Theorem c1liplem1 21310

Description: Lemma for c1lip1 21311. (Contributed by Stefan O'Rear, 15-Nov-2014.)

Hypotheses

Ref	Expression
c1liplem1.a	|- ( ph -> A e. RR )
c1liplem1.b	|- ( ph -> B e. RR )
c1liplem1.le	|- ( ph -> A <_ B )
c1liplem1.f	|- ( ph -> F e. ( CC ^pm RR ) )
c1liplem1.dv	|- ( ph -> ( ( RR _D F ) |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> RR ) )
c1liplem1.cn	|- ( ph -> ( F |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> RR ) )
c1liplem1.k	|- K = sup ( ( abs " ( ( RR _D F ) " ( A [,] B ) ) ) , RR , < )

Assertion

Ref	Expression
c1liplem1	|- ( ph -> ( K e. RR /\ A. x e. ( A [,] B ) A. y e. ( A [,] B ) ( x < y -> ( abs ` ( ( F ` y ) - ( F ` x ) ) ) <_ ( K x. ( abs ` ( y - x ) ) ) ) ) )

Distinct variable groups:   ph, x, y   x, A, y   x, B, y   x, F, y

Allowed substitution hints:   K( x, y)

Proof of Theorem c1liplem1

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		c1liplem1.k	. . 3 |- K = sup ( ( abs " ( ( RR _D F ) " ( A [,] B ) ) ) , RR , < )
2		imassrn 5168	. . . . . 6 |- ( abs " ( ( RR _D F ) " ( A [,] B ) ) ) C_ ran abs
3		absf 12809	. . . . . . 7 |- abs : CC --> RR
4		frn 5553	. . . . . . 7 |- ( abs : CC --> RR -> ran abs C_ RR )
5	3, 4	ax-mp 5	. . . . . 6 |- ran abs C_ RR
6	2, 5	sstri 3353	. . . . 5 |- ( abs " ( ( RR _D F ) " ( A [,] B ) ) ) C_ RR
7	6	a1i 11	. . . 4 |- ( ph -> ( abs " ( ( RR _D F ) " ( A [,] B ) ) ) C_ RR )
8		dvf 21224	. . . . . . . 8 |- ( RR _D F ) : dom ( RR _D F ) --> CC
9		ffun 5549	. . . . . . . 8 |- ( ( RR _D F ) : dom ( RR _D F ) --> CC -> Fun ( RR _D F ) )
10	8, 9	ax-mp 5	. . . . . . 7 |- Fun ( RR _D F )
11	10	a1i 11	. . . . . 6 |- ( ph -> Fun ( RR _D F ) )
12		c1liplem1.dv	. . . . . . . 8 |- ( ph -> ( ( RR _D F ) |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> RR ) )
13		cncff 20311	. . . . . . . 8 |- ( ( ( RR _D F ) |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> RR ) -> ( ( RR _D F ) |` ( A [,] B ) ) : ( A [,] B ) --> RR )
14		fdm 5551	. . . . . . . 8 |- ( ( ( RR _D F ) |` ( A [,] B ) ) : ( A [,] B ) --> RR -> dom ( ( RR _D F ) |` ( A [,] B ) ) = ( A [,] B ) )
15	12, 13, 14	3syl 20	. . . . . . 7 |- ( ph -> dom ( ( RR _D F ) |` ( A [,] B ) ) = ( A [,] B ) )
16		ssdmres 5120	. . . . . . 7 |- ( ( A [,] B ) C_ dom ( RR _D F ) <-> dom ( ( RR _D F ) |` ( A [,] B ) ) = ( A [,] B ) )
17	15, 16	sylibr 212	. . . . . 6 |- ( ph -> ( A [,] B ) C_ dom ( RR _D F ) )
18		c1liplem1.a	. . . . . . . 8 |- ( ph -> A e. RR )
19	18	rexrd 9421	. . . . . . 7 |- ( ph -> A e. RR* )
20		c1liplem1.b	. . . . . . . 8 |- ( ph -> B e. RR )
21	20	rexrd 9421	. . . . . . 7 |- ( ph -> B e. RR* )
22		c1liplem1.le	. . . . . . 7 |- ( ph -> A <_ B )
23		lbicc2 11388	. . . . . . 7 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> A e. ( A [,] B ) )
24	19, 21, 22, 23	syl3anc 1211	. . . . . 6 |- ( ph -> A e. ( A [,] B ) )
25		funfvima2 5940	. . . . . . 7 |- ( ( Fun ( RR _D F ) /\ ( A [,] B ) C_ dom ( RR _D F ) ) -> ( A e. ( A [,] B ) -> ( ( RR _D F ) ` A ) e. ( ( RR _D F ) " ( A [,] B ) ) ) )
26	25	imp 429	. . . . . 6 |- ( ( ( Fun ( RR _D F ) /\ ( A [,] B ) C_ dom ( RR _D F ) ) /\ A e. ( A [,] B ) ) -> ( ( RR _D F ) ` A ) e. ( ( RR _D F ) " ( A [,] B ) ) )
27	11, 17, 24, 26	syl21anc 1210	. . . . 5 |- ( ph -> ( ( RR _D F ) ` A ) e. ( ( RR _D F ) " ( A [,] B ) ) )
28		ffun 5549	. . . . . . 7 |- ( abs : CC --> RR -> Fun abs )
29	3, 28	ax-mp 5	. . . . . 6 |- Fun abs
30		imassrn 5168	. . . . . . . 8 |- ( ( RR _D F ) " ( A [,] B ) ) C_ ran ( RR _D F )
31		frn 5553	. . . . . . . . 9 |- ( ( RR _D F ) : dom ( RR _D F ) --> CC -> ran ( RR _D F ) C_ CC )
32	8, 31	ax-mp 5	. . . . . . . 8 |- ran ( RR _D F ) C_ CC
