Theorem logsqvma 23925

Description: A formula for log ^ 2 ( N ) in terms of the primes. Equation 10.4.6 of [Shapiro], p. 418. (Contributed by Mario Carneiro, 13-May-2016.)

Assertion

Ref	Expression
logsqvma	|- ( N e. NN -> sum_ d e. { x e. NN | x || N } ( sum_ u e. { x e. NN | x || d } ( (Λ ` u ) x. (Λ ` ( d / u ) ) ) + ( (Λ ` d ) x. ( log ` d ) ) ) = ( ( log ` N ) ^ 2 ) )

Distinct variable group:   u, d, x, N

Proof of Theorem logsqvma

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fzfid 12065	. . . 4 |- ( N e. NN -> ( 1 ... N ) e. Fin )
2		sgmss 23578	. . . 4 |- ( N e. NN -> { x e. NN | x || N } C_ ( 1 ... N ) )
3		ssfi 7733	. . . 4 |- ( ( ( 1 ... N ) e. Fin /\ { x e. NN | x || N } C_ ( 1 ... N ) ) -> { x e. NN | x || N } e. Fin )
4	1, 2, 3	syl2anc 659	. . 3 |- ( N e. NN -> { x e. NN | x || N } e. Fin )
5		fzfid 12065	. . . . 5 |- ( ( N e. NN /\ d e. { x e. NN | x || N } ) -> ( 1 ... d ) e. Fin )
6		elrabi 3251	. . . . . . 7 |- ( d e. { x e. NN | x || N } -> d e. NN )
7	6	adantl 464	. . . . . 6 |- ( ( N e. NN /\ d e. { x e. NN | x || N } ) -> d e. NN )
8		sgmss 23578	. . . . . 6 |- ( d e. NN -> { x e. NN | x || d } C_ ( 1 ... d ) )
9	7, 8	syl 16	. . . . 5 |- ( ( N e. NN /\ d e. { x e. NN | x || N } ) -> { x e. NN | x || d } C_ ( 1 ... d ) )
10		ssfi 7733	. . . . 5 |- ( ( ( 1 ... d ) e. Fin /\ { x e. NN | x || d } C_ ( 1 ... d ) ) -> { x e. NN | x || d } e. Fin )
11	5, 9, 10	syl2anc 659	. . . 4 |- ( ( N e. NN /\ d e. { x e. NN | x || N } ) -> { x e. NN | x || d } e. Fin )
12		elrabi 3251	. . . . . . . . 9 |- ( u e. { x e. NN | x || d } -> u e. NN )
13	12	ad2antll 726	. . . . . . . 8 |- ( ( N e. NN /\ ( d e. { x e. NN | x || N } /\ u e. { x e. NN | x || d } ) ) -> u e. NN )
14		vmacl 23590	. . . . . . . 8 |- ( u e. NN -> (Λ ` u ) e. RR )
15	13, 14	syl 16	. . . . . . 7 |- ( ( N e. NN /\ ( d e. { x e. NN | x || N } /\ u e. { x e. NN | x || d } ) ) -> (Λ ` u ) e. RR )
16		breq1 4442	. . . . . . . . . . . 12 |- ( x = u -> ( x || d <-> u || d ) )
17	16	elrab 3254	. . . . . . . . . . 11 |- ( u e. { x e. NN | x || d } <-> ( u e. NN /\ u || d ) )
18	17	simprbi 462	. . . . . . . . . 10 |- ( u e. { x e. NN | x || d } -> u || d )
19	18	ad2antll 726	. . . . . . . . 9 |- ( ( N e. NN /\ ( d e. { x e. NN | x || N } /\ u e. { x e. NN | x || d } ) ) -> u || d )
20	6	ad2antrl 725	. . . . . . . . . 10 |- ( ( N e. NN /\ ( d e. { x e. NN | x || N } /\ u e. { x e. NN | x || d } ) ) -> d e. NN )
21		nndivdvds 14076	. . . . . . . . . 10 |- ( ( d e. NN /\ u e. NN ) -> ( u || d <-> ( d / u ) e. NN ) )
22	20, 13, 21	syl2anc 659	. . . . . . . . 9 |- ( ( N e. NN /\ ( d e. { x e. NN | x || N } /\ u e. { x e. NN | x || d } ) ) -> ( u || d <-> ( d / u ) e. NN ) )
23	19, 22	mpbid 210	. . . . . . . 8 |- ( ( N e. NN /\ ( d e. { x e. NN | x || N } /\ u e. { x e. NN | x || d } ) ) -> ( d / u ) e. NN )
