Theorem c1liplem1 23759

Description: Lemma for c1lip1 23760. (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 5477	. . . . . 6 |- ( abs " ( ( RR _D F ) " ( A [,] B ) ) ) C_ ran abs
3		absf 14077	. . . . . . 7 |- abs : CC --> RR
4		frn 6053	. . . . . . 7 |- ( abs : CC --> RR -> ran abs C_ RR )
5	3, 4	ax-mp 5	. . . . . 6 |- ran abs C_ RR
6	2, 5	sstri 3612	. . . . 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 23671	. . . . . . . 8 |- ( RR _D F ) : dom ( RR _D F ) --> CC
9		ffun 6048	. . . . . . . 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 22696	. . . . . . . 8 |- ( ( ( RR _D F ) |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> RR ) -> ( ( RR _D F ) |` ( A [,] B ) ) : ( A [,] B ) --> RR )
14		fdm 6051	. . . . . . . 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 5420	. . . . . . 7 |- ( ( A [,] B ) C_ dom ( RR _D F ) <-> dom ( ( RR _D F ) |` ( A [,] B ) ) = ( A [,] B ) )
17	15, 16	sylibr 224	. . . . . 6 |- ( ph -> ( A [,] B ) C_ dom ( RR _D F ) )
18		c1liplem1.a	. . . . . . . 8 |- ( ph -> A e. RR )
19	18	rexrd 10089	. . . . . . 7 |- ( ph -> A e. RR* )
20		c1liplem1.b	. . . . . . . 8 |- ( ph -> B e. RR )
21	20	rexrd 10089	. . . . . . 7 |- ( ph -> B e. RR* )
22		c1liplem1.le	. . . . . . 7 |- ( ph -> A <_ B )
23		lbicc2 12288	. . . . . . 7 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> A e. ( A [,] B ) )
24	19, 21, 22, 23	syl3anc 1326	. . . . . 6 |- ( ph -> A e. ( A [,] B ) )
25		funfvima2 6493	. . . . . . 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 445	. . . . . 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 1325	. . . . 5 |- ( ph -> ( ( RR _D F ) ` A ) e. ( ( RR _D F ) " ( A [,] B ) ) )
28		ffun 6048	. . . . . . 7 |- ( abs : CC --> RR -> Fun abs )
29	3, 28	ax-mp 5	. . . . . 6 |- Fun abs
30		imassrn 5477	. . . . . . . 8 |- ( ( RR _D F ) " ( A [,] B ) ) C_ ran ( RR _D F )
31		frn 6053	. . . . . . . . 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 3612	. . . . . . 7 |- ( ( RR _D F ) " ( A [,] B ) ) C_ CC
34	3	fdmi 6052	. . . . . . 7 |- dom abs = CC
35	33, 34	sseqtr4i 3638	. . . . . 6 |- ( ( RR _D F ) " ( A [,] B ) ) C_ dom abs
36		funfvima2 6493	. . . . . 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 708	. . . . 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 3921	. . . . 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 9993	. . . . . . . 8 |- RR C_ CC
41		ssid 3624	. . . . . . . 8 |- CC C_ CC
42		cncfss 22702	. . . . . . . 8 |- ( ( RR C_ CC /\ CC C_ CC ) -> ( ( A [,] B ) -cn-> RR ) C_ ( ( A [,] B ) -cn-> CC ) )
43	40, 41, 42	mp2an 708	. . . . . . 7 |- ( ( A [,] B ) -cn-> RR ) C_ ( ( A [,] B ) -cn-> CC )
44	43, 12	sseldi 3601	. . . . . 6 |- ( ph -> ( ( RR _D F ) |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> CC ) )
45		cniccbdd 23230	. . . . . 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 1326	. . . . 5 |- ( ph -> E. a e. RR A. x e. ( A [,] B ) ( abs ` ( ( ( RR _D F ) |` ( A [,] B ) ) ` x ) ) <_ a )
47		fvelima 6248	. . . . . . . . . 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 706	. . . . . . . . 9 |- ( b e. ( abs " ( ( RR _D F ) " ( A [,] B ) ) ) -> E. y e. ( ( RR _D F ) " ( A [,] B ) ) ( abs ` y ) = b )
49		fvelima 6248	. . . . . . . . . . . . . 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 706	. . . . . . . . . . . . 13 |- ( y e. ( ( RR _D F ) " ( A [,] B ) ) -> E. b e. ( A [,] B ) ( ( RR _D F ) ` b ) = y )
51		fvres 6207	. . . . . . . . . . . . . . . . . . 19 |- ( b e. ( A [,] B ) -> ( ( ( RR _D F ) |` ( A [,] B ) ) ` b ) = ( ( RR _D F ) ` b ) )
52	51	adantl 482	. . . . . . . . . . . . . . . . . 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 6195	. . . . . . . . . . . . . . . . 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 6191	. . . . . . . . . . . . . . . . . . . 20 |- ( x = b -> ( ( ( RR _D F ) |` ( A [,] B ) ) ` x ) = ( ( ( RR _D F ) |` ( A [,] B ) ) ` b ) )
55	54	fveq2d 6195	. . . . . . . . . . . . . . . . . . 19 |- ( x = b -> ( abs ` ( ( ( RR _D F ) |` ( A [,] B ) ) ` x ) ) = ( abs ` ( ( ( RR _D F ) |` ( A [,] B ) ) ` b ) ) )
56	55	breq1d 4663	. . . . . . . . . . . . . . . . . 18 |- ( x = b -> ( ( abs ` ( ( ( RR _D F ) |` ( A [,] B ) ) ` x ) ) <_ a <-> ( abs ` ( ( ( RR _D F ) |` ( A [,] B ) ) ` b ) ) <_ a ) )
