Theorem knoppndvlem17 32519

Description: Lemma for knoppndv 32525. (Contributed by Asger C. Ipsen, 12-Aug-2021.)

Hypotheses

Ref	Expression
knoppndvlem17.t	|- T = ( x e. RR |-> ( abs ` ( ( |_ ` ( x + ( 1 / 2 ) ) ) - x ) ) )
knoppndvlem17.f	|- F = ( y e. RR |-> ( n e. NN0 |-> ( ( C ^ n ) x. ( T ` ( ( ( 2 x. N ) ^ n ) x. y ) ) ) ) )
knoppndvlem17.w	|- W = ( w e. RR |-> sum_ i e. NN0 ( ( F ` w ) ` i ) )
knoppndvlem17.a	|- A = ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. M )
knoppndvlem17.b	|- B = ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. ( M + 1 ) )
knoppndvlem17.c	|- ( ph -> C e. ( -u 1 (,) 1 ) )
knoppndvlem17.j	|- ( ph -> J e. NN0 )
knoppndvlem17.m	|- ( ph -> M e. ZZ )
knoppndvlem17.n	|- ( ph -> N e. NN )
knoppndvlem17.1	|- ( ph -> 1 < ( N x. ( abs ` C ) ) )

Assertion

Ref	Expression
knoppndvlem17	|- ( ph -> ( ( ( ( 2 x. N ) x. ( abs ` C ) ) ^ J ) x. ( 1 - ( 1 / ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) ) ) ) <_ ( ( abs ` ( ( W ` B ) - ( W ` A ) ) ) / ( B - A ) ) )

Distinct variable groups:   A, i, n, w, y   x, A, i, w   B, i, n, w, y   x, B   C, i, n, y   i, F, w   i, J, n, y   x, J   n, M, y   x, M   i, N, n, y   x, N   T, n, y   ph, i, n, w, y

Allowed substitution hints:   ph( x)   C( x, w)   T( x, w, i)   F( x, y, n)   J( w)   M( w, i)   N( w)   W( x, y, w, i, n)