24		vmacl 23590	. . . . . . . 8 |- ( ( d / u ) e. NN -> (Λ ` ( d / u ) ) e. RR )
25	23, 24	syl 16	. . . . . . 7 |- ( ( N e. NN /\ ( d e. { x e. NN | x || N } /\ u e. { x e. NN | x || d } ) ) -> (Λ ` ( d / u ) ) e. RR )
26	15, 25	remulcld 9613	. . . . . 6 |- ( ( N e. NN /\ ( d e. { x e. NN | x || N } /\ u e. { x e. NN | x || d } ) ) -> ( (Λ ` u ) x. (Λ ` ( d / u ) ) ) e. RR )
27	26	recnd 9611	. . . . 5 |- ( ( N e. NN /\ ( d e. { x e. NN | x || N } /\ u e. { x e. NN | x || d } ) ) -> ( (Λ ` u ) x. (Λ ` ( d / u ) ) ) e. CC )
28	27	anassrs 646	. . . 4 |- ( ( ( N e. NN /\ d e. { x e. NN | x || N } ) /\ u e. { x e. NN | x || d } ) -> ( (Λ ` u ) x. (Λ ` ( d / u ) ) ) e. CC )
29	11, 28	fsumcl 13637	. . 3 |- ( ( N e. NN /\ d e. { x e. NN | x || N } ) -> sum_ u e. { x e. NN | x || d } ( (Λ ` u ) x. (Λ ` ( d / u ) ) ) e. CC )
30		vmacl 23590	. . . . . 6 |- ( d e. NN -> (Λ ` d ) e. RR )
31	7, 30	syl 16	. . . . 5 |- ( ( N e. NN /\ d e. { x e. NN | x || N } ) -> (Λ ` d ) e. RR )
32	7	nnrpd 11257	. . . . . 6 |- ( ( N e. NN /\ d e. { x e. NN | x || N } ) -> d e. RR+ )
33	32	relogcld 23176	. . . . 5 |- ( ( N e. NN /\ d e. { x e. NN | x || N } ) -> ( log ` d ) e. RR )
34	31, 33	remulcld 9613	. . . 4 |- ( ( N e. NN /\ d e. { x e. NN | x || N } ) -> ( (Λ ` d ) x. ( log ` d ) ) e. RR )
35	34	recnd 9611	. . 3 |- ( ( N e. NN /\ d e. { x e. NN | x || N } ) -> ( (Λ ` d ) x. ( log ` d ) ) e. CC )
36	4, 29, 35	fsumadd 13643	. 2 |- ( N e. NN -> sum_ d e. { x e. NN | x || N } ( sum_ u e. { x e. NN | x || d } ( (Λ ` u ) x. (Λ ` ( d / u ) ) ) + ( (Λ ` d ) x. ( log ` d ) ) ) = ( sum_ d e. { x e. NN | x || N } sum_ u e. { x e. NN | x || d } ( (Λ ` u ) x. (Λ ` ( d / u ) ) ) + sum_ d e. { x e. NN | x || N } ( (Λ ` d ) x. ( log ` d ) ) ) )
37		id 22	. . . . 5 |- ( N e. NN -> N e. NN )
38		oveq1 6277	. . . . . . 7 |- ( d = ( u x. k ) -> ( d / u ) = ( ( u x. k ) / u ) )
39	38	fveq2d 5852	. . . . . 6 |- ( d = ( u x. k ) -> (Λ ` ( d / u ) ) = (Λ ` ( ( u x. k ) / u ) ) )
40	39	oveq2d 6286	. . . . 5 |- ( d = ( u x. k ) -> ( (Λ ` u ) x. (Λ ` ( d / u ) ) ) = ( (Λ ` u ) x. (Λ ` ( ( u x. k ) / u ) ) ) )
41	37, 40, 27	fsumdvdscom 23659	. . . 4 |- ( N e. NN -> sum_ d e. { x e. NN | x || N } sum_ u e. { x e. NN | x || d } ( (Λ ` u ) x. (Λ ` ( d / u ) ) ) = sum_ u e. { x e. NN | x || N } sum_ k e. { x e. NN | x || ( N / u ) } ( (Λ ` u ) x. (Λ ` ( ( u x. k ) / u ) ) ) )
42		ssrab2 3571	. . . . . . . . . . . . 13 |- { x e. NN | x || ( N / u ) } C_ NN
43		simpr 459	. . . . . . . . . . . . 13 |- ( ( ( N e. NN /\ u e. { x e. NN | x || N } ) /\ k e. { x e. NN | x || ( N / u ) } ) -> k e. { x e. NN | x || ( N / u ) } )
44	42, 43	sseldi 3487	. . . . . . . . . . . 12 |- ( ( ( N e. NN /\ u e. { x e. NN | x || N } ) /\ k e. { x e. NN | x || ( N / u ) } ) -> k e. NN )
45	44	nncnd 10547	. . . . . . . . . . 11 |- ( ( ( N e. NN /\ u e. { x e. NN | x || N } ) /\ k e. { x e. NN | x || ( N / u ) } ) -> k e. CC )
