Theorem pntlemg 25287

Description: Lemma for pnt 25303. Closure for the constants used in the proof. For comparison with Equation 10.6.27 of [Shapiro], p. 434, M is j^* and N is ĵ. (Contributed by Mario Carneiro, 13-Apr-2016.)

Hypotheses

Ref	Expression
pntlem1.r	|- R = ( a e. RR+ |-> ( (ψ ` a ) - a ) )
pntlem1.a	|- ( ph -> A e. RR+ )
pntlem1.b	|- ( ph -> B e. RR+ )
pntlem1.l	|- ( ph -> L e. ( 0 (,) 1 ) )
pntlem1.d	|- D = ( A + 1 )
pntlem1.f	|- F = ( ( 1 - ( 1 / D ) ) x. ( ( L / (; 3 2 x. B ) ) / ( D ^ 2 ) ) )
pntlem1.u	|- ( ph -> U e. RR+ )
pntlem1.u2	|- ( ph -> U <_ A )
pntlem1.e	|- E = ( U / D )
pntlem1.k	|- K = ( exp ` ( B / E ) )
pntlem1.y	|- ( ph -> ( Y e. RR+ /\ 1 <_ Y ) )
pntlem1.x	|- ( ph -> ( X e. RR+ /\ Y < X ) )
pntlem1.c	|- ( ph -> C e. RR+ )
pntlem1.w	|- W = ( ( ( Y + ( 4 / ( L x. E ) ) ) ^ 2 ) + ( ( ( X x. ( K ^ 2 ) ) ^ 4 ) + ( exp ` ( ( (; 3 2 x. B ) / ( ( U - E ) x. ( L x. ( E ^ 2 ) ) ) ) x. ( ( U x. 3 ) + C ) ) ) ) )
pntlem1.z	|- ( ph -> Z e. ( W [,) +oo ) )
pntlem1.m	|- M = ( ( |_ ` ( ( log ` X ) / ( log ` K ) ) ) + 1 )
pntlem1.n	|- N = ( |_ ` ( ( ( log ` Z ) / ( log ` K ) ) / 2 ) )

Assertion

Ref	Expression
pntlemg	|- ( ph -> ( M e. NN /\ N e. ( ZZ>= ` M ) /\ ( ( ( log ` Z ) / ( log ` K ) ) / 4 ) <_ ( N - M ) ) )

Distinct variable group:   E, a

Allowed substitution hints:   ph( a)   A( a)   B( a)   C( a)   D( a)   R( a)   U( a)   F( a)   K( a)   L( a)   M( a)   N( a)   W( a)   X( a)   Y( a)   Z( a)

