Theorem c1liplem1 21600

Description: Lemma for c1lip1 21601. (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 5287	. . . . . 6 |- ( abs " ( ( RR _D F ) " ( A [,] B ) ) ) C_ ran abs
3		absf 12942	. . . . . . 7 |- abs : CC --> RR
4		frn 5672	. . . . . . 7 |- ( abs : CC --> RR -> ran abs C_ RR )
5	3, 4	ax-mp 5	. . . . . 6 |- ran abs C_ RR
6	2, 5	sstri 3472	. . . . 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 21514	. . . . . . . 8 |- ( RR _D F ) : dom ( RR _D F ) --> CC
9		ffun 5668	. . . . . . . 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 20600	. . . . . . . 8 |- ( ( ( RR _D F ) |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> RR ) -> ( ( RR _D F ) |` ( A [,] B ) ) : ( A [,] B ) --> RR )
14		fdm 5670	. . . . . . . 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 5239	. . . . . . 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 9543	. . . . . . 7 |- ( ph -> A e. RR* )
20		c1liplem1.b	. . . . . . . 8 |- ( ph -> B e. RR )
21	20	rexrd 9543	. . . . . . 7 |- ( ph -> B e. RR* )
22		c1liplem1.le	. . . . . . 7 |- ( ph -> A <_ B )
23		lbicc2 11517	. . . . . . 7 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> A e. ( A [,] B ) )
24	19, 21, 22, 23	syl3anc 1219	. . . . . 6 |- ( ph -> A e. ( A [,] B ) )
25		funfvima2 6061	. . . . . . 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 1218	. . . . 5 |- ( ph -> ( ( RR _D F ) ` A ) e. ( ( RR _D F ) " ( A [,] B ) ) )
28		ffun 5668	. . . . . . 7 |- ( abs : CC --> RR -> Fun abs )
29	3, 28	ax-mp 5	. . . . . 6 |- Fun abs
30		imassrn 5287	. . . . . . . 8 |- ( ( RR _D F ) " ( A [,] B ) ) C_ ran ( RR _D F )
31		frn 5672	. . . . . . . . 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 3472	. . . . . . 7 |- ( ( RR _D F ) " ( A [,] B ) ) C_ CC
34	3	fdmi 5671	. . . . . . 7 |- dom abs = CC
35	33, 34	sseqtr4i 3496	. . . . . 6 |- ( ( RR _D F ) " ( A [,] B ) ) C_ dom abs
36		funfvima2 6061	. . . . . 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 672	. . . . 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 3750	. . . . 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 9449	. . . . . . . 8 |- RR C_ CC
41		ssid 3482	. . . . . . . 8 |- CC C_ CC
42		cncfss 20606	. . . . . . . 8 |- ( ( RR C_ CC /\ CC C_ CC ) -> ( ( A [,] B ) -cn-> RR ) C_ ( ( A [,] B ) -cn-> CC ) )
43	40, 41, 42	mp2an 672	. . . . . . 7 |- ( ( A [,] B ) -cn-> RR ) C_ ( ( A [,] B ) -cn-> CC )
44	43, 12	sseldi 3461	. . . . . 6 |- ( ph -> ( ( RR _D F ) |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> CC ) )
45		cniccbdd 21076	. . . . . 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 1219	. . . . 5 |- ( ph -> E. a e. RR A. x e. ( A [,] B ) ( abs ` ( ( ( RR _D F ) |` ( A [,] B ) ) ` x ) ) <_ a )
47		fvelima 5851	. . . . . . . . . 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 670	. . . . . . . . 9 |- ( b e. ( abs " ( ( RR _D F ) " ( A [,] B ) ) ) -> E. y e. ( ( RR _D F ) " ( A [,] B ) ) ( abs ` y ) = b )
49		fvelima 5851	. . . . . . . . . . . . . 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 670	. . . . . . . . . . . . 13 |- ( y e. ( ( RR _D F ) " ( A [,] B ) ) -> E. b e. ( A [,] B ) ( ( RR _D F ) ` b ) = y )
51		fvres 5812	. . . . . . . . . . . . . . . . . . 19 |- ( b e. ( A [,] B ) -> ( ( ( RR _D F ) |` ( A [,] B ) ) ` b ) = ( ( RR _D F ) ` b ) )
52	51	adantl 466	. . . . . . . . . . . . . . . . . 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 5802	. . . . . . . . . . . . . . . . 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 5798	. . . . . . . . . . . . . . . . . . . 20 |- ( x = b -> ( ( ( RR _D F ) |` ( A [,] B ) ) ` x ) = ( ( ( RR _D F ) |` ( A [,] B ) ) ` b ) )
55	54	fveq2d 5802	. . . . . . . . . . . . . . . . . . 19 |- ( x = b -> ( abs ` ( ( ( RR _D F ) |` ( A [,] B ) ) ` x ) ) = ( abs ` ( ( ( RR _D F ) |` ( A [,] B ) ) ` b ) ) )
56	55	breq1d 4409	. . . . . . . . . . . . . . . . . 18 |- ( x = b -> ( ( abs ` ( ( ( RR _D F ) |` ( A [,] B ) ) ` x ) ) <_ a <-> ( abs ` ( ( ( RR _D F ) |` ( A [,] B ) ) ` b ) ) <_ a ) )
57	56	rspccva 3176	. . . . . . . . . . . . . . . . 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 4421	. . . . . . . . . . . . . . . 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 713	. . . . . . . . . . . . . . 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 5798	. . . . . . . . . . . . . . . 16 |- ( ( ( RR _D F ) ` b ) = y -> ( abs ` ( ( RR _D F ) ` b ) ) = ( abs ` y ) )