Proof of Theorem knoppndvlem17

Step	Hyp	Ref	Expression
1		knoppndvlem17.c	. . . . . . . . . . . . . 14 |- ( ph -> C e. ( -u 1 (,) 1 ) )
2	1	knoppndvlem3 32505	. . . . . . . . . . . . 13 |- ( ph -> ( C e. RR /\ ( abs ` C ) < 1 ) )
3	2	simpld 475	. . . . . . . . . . . 12 |- ( ph -> C e. RR )
4	3	recnd 10068	. . . . . . . . . . 11 |- ( ph -> C e. CC )
5	4	abscld 14175	. . . . . . . . . 10 |- ( ph -> ( abs ` C ) e. RR )
6		knoppndvlem17.j	. . . . . . . . . 10 |- ( ph -> J e. NN0 )
7	5, 6	reexpcld 13025	. . . . . . . . 9 |- ( ph -> ( ( abs ` C ) ^ J ) e. RR )
8		2re 11090	. . . . . . . . . 10 |- 2 e. RR
9	8	a1i 11	. . . . . . . . 9 |- ( ph -> 2 e. RR )
10		2ne0 11113	. . . . . . . . . 10 |- 2 =/= 0
11	10	a1i 11	. . . . . . . . 9 |- ( ph -> 2 =/= 0 )
12	7, 9, 11	redivcld 10853	. . . . . . . 8 |- ( ph -> ( ( ( abs ` C ) ^ J ) / 2 ) e. RR )
13	12	recnd 10068	. . . . . . 7 |- ( ph -> ( ( ( abs ` C ) ^ J ) / 2 ) e. CC )
14		1red 10055	. . . . . . . . 9 |- ( ph -> 1 e. RR )
15		knoppndvlem17.n	. . . . . . . . . . . . . 14 |- ( ph -> N e. NN )
16	15	nnred 11035	. . . . . . . . . . . . 13 |- ( ph -> N e. RR )
17	9, 16	remulcld 10070	. . . . . . . . . . . 12 |- ( ph -> ( 2 x. N ) e. RR )
18	17, 5	remulcld 10070	. . . . . . . . . . 11 |- ( ph -> ( ( 2 x. N ) x. ( abs ` C ) ) e. RR )
19	18, 14	resubcld 10458	. . . . . . . . . 10 |- ( ph -> ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) e. RR )
20		0red 10041	. . . . . . . . . . . . 13 |- ( ph -> 0 e. RR )
21		0lt1 10550	. . . . . . . . . . . . . 14 |- 0 < 1
22	21	a1i 11	. . . . . . . . . . . . 13 |- ( ph -> 0 < 1 )
23		knoppndvlem17.1	. . . . . . . . . . . . . . 15 |- ( ph -> 1 < ( N x. ( abs ` C ) ) )
24	1, 15, 23	knoppndvlem12 32514	. . . . . . . . . . . . . 14 |- ( ph -> ( ( ( 2 x. N ) x. ( abs ` C ) ) =/= 1 /\ 1 < ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) ) )
25	24	simprd 479	. . . . . . . . . . . . 13 |- ( ph -> 1 < ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) )
26	20, 14, 19, 22, 25	lttrd 10198	. . . . . . . . . . . 12 |- ( ph -> 0 < ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) )
27	19, 26	jca 554	. . . . . . . . . . 11 |- ( ph -> ( ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) e. RR /\ 0 < ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) ) )
28		gt0ne0 10493	. . . . . . . . . . 11 |- ( ( ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) e. RR /\ 0 < ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) ) -> ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) =/= 0 )
29	27, 28	syl 17	. . . . . . . . . 10 |- ( ph -> ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) =/= 0 )
30	14, 19, 29	redivcld 10853	. . . . . . . . 9 |- ( ph -> ( 1 / ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) ) e. RR )
31	14, 30	resubcld 10458	. . . . . . . 8 |- ( ph -> ( 1 - ( 1 / ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) ) ) e. RR )
32	31	recnd 10068	. . . . . . 7 |- ( ph -> ( 1 - ( 1 / ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) ) ) e. CC )
33	13, 32	mulcomd 10061	. . . . . 6 |- ( ph -> ( ( ( ( abs ` C ) ^ J ) / 2 ) x. ( 1 - ( 1 / ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) ) ) ) = ( ( 1 - ( 1 / ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) ) ) x. ( ( ( abs ` C ) ^ J ) / 2 ) ) )
34	33	oveq1d 6665	. . . . 5 |- ( ph -> ( ( ( ( ( abs ` C ) ^ J ) / 2 ) x. ( 1 - ( 1 / ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) ) ) ) / ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) = ( ( ( 1 - ( 1 / ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) ) ) x. ( ( ( abs ` C ) ^ J ) / 2 ) ) / ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) )
