) = ( ( ( F ` D ) - ( F ` C ) ) / ( D - C ) ) ) -> ( abs ` ( ( F ` D ) - ( F ` C ) ) ) = ( abs ` ( ( ( RR _D F ) ` x ) x. ( D - C ) ) ) )
99	90	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ x e. ( C (,) D ) ) -> ( D - C ) e. CC )
100	87, 99	absmuld 14193	. . . . . . . . 9 |- ( ( ph /\ x e. ( C (,) D ) ) -> ( abs ` ( ( ( RR _D F ) ` x ) x. ( D - C ) ) ) = ( ( abs ` ( ( RR _D F ) ` x ) ) x. ( abs ` ( D - C ) ) ) )
101	100	3adant3 1081	. . . . . . . 8 |- ( ( ph /\ x e. ( C (,) D ) /\ ( ( RR _D F ) ` x ) = ( ( ( F ` D ) - ( F ` C ) ) / ( D - C ) ) ) -> ( abs ` ( ( ( RR _D F ) ` x ) x. ( D - C ) ) ) = ( ( abs ` ( ( RR _D F ) ` x ) ) x. ( abs ` ( D - C ) ) ) )
102	98, 101	eqtrd 2656	. . . . . . 7 |- ( ( ph /\ x e. ( C (,) D ) /\ ( ( RR _D F ) ` x ) = ( ( ( F ` D ) - ( F ` C ) ) / ( D - C ) ) ) -> ( abs ` ( ( F ` D ) - ( F ` C ) ) ) = ( ( abs ` ( ( RR _D F ) ` x ) ) x. ( abs ` ( D - C ) ) ) )
103	3, 6, 59	abssubge0d 14170	. . . . . . . . 9 |- ( ph -> ( abs ` ( D - C ) ) = ( D - C ) )
104	103	oveq2d 6666	. . . . . . . 8 |- ( ph -> ( ( abs ` ( ( RR _D F ) ` x ) ) x. ( abs ` ( D - C ) ) ) = ( ( abs ` ( ( RR _D F ) ` x ) ) x. ( D - C ) ) )
105	104	3ad2ant1 1082	. . . . . . 7 |- ( ( ph /\ x e. ( C (,) D ) /\ ( ( RR _D F ) ` x ) = ( ( ( F ` D ) - ( F ` C ) ) / ( D - C ) ) ) -> ( ( abs ` ( ( RR _D F ) ` x ) ) x. ( abs ` ( D - C ) ) ) = ( ( abs ` ( ( RR _D F ) ` x ) ) x. ( D - C ) ) )
106	102, 105	eqtrd 2656	. . . . . 6 |- ( ( ph /\ x e. ( C (,) D ) /\ ( ( RR _D F ) ` x ) = ( ( ( F ` D ) - ( F ` C ) ) / ( D - C ) ) ) -> ( abs ` ( ( F ` D ) - ( F ` C ) ) ) = ( ( abs ` ( ( RR _D F ) ` x ) ) x. ( D - C ) ) )
107	87	abscld 14175	. . . . . . . 8 |- ( ( ph /\ x e. ( C (,) D ) ) -> ( abs ` ( ( RR _D F ) ` x ) ) e. RR )
108		dvbdfbdioolem1.k	. . . . . . . . 9 |- ( ph -> K e. RR )
109	108	adantr 481	. . . . . . . 8 |- ( ( ph /\ x e. ( C (,) D ) ) -> K e. RR )
110	89	adantr 481	. . . . . . . 8 |- ( ( ph /\ x e. ( C (,) D ) ) -> ( D - C ) e. RR )
111		0red 10041	. . . . . . . . . 10 |- ( ph -> 0 e. RR )
112	111, 89, 93	ltled 10185	. . . . . . . . 9 |- ( ph -> 0 <_ ( D - C ) )
113	112	adantr 481	. . . . . . . 8 |- ( ( ph /\ x e. ( C (,) D ) ) -> 0 <_ ( D - C ) )