33	30, 32	sstri 3353	. . . . . . 7 |- ( ( RR _D F ) " ( A [,] B ) ) C_ CC
34	3	fdmi 5552	. . . . . . 7 |- dom abs = CC
35	33, 34	sseqtr4i 3377	. . . . . 6 |- ( ( RR _D F ) " ( A [,] B ) ) C_ dom abs
36		funfvima2 5940	. . . . . 6 |- ( ( Fun abs /\ ( ( RR _D F ) " ( A [,] B ) ) C_ dom abs ) -> ( ( ( RR _D F ) ` A ) e. ( ( RR _D F ) " ( A [,] B ) ) -> ( abs ` ( ( RR _D F ) ` A ) ) e. ( abs " ( ( RR _D F ) " ( A [,] B ) ) ) ) )
37	29, 35, 36	mp2an 665	. . . . 5 |- ( ( ( RR _D F ) ` A ) e. ( ( RR _D F ) " ( A [,] B ) ) -> ( abs ` ( ( RR _D F ) ` A ) ) e. ( abs " ( ( RR _D F ) " ( A [,] B ) ) ) )
38		ne0i 3631	. . . . 5 |- ( ( abs ` ( ( RR _D F ) ` A ) ) e. ( abs " ( ( RR _D F ) " ( A [,] B ) ) ) -> ( abs " ( ( RR _D F ) " ( A [,] B ) ) ) =/= (/) )
39	27, 37, 38	3syl 20	. . . 4 |- ( ph -> ( abs " ( ( RR _D F ) " ( A [,] B ) ) ) =/= (/) )
40		ax-resscn 9327	. . . . . . . 8 |- RR C_ CC
41		ssid 3363	. . . . . . . 8 |- CC C_ CC
42		cncfss 20317	. . . . . . . 8 |- ( ( RR C_ CC /\ CC C_ CC ) -> ( ( A [,] B ) -cn-> RR ) C_ ( ( A [,] B ) -cn-> CC ) )
43	40, 41, 42	mp2an 665	. . . . . . 7 |- ( ( A [,] B ) -cn-> RR ) C_ ( ( A [,] B ) -cn-> CC )
44	43, 12	sseldi 3342	. . . . . 6 |- ( ph -> ( ( RR _D F ) |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> CC ) )
45		cniccbdd 20787	. . . . . 6 |- ( ( A e. RR /\ B e. RR /\ ( ( RR _D F ) |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> CC ) ) -> E. a e. RR A. x e. ( A [,] B ) ( abs ` ( ( ( RR _D F ) |` ( A [,] B ) ) ` x ) ) <_ a )
46	18, 20, 44, 45	syl3anc 1211	. . . . 5 |- ( ph -> E. a e. RR A. x e. ( A [,] B ) ( abs ` ( ( ( RR _D F ) |` ( A [,] B ) ) ` x ) ) <_ a )
47		fvelima 5731	. . . . . . . . . 10 |- ( ( Fun abs /\ b e. ( abs " ( ( RR _D F ) " ( A [,] B ) ) ) ) -> E. y e. ( ( RR _D F ) " ( A [,] B ) ) ( abs ` y ) = b )
48	29, 47	mpan 663	. . . . . . . . 9 |- ( b e. ( abs " ( ( RR _D F ) " ( A [,] B ) ) ) -> E. y e. ( ( RR _D F ) " ( A [,] B ) ) ( abs ` y ) = b )
49		fvelima 5731	. . . . . . . . . . . . . 14 |- ( ( Fun ( RR _D F ) /\ y e. ( ( RR _D F ) " ( A [,] B ) ) ) -> E. b e. ( A [,] B ) ( ( RR _D F ) ` b ) = y )
50	10, 49	mpan 663	. . . . . . . . . . . . 13 |- ( y e. ( ( RR _D F ) " ( A [,] B ) ) -> E. b e. ( A [,] B ) ( ( RR _D F ) ` b ) = y )
51		fvres 5692	. . . . . . . . . . . . . . . . . . 19 |- ( b e. ( A [,] B ) -> ( ( ( RR _D F ) |` ( A [,] B ) ) ` b ) = ( ( RR _D F ) ` b ) )
52	51	adantl 463	. . . . . . . . . . . . . . . . . 18 |- ( ( A. x e. ( A [,] B ) ( abs ` ( ( ( RR _D F ) |` ( A [,] B ) ) ` x ) ) <_ a /\ b e. ( A [,] B ) ) -> ( ( ( RR _D F ) |` ( A [,] B ) ) ` b ) = ( ( RR _D F ) ` b ) )
53	52	fveq2d 5683	. . . . . . . . . . . . . . . . 17 |- ( ( A. x e. ( A [,] B ) ( abs ` ( ( ( RR _D F ) |` ( A [,] B ) ) ` x ) ) <_ a /\ b e. ( A [,] B ) ) -> ( abs ` ( ( ( RR _D F ) |` ( A [,] B ) ) ` b ) ) = ( abs ` ( ( RR _D F ) ` b ) ) )
54		fveq2 5679	. . . . . . . . . . . . . . . . . . . 20 |- ( x = b -> ( ( ( RR _D F ) |` ( A [,] B ) ) ` x ) = ( ( ( RR _D F ) |` ( A [,] B ) ) ` b ) )
55	54	fveq2d 5683	. . . . . . . . . . . . . . . . . . 19 |- ( x = b -> ( abs ` ( ( ( RR _D F ) |` ( A [,] B ) ) ` x ) ) = ( abs ` ( ( ( RR _D F ) |` ( A [,] B ) ) ` b ) ) )
56	55	breq1d 4290	. . . . . . . . . . . . . . . . . 18 |- ( x = b -> ( ( abs ` ( ( ( RR _D F ) |` ( A [,] B ) ) ` x ) ) <_ a <-> ( abs ` ( ( ( RR _D F ) |` ( A [,] B ) ) ` b ) ) <_ a ) )
57	56	rspccva 3061	. . . . . . . . . . . . . . . . 17 |- ( ( A. x e. ( A [,] B ) ( abs ` ( ( ( RR _D F ) |` ( A [,] B ) ) ` x ) ) <_ a /\ b e. ( A [,] B ) ) -> ( abs ` ( ( ( RR _D F ) |` ( A [,] B ) ) ` b ) ) <_ a )