46		ssrab2 3571	. . . . . . . . . . . . . 14 |- { x e. NN | x || N } C_ NN
47		simpr 459	. . . . . . . . . . . . . 14 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> u e. { x e. NN | x || N } )
48	46, 47	sseldi 3487	. . . . . . . . . . . . 13 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> u e. NN )
49	48	nncnd 10547	. . . . . . . . . . . 12 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> u e. CC )
50	49	adantr 463	. . . . . . . . . . 11 |- ( ( ( N e. NN /\ u e. { x e. NN | x || N } ) /\ k e. { x e. NN | x || ( N / u ) } ) -> u e. CC )
51	48	nnne0d 10576	. . . . . . . . . . . 12 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> u =/= 0 )
52	51	adantr 463	. . . . . . . . . . 11 |- ( ( ( N e. NN /\ u e. { x e. NN | x || N } ) /\ k e. { x e. NN | x || ( N / u ) } ) -> u =/= 0 )
53	45, 50, 52	divcan3d 10321	. . . . . . . . . 10 |- ( ( ( N e. NN /\ u e. { x e. NN | x || N } ) /\ k e. { x e. NN | x || ( N / u ) } ) -> ( ( u x. k ) / u ) = k )
54	53	fveq2d 5852	. . . . . . . . 9 |- ( ( ( N e. NN /\ u e. { x e. NN | x || N } ) /\ k e. { x e. NN | x || ( N / u ) } ) -> (Λ ` ( ( u x. k ) / u ) ) = (Λ ` k ) )
55	54	sumeq2dv 13607	. . . . . . . 8 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> sum_ k e. { x e. NN | x || ( N / u ) } (Λ ` ( ( u x. k ) / u ) ) = sum_ k e. { x e. NN | x || ( N / u ) } (Λ ` k ) )
56		dvdsdivcl 23655	. . . . . . . . . 10 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> ( N / u ) e. { x e. NN | x || N } )
57	46, 56	sseldi 3487	. . . . . . . . 9 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> ( N / u ) e. NN )
58		vmasum 23689	. . . . . . . . 9 |- ( ( N / u ) e. NN -> sum_ k e. { x e. NN | x || ( N / u ) } (Λ ` k ) = ( log ` ( N / u ) ) )
59	57, 58	syl 16	. . . . . . . 8 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> sum_ k e. { x e. NN | x || ( N / u ) } (Λ ` k ) = ( log ` ( N / u ) ) )
60		nnrp 11230	. . . . . . . . . 10 |- ( N e. NN -> N e. RR+ )
61	60	adantr 463	. . . . . . . . 9 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> N e. RR+ )
62	48	nnrpd 11257	. . . . . . . . 9 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> u e. RR+ )
63	61, 62	relogdivd 23179	. . . . . . . 8 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> ( log ` ( N / u ) ) = ( ( log ` N ) - ( log ` u ) ) )
64	55, 59, 63	3eqtrd 2499	. . . . . . 7 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> sum_ k e. { x e. NN | x || ( N / u ) } (Λ ` ( ( u x. k ) / u ) ) = ( ( log ` N ) - ( log ` u ) ) )
65	64	oveq2d 6286	. . . . . 6 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> ( (Λ ` u ) x. sum_ k e. { x e. NN | x || ( N / u ) } (Λ ` ( ( u x. k ) / u ) ) ) = ( (Λ ` u ) x. ( ( log ` N ) - ( log ` u ) ) ) )
66		fzfid 12065	. . . . . . . 8 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> ( 1 ... ( N / u ) ) e. Fin )
