Theorem c1liplem1 22129

Description: Lemma for c1lip1 22130. (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 5346	. . . . . 6 |- ( abs " ( ( RR _D F ) " ( A [,] B ) ) ) C_ ran abs
3		absf 13126	. . . . . . 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 3513	. . . . 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 22043	. . . . . . . 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 21129	. . . . . . . 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 20	. . . . . . 7 |- ( ph -> dom ( ( RR _D F ) |` ( A [,] B ) ) = ( A [,] B ) )
16		ssdmres 5293	. . . . . . 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 9639	. . . . . . 7 |- ( ph -> A e. RR* )
20		c1liplem1.b	. . . . . . . 8 |- ( ph -> B e. RR )
21	20	rexrd 9639	. . . . . . 7 |- ( ph -> B e. RR* )
22		c1liplem1.le	. . . . . . 7 |- ( ph -> A <_ B )
23		lbicc2 11632	. . . . . . 7 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> A e. ( A [,] B ) )
24	19, 21, 22, 23	syl3anc 1228	. . . . . 6 |- ( ph -> A e. ( A [,] B ) )
25		funfvima2 6134	. . . . . . 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 1227	. . . . 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 5346	. . . . . . . 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 3513	. . . . . . 7 |- ( ( RR _D F ) " ( A [,] B ) ) C_ CC
34	3	fdmi 5734	. . . . . . 7 |- dom abs = CC
35	33, 34	sseqtr4i 3537	. . . . . 6 |- ( ( RR _D F ) " ( A [,] B ) ) C_ dom abs
36		funfvima2 6134	. . . . . 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 3791	. . . . 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 9545	. . . . . . . 8 |- RR C_ CC
41		ssid 3523	. . . . . . . 8 |- CC C_ CC
42		cncfss 21135	. . . . . . . 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 3502	. . . . . 6 |- ( ph -> ( ( RR _D F ) |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> CC ) )
45		cniccbdd 21605	. . . . . 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 1228	. . . . 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 670	. . . . . . . . 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 670	. . . . . . . . . . . . 13 |- ( y e. ( ( RR _D F ) " ( A [,] B ) ) -> E. b e. ( A [,] B ) ( ( RR _D F ) ` b ) = y )
51		fvres 5878	. . . . . . . . . . . . . . . . . . 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 5868	. . . . . . . . . . . . . . . . 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 5864	. . . . . . . . . . . . . . . . . . . 20 |- ( x = b -> ( ( ( RR _D F ) |` ( A [,] B ) ) ` x ) = ( ( ( RR _D F ) |` ( A [,] B ) ) ` b ) )
55	54	fveq2d 5868	. . . . . . . . . . . . . . . . . . 19 |- ( x = b -> ( abs ` ( ( ( RR _D F ) |` ( A [,] B ) ) ` x ) ) = ( abs ` ( ( ( RR _D F ) |` ( A [,] B ) ) ` b ) ) )
56	55	breq1d 4457	. . . . . . . . . . . . . . . . . 18 |- ( x = b -> ( ( abs ` ( ( ( RR _D F ) |` ( A [,] B ) ) ` x ) ) <_ a <-> ( abs ` ( ( ( RR _D F ) |` ( A [,] B ) ) ` b ) ) <_ a ) )
57	56	rspccva 3213	. . . . . . . . . . . . . . . . 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 4469	. . . . . . . . . . . . . . . 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 5864	. . . . . . . . . . . . . . . 16 |- ( ( ( RR _D F ) ` b ) = y -> ( abs ` ( ( RR _D F ) ` b ) ) = ( abs ` y ) )
61	60	breq1d 4457	. . . . . . . . . . . . . . 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 2955	. . . . . . . . . . . . 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 4450	. . . . . . . . . . 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 2955	. . . . . . . . 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 2876	. . . . . . 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 2938	. . . . 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 10499	. . . 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 1228	. . 3 |- ( ph -> sup ( ( abs " ( ( RR _D F ) " ( A [,] B ) ) ) , RR , < ) e. RR )
76	1, 75	syl5eqel 2559	. 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 5878	. . . . . . . . . . 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 21129	. . . . . . . . . . . . . 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 6020	. . . . . . . . . . 11 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( F |` ( A [,] B ) ) ` y ) e. RR )
85	84	recnd 9618	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( F |` ( A [,] B ) ) ` y ) e. CC )
86	79, 85	eqeltrrd 2556	. . . . . . . . 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 5878	. . . . . . . . . . 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 6020	. . . . . . . . . . 11 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( F |` ( A [,] B ) ) ` x ) e. RR )