114		dvbdfbdioolem1.dvbd	. . . . . . . . . 10 |- ( ph -> A. x e. ( A (,) B ) ( abs ` ( ( RR _D F ) ` x ) ) <_ K )
115	114	adantr 481	. . . . . . . . 9 |- ( ( ph /\ x e. ( C (,) D ) ) -> A. x e. ( A (,) B ) ( abs ` ( ( RR _D F ) ` x ) ) <_ K )
116		rspa 2930	. . . . . . . . 9 |- ( ( A. x e. ( A (,) B ) ( abs ` ( ( RR _D F ) ` x ) ) <_ K /\ x e. ( A (,) B ) ) -> ( abs ` ( ( RR _D F ) ` x ) ) <_ K )
117	115, 85, 116	syl2anc 693	. . . . . . . 8 |- ( ( ph /\ x e. ( C (,) D ) ) -> ( abs ` ( ( RR _D F ) ` x ) ) <_ K )
118	107, 109, 110, 113, 117	lemul1ad 10963	. . . . . . 7 |- ( ( ph /\ x e. ( C (,) D ) ) -> ( ( abs ` ( ( RR _D F ) ` x ) ) x. ( D - C ) ) <_ ( K x. ( D - C ) ) )
119	118	3adant3 1081	. . . . . 6 |- ( ( ph /\ x e. ( C (,) D ) /\ ( ( RR _D F ) ` x ) = ( ( ( F ` D ) - ( F ` C ) ) / ( D - C ) ) ) -> ( ( abs ` ( ( RR _D F ) ` x ) ) x. ( D - C ) ) <_ ( K x. ( D - C ) ) )
120	106, 119	eqbrtrd 4675	. . . . 5 |- ( ( ph /\ x e. ( C (,) D ) /\ ( ( RR _D F ) ` x ) = ( ( ( F ` D ) - ( F ` C ) ) / ( D - C ) ) ) -> ( abs ` ( ( F ` D ) - ( F ` C ) ) ) <_ ( K x. ( D - C ) ) )
121	71, 120	syld3an3 1371	. . . 4 |- ( ( ph /\ x e. ( C (,) D ) /\ ( ( RR _D ( F |` ( C [,] D ) ) ) ` x ) = ( ( ( ( F |` ( C [,] D ) ) ` D ) - ( ( F |` ( C [,] D ) ) ` C ) ) / ( D - C ) ) ) -> ( abs ` ( ( F ` D ) - ( F ` C ) ) ) <_ ( K x. ( D - C ) ) )
122	99	abscld 14175	. . . . . . . 8 |- ( ( ph /\ x e. ( C (,) D ) ) -> ( abs ` ( D - C ) ) e. RR )
123	8, 12	resubcld 10458	. . . . . . . . 9 |- ( ph -> ( B - A ) e. RR )
124	123	adantr 481	. . . . . . . 8 |- ( ( ph /\ x e. ( C (,) D ) ) -> ( B - A ) e. RR )
125	87	absge0d 14183	. . . . . . . 8 |- ( ( ph /\ x e. ( C (,) D ) ) -> 0 <_ ( abs ` ( ( RR _D F ) ` x ) ) )
126	99	absge0d 14183	. . . . . . . 8 |- ( ( ph /\ x e. ( C (,) D ) ) -> 0 <_ ( abs ` ( D - C ) ) )
127	6, 12, 8, 3, 44, 43	le2subd 10647	. . . . . . . . . 10 |- ( ph -> ( D - C ) <_ ( B - A ) )
128	103, 127	eqbrtrd 4675	. . . . . . . . 9 |- ( ph -> ( abs ` ( D - C ) ) <_ ( B - A ) )
129	128	adantr 481	. . . . . . . 8 |- ( ( ph /\ x e. ( C (,) D ) ) -> ( abs ` ( D - C ) ) <_ ( B - A ) )