61	60	breq1d 4409	. . . . . . . . . . . . . . 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 2945	. . . . . . . . . . . . 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 4402	. . . . . . . . . . 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 2945	. . . . . . . . 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 2827	. . . . . . 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 2932	. . . . 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 10400	. . . 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 1219	. . 3 |- ( ph -> sup ( ( abs " ( ( RR _D F ) " ( A [,] B ) ) ) , RR , < ) e. RR )
76	1, 75	syl5eqel 2546	. 2 |- ( ph -> K e. RR )
77		simplrr 760	. . . . . . . . . . 11 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> y e. ( A [,] B ) )
78		fvres 5812	. . . . . . . . . . 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 20600	. . . . . . . . . . . . . 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 725	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( F |` ( A [,] B ) ) : ( A [,] B ) --> RR )
84	83, 77	ffvelrnd 5952	. . . . . . . . . . 11 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( F |` ( A [,] B ) ) ` y ) e. RR )
85	84	recnd 9522	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( F |` ( A [,] B ) ) ` y ) e. CC )
86	79, 85	eqeltrrd 2543	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( F ` y ) e. CC )
87		simplrl 759	. . . . . . . . . . 11 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> x e. ( A [,] B ) )
88		fvres 5812	. . . . . . . . . . 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 5952	. . . . . . . . . . 11 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( F |` ( A [,] B ) ) ` x ) e. RR )
91	90	recnd 9522	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( F |` ( A [,] B ) ) ` x ) e. CC )
92	89, 91	eqeltrrd 2543	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( F ` x ) e. CC )
93	86, 92	subcld 9829	. . . . . . . 8 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( F ` y ) - ( F ` x ) ) e. CC )
94		iccssre 11487	. . . . . . . . . . . . 13 |- ( ( A e. RR /\ B e. RR ) -> ( A [,] B ) C_ RR )
95	18, 20, 94	syl2anc 661	. . . . . . . . . . . 12 |- ( ph -> ( A [,] B ) C_ RR )
96	95	ad2antrr 725	. . . . . . . . . . 11 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( A [,] B ) C_ RR )
97	96, 77	sseldd 3464	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> y e. RR )
98	96, 87	sseldd 3464	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> x e. RR )
99	97, 98	resubcld 9886	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( y - x ) e. RR )
100	99	recnd 9522	. . . . . . . 8 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( y - x ) e. CC )
101		simpr 461	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> x < y )
102		difrp 11134	. . . . . . . . . . 11 |- ( ( x e. RR /\ y e. RR ) -> ( x < y <-> ( y - x ) e. RR+ ) )
103	98, 97, 102	syl2anc 661	. . . . . . . . . 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 11142	. . . . . . . 8 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( y - x ) =/= 0 )
106	93, 100, 105	absdivd 13058	. . . . . . 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 725	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( abs " ( ( RR _D F ) " ( A [,] B ) ) ) =/= (/) )
109	73	ad2antrr 725	. . . . . . . . 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 10217	. . . . . . . . . . 11 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( ( F ` y ) - ( F ` x ) ) / ( y - x ) ) e. CC )
112	111, 34	syl6eleqr 2553	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( ( F ` y ) - ( F ` x ) ) / ( y - x ) ) e. dom abs )
113	98	rexrd 9543	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> x e. RR* )
114	97	rexrd 9543	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> y e. RR* )
115	98, 97, 101	ltled 9632	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> x <_ y )
116		ubicc2 11518	. . . . . . . . . . . . . . 15 |- ( ( x e. RR* /\ y e. RR* /\ x <_ y ) -> y e. ( x [,] y ) )
117	113, 114, 115, 116	syl3anc 1219	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> y e. ( x [,] y ) )
118		fvres 5812	. . . . . . . . . . . . . 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 11517	. . . . . . . . . . . . . . 15 |- ( ( x e. RR* /\ y e. RR* /\ x <_ y ) -> x e. ( x [,] y ) )
121	113, 114, 115, 120	syl3anc 1219	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> x e. ( x [,] y ) )