35		2rp 11837	. . . . . . . . . . 11 |- 2 e. RR+
36	35	a1i 11	. . . . . . . . . 10 |- ( ph -> 2 e. RR+ )
37	15	nnrpd 11870	. . . . . . . . . 10 |- ( ph -> N e. RR+ )
38	36, 37	rpmulcld 11888	. . . . . . . . 9 |- ( ph -> ( 2 x. N ) e. RR+ )
39	6	nn0zd 11480	. . . . . . . . . 10 |- ( ph -> J e. ZZ )
40	39	znegcld 11484	. . . . . . . . 9 |- ( ph -> -u J e. ZZ )
41	38, 40	rpexpcld 13032	. . . . . . . 8 |- ( ph -> ( ( 2 x. N ) ^ -u J ) e. RR+ )
42	41	rphalfcld 11884	. . . . . . 7 |- ( ph -> ( ( ( 2 x. N ) ^ -u J ) / 2 ) e. RR+ )
43	42	rpcnd 11874	. . . . . 6 |- ( ph -> ( ( ( 2 x. N ) ^ -u J ) / 2 ) e. CC )
44	42	rpne0d 11877	. . . . . 6 |- ( ph -> ( ( ( 2 x. N ) ^ -u J ) / 2 ) =/= 0 )
45	32, 13, 43, 44	divassd 10836	. . . . 5 |- ( ph -> ( ( ( 1 - ( 1 / ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) ) ) x. ( ( ( abs ` C ) ^ J ) / 2 ) ) / ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) = ( ( 1 - ( 1 / ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) ) ) x. ( ( ( ( abs ` C ) ^ J ) / 2 ) / ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) ) )
46	13, 43, 44	divcld 10801	. . . . . . 7 |- ( ph -> ( ( ( ( abs ` C ) ^ J ) / 2 ) / ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) e. CC )
47	32, 46	mulcomd 10061	. . . . . 6 |- ( ph -> ( ( 1 - ( 1 / ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) ) ) x. ( ( ( ( abs ` C ) ^ J ) / 2 ) / ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) ) = ( ( ( ( ( abs ` C ) ^ J ) / 2 ) / ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) x. ( 1 - ( 1 / ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) ) ) ) )
48	7	recnd 10068	. . . . . . . . 9 |- ( ph -> ( ( abs ` C ) ^ J ) e. CC )
49	41	rpcnd 11874	. . . . . . . . 9 |- ( ph -> ( ( 2 x. N ) ^ -u J ) e. CC )
50	9	recnd 10068	. . . . . . . . 9 |- ( ph -> 2 e. CC )
51	41	rpne0d 11877	. . . . . . . . 9 |- ( ph -> ( ( 2 x. N ) ^ -u J ) =/= 0 )
52	48, 49, 50, 51, 11	divcan7d 10829	. . . . . . . 8 |- ( ph -> ( ( ( ( abs ` C ) ^ J ) / 2 ) / ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) = ( ( ( abs ` C ) ^ J ) / ( ( 2 x. N ) ^ -u J ) ) )
53	17	recnd 10068	. . . . . . . . . . 11 |- ( ph -> ( 2 x. N ) e. CC )
54	38	rpne0d 11877	. . . . . . . . . . 11 |- ( ph -> ( 2 x. N ) =/= 0 )
55	53, 54, 39	expnegd 13015	. . . . . . . . . 10 |- ( ph -> ( ( 2 x. N ) ^ -u J ) = ( 1 / ( ( 2 x. N ) ^ J ) ) )
56	55	oveq2d 6666	. . . . . . . . 9 |- ( ph -> ( ( ( abs ` C ) ^ J ) / ( ( 2 x. N ) ^ -u J ) ) = ( ( ( abs ` C ) ^ J ) / ( 1 / ( ( 2 x. N ) ^ J ) ) ) )
57		1cnd 10056	. . . . . . . . . 10 |- ( ph -> 1 e. CC )
58	53, 6	expcld 13008	. . . . . . . . . 10 |- ( ph -> ( ( 2 x. N ) ^ J ) e. CC )
59	20, 22	gtned 10172	. . . . . . . . . 10 |- ( ph -> 1 =/= 0 )
60	53, 54, 39	expne0d 13014	. . . . . . . . . 10 |- ( ph -> ( ( 2 x. N ) ^ J ) =/= 0 )
61	48, 57, 58, 59, 60	divdiv2d 10833	. . . . . . . . 9 |- ( ph -> ( ( ( abs ` C ) ^ J ) / ( 1 / ( ( 2 x. N ) ^ J ) ) ) = ( ( ( ( abs ` C ) ^ J ) x. ( ( 2 x. N ) ^ J ) ) / 1 ) )
62	48, 58	mulcld 10060	. . . . . . . . . . 11 |- ( ph -> ( ( ( abs ` C ) ^ J ) x. ( ( 2 x. N ) ^ J ) ) e. CC )
63	62	div1d 10793	. . . . . . . . . 10 |- ( ph -> ( ( ( ( abs ` C ) ^ J ) x. ( ( 2 x. N ) ^ J ) ) / 1 ) = ( ( ( abs ` C ) ^ J ) x. ( ( 2 x. N ) ^ J ) ) )