58	53, 57	eqbrtrrd 4302	. . . . . . . . . . . . . . . 16 |- ( ( A. x e. ( A [,] B ) ( abs ` ( ( ( RR _D F ) |` ( A [,] B ) ) ` x ) ) <_ a /\ b e. ( A [,] B ) ) -> ( abs ` ( ( RR _D F ) ` b ) ) <_ a )
59	58	adantll 706	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ a e. RR ) /\ A. x e. ( A [,] B ) ( abs ` ( ( ( RR _D F ) |` ( A [,] B ) ) ` x ) ) <_ a ) /\ b e. ( A [,] B ) ) -> ( abs ` ( ( RR _D F ) ` b ) ) <_ a )
60		fveq2 5679	. . . . . . . . . . . . . . . 16 |- ( ( ( RR _D F ) ` b ) = y -> ( abs ` ( ( RR _D F ) ` b ) ) = ( abs ` y ) )
61	60	breq1d 4290	. . . . . . . . . . . . . . 15 |- ( ( ( RR _D F ) ` b ) = y -> ( ( abs ` ( ( RR _D F ) ` b ) ) <_ a <-> ( abs ` y ) <_ a ) )
62	59, 61	syl5ibcom 220	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ a e. RR ) /\ A. x e. ( A [,] B ) ( abs ` ( ( ( RR _D F ) |` ( A [,] B ) ) ` x ) ) <_ a ) /\ b e. ( A [,] B ) ) -> ( ( ( RR _D F ) ` b ) = y -> ( abs ` y ) <_ a ) )
63	62	rexlimdva 2831	. . . . . . . . . . . . 13 |- ( ( ( ph /\ a e. RR ) /\ A. x e. ( A [,] B ) ( abs ` ( ( ( RR _D F ) |` ( A [,] B ) ) ` x ) ) <_ a ) -> ( E. b e. ( A [,] B ) ( ( RR _D F ) ` b ) = y -> ( abs ` y ) <_ a ) )
64	50, 63	syl5 32	. . . . . . . . . . . 12 |- ( ( ( ph /\ a e. RR ) /\ A. x e. ( A [,] B ) ( abs ` ( ( ( RR _D F ) |` ( A [,] B ) ) ` x ) ) <_ a ) -> ( y e. ( ( RR _D F ) " ( A [,] B ) ) -> ( abs ` y ) <_ a ) )
65	64	imp 429	. . . . . . . . . . 11 |- ( ( ( ( ph /\ a e. RR ) /\ A. x e. ( A [,] B ) ( abs ` ( ( ( RR _D F ) |` ( A [,] B ) ) ` x ) ) <_ a ) /\ y e. ( ( RR _D F ) " ( A [,] B ) ) ) -> ( abs ` y ) <_ a )
66		breq1 4283	. . . . . . . . . . 11 |- ( ( abs ` y ) = b -> ( ( abs ` y ) <_ a <-> b <_ a ) )
67	65, 66	syl5ibcom 220	. . . . . . . . . 10 |- ( ( ( ( ph /\ a e. RR ) /\ A. x e. ( A [,] B ) ( abs ` ( ( ( RR _D F ) |` ( A [,] B ) ) ` x ) ) <_ a ) /\ y e. ( ( RR _D F ) " ( A [,] B ) ) ) -> ( ( abs ` y ) = b -> b <_ a ) )
68	67	rexlimdva 2831	. . . . . . . . 9 |- ( ( ( ph /\ a e. RR ) /\ A. x e. ( A [,] B ) ( abs ` ( ( ( RR _D F ) |` ( A [,] B ) ) ` x ) ) <_ a ) -> ( E. y e. ( ( RR _D F ) " ( A [,] B ) ) ( abs ` y ) = b -> b <_ a ) )
69	48, 68	syl5 32	. . . . . . . 8 |- ( ( ( ph /\ a e. RR ) /\ A. x e. ( A [,] B ) ( abs ` ( ( ( RR _D F ) |` ( A [,] B ) ) ` x ) ) <_ a ) -> ( b e. ( abs " ( ( RR _D F ) " ( A [,] B ) ) ) -> b <_ a ) )
70	69	ralrimiv 2788	. . . . . . 7 |- ( ( ( ph /\ a e. RR ) /\ A. x e. ( A [,] B ) ( abs ` ( ( ( RR _D F ) |` ( A [,] B ) ) ` x ) ) <_ a ) -> A. b e. ( abs " ( ( RR _D F ) " ( A [,] B ) ) ) b <_ a )
71	70	ex 434	. . . . . 6 |- ( ( ph /\ a e. RR ) -> ( A. x e. ( A [,] B ) ( abs ` ( ( ( RR _D F ) |` ( A [,] B ) ) ` x ) ) <_ a -> A. b e. ( abs " ( ( RR _D F ) " ( A [,] B ) ) ) b <_ a ) )
72	71	reximdva 2818	. . . . 5 |- ( ph -> ( E. a e. RR A. x e. ( A [,] B ) ( abs ` ( ( ( RR _D F ) |` ( A [,] B ) ) ` x ) ) <_ a -> E. a e. RR A. b e. ( abs " ( ( RR _D F ) " ( A [,] B ) ) ) b <_ a ) )
73	46, 72	mpd 15	. . . 4 |- ( ph -> E. a e. RR A. b e. ( abs " ( ( RR _D F ) " ( A [,] B ) ) ) b <_ a )
74		suprcl 10278	. . . 4 |- ( ( ( abs " ( ( RR _D F ) " ( A [,] B ) ) ) C_ RR /\ ( abs " ( ( RR _D F ) " ( A [,] B ) ) ) =/= (/) /\ E. a e. RR A. b e. ( abs " ( ( RR _D F ) " ( A [,] B ) ) ) b <_ a ) -> sup ( ( abs " ( ( RR _D F ) " ( A [,] B ) ) ) , RR , < ) e. RR )
75	7, 39, 73, 74	syl3anc 1211	. . 3 |- ( ph -> sup ( ( abs " ( ( RR _D F ) " ( A [,] B ) ) ) , RR , < ) e. RR )
76	1, 75	syl5eqel 2517	. 2 |- ( ph -> K e. RR )
77		simplrr 753	. . . . . . . . . . 11 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> y e. ( A [,] B ) )
78		fvres 5692	. . . . . . . . . . 11 |- ( y e. ( A [,] B ) -> ( ( F |` ( A [,] B ) ) ` y ) = ( F ` y ) )
79	77, 78	syl 16	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( F |` ( A [,] B ) ) ` y ) = ( F ` y ) )