122		fvres 5812	. . . . . . . . . . . . . 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 6217	. . . . . . . . . . . 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 6214	. . . . . . . . . . 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 11476	. . . . . . . . . . . . . . . 16 |- ( ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) -> ( x [,] y ) C_ ( A [,] B ) )
127	126	ad2antlr 726	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( x [,] y ) C_ ( A [,] B ) )
128		resabs1 5246	. . . . . . . . . . . . . . 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 725	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( F |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> RR ) )
131		rescncf 20604	. . . . . . . . . . . . . . 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 2543	. . . . . . . . . . . . 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 725	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> F e. ( CC ^pm RR ) )
137		cnex 9473	. . . . . . . . . . . . . . . . . . . 20 |- CC e. _V
138		reex 9483	. . . . . . . . . . . . . . . . . . . 20 |- RR e. _V
139	137, 138	elpm2 7353	. . . . . . . . . . . . . . . . . . 19 |- ( F e. ( CC ^pm RR ) <-> ( F : dom F --> CC /\ dom F C_ RR ) )
140	139	simplbi 460	. . . . . . . . . . . . . . . . . 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 464	. . . . . . . . . . . . . . . . . 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 11487	. . . . . . . . . . . . . . . . . 18 |- ( ( x e. RR /\ y e. RR ) -> ( x [,] y ) C_ RR )
145	98, 97, 144	syl2anc 661	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( x [,] y ) C_ RR )
146		eqid 2454	. . . . . . . . . . . . . . . . . 18 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
147	146	tgioo2 20511	. . . . . . . . . . . . . . . . . 18 |- ( topGen ` ran (,) ) = ( ( TopOpen ` ℂfld ) ↾t RR )
148	146, 147	dvres 21518	. . . . . . . . . . . . . . . . 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 1220	. . . . . . . . . . . . . . . 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 20529	. . . . . . . . . . . . . . . . . 18 |- ( ( x e. RR /\ y e. RR ) -> ( ( int ` ( topGen ` ran (,) ) ) ` ( x [,] y ) ) = ( x (,) y ) )
151	98, 97, 150	syl2anc 661	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( int ` ( topGen ` ran (,) ) ) ` ( x [,] y ) ) = ( x (,) y ) )
152	151	reseq2d 5217	. . . . . . . . . . . . . . . 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 2495	. . . . . . . . . . . . . . 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 5149	. . . . . . . . . . . . . 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 11491	. . . . . . . . . . . . . . . . 17 |- ( x (,) y ) C_ ( x [,] y )
156	155, 127	syl5ss 3474	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( x (,) y ) C_ ( A [,] B ) )
157	17	ad2antrr 725	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( A [,] B ) C_ dom ( RR _D F ) )
158	156, 157	sstrd 3473	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( x (,) y ) C_ dom ( RR _D F ) )
159		ssdmres 5239	. . . . . . . . . . . . . . 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 2495	. . . . . . . . . . . . 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 21596	. . . . . . . . . . . 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 5800	. . . . . . . . . . . . . . . . . 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 716	. . . . . . . . . . . . . . . . 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 5812	. . . . . . . . . . . . . . . . . 18 |- ( a e. ( x (,) y ) -> ( ( ( RR _D F ) |` ( x (,) y ) ) ` a ) = ( ( RR _D F ) ` a ) )
166	165	ad2antll 728	. . . . . . . . . . . . . . . . 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 2495	. . . . . . . . . . . . . . . 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 725	. . . . . . . . . . . . . . . . 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 3462	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( a e. ( x (,) y ) -> a e. ( A [,] B ) ) )
171	170	impr 619	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ ( x < y /\ a e. ( x (,) y ) ) ) -> a e. ( A [,] B ) )