64	48, 58	mulcomd 10061	. . . . . . . . . 10 |- ( ph -> ( ( ( abs ` C ) ^ J ) x. ( ( 2 x. N ) ^ J ) ) = ( ( ( 2 x. N ) ^ J ) x. ( ( abs ` C ) ^ J ) ) )
65	53, 54	jca 554	. . . . . . . . . . . . 13 |- ( ph -> ( ( 2 x. N ) e. CC /\ ( 2 x. N ) =/= 0 ) )
66	5	recnd 10068	. . . . . . . . . . . . . 14 |- ( ph -> ( abs ` C ) e. CC )
67	1, 15, 23	knoppndvlem13 32515	. . . . . . . . . . . . . . 15 |- ( ph -> C =/= 0 )
68	4, 67	absne0d 14186	. . . . . . . . . . . . . 14 |- ( ph -> ( abs ` C ) =/= 0 )
69	66, 68	jca 554	. . . . . . . . . . . . 13 |- ( ph -> ( ( abs ` C ) e. CC /\ ( abs ` C ) =/= 0 ) )
70	65, 69, 39	3jca 1242	. . . . . . . . . . . 12 |- ( ph -> ( ( ( 2 x. N ) e. CC /\ ( 2 x. N ) =/= 0 ) /\ ( ( abs ` C ) e. CC /\ ( abs ` C ) =/= 0 ) /\ J e. ZZ ) )
71		mulexpz 12900	. . . . . . . . . . . 12 |- ( ( ( ( 2 x. N ) e. CC /\ ( 2 x. N ) =/= 0 ) /\ ( ( abs ` C ) e. CC /\ ( abs ` C ) =/= 0 ) /\ J e. ZZ ) -> ( ( ( 2 x. N ) x. ( abs ` C ) ) ^ J ) = ( ( ( 2 x. N ) ^ J ) x. ( ( abs ` C ) ^ J ) ) )
72	70, 71	syl 17	. . . . . . . . . . 11 |- ( ph -> ( ( ( 2 x. N ) x. ( abs ` C ) ) ^ J ) = ( ( ( 2 x. N ) ^ J ) x. ( ( abs ` C ) ^ J ) ) )
73	72	eqcomd 2628	. . . . . . . . . 10 |- ( ph -> ( ( ( 2 x. N ) ^ J ) x. ( ( abs ` C ) ^ J ) ) = ( ( ( 2 x. N ) x. ( abs ` C ) ) ^ J ) )
74	63, 64, 73	3eqtrd 2660	. . . . . . . . 9 |- ( ph -> ( ( ( ( abs ` C ) ^ J ) x. ( ( 2 x. N ) ^ J ) ) / 1 ) = ( ( ( 2 x. N ) x. ( abs ` C ) ) ^ J ) )
75	56, 61, 74	3eqtrd 2660	. . . . . . . 8 |- ( ph -> ( ( ( abs ` C ) ^ J ) / ( ( 2 x. N ) ^ -u J ) ) = ( ( ( 2 x. N ) x. ( abs ` C ) ) ^ J ) )
76	52, 75	eqtrd 2656	. . . . . . 7 |- ( ph -> ( ( ( ( abs ` C ) ^ J ) / 2 ) / ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) = ( ( ( 2 x. N ) x. ( abs ` C ) ) ^ J ) )
77	76	oveq1d 6665	. . . . . 6 |- ( ph -> ( ( ( ( ( abs ` C ) ^ J ) / 2 ) / ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) x. ( 1 - ( 1 / ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) ) ) ) = ( ( ( ( 2 x. N ) x. ( abs ` C ) ) ^ J ) x. ( 1 - ( 1 / ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) ) ) ) )
78	47, 77	eqtrd 2656	. . . . 5 |- ( ph -> ( ( 1 - ( 1 / ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) ) ) x. ( ( ( ( abs ` C ) ^ J ) / 2 ) / ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) ) = ( ( ( ( 2 x. N ) x. ( abs ` C ) ) ^ J ) x. ( 1 - ( 1 / ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) ) ) ) )
79	34, 45, 78	3eqtrd 2660	. . . 4 |- ( ph -> ( ( ( ( ( abs ` C ) ^ J ) / 2 ) x. ( 1 - ( 1 / ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) ) ) ) / ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) = ( ( ( ( 2 x. N ) x. ( abs ` C ) ) ^ J ) x. ( 1 - ( 1 / ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) ) ) ) )
80	79	eqcomd 2628	. . 3 |- ( ph -> ( ( ( ( 2 x. N ) x. ( abs ` C ) ) ^ J ) x. ( 1 - ( 1 / ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) ) ) ) = ( ( ( ( ( abs ` C ) ^ J ) / 2 ) x. ( 1 - ( 1 / ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) ) ) ) / ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) )
81	12, 31	remulcld 10070	. . . 4 |- ( ph -> ( ( ( ( abs ` C ) ^ J ) / 2 ) x. ( 1 - ( 1 / ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) ) ) ) e. RR )
82		knoppndvlem17.t	. . . . . . 7 |- T = ( x e. RR |-> ( abs ` ( ( |_ ` ( x + ( 1 / 2 ) ) ) - x ) ) )
83		knoppndvlem17.f	. . . . . . 7 |- F = ( y e. RR |-> ( n e. NN0 |-> ( ( C ^ n ) x. ( T ` ( ( ( 2 x. N ) ^ n ) x. y ) ) ) ) )
84		knoppndvlem17.w	. . . . . . 7 |- W = ( w e. RR |-> sum_ i e. NN0 ( ( F ` w ) ` i ) )