80		c1liplem1.cn	. . . . . . . . . . . . . 14 |- ( ph -> ( F |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> RR ) )
81		cncff 20311	. . . . . . . . . . . . . 14 |- ( ( F |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> RR ) -> ( F |` ( A [,] B ) ) : ( A [,] B ) --> RR )
82	80, 81	syl 16	. . . . . . . . . . . . 13 |- ( ph -> ( F |` ( A [,] B ) ) : ( A [,] B ) --> RR )
83	82	ad2antrr 718	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( F |` ( A [,] B ) ) : ( A [,] B ) --> RR )
84	83, 77	ffvelrnd 5832	. . . . . . . . . . 11 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( F |` ( A [,] B ) ) ` y ) e. RR )
85	84	recnd 9400	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( F |` ( A [,] B ) ) ` y ) e. CC )
86	79, 85	eqeltrrd 2508	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( F ` y ) e. CC )
87		simplrl 752	. . . . . . . . . . 11 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> x e. ( A [,] B ) )
88		fvres 5692	. . . . . . . . . . 11 |- ( x e. ( A [,] B ) -> ( ( F |` ( A [,] B ) ) ` x ) = ( F ` x ) )
89	87, 88	syl 16	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( F |` ( A [,] B ) ) ` x ) = ( F ` x ) )
90	83, 87	ffvelrnd 5832	. . . . . . . . . . 11 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( F |` ( A [,] B ) ) ` x ) e. RR )
91	90	recnd 9400	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( F |` ( A [,] B ) ) ` x ) e. CC )
92	89, 91	eqeltrrd 2508	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( F ` x ) e. CC )
93	86, 92	subcld 9707	. . . . . . . 8 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( F ` y ) - ( F ` x ) ) e. CC )
94		iccssre 11365	. . . . . . . . . . . . 13 |- ( ( A e. RR /\ B e. RR ) -> ( A [,] B ) C_ RR )
95	18, 20, 94	syl2anc 654	. . . . . . . . . . . 12 |- ( ph -> ( A [,] B ) C_ RR )
96	95	ad2antrr 718	. . . . . . . . . . 11 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( A [,] B ) C_ RR )
97	96, 77	sseldd 3345	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> y e. RR )
98	96, 87	sseldd 3345	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> x e. RR )
99	97, 98	resubcld 9764	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( y - x ) e. RR )
100	99	recnd 9400	. . . . . . . 8 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( y - x ) e. CC )
101		simpr 458	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> x < y )
102		difrp 11012	. . . . . . . . . . 11 |- ( ( x e. RR /\ y e. RR ) -> ( x < y <-> ( y - x ) e. RR+ ) )
103	98, 97, 102	syl2anc 654	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( x < y <-> ( y - x ) e. RR+ ) )
104	101, 103	mpbid 210	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( y - x ) e. RR+ )
105	104	rpne0d 11020	. . . . . . . 8 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( y - x ) =/= 0 )
106	93, 100, 105	absdivd 12925	. . . . . . 7 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( abs ` ( ( ( F ` y ) - ( F ` x ) ) / ( y - x ) ) ) = ( ( abs ` ( ( F ` y ) - ( F ` x ) ) ) / ( abs ` ( y - x ) ) ) )
107	6	a1i 11	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( abs " ( ( RR _D F ) " ( A [,] B ) ) ) C_ RR )
108	39	ad2antrr 718	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( abs " ( ( RR _D F ) " ( A [,] B ) ) ) =/= (/) )
109	73	ad2antrr 718	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> E. a e. RR A. b e. ( abs " ( ( RR _D F ) " ( A [,] B ) ) ) b <_ a )
110	29	a1i 11	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> Fun abs )
111	93, 100, 105	divcld 10095	. . . . . . . . . . 11 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( ( F ` y ) - ( F ` x ) ) / ( y - x ) ) e. CC )