Proof of Theorem pntlemg

Step	Hyp	Ref	Expression
1		pntlem1.m	. . 3 |- M = ( ( |_ ` ( ( log ` X ) / ( log ` K ) ) ) + 1 )
2		pntlem1.x	. . . . . . . . 9 |- ( ph -> ( X e. RR+ /\ Y < X ) )
3	2	simpld 475	. . . . . . . 8 |- ( ph -> X e. RR+ )
4	3	rpred 11872	. . . . . . 7 |- ( ph -> X e. RR )
5		1red 10055	. . . . . . . 8 |- ( ph -> 1 e. RR )
6		pntlem1.y	. . . . . . . . . 10 |- ( ph -> ( Y e. RR+ /\ 1 <_ Y ) )
7	6	simpld 475	. . . . . . . . 9 |- ( ph -> Y e. RR+ )
8	7	rpred 11872	. . . . . . . 8 |- ( ph -> Y e. RR )
9	6	simprd 479	. . . . . . . 8 |- ( ph -> 1 <_ Y )
10	2	simprd 479	. . . . . . . 8 |- ( ph -> Y < X )
11	5, 8, 4, 9, 10	lelttrd 10195	. . . . . . 7 |- ( ph -> 1 < X )
12	4, 11	rplogcld 24375	. . . . . 6 |- ( ph -> ( log ` X ) e. RR+ )
13		pntlem1.r	. . . . . . . . . 10 |- R = ( a e. RR+ |-> ( (ψ ` a ) - a ) )
14		pntlem1.a	. . . . . . . . . 10 |- ( ph -> A e. RR+ )
15		pntlem1.b	. . . . . . . . . 10 |- ( ph -> B e. RR+ )
16		pntlem1.l	. . . . . . . . . 10 |- ( ph -> L e. ( 0 (,) 1 ) )
17		pntlem1.d	. . . . . . . . . 10 |- D = ( A + 1 )
18		pntlem1.f	. . . . . . . . . 10 |- F = ( ( 1 - ( 1 / D ) ) x. ( ( L / (; 3 2 x. B ) ) / ( D ^ 2 ) ) )
19		pntlem1.u	. . . . . . . . . 10 |- ( ph -> U e. RR+ )
20		pntlem1.u2	. . . . . . . . . 10 |- ( ph -> U <_ A )
21		pntlem1.e	. . . . . . . . . 10 |- E = ( U / D )
22		pntlem1.k	. . . . . . . . . 10 |- K = ( exp ` ( B / E ) )
23	13, 14, 15, 16, 17, 18, 19, 20, 21, 22	pntlemc 25284	. . . . . . . . 9 |- ( ph -> ( E e. RR+ /\ K e. RR+ /\ ( E e. ( 0 (,) 1 ) /\ 1 < K /\ ( U - E ) e. RR+ ) ) )
24	23	simp2d 1074	. . . . . . . 8 |- ( ph -> K e. RR+ )
25	24	rpred 11872	. . . . . . 7 |- ( ph -> K e. RR )
26	23	simp3d 1075	. . . . . . . 8 |- ( ph -> ( E e. ( 0 (,) 1 ) /\ 1 < K /\ ( U - E ) e. RR+ ) )
27	26	simp2d 1074	. . . . . . 7 |- ( ph -> 1 < K )
28	25, 27	rplogcld 24375	. . . . . 6 |- ( ph -> ( log ` K ) e. RR+ )
29	12, 28	rpdivcld 11889	. . . . 5 |- ( ph -> ( ( log ` X ) / ( log ` K ) ) e. RR+ )
30	29	rprege0d 11879	. . . 4 |- ( ph -> ( ( ( log ` X ) / ( log ` K ) ) e. RR /\ 0 <_ ( ( log ` X ) / ( log ` K ) ) ) )
31		flge0nn0 12621	. . . 4 |- ( ( ( ( log ` X ) / ( log ` K ) ) e. RR /\ 0 <_ ( ( log ` X ) / ( log ` K ) ) ) -> ( |_ ` ( ( log ` X ) / ( log ` K ) ) ) e. NN0 )
32		nn0p1nn 11332	. . . 4 |- ( ( |_ ` ( ( log ` X ) / ( log ` K ) ) ) e. NN0 -> ( ( |_ ` ( ( log ` X ) / ( log ` K ) ) ) + 1 ) e. NN )
33	30, 31, 32	3syl 18	. . 3 |- ( ph -> ( ( |_ ` ( ( log ` X ) / ( log ` K ) ) ) + 1 ) e. NN )
34	1, 33	syl5eqel 2705	. 2 |- ( ph -> M e. NN )
35	34	nnzd 11481	. . 3 |- ( ph -> M e. ZZ )
36		pntlem1.n	. . . 4 |- N = ( |_ ` ( ( ( log ` Z ) / ( log ` K ) ) / 2 ) )