85		knoppndvlem17.b	. . . . . . . . 9 |- B = ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. ( M + 1 ) )
86	85	a1i 11	. . . . . . . 8 |- ( ph -> B = ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. ( M + 1 ) ) )
87		knoppndvlem17.m	. . . . . . . . . 10 |- ( ph -> M e. ZZ )
88	87	peano2zd 11485	. . . . . . . . 9 |- ( ph -> ( M + 1 ) e. ZZ )
89	15, 39, 88	knoppndvlem1 32503	. . . . . . . 8 |- ( ph -> ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. ( M + 1 ) ) e. RR )
90	86, 89	eqeltrd 2701	. . . . . . 7 |- ( ph -> B e. RR )
91	2	simprd 479	. . . . . . 7 |- ( ph -> ( abs ` C ) < 1 )
92	82, 83, 84, 90, 15, 3, 91	knoppcld 32495	. . . . . 6 |- ( ph -> ( W ` B ) e. CC )
93		knoppndvlem17.a	. . . . . . . . 9 |- A = ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. M )
94	93	a1i 11	. . . . . . . 8 |- ( ph -> A = ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. M ) )
95	15, 39, 87	knoppndvlem1 32503	. . . . . . . 8 |- ( ph -> ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. M ) e. RR )
96	94, 95	eqeltrd 2701	. . . . . . 7 |- ( ph -> A e. RR )
97	82, 83, 84, 96, 15, 3, 91	knoppcld 32495	. . . . . 6 |- ( ph -> ( W ` A ) e. CC )
98	92, 97	subcld 10392	. . . . 5 |- ( ph -> ( ( W ` B ) - ( W ` A ) ) e. CC )
99	98	abscld 14175	. . . 4 |- ( ph -> ( abs ` ( ( W ` B ) - ( W ` A ) ) ) e. RR )
100	82, 83, 84, 93, 85, 1, 6, 87, 15, 23	knoppndvlem15 32517	. . . 4 |- ( ph -> ( ( ( ( abs ` C ) ^ J ) / 2 ) x. ( 1 - ( 1 / ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) ) ) ) <_ ( abs ` ( ( W ` B ) - ( W ` A ) ) ) )
101	81, 99, 42, 100	lediv1dd 11930	. . 3 |- ( ph -> ( ( ( ( ( abs ` C ) ^ J ) / 2 ) x. ( 1 - ( 1 / ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) ) ) ) / ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) <_ ( ( abs ` ( ( W ` B ) - ( W ` A ) ) ) / ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) )
102	80, 101	eqbrtrd 4675	. 2 |- ( ph -> ( ( ( ( 2 x. N ) x. ( abs ` C ) ) ^ J ) x. ( 1 - ( 1 / ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) ) ) ) <_ ( ( abs ` ( ( W ` B ) - ( W ` A ) ) ) / ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) )
103	93, 85, 6, 87, 15	knoppndvlem16 32518	. . . 4 |- ( ph -> ( B - A ) = ( ( ( 2 x. N ) ^ -u J ) / 2 ) )
104	103	eqcomd 2628	. . 3 |- ( ph -> ( ( ( 2 x. N ) ^ -u J ) / 2 ) = ( B - A ) )
105	104	oveq2d 6666	. 2 |- ( ph -> ( ( abs ` ( ( W ` B ) - ( W ` A ) ) ) / ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) = ( ( abs ` ( ( W ` B ) - ( W ` A ) ) ) / ( B - A ) ) )
106	102, 105	breqtrd 4679	1 |- ( ph -> ( ( ( ( 2 x. N ) x. ( abs ` C ) ) ^ J ) x. ( 1 - ( 1 / ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) ) ) ) <_ ( ( abs ` ( ( W ` B ) - ( W ` A ) ) ) / ( B - A ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   <_ cle 10075   - cmin 10266   -ucneg 10267   / cdiv 10684   NNcn 11020   2c2 11070   NN0cn0 11292   ZZcz 11377   RR+crp 11832   (,)cioo 12175   |_cfl 12591   ^cexp 12860   abscabs 13974   sum_csu 14416

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-fal 1489  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-fl 12593  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-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-dvds 14984  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-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cn 21031  df-cnp 21032  df-tx 21365  df-hmeo 21558  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-ulm 24131

This theorem is referenced by:  knoppndvlem21  32523