91	90	recnd 9618	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( F |` ( A [,] B ) ) ` x ) e. CC )
92	89, 91	eqeltrrd 2556	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( F ` x ) e. CC )
93	86, 92	subcld 9926	. . . . . . . 8 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( F ` y ) - ( F ` x ) ) e. CC )
94		iccssre 11602	. . . . . . . . . . . . 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 3505	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> y e. RR )
98	96, 87	sseldd 3505	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> x e. RR )
99	97, 98	resubcld 9983	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( y - x ) e. RR )
100	99	recnd 9618	. . . . . . . 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 11249	. . . . . . . . . . 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 11257	. . . . . . . 8 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( y - x ) =/= 0 )
106	93, 100, 105	absdivd 13242	. . . . . . 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 10316	. . . . . . . . . . 11 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( ( F ` y ) - ( F ` x ) ) / ( y - x ) ) e. CC )
112	111, 34	syl6eleqr 2566	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( ( F ` y ) - ( F ` x ) ) / ( y - x ) ) e. dom abs )
113	98	rexrd 9639	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> x e. RR* )
114	97	rexrd 9639	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> y e. RR* )
115	98, 97, 101	ltled 9728	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> x <_ y )
116		ubicc2 11633	. . . . . . . . . . . . . . 15 |- ( ( x e. RR* /\ y e. RR* /\ x <_ y ) -> y e. ( x [,] y ) )
117	113, 114, 115, 116	syl3anc 1228	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> y e. ( x [,] y ) )
118		fvres 5878	. . . . . . . . . . . . . 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 11632	. . . . . . . . . . . . . . 15 |- ( ( x e. RR* /\ y e. RR* /\ x <_ y ) -> x e. ( x [,] y ) )
121	113, 114, 115, 120	syl3anc 1228	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> x e. ( x [,] y ) )
122		fvres 5878	. . . . . . . . . . . . . 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 6300	. . . . . . . . . . . 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 6297	. . . . . . . . . . 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 11591	. . . . . . . . . . . . . . . 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 5300	. . . . . . . . . . . . . . 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 21133	. . . . . . . . . . . . . . 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 2556	. . . . . . . . . . . . 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 9569	. . . . . . . . . . . . . . . . . . . 20 |- CC e. _V
138		reex 9579	. . . . . . . . . . . . . . . . . . . 20 |- RR e. _V
139	137, 138	elpm2 7447	. . . . . . . . . . . . . . . . . . 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 11602	. . . . . . . . . . . . . . . . . 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 2467	. . . . . . . . . . . . . . . . . 18 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
147	146	tgioo2 21040	. . . . . . . . . . . . . . . . . 18 |- ( topGen ` ran (,) ) = ( ( TopOpen ` ℂfld ) ↾t RR )
148	146, 147	dvres 22047	. . . . . . . . . . . . . . . . 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 1229	. . . . . . . . . . . . . . . 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 21058	. . . . . . . . . . . . . . . . . 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 5271	. . . . . . . . . . . . . . . 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 2508	. . . . . . . . . . . . . . 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 5203	. . . . . . . . . . . . . 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 11606	. . . . . . . . . . . . . . . . 17 |- ( x (,) y ) C_ ( x [,] y )
156	155, 127	syl5ss 3515	. . . . . . . . . . . . . . . 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 3514	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( x (,) y ) C_ dom ( RR _D F ) )