130	107, 109, 122, 124, 125, 126, 117, 129	lemul12ad 10966	. . . . . . 7 |- ( ( ph /\ x e. ( C (,) D ) ) -> ( ( abs ` ( ( RR _D F ) ` x ) ) x. ( abs ` ( D - C ) ) ) <_ ( K x. ( B - A ) ) )
131	130	3adant3 1081	. . . . . 6 |- ( ( ph /\ x e. ( C (,) D ) /\ ( ( RR _D F ) ` x ) = ( ( ( F ` D ) - ( F ` C ) ) / ( D - C ) ) ) -> ( ( abs ` ( ( RR _D F ) ` x ) ) x. ( abs ` ( D - C ) ) ) <_ ( K x. ( B - A ) ) )
132	102, 131	eqbrtrd 4675	. . . . 5 |- ( ( ph /\ x e. ( C (,) D ) /\ ( ( RR _D F ) ` x ) = ( ( ( F ` D ) - ( F ` C ) ) / ( D - C ) ) ) -> ( abs ` ( ( F ` D ) - ( F ` C ) ) ) <_ ( K x. ( B - A ) ) )
133	71, 132	syld3an3 1371	. . . 4 |- ( ( ph /\ x e. ( C (,) D ) /\ ( ( RR _D ( F |` ( C [,] D ) ) ) ` x ) = ( ( ( ( F |` ( C [,] D ) ) ` D ) - ( ( F |` ( C [,] D ) ) ` C ) ) / ( D - C ) ) ) -> ( abs ` ( ( F ` D ) - ( F ` C ) ) ) <_ ( K x. ( B - A ) ) )
134	121, 133	jca 554	. . 3 |- ( ( ph /\ x e. ( C (,) D ) /\ ( ( RR _D ( F |` ( C [,] D ) ) ) ` x ) = ( ( ( ( F |` ( C [,] D ) ) ` D ) - ( ( F |` ( C [,] D ) ) ` C ) ) / ( D - C ) ) ) -> ( ( abs ` ( ( F ` D ) - ( F ` C ) ) ) <_ ( K x. ( D - C ) ) /\ ( abs ` ( ( F ` D ) - ( F ` C ) ) ) <_ ( K x. ( B - A ) ) ) )
135	134	rexlimdv3a 3033	. 2 |- ( ph -> ( E. x e. ( C (,) D ) ( ( RR _D ( F |` ( C [,] D ) ) ) ` x ) = ( ( ( ( F |` ( C [,] D ) ) ` D ) - ( ( F |` ( C [,] D ) ) ` C ) ) / ( D - C ) ) -> ( ( abs ` ( ( F ` D ) - ( F ` C ) ) ) <_ ( K x. ( D - C ) ) /\ ( abs ` ( ( F ` D ) - ( F ` C ) ) ) <_ ( K x. ( B - A ) ) ) ) )
136	51, 135	mpd 15	1 |- ( ph -> ( ( abs ` ( ( F ` D ) - ( F ` C ) ) ) <_ ( K x. ( D - C ) ) /\ ( abs ` ( ( F ` D ) - ( F ` C ) ) ) <_ ( K x. ( B - A ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   C_ wss 3574   class class class wbr 4653   dom cdm 5114   ran crn 5115   |` cres 5116   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   x. cmul 9941   RR*cxr 10073   < clt 10074   <_ cle 10075   - cmin 10266   / cdiv 10684   (,)cioo 12175   [,]cicc 12178   abscabs 13974   TopOpenctopn 16082   topGenctg 16098  ℂ_fldccnfld 19746   intcnt 20821   -cn->ccncf 22679   _D cdv 23627

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

This theorem is referenced by:  dvbdfbdioolem2  40144  ioodvbdlimc1lem1  40146  ioodvbdlimc1lem2  40147  ioodvbdlimc2lem  40149