172		funfvima2 6061	. . . . . . . . . . . . . . . . . 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 1218	. . . . . . . . . . . . . . . 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 2542	. . . . . . . . . . . . . . 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 2526	. . . . . . . . . . . . . . 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 615	. . . . . . . . . . . . 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 2944	. . . . . . . . . . . 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 2543	. . . . . . . . . 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 6060	. . . . . . . . . . 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 1218	. . . . . . . . 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 10401	. . . . . . . . 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 1222	. . . . . . . 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 4439	. . . . . . 7 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( abs ` ( ( ( F ` y ) - ( F ` x ) ) / ( y - x ) ) ) <_ K )
188	106, 187	eqbrtrrd 4421	. . . . . 6 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( abs ` ( ( F ` y ) - ( F ` x ) ) ) / ( abs ` ( y - x ) ) ) <_ K )
189	93	abscld 13039	. . . . . . 7 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( abs ` ( ( F ` y ) - ( F ` x ) ) ) e. RR )
190	76	ad2antrr 725	. . . . . . 7 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> K e. RR )
191	100, 105	absrpcld 13051	. . . . . . 7 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( abs ` ( y - x ) ) e. RR+ )
192	189, 190, 191	ledivmuld 11186	. . . . . 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 11139	. . . . . 6 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( abs ` ( y - x ) ) e. CC )
195	190	recnd 9522	. . . . . 6 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> K e. CC )
196	194, 195	mulcomd 9517	. . . . 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 4423	. . . 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 2912	. 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 532	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 1370   e. wcel 1758   =/= wne 2647   A.wral 2798   E.wrex 2799   C_ wss 3435   (/)c0 3744   class class class wbr 4399   dom cdm 4947   ran crn 4948   |` cres 4949   "cima 4950   Fun wfun 5519   -->wf 5521   ` cfv 5525  (class class class)co 6199   ^pm cpm 7324   supcsup 7800   CCcc 9390   RRcr 9391   x. cmul 9397   RR*cxr 9527   < clt 9528   <_ cle 9529   - cmin 9705   / cdiv 10103   RR+crp 11101   (,)cioo 11410   [,]cicc 11413   abscabs 12840   TopOpenctopn 14478   topGenctg 14494  ℂ_fldccnfld 17942   intcnt 18752   -cn->ccncf 20583   _D cdv 21470

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-inf2 7957  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469  ax-pre-sup 9470  ax-addf 9471  ax-mulf 9472

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-int 4236  df-iun 4280  df-iin 4281  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-se 4787  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-isom 5534  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-of 6429  df-om 6586  df-1st 6686  df-2nd 6687  df-supp 6800  df-recs 6941  df-rdg 6975  df-1o 7029  df-2o 7030  df-oadd 7033  df-er 7210  df-map 7325  df-pm 7326  df-ixp 7373  df-en 7420  df-dom 7421  df-sdom 7422  df-fin 7423  df-fsupp 7731  df-fi 7771  df-sup 7801  df-oi 7834  df-card 8219  df-cda 8447  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-div 10104  df-nn 10433  df-2 10490  df-3 10491  df-4 10492  df-5 10493  df-6 10494  df-7 10495  df-8 10496  df-9 10497  df-10 10498  df-n0 10690  df-z 10757  df-dec 10866  df-uz 10972  df-q 11064  df-rp 11102  df-xneg 11199  df-xadd 11200  df-xmul 11201  df-ioo 11414  df-ico 11416  df-icc 11417  df-fz 11554  df-fzo 11665  df-seq 11923  df-exp 11982  df-hash 12220  df-cj 12705  df-re 12706  df-im 12707  df-sqr 12841  df-abs 12842  df-struct 14293  df-ndx 14294  df-slot 14295  df-base 14296  df-sets 14297  df-ress 14298  df-plusg 14369  df-mulr 14370  df-starv 14371  df-sca 14372  df-vsca 14373  df-ip 14374  df-tset 14375  df-ple 14376  df-ds 14378  df-unif 14379  df-hom 14380  df-cco 14381  df-rest 14479  df-topn 14480  df-0g 14498  df-gsum 14499  df-topgen 14500  df-pt 14501  df-prds 14504  df-xrs 14558  df-qtop 14563  df-imas 14564  df-xps 14566  df-mre 14642  df-mrc 14643  df-acs 14645  df-mnd 15533  df-submnd 15583  df-mulg 15666  df-cntz 15953  df-cmn 16399  df-psmet 17933  df-xmet 17934  df-met 17935  df-bl 17936  df-mopn 17937  df-fbas 17938  df-fg 17939  df-cnfld 17943  df-top 18634  df-bases 18636  df-topon 18637  df-topsp 18638  df-cld 18754  df-ntr 18755  df-cls 18756  df-nei 18833  df-lp 18871  df-perf 18872  df-cn 18962  df-cnp 18963  df-haus 19050  df-cmp 19121  df-tx 19266  df-hmeo 19459  df-fil 19550  df-fm 19642  df-flim 19643  df-flf 19644  df-xms 20026  df-ms 20027  df-tms 20028  df-cncf 20585  df-limc 21473  df-dv 21474

This theorem is referenced by:  c1lip1  21601