112	111, 34	syl6eleqr 2524	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( ( F ` y ) - ( F ` x ) ) / ( y - x ) ) e. dom abs )
113	98	rexrd 9421	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> x e. RR* )
114	97	rexrd 9421	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> y e. RR* )
115	98, 97, 101	ltled 9510	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> x <_ y )
116		ubicc2 11389	. . . . . . . . . . . . . . 15 |- ( ( x e. RR* /\ y e. RR* /\ x <_ y ) -> y e. ( x [,] y ) )
117	113, 114, 115, 116	syl3anc 1211	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> y e. ( x [,] y ) )
118		fvres 5692	. . . . . . . . . . . . . 14 |- ( y e. ( x [,] y ) -> ( ( F |` ( x [,] y ) ) ` y ) = ( F ` y ) )
119	117, 118	syl 16	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( F |` ( x [,] y ) ) ` y ) = ( F ` y ) )
120		lbicc2 11388	. . . . . . . . . . . . . . 15 |- ( ( x e. RR* /\ y e. RR* /\ x <_ y ) -> x e. ( x [,] y ) )
121	113, 114, 115, 120	syl3anc 1211	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> x e. ( x [,] y ) )
122		fvres 5692	. . . . . . . . . . . . . 14 |- ( x e. ( x [,] y ) -> ( ( F |` ( x [,] y ) ) ` x ) = ( F ` x ) )
123	121, 122	syl 16	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( F |` ( x [,] y ) ) ` x ) = ( F ` x ) )
124	119, 123	oveq12d 6098	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( ( F |` ( x [,] y ) ) ` y ) - ( ( F |` ( x [,] y ) ) ` x ) ) = ( ( F ` y ) - ( F ` x ) ) )
125	124	oveq1d 6095	. . . . . . . . . . 11 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( ( ( F |` ( x [,] y ) ) ` y ) - ( ( F |` ( x [,] y ) ) ` x ) ) / ( y - x ) ) = ( ( ( F ` y ) - ( F ` x ) ) / ( y - x ) ) )
126		iccss2 11354	. . . . . . . . . . . . . . . 16 |- ( ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) -> ( x [,] y ) C_ ( A [,] B ) )
127	126	ad2antlr 719	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( x [,] y ) C_ ( A [,] B ) )
128		resabs1 5127	. . . . . . . . . . . . . . 15 |- ( ( x [,] y ) C_ ( A [,] B ) -> ( ( F |` ( A [,] B ) ) |` ( x [,] y ) ) = ( F |` ( x [,] y ) ) )
129	127, 128	syl 16	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( F |` ( A [,] B ) ) |` ( x [,] y ) ) = ( F |` ( x [,] y ) ) )
130	80	ad2antrr 718	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( F |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> RR ) )
131		rescncf 20315	. . . . . . . . . . . . . . 15 |- ( ( x [,] y ) C_ ( A [,] B ) -> ( ( F |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> RR ) -> ( ( F |` ( A [,] B ) ) |` ( x [,] y ) ) e. ( ( x [,] y ) -cn-> RR ) ) )
132	127, 130, 131	sylc 60	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( F |` ( A [,] B ) ) |` ( x [,] y ) ) e. ( ( x [,] y ) -cn-> RR ) )
133	129, 132	eqeltrrd 2508	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( F |` ( x [,] y ) ) e. ( ( x [,] y ) -cn-> RR ) )
134	40	a1i 11	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> RR C_ CC )
135		c1liplem1.f	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> F e. ( CC ^pm RR ) )
136	135	ad2antrr 718	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> F e. ( CC ^pm RR ) )
137		cnex 9351	. . . . . . . . . . . . . . . . . . . 20 |- CC e. _V
138		reex 9361	. . . . . . . . . . . . . . . . . . . 20 |- RR e. _V
139	137, 138	elpm2 7232	. . . . . . . . . . . . . . . . . . 19 |- ( F e. ( CC ^pm RR ) <-> ( F : dom F --> CC /\ dom F C_ RR ) )
140	139	simplbi 457	. . . . . . . . . . . . . . . . . 18 |- ( F e. ( CC ^pm RR ) -> F : dom F --> CC )
141	136, 140	syl 16	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> F : dom F --> CC )