37		pntlem1.c	. . . . . . . . . 10 |- ( ph -> C e. RR+ )
38		pntlem1.w	. . . . . . . . . 10 |- W = ( ( ( Y + ( 4 / ( L x. E ) ) ) ^ 2 ) + ( ( ( X x. ( K ^ 2 ) ) ^ 4 ) + ( exp ` ( ( (; 3 2 x. B ) / ( ( U - E ) x. ( L x. ( E ^ 2 ) ) ) ) x. ( ( U x. 3 ) + C ) ) ) ) )
39		pntlem1.z	. . . . . . . . . 10 |- ( ph -> Z e. ( W [,) +oo ) )
40	13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 6, 2, 37, 38, 39	pntlemb 25286	. . . . . . . . 9 |- ( ph -> ( Z e. RR+ /\ ( 1 < Z /\ _e <_ ( sqr ` Z ) /\ ( sqr ` Z ) <_ ( Z / Y ) ) /\ ( ( 4 / ( L x. E ) ) <_ ( sqr ` Z ) /\ ( ( ( log ` X ) / ( log ` K ) ) + 2 ) <_ ( ( ( log ` Z ) / ( log ` K ) ) / 4 ) /\ ( ( U x. 3 ) + C ) <_ ( ( ( U - E ) x. ( ( L x. ( E ^ 2 ) ) / (; 3 2 x. B ) ) ) x. ( log ` Z ) ) ) ) )
41	40	simp1d 1073	. . . . . . . 8 |- ( ph -> Z e. RR+ )
42	41	relogcld 24369	. . . . . . 7 |- ( ph -> ( log ` Z ) e. RR )
43	42, 28	rerpdivcld 11903	. . . . . 6 |- ( ph -> ( ( log ` Z ) / ( log ` K ) ) e. RR )
44	43	rehalfcld 11279	. . . . 5 |- ( ph -> ( ( ( log ` Z ) / ( log ` K ) ) / 2 ) e. RR )
45	44	flcld 12599	. . . 4 |- ( ph -> ( |_ ` ( ( ( log ` Z ) / ( log ` K ) ) / 2 ) ) e. ZZ )
46	36, 45	syl5eqel 2705	. . 3 |- ( ph -> N e. ZZ )
47		0red 10041	. . . . 5 |- ( ph -> 0 e. RR )
48		4nn 11187	. . . . . 6 |- 4 e. NN
49		nndivre 11056	. . . . . 6 |- ( ( ( ( log ` Z ) / ( log ` K ) ) e. RR /\ 4 e. NN ) -> ( ( ( log ` Z ) / ( log ` K ) ) / 4 ) e. RR )
50	43, 48, 49	sylancl 694	. . . . 5 |- ( ph -> ( ( ( log ` Z ) / ( log ` K ) ) / 4 ) e. RR )
51	46	zred 11482	. . . . . 6 |- ( ph -> N e. RR )
52	34	nnred 11035	. . . . . 6 |- ( ph -> M e. RR )
53	51, 52	resubcld 10458	. . . . 5 |- ( ph -> ( N - M ) e. RR )
54	41	rpred 11872	. . . . . . . . 9 |- ( ph -> Z e. RR )
55	40	simp2d 1074	. . . . . . . . . 10 |- ( ph -> ( 1 < Z /\ _e <_ ( sqr ` Z ) /\ ( sqr ` Z ) <_ ( Z / Y ) ) )
56	55	simp1d 1073	. . . . . . . . 9 |- ( ph -> 1 < Z )
57	54, 56	rplogcld 24375	. . . . . . . 8 |- ( ph -> ( log ` Z ) e. RR+ )
58	57, 28	rpdivcld 11889	. . . . . . 7 |- ( ph -> ( ( log ` Z ) / ( log ` K ) ) e. RR+ )
59		4re 11097	. . . . . . . 8 |- 4 e. RR
60		4pos 11116	. . . . . . . 8 |- 0 < 4
61	59, 60	elrpii 11835	. . . . . . 7 |- 4 e. RR+
62		rpdivcl 11856	. . . . . . 7 |- ( ( ( ( log ` Z ) / ( log ` K ) ) e. RR+ /\ 4 e. RR+ ) -> ( ( ( log ` Z ) / ( log ` K ) ) / 4 ) e. RR+ )
63	58, 61, 62	sylancl 694	. . . . . 6 |- ( ph -> ( ( ( log ` Z ) / ( log ` K ) ) / 4 ) e. RR+ )
64	63	rpge0d 11876	. . . . 5 |- ( ph -> 0 <_ ( ( ( log ` Z ) / ( log ` K ) ) / 4 ) )
65	50	recnd 10068	. . . . . . . . 9 |- ( ph -> ( ( ( log ` Z ) / ( log ` K ) ) / 4 ) e. CC )
66	34	nncnd 11036	. . . . . . . . 9 |- ( ph -> M e. CC )
67		1cnd 10056	. . . . . . . . 9 |- ( ph -> 1 e. CC )