57	56	rspccva 3308	. . . . . . . . . . . . . . . . 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 4677	. . . . . . . . . . . . . . . 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 750	. . . . . . . . . . . . . . 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 6191	. . . . . . . . . . . . . . . 16 |- ( ( ( RR _D F ) ` b ) = y -> ( abs ` ( ( RR _D F ) ` b ) ) = ( abs ` y ) )
61	60	breq1d 4663	. . . . . . . . . . . . . . 15 |- ( ( ( RR _D F ) ` b ) = y -> ( ( abs ` ( ( RR _D F ) ` b ) ) <_ a <-> ( abs ` y ) <_ a ) )
62	59, 61	syl5ibcom 235	. . . . . . . . . . . . . 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 3031	. . . . . . . . . . . . 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 34	. . . . . . . . . . . 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 445	. . . . . . . . . . 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 4656	. . . . . . . . . . 11 |- ( ( abs ` y ) = b -> ( ( abs ` y ) <_ a <-> b <_ a ) )
67	65, 66	syl5ibcom 235	. . . . . . . . . 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 3031	. . . . . . . . 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 34	. . . . . . . 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 2965	. . . . . . 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 450	. . . . . 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 3017	. . . . 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 10983	. . . 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 1326	. . 3 |- ( ph -> sup ( ( abs " ( ( RR _D F ) " ( A [,] B ) ) ) , RR , < ) e. RR )
76	1, 75	syl5eqel 2705	. 2 |- ( ph -> K e. RR )
77		simplrr 801	. . . . . . . . . . 11 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> y e. ( A [,] B ) )
78		fvres 6207	. . . . . . . . . . 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 22696	. . . . . . . . . . . . . 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 762	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( F |` ( A [,] B ) ) : ( A [,] B ) --> RR )
84	83, 77	ffvelrnd 6360	. . . . . . . . . . 11 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( F |` ( A [,] B ) ) ` y ) e. RR )
85	84	recnd 10068	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( F |` ( A [,] B ) ) ` y ) e. CC )
86	79, 85	eqeltrrd 2702	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( F ` y ) e. CC )
87		simplrl 800	. . . . . . . . . . 11 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> x e. ( A [,] B ) )
88		fvres 6207	. . . . . . . . . . 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 6360	. . . . . . . . . . 11 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( F |` ( A [,] B ) ) ` x ) e. RR )
91	90	recnd 10068	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( F |` ( A [,] B ) ) ` x ) e. CC )
92	89, 91	eqeltrrd 2702	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( F ` x ) e. CC )
93	86, 92	subcld 10392	. . . . . . . 8 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( F ` y ) - ( F ` x ) ) e. CC )
94		iccssre 12255	. . . . . . . . . . . . 13 |- ( ( A e. RR /\ B e. RR ) -> ( A [,] B ) C_ RR )
95	18, 20, 94	syl2anc 693	. . . . . . . . . . . 12 |- ( ph -> ( A [,] B ) C_ RR )
96	95	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( A [,] B ) C_ RR )
97	96, 77	sseldd 3604	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> y e. RR )
98	96, 87	sseldd 3604	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> x e. RR )
99	97, 98	resubcld 10458	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( y - x ) e. RR )
100	99	recnd 10068	. . . . . . . 8 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( y - x ) e. CC )
101		simpr 477	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> x < y )
102		difrp 11868	. . . . . . . . . . 11 |- ( ( x e. RR /\ y e. RR ) -> ( x < y <-> ( y - x ) e. RR+ ) )
103	98, 97, 102	syl2anc 693	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( x < y <-> ( y - x ) e. RR+ ) )
104	101, 103	mpbid 222	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( y - x ) e. RR+ )
105	104	rpne0d 11877	. . . . . . . 8 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( y - x ) =/= 0 )
106	93, 100, 105	absdivd 14194	. . . . . . 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 762	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( abs " ( ( RR _D F ) " ( A [,] B ) ) ) =/= (/) )
