Theorem c1liplem1 22948

Description: Lemma for c1lip1 22949. (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 5179	. . . . . 6 |- ( abs " ( ( RR _D F ) " ( A [,] B ) ) ) C_ ran abs
3		absf 13400	. . . . . . 7 |- abs : CC --> RR
4		frn 5735	. . . . . . 7 |- ( abs : CC --> RR -> ran abs C_ RR )
5	3, 4	ax-mp 5	. . . . . 6 |- ran abs C_ RR
6	2, 5	sstri 3441	. . . . 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 22862	. . . . . . . 8 |- ( RR _D F ) : dom ( RR _D F ) --> CC
9		ffun 5731	. . . . . . . 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 21925	. . . . . . . 8 |- ( ( ( RR _D F ) |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> RR ) -> ( ( RR _D F ) |` ( A [,] B ) ) : ( A [,] B ) --> RR )
14		fdm 5733	. . . . . . . 8 |- ( ( ( RR _D F ) |` ( A [,] B ) ) : ( A [,] B ) --> RR -> dom ( ( RR _D F ) |` ( A [,] B ) ) = ( A [,] B ) )
15	12, 13, 14	3syl 18	. . . . . . 7 |- ( ph -> dom ( ( RR _D F ) |` ( A [,] B ) ) = ( A [,] B ) )
16		ssdmres 5126	. . . . . . 7 |- ( ( A [,] B ) C_ dom ( RR _D F ) <-> dom ( ( RR _D F ) |` ( A [,] B ) ) = ( A [,] B ) )
17	15, 16	sylibr 216	. . . . . 6 |- ( ph -> ( A [,] B ) C_ dom ( RR _D F ) )
18		c1liplem1.a	. . . . . . . 8 |- ( ph -> A e. RR )
19	18	rexrd 9690	. . . . . . 7 |- ( ph -> A e. RR* )
20		c1liplem1.b	. . . . . . . 8 |- ( ph -> B e. RR )
21	20	rexrd 9690	. . . . . . 7 |- ( ph -> B e. RR* )
22		c1liplem1.le	. . . . . . 7 |- ( ph -> A <_ B )
23		lbicc2 11748	. . . . . . 7 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> A e. ( A [,] B ) )
24	19, 21, 22, 23	syl3anc 1268	. . . . . 6 |- ( ph -> A e. ( A [,] B ) )
25		funfvima2 6141	. . . . . . 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 431	. . . . . 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 1267	. . . . 5 |- ( ph -> ( ( RR _D F ) ` A ) e. ( ( RR _D F ) " ( A [,] B ) ) )
28		ffun 5731	. . . . . . 7 |- ( abs : CC --> RR -> Fun abs )
29	3, 28	ax-mp 5	. . . . . 6 |- Fun abs
30		imassrn 5179	. . . . . . . 8 |- ( ( RR _D F ) " ( A [,] B ) ) C_ ran ( RR _D F )
31		frn 5735	. . . . . . . . 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 3441	. . . . . . 7 |- ( ( RR _D F ) " ( A [,] B ) ) C_ CC
34	3	fdmi 5734	. . . . . . 7 |- dom abs = CC
35	33, 34	sseqtr4i 3465	. . . . . 6 |- ( ( RR _D F ) " ( A [,] B ) ) C_ dom abs
36		funfvima2 6141	. . . . . 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 678	. . . . 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 3737	. . . . 5 |- ( ( abs ` ( ( RR _D F ) ` A ) ) e. ( abs " ( ( RR _D F ) " ( A [,] B ) ) ) -> ( abs " ( ( RR _D F ) " ( A [,] B ) ) ) =/= (/) )
39	27, 37, 38	3syl 18	. . . 4 |- ( ph -> ( abs " ( ( RR _D F ) " ( A [,] B ) ) ) =/= (/) )
40		ax-resscn 9596	. . . . . . . 8 |- RR C_ CC
41		ssid 3451	. . . . . . . 8 |- CC C_ CC
42		cncfss 21931	. . . . . . . 8 |- ( ( RR C_ CC /\ CC C_ CC ) -> ( ( A [,] B ) -cn-> RR ) C_ ( ( A [,] B ) -cn-> CC ) )
43	40, 41, 42	mp2an 678	. . . . . . 7 |- ( ( A [,] B ) -cn-> RR ) C_ ( ( A [,] B ) -cn-> CC )
44	43, 12	sseldi 3430	. . . . . 6 |- ( ph -> ( ( RR _D F ) |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> CC ) )
45		cniccbdd 22412	. . . . . 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 1268	. . . . 5 |- ( ph -> E. a e. RR A. x e. ( A [,] B ) ( abs ` ( ( ( RR _D F ) |` ( A [,] B ) ) ` x ) ) <_ a )
47		fvelima 5917	. . . . . . . . . 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 676	. . . . . . . . 9 |- ( b e. ( abs " ( ( RR _D F ) " ( A [,] B ) ) ) -> E. y e. ( ( RR _D F ) " ( A [,] B ) ) ( abs ` y ) = b )
49		fvelima 5917	. . . . . . . . . . . . . 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 676	. . . . . . . . . . . . 13 |- ( y e. ( ( RR _D F ) " ( A [,] B ) ) -> E. b e. ( A [,] B ) ( ( RR _D F ) ` b ) = y )