68	65, 66, 67	addassd 10062	. . . . . . . 8 |- ( ph -> ( ( ( ( ( log ` Z ) / ( log ` K ) ) / 4 ) + M ) + 1 ) = ( ( ( ( log ` Z ) / ( log ` K ) ) / 4 ) + ( M + 1 ) ) )
69	52, 5	readdcld 10069	. . . . . . . . . 10 |- ( ph -> ( M + 1 ) e. RR )
70	50, 69	readdcld 10069	. . . . . . . . 9 |- ( ph -> ( ( ( ( log ` Z ) / ( log ` K ) ) / 4 ) + ( M + 1 ) ) e. RR )
71		peano2re 10209	. . . . . . . . . 10 |- ( N e. RR -> ( N + 1 ) e. RR )
72	51, 71	syl 17	. . . . . . . . 9 |- ( ph -> ( N + 1 ) e. RR )
73	29	rpred 11872	. . . . . . . . . . . . 13 |- ( ph -> ( ( log ` X ) / ( log ` K ) ) e. RR )
74		2re 11090	. . . . . . . . . . . . . 14 |- 2 e. RR
75	74	a1i 11	. . . . . . . . . . . . 13 |- ( ph -> 2 e. RR )
76	73, 75	readdcld 10069	. . . . . . . . . . . 12 |- ( ph -> ( ( ( log ` X ) / ( log ` K ) ) + 2 ) e. RR )
77		reflcl 12597	. . . . . . . . . . . . . . . . 17 |- ( ( ( log ` X ) / ( log ` K ) ) e. RR -> ( |_ ` ( ( log ` X ) / ( log ` K ) ) ) e. RR )
78	73, 77	syl 17	. . . . . . . . . . . . . . . 16 |- ( ph -> ( |_ ` ( ( log ` X ) / ( log ` K ) ) ) e. RR )
79	78	recnd 10068	. . . . . . . . . . . . . . 15 |- ( ph -> ( |_ ` ( ( log ` X ) / ( log ` K ) ) ) e. CC )
80	79, 67, 67	addassd 10062	. . . . . . . . . . . . . 14 |- ( ph -> ( ( ( |_ ` ( ( log ` X ) / ( log ` K ) ) ) + 1 ) + 1 ) = ( ( |_ ` ( ( log ` X ) / ( log ` K ) ) ) + ( 1 + 1 ) ) )
81	1	oveq1i 6660	. . . . . . . . . . . . . 14 |- ( M + 1 ) = ( ( ( |_ ` ( ( log ` X ) / ( log ` K ) ) ) + 1 ) + 1 )
82		df-2 11079	. . . . . . . . . . . . . . 15 |- 2 = ( 1 + 1 )
83	82	oveq2i 6661	. . . . . . . . . . . . . 14 |- ( ( |_ ` ( ( log ` X ) / ( log ` K ) ) ) + 2 ) = ( ( |_ ` ( ( log ` X ) / ( log ` K ) ) ) + ( 1 + 1 ) )
84	80, 81, 83	3eqtr4g 2681	. . . . . . . . . . . . 13 |- ( ph -> ( M + 1 ) = ( ( |_ ` ( ( log ` X ) / ( log ` K ) ) ) + 2 ) )
85		flle 12600	. . . . . . . . . . . . . . 15 |- ( ( ( log ` X ) / ( log ` K ) ) e. RR -> ( |_ ` ( ( log ` X ) / ( log ` K ) ) ) <_ ( ( log ` X ) / ( log ` K ) ) )
86	73, 85	syl 17	. . . . . . . . . . . . . 14 |- ( ph -> ( |_ ` ( ( log ` X ) / ( log ` K ) ) ) <_ ( ( log ` X ) / ( log ` K ) ) )
87	78, 73, 75, 86	leadd1dd 10641	. . . . . . . . . . . . 13 |- ( ph -> ( ( |_ ` ( ( log ` X ) / ( log ` K ) ) ) + 2 ) <_ ( ( ( log ` X ) / ( log ` K ) ) + 2 ) )
88	84, 87	eqbrtrd 4675	. . . . . . . . . . . 12 |- ( ph -> ( M + 1 ) <_ ( ( ( log ` X ) / ( log ` K ) ) + 2 ) )
89	40	simp3d 1075	. . . . . . . . . . . . 13 |- ( ph -> ( ( 4 / ( L x. E ) ) <_ ( sqr ` Z ) /\ ( ( ( log ` X ) / ( log ` K ) ) + 2 ) <_ ( ( ( log ` Z ) / ( log ` K ) ) / 4 ) /\ ( ( U x. 3 ) + C ) <_ ( ( ( U - E ) x. ( ( L x. ( E ^ 2 ) ) / (; 3 2 x. B ) ) ) x. ( log ` Z ) ) ) )