159		ssdmres 5293	. . . . . . . . . . . . . . 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 2508	. . . . . . . . . . . . 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 22125	. . . . . . . . . . . 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 5866	. . . . . . . . . . . . . . . . . 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 5878	. . . . . . . . . . . . . . . . . 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 2508	. . . . . . . . . . . . . . . 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 3503	. . . . . . . . . . . . . . . . . 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 6134	. . . . . . . . . . . . . . . . . 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 1227	. . . . . . . . . . . . . . . 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 2555	. . . . . . . . . . . . . . 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 2539	. . . . . . . . . . . . . . 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 2953	. . . . . . . . . . . 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 2556	. . . . . . . . . 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 6133	. . . . . . . . . . 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 1227	. . . . . . . . 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 10500	. . . . . . . . 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 1231	. . . . . . . 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 4487	. . . . . . 7 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( abs ` ( ( ( F ` y ) - ( F ` x ) ) / ( y - x ) ) ) <_ K )
188	106, 187	eqbrtrrd 4469	. . . . . 6 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( abs ` ( ( F ` y ) - ( F ` x ) ) ) / ( abs ` ( y - x ) ) ) <_ K )
189	93	abscld 13223	. . . . . . 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 13235	. . . . . . 7 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( abs ` ( y - x ) ) e. RR+ )
192	189, 190, 191	ledivmuld 11301	. . . . . 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 11254	. . . . . 6 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( abs ` ( y - x ) ) e. CC )
195	190	recnd 9618	. . . . . 6 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> K e. CC )
196	194, 195	mulcomd 9613	. . . . 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 4471	. . . 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 2885	. 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 1379   e. wcel 1767   =/= wne 2662   A.wral 2814   E.wrex 2815   C_ wss 3476   (/)c0 3785   class class class wbr 4447   dom cdm 4999   ran crn 5000   |` cres 5001   "cima 5002   Fun wfun 5580   -->wf 5582   ` cfv 5586  (class class class)co 6282   ^pm cpm 7418   supcsup 7896   CCcc 9486   RRcr 9487   x. cmul 9493   RR*cxr 9623   < clt 9624   <_ cle 9625   - cmin 9801   / cdiv 10202   RR+crp 11216   (,)cioo 11525   [,]cicc 11528   abscabs 13024   TopOpenctopn 14670   topGenctg 14686  ℂ_fldccnfld 18188   intcnt 19281   -cn->ccncf 21112   _D cdv 21999

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566  ax-addf 9567  ax-mulf 9568

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-of 6522  df-om 6679  df-1st 6781  df-2nd 6782  df-supp 6899  df-recs 7039  df-rdg 7073  df-1o 7127  df-2o 7128  df-oadd 7131  df-er 7308  df-map 7419  df-pm 7420  df-ixp 7467  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-fsupp 7826  df-fi 7867  df-sup 7897  df-oi 7931  df-card 8316  df-cda 8544  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10973  df-uz 11079  df-q 11179  df-rp 11217  df-xneg 11314  df-xadd 11315  df-xmul 11316  df-ioo 11529  df-ico 11531  df-icc 11532  df-fz 11669  df-fzo 11789  df-seq 12071  df-exp 12130  df-hash 12368  df-cj 12889  df-re 12890  df-im 12891  df-sqrt 13025  df-abs 13026  df-struct 14485  df-ndx 14486  df-slot 14487  df-base 14488  df-sets 14489  df-ress 14490  df-plusg 14561  df-mulr 14562  df-starv 14563  df-sca 14564  df-vsca 14565  df-ip 14566  df-tset 14567  df-ple 14568  df-ds 14570  df-unif 14571  df-hom 14572  df-cco 14573  df-rest 14671  df-topn 14672  df-0g 14690  df-gsum 14691  df-topgen 14692  df-pt 14693  df-prds 14696  df-xrs 14750  df-qtop 14755  df-imas 14756  df-xps 14758  df-mre 14834  df-mrc 14835  df-acs 14837  df-mnd 15725  df-submnd 15775  df-mulg 15858  df-cntz 16147  df-cmn 16593  df-psmet 18179  df-xmet 18180  df-met 18181  df-bl 18182  df-mopn 18183  df-fbas 18184  df-fg 18185  df-cnfld 18189  df-top 19163  df-bases 19165  df-topon 19166  df-topsp 19167  df-cld 19283  df-ntr 19284  df-cls 19285  df-nei 19362  df-lp 19400  df-perf 19401  df-cn 19491  df-cnp 19492  df-haus 19579  df-cmp 19650  df-tx 19795  df-hmeo 19988  df-fil 20079  df-fm 20171  df-flim 20172  df-flf 20173  df-xms 20555  df-ms 20556  df-tms 20557  df-cncf 21114  df-limc 22002  df-dv 22003

This theorem is referenced by:  c1lip1  22130