51		fvres 5879	. . . . . . . . . . . . . . . . . . 19 |- ( b e. ( A [,] B ) -> ( ( ( RR _D F ) |` ( A [,] B ) ) ` b ) = ( ( RR _D F ) ` b ) )
52	51	adantl 468	. . . . . . . . . . . . . . . . . 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 5869	. . . . . . . . . . . . . . . . 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 5865	. . . . . . . . . . . . . . . . . . . 20 |- ( x = b -> ( ( ( RR _D F ) |` ( A [,] B ) ) ` x ) = ( ( ( RR _D F ) |` ( A [,] B ) ) ` b ) )
55	54	fveq2d 5869	. . . . . . . . . . . . . . . . . . 19 |- ( x = b -> ( abs ` ( ( ( RR _D F ) |` ( A [,] B ) ) ` x ) ) = ( abs ` ( ( ( RR _D F ) |` ( A [,] B ) ) ` b ) ) )
56	55	breq1d 4412	. . . . . . . . . . . . . . . . . 18 |- ( x = b -> ( ( abs ` ( ( ( RR _D F ) |` ( A [,] B ) ) ` x ) ) <_ a <-> ( abs ` ( ( ( RR _D F ) |` ( A [,] B ) ) ` b ) ) <_ a ) )
57	56	rspccva 3149	. . . . . . . . . . . . . . . . 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 4425	. . . . . . . . . . . . . . . 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 720	. . . . . . . . . . . . . . 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 5865	. . . . . . . . . . . . . . . 16 |- ( ( ( RR _D F ) ` b ) = y -> ( abs ` ( ( RR _D F ) ` b ) ) = ( abs ` y ) )
61	60	breq1d 4412	. . . . . . . . . . . . . . 15 |- ( ( ( RR _D F ) ` b ) = y -> ( ( abs ` ( ( RR _D F ) ` b ) ) <_ a <-> ( abs ` y ) <_ a ) )
62	59, 61	syl5ibcom 224	. . . . . . . . . . . . . 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 2879	. . . . . . . . . . . . 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 33	. . . . . . . . . . . 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 431	. . . . . . . . . . 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 4405	. . . . . . . . . . 11 |- ( ( abs ` y ) = b -> ( ( abs ` y ) <_ a <-> b <_ a ) )
67	65, 66	syl5ibcom 224	. . . . . . . . . 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 2879	. . . . . . . . 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 33	. . . . . . . 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 2800	. . . . . . 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 436	. . . . . 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 2862	. . . . 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 10569	. . . 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 1268	. . 3 |- ( ph -> sup ( ( abs " ( ( RR _D F ) " ( A [,] B ) ) ) , RR , < ) e. RR )
76	1, 75	syl5eqel 2533	. 2 |- ( ph -> K e. RR )
77		simplrr 771	. . . . . . . . . . 11 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> y e. ( A [,] B ) )
78		fvres 5879	. . . . . . . . . . 11 |- ( y e. ( A [,] B ) -> ( ( F |` ( A [,] B ) ) ` y ) = ( F ` y ) )
79	77, 78	syl 17	. . . . . . . . . 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 21925	. . . . . . . . . . . . . 14 |- ( ( F |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> RR ) -> ( F |` ( A [,] B ) ) : ( A [,] B ) --> RR )
82	80, 81	syl 17	. . . . . . . . . . . . 13 |- ( ph -> ( F |` ( A [,] B ) ) : ( A [,] B ) --> RR )
83	82	ad2antrr 732	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( F |` ( A [,] B ) ) : ( A [,] B ) --> RR )
84	83, 77	ffvelrnd 6023	. . . . . . . . . . 11 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( F |` ( A [,] B ) ) ` y ) e. RR )
85	84	recnd 9669	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( F |` ( A [,] B ) ) ` y ) e. CC )
86	79, 85	eqeltrrd 2530	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( F ` y ) e. CC )
87		simplrl 770	. . . . . . . . . . 11 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> x e. ( A [,] B ) )
88		fvres 5879	. . . . . . . . . . 11 |- ( x e. ( A [,] B ) -> ( ( F |` ( A [,] B ) ) ` x ) = ( F ` x ) )
