Theorem logsqvma 24373

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 12183	. . . 4 |- ( N e. NN -> ( 1 ... N ) e. Fin )
2		sgmss 24026	. . . 4 |- ( N e. NN -> { x e. NN | x || N } C_ ( 1 ... N ) )
3		ssfi 7789	. . . 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 666	. . 3 |- ( N e. NN -> { x e. NN | x || N } e. Fin )
5		fzfid 12183	. . . . 5 |- ( ( N e. NN /\ d e. { x e. NN | x || N } ) -> ( 1 ... d ) e. Fin )
6		elrabi 3192	. . . . . . 7 |- ( d e. { x e. NN | x || N } -> d e. NN )
7	6	adantl 468	. . . . . 6 |- ( ( N e. NN /\ d e. { x e. NN | x || N } ) -> d e. NN )
8		sgmss 24026	. . . . . 6 |- ( d e. NN -> { x e. NN | x || d } C_ ( 1 ... d ) )
9	7, 8	syl 17	. . . . 5 |- ( ( N e. NN /\ d e. { x e. NN | x || N } ) -> { x e. NN | x || d } C_ ( 1 ... d ) )
10		ssfi 7789	. . . . 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 666	. . . 4 |- ( ( N e. NN /\ d e. { x e. NN | x || N } ) -> { x e. NN | x || d } e. Fin )
12		elrabi 3192	. . . . . . . . 9 |- ( u e. { x e. NN | x || d } -> u e. NN )
13	12	ad2antll 734	. . . . . . . 8 |- ( ( N e. NN /\ ( d e. { x e. NN | x || N } /\ u e. { x e. NN | x || d } ) ) -> u e. NN )
14		vmacl 24038	. . . . . . . 8 |- ( u e. NN -> (Λ ` u ) e. RR )
15	13, 14	syl 17	. . . . . . 7 |- ( ( N e. NN /\ ( d e. { x e. NN | x || N } /\ u e. { x e. NN | x || d } ) ) -> (Λ ` u ) e. RR )
16		breq1 4404	. . . . . . . . . . . 12 |- ( x = u -> ( x || d <-> u || d ) )
17	16	elrab 3195	. . . . . . . . . . 11 |- ( u e. { x e. NN | x || d } <-> ( u e. NN /\ u || d ) )
18	17	simprbi 466	. . . . . . . . . 10 |- ( u e. { x e. NN | x || d } -> u || d )
19	18	ad2antll 734	. . . . . . . . 9 |- ( ( N e. NN /\ ( d e. { x e. NN | x || N } /\ u e. { x e. NN | x || d } ) ) -> u || d )
20	6	ad2antrl 733	. . . . . . . . . 10 |- ( ( N e. NN /\ ( d e. { x e. NN | x || N } /\ u e. { x e. NN | x || d } ) ) -> d e. NN )
21		nndivdvds 14304	. . . . . . . . . 10 |- ( ( d e. NN /\ u e. NN ) -> ( u || d <-> ( d / u ) e. NN ) )
22	20, 13, 21	syl2anc 666	. . . . . . . . 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 214	. . . . . . . 8 |- ( ( N e. NN /\ ( d e. { x e. NN | x || N } /\ u e. { x e. NN | x || d } ) ) -> ( d / u ) e. NN )
24		vmacl 24038	. . . . . . . 8 |- ( ( d / u ) e. NN -> (Λ ` ( d / u ) ) e. RR )
25	23, 24	syl 17	. . . . . . 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 9668	. . . . . 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 9666	. . . . 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 653	. . . 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 13792	. . 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 24038	. . . . . 6 |- ( d e. NN -> (Λ ` d ) e. RR )
31	7, 30	syl 17	. . . . 5 |- ( ( N e. NN /\ d e. { x e. NN | x || N } ) -> (Λ ` d ) e. RR )
32	7	nnrpd 11336	. . . . . 6 |- ( ( N e. NN /\ d e. { x e. NN | x || N } ) -> d e. RR+ )
33	32	relogcld 23565	. . . . 5 |- ( ( N e. NN /\ d e. { x e. NN | x || N } ) -> ( log ` d ) e. RR )
34	31, 33	remulcld 9668	. . . 4 |- ( ( N e. NN /\ d e. { x e. NN | x || N } ) -> ( (Λ ` d ) x. ( log ` d ) ) e. RR )
35	34	recnd 9666	. . 3 |- ( ( N e. NN /\ d e. { x e. NN | x || N } ) -> ( (Λ ` d ) x. ( log ` d ) ) e. CC )
36	4, 29, 35	fsumadd 13798	. 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 6295	. . . . . . 7 |- ( d = ( u x. k ) -> ( d / u ) = ( ( u x. k ) / u ) )