109	73	ad2antrr 762	. . . . . . . . 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 10801	. . . . . . . . . . 11 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( ( F ` y ) - ( F ` x ) ) / ( y - x ) ) e. CC )
112	111, 34	syl6eleqr 2712	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( ( F ` y ) - ( F ` x ) ) / ( y - x ) ) e. dom abs )
113	98	rexrd 10089	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> x e. RR* )
114	97	rexrd 10089	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> y e. RR* )
115	98, 97, 101	ltled 10185	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> x <_ y )
116		ubicc2 12289	. . . . . . . . . . . . . . 15 |- ( ( x e. RR* /\ y e. RR* /\ x <_ y ) -> y e. ( x [,] y ) )
117	113, 114, 115, 116	syl3anc 1326	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> y e. ( x [,] y ) )
118		fvres 6207	. . . . . . . . . . . . . 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 12288	. . . . . . . . . . . . . . 15 |- ( ( x e. RR* /\ y e. RR* /\ x <_ y ) -> x e. ( x [,] y ) )
121	113, 114, 115, 120	syl3anc 1326	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> x e. ( x [,] y ) )
122		fvres 6207	. . . . . . . . . . . . . 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 6668	. . . . . . . . . . . 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 6665	. . . . . . . . . . 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 12244	. . . . . . . . . . . . . . . 16 |- ( ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) -> ( x [,] y ) C_ ( A [,] B ) )
127	126	ad2antlr 763	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( x [,] y ) C_ ( A [,] B ) )
128	127	resabs1d 5428	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( F |` ( A [,] B ) ) |` ( x [,] y ) ) = ( F |` ( x [,] y ) ) )
129	80	ad2antrr 762	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( F |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> RR ) )
130		rescncf 22700	. . . . . . . . . . . . . . 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 65	. . . . . . . . . . . . . 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 2702	. . . . . . . . . . . . 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 762	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> F e. ( CC ^pm RR ) )
136		cnex 10017	. . . . . . . . . . . . . . . . . . . 20 |- CC e. _V
137		reex 10027	. . . . . . . . . . . . . . . . . . . 20 |- RR e. _V
138	136, 137	elpm2 7889	. . . . . . . . . . . . . . . . . . 19 |- ( F e. ( CC ^pm RR ) <-> ( F : dom F --> CC /\ dom F C_ RR ) )
139	138	simplbi 476	. . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . 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 12255	. . . . . . . . . . . . . . . . . 18 |- ( ( x e. RR /\ y e. RR ) -> ( x [,] y ) C_ RR )
144	98, 97, 143	syl2anc 693	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( x [,] y ) C_ RR )
145		eqid 2622	. . . . . . . . . . . . . . . . . 18 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
146	145	tgioo2 22606	. . . . . . . . . . . . . . . . . 18 |- ( topGen ` ran (,) ) = ( ( TopOpen ` ℂfld ) ↾t RR )
147	145, 146	dvres 23675	. . . . . . . . . . . . . . . . 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 1327	. . . . . . . . . . . . . . . 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 22624	. . . . . . . . . . . . . . . . . 18 |- ( ( x e. RR /\ y e. RR ) -> ( ( int ` ( topGen ` ran (,) ) ) ` ( x [,] y ) ) = ( x (,) y ) )
150	98, 97, 149	syl2anc 693	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( int ` ( topGen ` ran (,) ) ) ` ( x [,] y ) ) = ( x (,) y ) )
151	150	reseq2d 5396	. . . . . . . . . . . . . . . 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 2656	. . . . . . . . . . . . . . 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 5326	. . . . . . . . . . . . . 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 12259	. . . . . . . . . . . . . . . . 17 |- ( x (,) y ) C_ ( x [,] y )
155	154, 127	syl5ss 3614	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( x (,) y ) C_ ( A [,] B ) )
156	17	ad2antrr 762	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( A [,] B ) C_ dom ( RR _D F ) )
157	155, 156	sstrd 3613	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( x (,) y ) C_ dom ( RR _D F ) )