142	139	simprbi 461	. . . . . . . . . . . . . . . . . 18 |- ( F e. ( CC ^pm RR ) -> dom F C_ RR )
143	136, 142	syl 16	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> dom F C_ RR )
144		iccssre 11365	. . . . . . . . . . . . . . . . . 18 |- ( ( x e. RR /\ y e. RR ) -> ( x [,] y ) C_ RR )
145	98, 97, 144	syl2anc 654	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( x [,] y ) C_ RR )
146		eqid 2433	. . . . . . . . . . . . . . . . . 18 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
147	146	tgioo2 20222	. . . . . . . . . . . . . . . . . 18 |- ( topGen ` ran (,) ) = ( ( TopOpen ` ℂfld ) ↾t RR )
148	146, 147	dvres 21228	. . . . . . . . . . . . . . . . 17 |- ( ( ( RR C_ CC /\ F : dom F --> CC ) /\ ( dom F C_ RR /\ ( x [,] y ) C_ RR ) ) -> ( RR _D ( F |` ( x [,] y ) ) ) = ( ( RR _D F ) |` ( ( int ` ( topGen ` ran (,) ) ) ` ( x [,] y ) ) ) )
149	134, 141, 143, 145, 148	syl22anc 1212	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( RR _D ( F |` ( x [,] y ) ) ) = ( ( RR _D F ) |` ( ( int ` ( topGen ` ran (,) ) ) ` ( x [,] y ) ) ) )
150		iccntr 20240	. . . . . . . . . . . . . . . . . 18 |- ( ( x e. RR /\ y e. RR ) -> ( ( int ` ( topGen ` ran (,) ) ) ` ( x [,] y ) ) = ( x (,) y ) )
151	98, 97, 150	syl2anc 654	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( int ` ( topGen ` ran (,) ) ) ` ( x [,] y ) ) = ( x (,) y ) )
152	151	reseq2d 5097	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( RR _D F ) |` ( ( int ` ( topGen ` ran (,) ) ) ` ( x [,] y ) ) ) = ( ( RR _D F ) |` ( x (,) y ) ) )
153	149, 152	eqtrd 2465	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( RR _D ( F |` ( x [,] y ) ) ) = ( ( RR _D F ) |` ( x (,) y ) ) )
154	153	dmeqd 5029	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> dom ( RR _D ( F |` ( x [,] y ) ) ) = dom ( ( RR _D F ) |` ( x (,) y ) ) )
155		ioossicc 11369	. . . . . . . . . . . . . . . . 17 |- ( x (,) y ) C_ ( x [,] y )
156	155, 127	syl5ss 3355	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( x (,) y ) C_ ( A [,] B ) )
157	17	ad2antrr 718	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( A [,] B ) C_ dom ( RR _D F ) )
158	156, 157	sstrd 3354	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( x (,) y ) C_ dom ( RR _D F ) )
159		ssdmres 5120	. . . . . . . . . . . . . . 15 |- ( ( x (,) y ) C_ dom ( RR _D F ) <-> dom ( ( RR _D F ) |` ( x (,) y ) ) = ( x (,) y ) )
160	158, 159	sylib 196	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> dom ( ( RR _D F ) |` ( x (,) y ) ) = ( x (,) y ) )
161	154, 160	eqtrd 2465	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> dom ( RR _D ( F |` ( x [,] y ) ) ) = ( x (,) y ) )
162	98, 97, 101, 133, 161	mvth 21306	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> E. a e. ( x (,) y ) ( ( RR _D ( F |` ( x [,] y ) ) ) ` a ) = ( ( ( ( F |` ( x [,] y ) ) ` y ) - ( ( F |` ( x [,] y ) ) ` x ) ) / ( y - x ) ) )
163	153	fveq1d 5681	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( RR _D ( F |` ( x [,] y ) ) ) ` a ) = ( ( ( RR _D F ) |` ( x (,) y ) ) ` a ) )
164	163	adantrr 709	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ ( x < y /\ a e. ( x (,) y ) ) ) -> ( ( RR _D ( F |` ( x [,] y ) ) ) ` a ) = ( ( ( RR _D F ) |` ( x (,) y ) ) ` a ) )
165		fvres 5692	. . . . . . . . . . . . . . . . . 18 |- ( a e. ( x (,) y ) -> ( ( ( RR _D F ) |` ( x (,) y ) ) ` a ) = ( ( RR _D F ) ` a ) )
166	165	ad2antll 721	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ ( x < y /\ a e. ( x (,) y ) ) ) -> ( ( ( RR _D F ) |` ( x (,) y ) ) ` a ) = ( ( RR _D F ) ` a ) )
167	164, 166	eqtrd 2465	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ ( x < y /\ a e. ( x (,) y ) ) ) -> ( ( RR _D ( F |` ( x [,] y ) ) ) ` a ) = ( ( RR _D F ) ` a ) )