89	87, 88	syl 17	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( F |` ( A [,] B ) ) ` x ) = ( F ` x ) )
90	83, 87	ffvelrnd 6023	. . . . . . . . . . 11 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( F |` ( A [,] B ) ) ` x ) e. RR )
91	90	recnd 9669	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( F |` ( A [,] B ) ) ` x ) e. CC )
92	89, 91	eqeltrrd 2530	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( F ` x ) e. CC )
93	86, 92	subcld 9986	. . . . . . . 8 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( F ` y ) - ( F ` x ) ) e. CC )
94		iccssre 11716	. . . . . . . . . . . . 13 |- ( ( A e. RR /\ B e. RR ) -> ( A [,] B ) C_ RR )
95	18, 20, 94	syl2anc 667	. . . . . . . . . . . 12 |- ( ph -> ( A [,] B ) C_ RR )
96	95	ad2antrr 732	. . . . . . . . . . 11 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( A [,] B ) C_ RR )
97	96, 77	sseldd 3433	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> y e. RR )
98	96, 87	sseldd 3433	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> x e. RR )
99	97, 98	resubcld 10047	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( y - x ) e. RR )
100	99	recnd 9669	. . . . . . . 8 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( y - x ) e. CC )
101		simpr 463	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> x < y )
102		difrp 11337	. . . . . . . . . . 11 |- ( ( x e. RR /\ y e. RR ) -> ( x < y <-> ( y - x ) e. RR+ ) )
103	98, 97, 102	syl2anc 667	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( x < y <-> ( y - x ) e. RR+ ) )
104	101, 103	mpbid 214	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( y - x ) e. RR+ )
105	104	rpne0d 11346	. . . . . . . 8 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( y - x ) =/= 0 )
106	93, 100, 105	absdivd 13517	. . . . . . 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 732	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( abs " ( ( RR _D F ) " ( A [,] B ) ) ) =/= (/) )
109	73	ad2antrr 732	. . . . . . . . 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 10383	. . . . . . . . . . 11 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( ( F ` y ) - ( F ` x ) ) / ( y - x ) ) e. CC )
112	111, 34	syl6eleqr 2540	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( ( F ` y ) - ( F ` x ) ) / ( y - x ) ) e. dom abs )
113	98	rexrd 9690	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> x e. RR* )
114	97	rexrd 9690	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> y e. RR* )
115	98, 97, 101	ltled 9783	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> x <_ y )
116		ubicc2 11749	. . . . . . . . . . . . . . 15 |- ( ( x e. RR* /\ y e. RR* /\ x <_ y ) -> y e. ( x [,] y ) )
117	113, 114, 115, 116	syl3anc 1268	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> y e. ( x [,] y ) )
118		fvres 5879	. . . . . . . . . . . . . 14 |- ( y e. ( x [,] y ) -> ( ( F |` ( x [,] y ) ) ` y ) = ( F ` y ) )
119	117, 118	syl 17	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( F |` ( x [,] y ) ) ` y ) = ( F ` y ) )
120		lbicc2 11748	. . . . . . . . . . . . . . 15 |- ( ( x e. RR* /\ y e. RR* /\ x <_ y ) -> x e. ( x [,] y ) )
121	113, 114, 115, 120	syl3anc 1268	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> x e. ( x [,] y ) )
122		fvres 5879	. . . . . . . . . . . . . 14 |- ( x e. ( x [,] y ) -> ( ( F |` ( x [,] y ) ) ` x ) = ( F ` x ) )
123	121, 122	syl 17	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( F |` ( x [,] y ) ) ` x ) = ( F ` x ) )
124	119, 123	oveq12d 6308	. . . . . . . . . . . 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 6305	. . . . . . . . . . 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 11705	. . . . . . . . . . . . . . . 16 |- ( ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) -> ( x [,] y ) C_ ( A [,] B ) )