90	89	simp2d 1074	. . . . . . . . . . . 12 |- ( ph -> ( ( ( log ` X ) / ( log ` K ) ) + 2 ) <_ ( ( ( log ` Z ) / ( log ` K ) ) / 4 ) )
91	69, 76, 50, 88, 90	letrd 10194	. . . . . . . . . . 11 |- ( ph -> ( M + 1 ) <_ ( ( ( log ` Z ) / ( log ` K ) ) / 4 ) )
92	69, 50, 50, 91	leadd2dd 10642	. . . . . . . . . 10 |- ( ph -> ( ( ( ( log ` Z ) / ( log ` K ) ) / 4 ) + ( M + 1 ) ) <_ ( ( ( ( log ` Z ) / ( log ` K ) ) / 4 ) + ( ( ( log ` Z ) / ( log ` K ) ) / 4 ) ) )
93	43	recnd 10068	. . . . . . . . . . . . . 14 |- ( ph -> ( ( log ` Z ) / ( log ` K ) ) e. CC )
94		2cnd 11093	. . . . . . . . . . . . . 14 |- ( ph -> 2 e. CC )
95		2ne0 11113	. . . . . . . . . . . . . . 15 |- 2 =/= 0
96	95	a1i 11	. . . . . . . . . . . . . 14 |- ( ph -> 2 =/= 0 )
97	93, 94, 94, 96, 96	divdiv1d 10832	. . . . . . . . . . . . 13 |- ( ph -> ( ( ( ( log ` Z ) / ( log ` K ) ) / 2 ) / 2 ) = ( ( ( log ` Z ) / ( log ` K ) ) / ( 2 x. 2 ) ) )
98		2t2e4 11177	. . . . . . . . . . . . . 14 |- ( 2 x. 2 ) = 4
99	98	oveq2i 6661	. . . . . . . . . . . . 13 |- ( ( ( log ` Z ) / ( log ` K ) ) / ( 2 x. 2 ) ) = ( ( ( log ` Z ) / ( log ` K ) ) / 4 )
100	97, 99	syl6eq 2672	. . . . . . . . . . . 12 |- ( ph -> ( ( ( ( log ` Z ) / ( log ` K ) ) / 2 ) / 2 ) = ( ( ( log ` Z ) / ( log ` K ) ) / 4 ) )
101	100	oveq2d 6666	. . . . . . . . . . 11 |- ( ph -> ( 2 x. ( ( ( ( log ` Z ) / ( log ` K ) ) / 2 ) / 2 ) ) = ( 2 x. ( ( ( log ` Z ) / ( log ` K ) ) / 4 ) ) )
102	44	recnd 10068	. . . . . . . . . . . 12 |- ( ph -> ( ( ( log ` Z ) / ( log ` K ) ) / 2 ) e. CC )
103	102, 94, 96	divcan2d 10803	. . . . . . . . . . 11 |- ( ph -> ( 2 x. ( ( ( ( log ` Z ) / ( log ` K ) ) / 2 ) / 2 ) ) = ( ( ( log ` Z ) / ( log ` K ) ) / 2 ) )
104	65	2timesd 11275	. . . . . . . . . . 11 |- ( ph -> ( 2 x. ( ( ( log ` Z ) / ( log ` K ) ) / 4 ) ) = ( ( ( ( log ` Z ) / ( log ` K ) ) / 4 ) + ( ( ( log ` Z ) / ( log ` K ) ) / 4 ) ) )
105	101, 103, 104	3eqtr3d 2664	. . . . . . . . . 10 |- ( ph -> ( ( ( log ` Z ) / ( log ` K ) ) / 2 ) = ( ( ( ( log ` Z ) / ( log ` K ) ) / 4 ) + ( ( ( log ` Z ) / ( log ` K ) ) / 4 ) ) )
106	92, 105	breqtrrd 4681	. . . . . . . . 9 |- ( ph -> ( ( ( ( log ` Z ) / ( log ` K ) ) / 4 ) + ( M + 1 ) ) <_ ( ( ( log ` Z ) / ( log ` K ) ) / 2 ) )
107		fllep1 12602	. . . . . . . . . . 11 |- ( ( ( ( log ` Z ) / ( log ` K ) ) / 2 ) e. RR -> ( ( ( log ` Z ) / ( log ` K ) ) / 2 ) <_ ( ( |_ ` ( ( ( log ` Z ) / ( log ` K ) ) / 2 ) ) + 1 ) )
108	44, 107	syl 17	. . . . . . . . . 10 |- ( ph -> ( ( ( log ` Z ) / ( log ` K ) ) / 2 ) <_ ( ( |_ ` ( ( ( log ` Z ) / ( log ` K ) ) / 2 ) ) + 1 ) )
109	36	oveq1i 6660	. . . . . . . . . 10 |- ( N + 1 ) = ( ( |_ ` ( ( ( log ` Z ) / ( log ` K ) ) / 2 ) ) + 1 )