158		ssdmres 5420	. . . . . . . . . . . . . . 15 |- ( ( x (,) y ) C_ dom ( RR _D F ) <-> dom ( ( RR _D F ) |` ( x (,) y ) ) = ( x (,) y ) )
159	157, 158	sylib 208	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> dom ( ( RR _D F ) |` ( x (,) y ) ) = ( x (,) y ) )
160	153, 159	eqtrd 2656	. . . . . . . . . . . . 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 23755	. . . . . . . . . . . 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 6193	. . . . . . . . . . . . . . . . . 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 753	. . . . . . . . . . . . . . . . 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 6207	. . . . . . . . . . . . . . . . . 18 |- ( a e. ( x (,) y ) -> ( ( ( RR _D F ) |` ( x (,) y ) ) ` a ) = ( ( RR _D F ) ` a ) )
165	164	ad2antll 765	. . . . . . . . . . . . . . . . 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 2656	. . . . . . . . . . . . . . . 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 762	. . . . . . . . . . . . . . . . 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 3602	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( a e. ( x (,) y ) -> a e. ( A [,] B ) ) )
170	169	impr 649	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ ( x < y /\ a e. ( x (,) y ) ) ) -> a e. ( A [,] B ) )
171		funfvima2 6493	. . . . . . . . . . . . . . . . . 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 445	. . . . . . . . . . . . . . . . 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 1325	. . . . . . . . . . . . . . . 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 2701	. . . . . . . . . . . . . . 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 2689	. . . . . . . . . . . . . . 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 235	. . . . . . . . . . . . . 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 643	. . . . . . . . . . . . 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 3030	. . . . . . . . . . . 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 2702	. . . . . . . . . 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 6492	. . . . . . . . . . 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 445	. . . . . . . . . 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 1325	. . . . . . . . 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 10984	. . . . . . . . 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 1329	. . . . . . . 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 4695	. . . . . . 7 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( abs ` ( ( ( F ` y ) - ( F ` x ) ) / ( y - x ) ) ) <_ K )
187	106, 186	eqbrtrrd 4677	. . . . . 6 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( abs ` ( ( F ` y ) - ( F ` x ) ) ) / ( abs ` ( y - x ) ) ) <_ K )
188	93	abscld 14175	. . . . . . 7 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( abs ` ( ( F ` y ) - ( F ` x ) ) ) e. RR )
189	76	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> K e. RR )
190	100, 105	absrpcld 14187	. . . . . . 7 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( abs ` ( y - x ) ) e. RR+ )
191	188, 189, 190	ledivmuld 11925	. . . . . 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 222	. . . . 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 11874	. . . . . 6 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( abs ` ( y - x ) ) e. CC )
194	189	recnd 10068	. . . . . 6 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> K e. CC )
195	193, 194	mulcomd 10061	. . . . 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 4679	. . . 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 450	. . 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 2971	. 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 554	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 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   C_ wss 3574   (/)c0 3915   class class class wbr 4653   dom cdm 5114   ran crn 5115   |` cres 5116   "cima 5117   Fun wfun 5882   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^pm cpm 7858   supcsup 8346   CCcc 9934   RRcr 9935   x. cmul 9941   RR*cxr 10073   < clt 10074   <_ cle 10075   - cmin 10266   / cdiv 10684   RR+crp 11832   (,)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:  c1lip1  23760