168	10	a1i 11	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ ( x < y /\ a e. ( x (,) y ) ) ) -> Fun ( RR _D F ) )
169	17	ad2antrr 718	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ ( x < y /\ a e. ( x (,) y ) ) ) -> ( A [,] B ) C_ dom ( RR _D F ) )
170	156	sseld 3343	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( a e. ( x (,) y ) -> a e. ( A [,] B ) ) )
171	170	impr 614	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ ( x < y /\ a e. ( x (,) y ) ) ) -> a e. ( A [,] B ) )
172		funfvima2 5940	. . . . . . . . . . . . . . . . . 18 |- ( ( Fun ( RR _D F ) /\ ( A [,] B ) C_ dom ( RR _D F ) ) -> ( a e. ( A [,] B ) -> ( ( RR _D F ) ` a ) e. ( ( RR _D F ) " ( A [,] B ) ) ) )
173	172	imp 429	. . . . . . . . . . . . . . . . 17 |- ( ( ( Fun ( RR _D F ) /\ ( A [,] B ) C_ dom ( RR _D F ) ) /\ a e. ( A [,] B ) ) -> ( ( RR _D F ) ` a ) e. ( ( RR _D F ) " ( A [,] B ) ) )
174	168, 169, 171, 173	syl21anc 1210	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ ( x < y /\ a e. ( x (,) y ) ) ) -> ( ( RR _D F ) ` a ) e. ( ( RR _D F ) " ( A [,] B ) ) )
175	167, 174	eqeltrd 2507	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ ( x < y /\ a e. ( x (,) y ) ) ) -> ( ( RR _D ( F |` ( x [,] y ) ) ) ` a ) e. ( ( RR _D F ) " ( A [,] B ) ) )
176		eleq1 2493	. . . . . . . . . . . . . . 15 |- ( ( ( RR _D ( F |` ( x [,] y ) ) ) ` a ) = ( ( ( ( F |` ( x [,] y ) ) ` y ) - ( ( F |` ( x [,] y ) ) ` x ) ) / ( y - x ) ) -> ( ( ( RR _D ( F |` ( x [,] y ) ) ) ` a ) e. ( ( RR _D F ) " ( A [,] B ) ) <-> ( ( ( ( F |` ( x [,] y ) ) ` y ) - ( ( F |` ( x [,] y ) ) ` x ) ) / ( y - x ) ) e. ( ( RR _D F ) " ( A [,] B ) ) ) )
177	175, 176	syl5ibcom 220	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ ( x < y /\ a e. ( x (,) y ) ) ) -> ( ( ( RR _D ( F |` ( x [,] y ) ) ) ` a ) = ( ( ( ( F |` ( x [,] y ) ) ` y ) - ( ( F |` ( x [,] y ) ) ` x ) ) / ( y - x ) ) -> ( ( ( ( F |` ( x [,] y ) ) ` y ) - ( ( F |` ( x [,] y ) ) ` x ) ) / ( y - x ) ) e. ( ( RR _D F ) " ( A [,] B ) ) ) )
178	177	expr 610	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( a e. ( x (,) y ) -> ( ( ( RR _D ( F |` ( x [,] y ) ) ) ` a ) = ( ( ( ( F |` ( x [,] y ) ) ` y ) - ( ( F |` ( x [,] y ) ) ` x ) ) / ( y - x ) ) -> ( ( ( ( F |` ( x [,] y ) ) ` y ) - ( ( F |` ( x [,] y ) ) ` x ) ) / ( y - x ) ) e. ( ( RR _D F ) " ( A [,] B ) ) ) ) )
179	178	rexlimdv 2830	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( E. a e. ( x (,) y ) ( ( RR _D ( F |` ( x [,] y ) ) ) ` a ) = ( ( ( ( F |` ( x [,] y ) ) ` y ) - ( ( F |` ( x [,] y ) ) ` x ) ) / ( y - x ) ) -> ( ( ( ( F |` ( x [,] y ) ) ` y ) - ( ( F |` ( x [,] y ) ) ` x ) ) / ( y - x ) ) e. ( ( RR _D F ) " ( A [,] B ) ) ) )
180	162, 179	mpd 15	. . . . . . . . . . 11 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( ( ( F |` ( x [,] y ) ) ` y ) - ( ( F |` ( x [,] y ) ) ` x ) ) / ( y - x ) ) e. ( ( RR _D F ) " ( A [,] B ) ) )
181	125, 180	eqeltrrd 2508	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( ( F ` y ) - ( F ` x ) ) / ( y - x ) ) e. ( ( RR _D F ) " ( A [,] B ) ) )
182		funfvima 5939	. . . . . . . . . . 11 |- ( ( Fun abs /\ ( ( ( F ` y ) - ( F ` x ) ) / ( y - x ) ) e. dom abs ) -> ( ( ( ( F ` y ) - ( F ` x ) ) / ( y - x ) ) e. ( ( RR _D F ) " ( A [,] B ) ) -> ( abs ` ( ( ( F ` y ) - ( F ` x ) ) / ( y - x ) ) ) e. ( abs " ( ( RR _D F ) " ( A [,] B ) ) ) ) )
183	182	imp 429	. . . . . . . . . 10 |- ( ( ( Fun abs /\ ( ( ( F ` y ) - ( F ` x ) ) / ( y - x ) ) e. dom abs ) /\ ( ( ( F ` y ) - ( F ` x ) ) / ( y - x ) ) e. ( ( RR _D F ) " ( A [,] B ) ) ) -> ( abs ` ( ( ( F ` y ) - ( F ` x ) ) / ( y - x ) ) ) e. ( abs " ( ( RR _D F ) " ( A [,] B ) ) ) )