110	108, 109	syl6breqr 4695	. . . . . . . . 9 |- ( ph -> ( ( ( log ` Z ) / ( log ` K ) ) / 2 ) <_ ( N + 1 ) )
111	70, 44, 72, 106, 110	letrd 10194	. . . . . . . 8 |- ( ph -> ( ( ( ( log ` Z ) / ( log ` K ) ) / 4 ) + ( M + 1 ) ) <_ ( N + 1 ) )
112	68, 111	eqbrtrd 4675	. . . . . . 7 |- ( ph -> ( ( ( ( ( log ` Z ) / ( log ` K ) ) / 4 ) + M ) + 1 ) <_ ( N + 1 ) )
113	50, 52	readdcld 10069	. . . . . . . 8 |- ( ph -> ( ( ( ( log ` Z ) / ( log ` K ) ) / 4 ) + M ) e. RR )
114	113, 51, 5	leadd1d 10621	. . . . . . 7 |- ( ph -> ( ( ( ( ( log ` Z ) / ( log ` K ) ) / 4 ) + M ) <_ N <-> ( ( ( ( ( log ` Z ) / ( log ` K ) ) / 4 ) + M ) + 1 ) <_ ( N + 1 ) ) )
115	112, 114	mpbird 247	. . . . . 6 |- ( ph -> ( ( ( ( log ` Z ) / ( log ` K ) ) / 4 ) + M ) <_ N )
116		leaddsub 10504	. . . . . . 7 |- ( ( ( ( ( log ` Z ) / ( log ` K ) ) / 4 ) e. RR /\ M e. RR /\ N e. RR ) -> ( ( ( ( ( log ` Z ) / ( log ` K ) ) / 4 ) + M ) <_ N <-> ( ( ( log ` Z ) / ( log ` K ) ) / 4 ) <_ ( N - M ) ) )
117	50, 52, 51, 116	syl3anc 1326	. . . . . 6 |- ( ph -> ( ( ( ( ( log ` Z ) / ( log ` K ) ) / 4 ) + M ) <_ N <-> ( ( ( log ` Z ) / ( log ` K ) ) / 4 ) <_ ( N - M ) ) )
118	115, 117	mpbid 222	. . . . 5 |- ( ph -> ( ( ( log ` Z ) / ( log ` K ) ) / 4 ) <_ ( N - M ) )
119	47, 50, 53, 64, 118	letrd 10194	. . . 4 |- ( ph -> 0 <_ ( N - M ) )
120	51, 52	subge0d 10617	. . . 4 |- ( ph -> ( 0 <_ ( N - M ) <-> M <_ N ) )
121	119, 120	mpbid 222	. . 3 |- ( ph -> M <_ N )
122		eluz2 11693	. . 3 |- ( N e. ( ZZ>= ` M ) <-> ( M e. ZZ /\ N e. ZZ /\ M <_ N ) )
123	35, 46, 121, 122	syl3anbrc 1246	. 2 |- ( ph -> N e. ( ZZ>= ` M ) )
124	34, 123, 118	3jca 1242	1 |- ( ph -> ( M e. NN /\ N e. ( ZZ>= ` M ) /\ ( ( ( log ` Z ) / ( log ` K ) ) / 4 ) <_ ( N - M ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ 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   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   +oocpnf 10071   < clt 10074   <_ cle 10075   - cmin 10266   / cdiv 10684   NNcn 11020   2c2 11070   3c3 11071   4c4 11072   NN0cn0 11292   ZZcz 11377  ;cdc 11493   ZZ>=cuz 11687   RR+crp 11832   (,)cioo 12175   [,)cico 12177   |_cfl 12591   ^cexp 12860   sqrcsqrt 13973   expce 14792   _eceu 14793   logclog 24301  ψcchp 24819

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-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  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-ef 14798  df-e 14799  df-sin 14800  df-cos 14801  df-pi 14803  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-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  df-log 24303

This theorem is referenced by:  pntlemh  25288  pntlemq  25290  pntlemr  25291  pntlemj  25292  pntlemf  25294