127	126	ad2antlr 733	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( x [,] y ) C_ ( A [,] B ) )
128	127	resabs1d 5134	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( F |` ( A [,] B ) ) |` ( x [,] y ) ) = ( F |` ( x [,] y ) ) )
129	80	ad2antrr 732	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( F |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> RR ) )
130		rescncf 21929	. . . . . . . . . . . . . . 15 |- ( ( x [,] y ) C_ ( A [,] B ) -> ( ( F |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> RR ) -> ( ( F |` ( A [,] B ) ) |` ( x [,] y ) ) e. ( ( x [,] y ) -cn-> RR ) ) )
131	127, 129, 130	sylc 62	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( F |` ( A [,] B ) ) |` ( x [,] y ) ) e. ( ( x [,] y ) -cn-> RR ) )
132	128, 131	eqeltrrd 2530	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( F |` ( x [,] y ) ) e. ( ( x [,] y ) -cn-> RR ) )
133	40	a1i 11	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> RR C_ CC )
134		c1liplem1.f	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> F e. ( CC ^pm RR ) )
135	134	ad2antrr 732	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> F e. ( CC ^pm RR ) )
136		cnex 9620	. . . . . . . . . . . . . . . . . . . 20 |- CC e. _V
137		reex 9630	. . . . . . . . . . . . . . . . . . . 20 |- RR e. _V
138	136, 137	elpm2 7503	. . . . . . . . . . . . . . . . . . 19 |- ( F e. ( CC ^pm RR ) <-> ( F : dom F --> CC /\ dom F C_ RR ) )
139	138	simplbi 462	. . . . . . . . . . . . . . . . . 18 |- ( F e. ( CC ^pm RR ) -> F : dom F --> CC )
140	135, 139	syl 17	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> F : dom F --> CC )
141	138	simprbi 466	. . . . . . . . . . . . . . . . . 18 |- ( F e. ( CC ^pm RR ) -> dom F C_ RR )
142	135, 141	syl 17	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> dom F C_ RR )
143		iccssre 11716	. . . . . . . . . . . . . . . . . 18 |- ( ( x e. RR /\ y e. RR ) -> ( x [,] y ) C_ RR )
144	98, 97, 143	syl2anc 667	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( x [,] y ) C_ RR )
145		eqid 2451	. . . . . . . . . . . . . . . . . 18 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
146	145	tgioo2 21821	. . . . . . . . . . . . . . . . . 18 |- ( topGen ` ran (,) ) = ( ( TopOpen ` ℂfld ) ↾t RR )
147	145, 146	dvres 22866	. . . . . . . . . . . . . . . . 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 ) ) ) )
148	133, 140, 142, 144, 147	syl22anc 1269	. . . . . . . . . . . . . . . 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 ) ) ) )
149		iccntr 21839	. . . . . . . . . . . . . . . . . 18 |- ( ( x e. RR /\ y e. RR ) -> ( ( int ` ( topGen ` ran (,) ) ) ` ( x [,] y ) ) = ( x (,) y ) )
150	98, 97, 149	syl2anc 667	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( int ` ( topGen ` ran (,) ) ) ` ( x [,] y ) ) = ( x (,) y ) )
151	150	reseq2d 5105	. . . . . . . . . . . . . . . 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 ) ) )
152	148, 151	eqtrd 2485	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( RR _D ( F |` ( x [,] y ) ) ) = ( ( RR _D F ) |` ( x (,) y ) ) )
153	152	dmeqd 5037	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> dom ( RR _D ( F |` ( x [,] y ) ) ) = dom ( ( RR _D F ) |` ( x (,) y ) ) )
154		ioossicc 11720	. . . . . . . . . . . . . . . . 17 |- ( x (,) y ) C_ ( x [,] y )
155	154, 127	syl5ss 3443	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( x (,) y ) C_ ( A [,] B ) )
156	17	ad2antrr 732	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( A [,] B ) C_ dom ( RR _D F ) )
157	155, 156	sstrd 3442	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( x (,) y ) C_ dom ( RR _D F ) )