184	110, 112, 181, 183	syl21anc 1210	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( abs ` ( ( ( F ` y ) - ( F ` x ) ) / ( y - x ) ) ) e. ( abs " ( ( RR _D F ) " ( A [,] B ) ) ) )
185		suprub 10279	. . . . . . . . 9 |- ( ( ( ( abs " ( ( RR _D F ) " ( A [,] B ) ) ) C_ RR /\ ( abs " ( ( RR _D F ) " ( A [,] B ) ) ) =/= (/) /\ E. a e. RR A. b e. ( abs " ( ( RR _D F ) " ( A [,] B ) ) ) b <_ a ) /\ ( abs ` ( ( ( F ` y ) - ( F ` x ) ) / ( y - x ) ) ) e. ( abs " ( ( RR _D F ) " ( A [,] B ) ) ) ) -> ( abs ` ( ( ( F ` y ) - ( F ` x ) ) / ( y - x ) ) ) <_ sup ( ( abs " ( ( RR _D F ) " ( A [,] B ) ) ) , RR , < ) )
186	107, 108, 109, 184, 185	syl31anc 1214	. . . . . . . 8 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( abs ` ( ( ( F ` y ) - ( F ` x ) ) / ( y - x ) ) ) <_ sup ( ( abs " ( ( RR _D F ) " ( A [,] B ) ) ) , RR , < ) )
187	186, 1	syl6breqr 4320	. . . . . . 7 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( abs ` ( ( ( F ` y ) - ( F ` x ) ) / ( y - x ) ) ) <_ K )
188	106, 187	eqbrtrrd 4302	. . . . . 6 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( abs ` ( ( F ` y ) - ( F ` x ) ) ) / ( abs ` ( y - x ) ) ) <_ K )
189	93	abscld 12906	. . . . . . 7 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( abs ` ( ( F ` y ) - ( F ` x ) ) ) e. RR )
190	76	ad2antrr 718	. . . . . . 7 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> K e. RR )
191	100, 105	absrpcld 12918	. . . . . . 7 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( abs ` ( y - x ) ) e. RR+ )
192	189, 190, 191	ledivmuld 11064	. . . . . 6 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( ( abs ` ( ( F ` y ) - ( F ` x ) ) ) / ( abs ` ( y - x ) ) ) <_ K <-> ( abs ` ( ( F ` y ) - ( F ` x ) ) ) <_ ( ( abs ` ( y - x ) ) x. K ) ) )
193	188, 192	mpbid 210	. . . . 5 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( abs ` ( ( F ` y ) - ( F ` x ) ) ) <_ ( ( abs ` ( y - x ) ) x. K ) )
194	191	rpcnd 11017	. . . . . 6 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( abs ` ( y - x ) ) e. CC )
195	190	recnd 9400	. . . . . 6 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> K e. CC )
196	194, 195	mulcomd 9395	. . . . 5 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( abs ` ( y - x ) ) x. K ) = ( K x. ( abs ` ( y - x ) ) ) )
197	193, 196	breqtrd 4304	. . . 4 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( abs ` ( ( F ` y ) - ( F ` x ) ) ) <_ ( K x. ( abs ` ( y - x ) ) ) )
198	197	ex 434	. . 3 |- ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) -> ( x < y -> ( abs ` ( ( F ` y ) - ( F ` x ) ) ) <_ ( K x. ( abs ` ( y - x ) ) ) ) )
199	198	ralrimivva 2798	. 2 |- ( ph -> A. x e. ( A [,] B ) A. y e. ( A [,] B ) ( x < y -> ( abs ` ( ( F ` y ) - ( F ` x ) ) ) <_ ( K x. ( abs ` ( y - x ) ) ) ) )
200	76, 199	jca 529	1 |- ( ph -> ( K e. RR /\ A. x e. ( A [,] B ) A. y e. ( A [,] B ) ( x < y -> ( abs ` ( ( F ` y ) - ( F ` x ) ) ) <_ ( K x. ( abs ` ( y - x ) ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1362   e. wcel 1755   =/= wne 2596   A.wral 2705   E.wrex 2706   C_ wss 3316   (/)c0 3625   class class class wbr 4280   dom cdm 4827   ran crn 4828   |` cres 4829   "cima 4830   Fun wfun 5400   -->wf 5402   ` cfv 5406  (class class class)co 6080   ^pm cpm 7203   supcsup 7678   CCcc 9268   RRcr 9269   x. cmul 9275   RR*cxr 9405   < clt 9406   <_ cle 9407   - cmin 9583   / cdiv 9981   RR+crp 10979   (,)cioo 11288   [,]cicc 11291   abscabs 12707   TopOpenctopn 14343   topGenctg 14359  ℂ_fldccnfld 17662   intcnt 18463   -cn->ccncf 20294   _D cdv 21180

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

This theorem is referenced by:  c1lip1  21311