67		sgmss 23578	. . . . . . . . 9 |- ( ( N / u ) e. NN -> { x e. NN | x || ( N / u ) } C_ ( 1 ... ( N / u ) ) )
68	57, 67	syl 16	. . . . . . . 8 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> { x e. NN | x || ( N / u ) } C_ ( 1 ... ( N / u ) ) )
69		ssfi 7733	. . . . . . . 8 |- ( ( ( 1 ... ( N / u ) ) e. Fin /\ { x e. NN | x || ( N / u ) } C_ ( 1 ... ( N / u ) ) ) -> { x e. NN | x || ( N / u ) } e. Fin )
70	66, 68, 69	syl2anc 659	. . . . . . 7 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> { x e. NN | x || ( N / u ) } e. Fin )
71	48, 14	syl 16	. . . . . . . 8 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> (Λ ` u ) e. RR )
72	71	recnd 9611	. . . . . . 7 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> (Λ ` u ) e. CC )
73		vmacl 23590	. . . . . . . . . 10 |- ( k e. NN -> (Λ ` k ) e. RR )
74	44, 73	syl 16	. . . . . . . . 9 |- ( ( ( N e. NN /\ u e. { x e. NN | x || N } ) /\ k e. { x e. NN | x || ( N / u ) } ) -> (Λ ` k ) e. RR )
75	74	recnd 9611	. . . . . . . 8 |- ( ( ( N e. NN /\ u e. { x e. NN | x || N } ) /\ k e. { x e. NN | x || ( N / u ) } ) -> (Λ ` k ) e. CC )
76	54, 75	eqeltrd 2542	. . . . . . 7 |- ( ( ( N e. NN /\ u e. { x e. NN | x || N } ) /\ k e. { x e. NN | x || ( N / u ) } ) -> (Λ ` ( ( u x. k ) / u ) ) e. CC )
77	70, 72, 76	fsummulc2 13681	. . . . . 6 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> ( (Λ ` u ) x. sum_ k e. { x e. NN | x || ( N / u ) } (Λ ` ( ( u x. k ) / u ) ) ) = sum_ k e. { x e. NN | x || ( N / u ) } ( (Λ ` u ) x. (Λ ` ( ( u x. k ) / u ) ) ) )
78		relogcl 23129	. . . . . . . . 9 |- ( N e. RR+ -> ( log ` N ) e. RR )
79	78	recnd 9611	. . . . . . . 8 |- ( N e. RR+ -> ( log ` N ) e. CC )
80	61, 79	syl 16	. . . . . . 7 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> ( log ` N ) e. CC )
81	62	relogcld 23176	. . . . . . . 8 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> ( log ` u ) e. RR )
82	81	recnd 9611	. . . . . . 7 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> ( log ` u ) e. CC )
83	72, 80, 82	subdid 10008	. . . . . 6 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> ( (Λ ` u ) x. ( ( log ` N ) - ( log ` u ) ) ) = ( ( (Λ ` u ) x. ( log ` N ) ) - ( (Λ ` u ) x. ( log ` u ) ) ) )
84	65, 77, 83	3eqtr3d 2503	. . . . 5 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> sum_ k e. { x e. NN | x || ( N / u ) } ( (Λ ` u ) x. (Λ ` ( ( u x. k ) / u ) ) ) = ( ( (Λ ` u ) x. ( log ` N ) ) - ( (Λ ` u ) x. ( log ` u ) ) ) )
85	84	sumeq2dv 13607	. . . 4 |- ( N e. NN -> sum_ u e. { x e. NN | x || N } sum_ k e. { x e. NN | x || ( N / u ) } ( (Λ ` u ) x. (Λ ` ( ( u x. k ) / u ) ) ) = sum_ u e. { x e. NN | x || N } ( ( (Λ ` u ) x. ( log ` N ) ) - ( (Λ ` u ) x. ( log ` u ) ) ) )
86	72, 80	mulcld 9605	. . . . . 6 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> ( (Λ ` u ) x. ( log ` N ) ) e. CC )
87	72, 82	mulcld 9605	. . . . . 6 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> ( (Λ ` u ) x. ( log ` u ) ) e. CC )