158		ssdmres 5126	. . . . . . . . . . . . . . 15 |- ( ( x (,) y ) C_ dom ( RR _D F ) <-> dom ( ( RR _D F ) |` ( x (,) y ) ) = ( x (,) y ) )
159	157, 158	sylib 200	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> dom ( ( RR _D F ) |` ( x (,) y ) ) = ( x (,) y ) )
160	153, 159	eqtrd 2485	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> dom ( RR _D ( F |` ( x [,] y ) ) ) = ( x (,) y ) )
161	98, 97, 101, 132, 160	mvth 22944	. . . . . . . . . . . 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 ) ) )
162	152	fveq1d 5867	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( RR _D ( F |` ( x [,] y ) ) ) ` a ) = ( ( ( RR _D F ) |` ( x (,) y ) ) ` a ) )
163	162	adantrr 723	. . . . . . . . . . . . . . . . 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 ) )
164		fvres 5879	. . . . . . . . . . . . . . . . . 18 |- ( a e. ( x (,) y ) -> ( ( ( RR _D F ) |` ( x (,) y ) ) ` a ) = ( ( RR _D F ) ` a ) )
165	164	ad2antll 735	. . . . . . . . . . . . . . . . 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 ) )
166	163, 165	eqtrd 2485	. . . . . . . . . . . . . . . 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 ) )
167	10	a1i 11	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ ( x < y /\ a e. ( x (,) y ) ) ) -> Fun ( RR _D F ) )
168	17	ad2antrr 732	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ ( x < y /\ a e. ( x (,) y ) ) ) -> ( A [,] B ) C_ dom ( RR _D F ) )
169	155	sseld 3431	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( a e. ( x (,) y ) -> a e. ( A [,] B ) ) )
170	169	impr 625	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ ( x < y /\ a e. ( x (,) y ) ) ) -> a e. ( A [,] B ) )
171		funfvima2 6141	. . . . . . . . . . . . . . . . . 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 ) ) ) )
172	171	imp 431	. . . . . . . . . . . . . . . . 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 ) ) )
173	167, 168, 170, 172	syl21anc 1267	. . . . . . . . . . . . . . . 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 ) ) )
174	166, 173	eqeltrd 2529	. . . . . . . . . . . . . . 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 ) ) )
175		eleq1 2517	. . . . . . . . . . . . . . 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 ) ) ) )
176	174, 175	syl5ibcom 224	. . . . . . . . . . . . . 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 ) ) ) )
177	176	expr 620	. . . . . . . . . . . . 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 ) ) ) ) )
178	177	rexlimdv 2877	. . . . . . . . . . . 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 ) ) ) )
179	161, 178	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 ) ) )
180	125, 179	eqeltrrd 2530	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( ( F ` y ) - ( F ` x ) ) / ( y - x ) ) e. ( ( RR _D F ) " ( A [,] B ) ) )
181		funfvima 6140	. . . . . . . . . . 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 ) ) ) ) )
182	181	imp 431	. . . . . . . . . 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 ) ) ) )
183	110, 112, 180, 182	syl21anc 1267	. . . . . . . . 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 ) ) ) )
184		suprub 10570	. . . . . . . . 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 , < ) )
185	107, 108, 109, 183, 184	syl31anc 1271	. . . . . . . 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 , < ) )
186	185, 1	syl6breqr 4443	. . . . . . 7 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( abs ` ( ( ( F ` y ) - ( F ` x ) ) / ( y - x ) ) ) <_ K )
187	106, 186	eqbrtrrd 4425	. . . . . 6 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( abs ` ( ( F ` y ) - ( F ` x ) ) ) / ( abs ` ( y - x ) ) ) <_ K )
188	93	abscld 13498	. . . . . . 7 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( abs ` ( ( F ` y ) - ( F ` x ) ) ) e. RR )