88	4, 86, 87	fsumsub 13685	. . . . 5 |- ( N e. NN -> sum_ u e. { x e. NN | x || N } ( ( (Λ ` u ) x. ( log ` N ) ) - ( (Λ ` u ) x. ( log ` u ) ) ) = ( sum_ u e. { x e. NN | x || N } ( (Λ ` u ) x. ( log ` N ) ) - sum_ u e. { x e. NN | x || N } ( (Λ ` u ) x. ( log ` u ) ) ) )
89	60, 79	syl 16	. . . . . . . 8 |- ( N e. NN -> ( log ` N ) e. CC )
90	89	sqvald 12289	. . . . . . 7 |- ( N e. NN -> ( ( log ` N ) ^ 2 ) = ( ( log ` N ) x. ( log ` N ) ) )
91		vmasum 23689	. . . . . . . 8 |- ( N e. NN -> sum_ u e. { x e. NN | x || N } (Λ ` u ) = ( log ` N ) )
92	91	oveq1d 6285	. . . . . . 7 |- ( N e. NN -> ( sum_ u e. { x e. NN | x || N } (Λ ` u ) x. ( log ` N ) ) = ( ( log ` N ) x. ( log ` N ) ) )
93	4, 89, 72	fsummulc1 13682	. . . . . . 7 |- ( N e. NN -> ( sum_ u e. { x e. NN | x || N } (Λ ` u ) x. ( log ` N ) ) = sum_ u e. { x e. NN | x || N } ( (Λ ` u ) x. ( log ` N ) ) )
94	90, 92, 93	3eqtr2rd 2502	. . . . . 6 |- ( N e. NN -> sum_ u e. { x e. NN | x || N } ( (Λ ` u ) x. ( log ` N ) ) = ( ( log ` N ) ^ 2 ) )
95		fveq2 5848	. . . . . . . . 9 |- ( u = d -> (Λ ` u ) = (Λ ` d ) )
96		fveq2 5848	. . . . . . . . 9 |- ( u = d -> ( log ` u ) = ( log ` d ) )
97	95, 96	oveq12d 6288	. . . . . . . 8 |- ( u = d -> ( (Λ ` u ) x. ( log ` u ) ) = ( (Λ ` d ) x. ( log ` d ) ) )
98	97	cbvsumv 13600	. . . . . . 7 |- sum_ u e. { x e. NN | x || N } ( (Λ ` u ) x. ( log ` u ) ) = sum_ d e. { x e. NN | x || N } ( (Λ ` d ) x. ( log ` d ) )
99	98	a1i 11	. . . . . 6 |- ( N e. NN -> sum_ u e. { x e. NN | x || N } ( (Λ ` u ) x. ( log ` u ) ) = sum_ d e. { x e. NN | x || N } ( (Λ ` d ) x. ( log ` d ) ) )
100	94, 99	oveq12d 6288	. . . . 5 |- ( N e. NN -> ( sum_ u e. { x e. NN | x || N } ( (Λ ` u ) x. ( log ` N ) ) - sum_ u e. { x e. NN | x || N } ( (Λ ` u ) x. ( log ` u ) ) ) = ( ( ( log ` N ) ^ 2 ) - sum_ d e. { x e. NN | x || N } ( (Λ ` d ) x. ( log ` d ) ) ) )
101	88, 100	eqtrd 2495	. . . 4 |- ( N e. NN -> sum_ u e. { x e. NN | x || N } ( ( (Λ ` u ) x. ( log ` N ) ) - ( (Λ ` u ) x. ( log ` u ) ) ) = ( ( ( log ` N ) ^ 2 ) - sum_ d e. { x e. NN | x || N } ( (Λ ` d ) x. ( log ` d ) ) ) )
102	41, 85, 101	3eqtrd 2499	. . 3 |- ( N e. NN -> sum_ d e. { x e. NN | x || N } sum_ u e. { x e. NN | x || d } ( (Λ ` u ) x. (Λ ` ( d / u ) ) ) = ( ( ( log ` N ) ^ 2 ) - sum_ d e. { x e. NN | x || N } ( (Λ ` d ) x. ( log ` d ) ) ) )
103	102	oveq1d 6285	. 2 |- ( N e. NN -> ( sum_ d e. { x e. NN | x || N } sum_ u e. { x e. NN | x || d } ( (Λ ` u ) x. (Λ ` ( d / u ) ) ) + sum_ d e. { x e. NN | x || N } ( (Λ ` d ) x. ( log ` d ) ) ) = ( ( ( ( log ` N ) ^ 2 ) - sum_ d e. { x e. NN | x || N } ( (Λ ` d ) x. ( log ` d ) ) ) + sum_ d e. { x e. NN | x || N } ( (Λ ` d ) x. ( log ` d ) ) ) )
104	89	sqcld 12290	. . 3 |- ( N e. NN -> ( ( log ` N ) ^ 2 ) e. CC )
105	4, 35	fsumcl 13637	. . 3 |- ( N e. NN -> sum_ d e. { x e. NN | x || N } ( (Λ ` d ) x. ( log ` d ) ) e. CC )
106	104, 105	npcand 9926	. 2 |- ( N e. NN -> ( ( ( ( log ` N ) ^ 2 ) - sum_ d e. { x e. NN | x || N } ( (Λ ` d ) x. ( log ` d ) ) ) + sum_ d e. { x e. NN | x || N } ( (Λ ` d ) x. ( log ` d ) ) ) = ( ( log ` N ) ^ 2 ) )
107	36, 103, 106	3eqtrd 2499	1 |- ( N e. NN -> sum_ d e. { x e. NN | x || N } ( sum_ u e. { x e. NN | x || d } ( (Λ ` u ) x. (Λ ` ( d / u ) ) ) + ( (Λ ` d ) x. ( log ` d ) ) ) = ( ( log ` N ) ^ 2 ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 367   = wceq 1398   e. wcel 1823   =/= wne 2649   {crab 2808   C_ wss 3461   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   Fincfn 7509   CCcc 9479   RRcr 9480   0cc0 9481   1c1 9482   + caddc 9484   x. cmul 9486   - cmin 9796   / cdiv 10202   NNcn 10531   2c2 10581   RR+crp 11221   ...cfz 11675   ^cexp 12148   sum_csu 13590   || cdvds 14070   logclog 23108  Λcvma 23563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560  ax-mulf 9561

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-ixp 7463  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-fi 7863  df-sup 7893  df-oi 7927  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  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 10977  df-uz 11083  df-q 11184  df-rp 11222  df-xneg 11321  df-xadd 11322  df-xmul 11323  df-ioo 11536  df-ioc 11537  df-ico 11538  df-icc 11539  df-fz 11676  df-fzo 11800  df-fl 11910  df-mod 11979  df-seq 12090  df-exp 12149  df-fac 12336  df-bc 12363  df-hash 12388  df-shft 12982  df-cj 13014  df-re 13015  df-im 13016  df-sqrt 13150  df-abs 13151  df-limsup 13376  df-clim 13393  df-rlim 13394  df-sum 13591  df-ef 13885  df-sin 13887  df-cos 13888  df-pi 13890  df-dvds 14071  df-gcd 14229  df-prm 14302  df-pc 14445  df-struct 14718  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ress 14723  df-plusg 14797  df-mulr 14798  df-starv 14799  df-sca 14800  df-vsca 14801  df-ip 14802  df-tset 14803  df-ple 14804  df-ds 14806  df-unif 14807  df-hom 14808  df-cco 14809  df-rest 14912  df-topn 14913  df-0g 14931  df-gsum 14932  df-topgen 14933  df-pt 14934  df-prds 14937  df-xrs 14991  df-qtop 14996  df-imas 14997  df-xps 14999  df-mre 15075  df-mrc 15076  df-acs 15078  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-submnd 16166  df-mulg 16259  df-cntz 16554  df-cmn 16999  df-psmet 18606  df-xmet 18607  df-met 18608  df-bl 18609  df-mopn 18610  df-fbas 18611  df-fg 18612  df-cnfld 18616  df-top 19566  df-bases 19568  df-topon 19569  df-topsp 19570  df-cld 19687  df-ntr 19688  df-cls 19689  df-nei 19766  df-lp 19804  df-perf 19805  df-cn 19895  df-cnp 19896  df-haus 19983  df-tx 20229  df-hmeo 20422  df-fil 20513  df-fm 20605  df-flim 20606  df-flf 20607  df-xms 20989  df-ms 20990  df-tms 20991  df-cncf 21548  df-limc 22436  df-dv 22437  df-log 23110  df-vma 23569

This theorem is referenced by:  logsqvma2  23926