189	76	ad2antrr 732	. . . . . . 7 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> K e. RR )
190	100, 105	absrpcld 13510	. . . . . . 7 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( abs ` ( y - x ) ) e. RR+ )
191	188, 189, 190	ledivmuld 11391	. . . . . 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 ) ) )
192	187, 191	mpbid 214	. . . . 5 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( abs ` ( ( F ` y ) - ( F ` x ) ) ) <_ ( ( abs ` ( y - x ) ) x. K ) )
193	190	rpcnd 11343	. . . . . 6 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( abs ` ( y - x ) ) e. CC )
194	189	recnd 9669	. . . . . 6 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> K e. CC )
195	193, 194	mulcomd 9664	. . . . 5 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( abs ` ( y - x ) ) x. K ) = ( K x. ( abs ` ( y - x ) ) ) )
196	192, 195	breqtrd 4427	. . . 4 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( abs ` ( ( F ` y ) - ( F ` x ) ) ) <_ ( K x. ( abs ` ( y - x ) ) ) )
197	196	ex 436	. . 3 |- ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) -> ( x < y -> ( abs ` ( ( F ` y ) - ( F ` x ) ) ) <_ ( K x. ( abs ` ( y - x ) ) ) ) )
198	197	ralrimivva 2809	. 2 |- ( ph -> A. x e. ( A [,] B ) A. y e. ( A [,] B ) ( x < y -> ( abs ` ( ( F ` y ) - ( F ` x ) ) ) <_ ( K x. ( abs ` ( y - x ) ) ) ) )
199	76, 198	jca 535	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 188   /\ wa 371   = wceq 1444   e. wcel 1887   =/= wne 2622   A.wral 2737   E.wrex 2738   C_ wss 3404   (/)c0 3731   class class class wbr 4402   dom cdm 4834   ran crn 4835   |` cres 4836   "cima 4837   Fun wfun 5576   -->wf 5578   ` cfv 5582  (class class class)co 6290   ^pm cpm 7473   supcsup 7954   CCcc 9537   RRcr 9538   x. cmul 9544   RR*cxr 9674   < clt 9675   <_ cle 9676   - cmin 9860   / cdiv 10269   RR+crp 11302   (,)cioo 11635   [,]cicc 11638   abscabs 13297   TopOpenctopn 15320   topGenctg 15336  ℂ_fldccnfld 18970   intcnt 20032   -cn->ccncf 21908   _D cdv 22818

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617  ax-addf 9618  ax-mulf 9619

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-iin 4281  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-of 6531  df-om 6693  df-1st 6793  df-2nd 6794  df-supp 6915  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-2o 7183  df-oadd 7186  df-er 7363  df-map 7474  df-pm 7475  df-ixp 7523  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-fsupp 7884  df-fi 7925  df-sup 7956  df-inf 7957  df-oi 8025  df-card 8373  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ioo 11639  df-ico 11641  df-icc 11642  df-fz 11785  df-fzo 11916  df-seq 12214  df-exp 12273  df-hash 12516  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-mulr 15204  df-starv 15205  df-sca 15206  df-vsca 15207  df-ip 15208  df-tset 15209  df-ple 15210  df-ds 15212  df-unif 15213  df-hom 15214  df-cco 15215  df-rest 15321  df-topn 15322  df-0g 15340  df-gsum 15341  df-topgen 15342  df-pt 15343  df-prds 15346  df-xrs 15400  df-qtop 15406  df-imas 15407  df-xps 15410  df-mre 15492  df-mrc 15493  df-acs 15495  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-submnd 16583  df-mulg 16676  df-cntz 16971  df-cmn 17432  df-psmet 18962  df-xmet 18963  df-met 18964  df-bl 18965  df-mopn 18966  df-fbas 18967  df-fg 18968  df-cnfld 18971  df-top 19921  df-bases 19922  df-topon 19923  df-topsp 19924  df-cld 20034  df-ntr 20035  df-cls 20036  df-nei 20114  df-lp 20152  df-perf 20153  df-cn 20243  df-cnp 20244  df-haus 20331  df-cmp 20402  df-tx 20577  df-hmeo 20770  df-fil 20861  df-fm 20953  df-flim 20954  df-flf 20955  df-xms 21335  df-ms 21336  df-tms 21337  df-cncf 21910  df-limc 22821  df-dv 22822

This theorem is referenced by:  c1lip1  22949