39	38	fveq2d 5867	. . . . . 6 |- ( d = ( u x. k ) -> (Λ ` ( d / u ) ) = (Λ ` ( ( u x. k ) / u ) ) )
40	39	oveq2d 6304	. . . . 5 |- ( d = ( u x. k ) -> ( (Λ ` u ) x. (Λ ` ( d / u ) ) ) = ( (Λ ` u ) x. (Λ ` ( ( u x. k ) / u ) ) ) )
41	37, 40, 27	fsumdvdscom 24107	. . . 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 3513	. . . . . . . . . . . . 13 |- { x e. NN | x || ( N / u ) } C_ NN
43		simpr 463	. . . . . . . . . . . . 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 3429	. . . . . . . . . . . 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 10622	. . . . . . . . . . 11 |- ( ( ( N e. NN /\ u e. { x e. NN | x || N } ) /\ k e. { x e. NN | x || ( N / u ) } ) -> k e. CC )
46		ssrab2 3513	. . . . . . . . . . . . . 14 |- { x e. NN | x || N } C_ NN
47		simpr 463	. . . . . . . . . . . . . 14 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> u e. { x e. NN | x || N } )
48	46, 47	sseldi 3429	. . . . . . . . . . . . 13 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> u e. NN )
49	48	nncnd 10622	. . . . . . . . . . . 12 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> u e. CC )
50	49	adantr 467	. . . . . . . . . . 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 10651	. . . . . . . . . . . 12 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> u =/= 0 )
52	51	adantr 467	. . . . . . . . . . 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 10385	. . . . . . . . . 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 5867	. . . . . . . . 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 13762	. . . . . . . 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 24103	. . . . . . . . . 10 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> ( N / u ) e. { x e. NN | x || N } )
57	46, 56	sseldi 3429	. . . . . . . . 9 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> ( N / u ) e. NN )
58		vmasum 24137	. . . . . . . . 9 |- ( ( N / u ) e. NN -> sum_ k e. { x e. NN | x || ( N / u ) } (Λ ` k ) = ( log ` ( N / u ) ) )
59	57, 58	syl 17	. . . . . . . 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 11308	. . . . . . . . . 10 |- ( N e. NN -> N e. RR+ )
61	60	adantr 467	. . . . . . . . 9 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> N e. RR+ )
62	48	nnrpd 11336	. . . . . . . . 9 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> u e. RR+ )
63	61, 62	relogdivd 23568	. . . . . . . 8 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> ( log ` ( N / u ) ) = ( ( log ` N ) - ( log ` u ) ) )
64	55, 59, 63	3eqtrd 2488	. . . . . . 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 6304	. . . . . 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 12183	. . . . . . . 8 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> ( 1 ... ( N / u ) ) e. Fin )
67		sgmss 24026	. . . . . . . . 9 |- ( ( N / u ) e. NN -> { x e. NN | x || ( N / u ) } C_ ( 1 ... ( N / u ) ) )
68	57, 67	syl 17	. . . . . . . 8 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> { x e. NN | x || ( N / u ) } C_ ( 1 ... ( N / u ) ) )
69		ssfi 7789	. . . . . . . 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 666	. . . . . . 7 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> { x e. NN | x || ( N / u ) } e. Fin )
71	48, 14	syl 17	. . . . . . . 8 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> (Λ ` u ) e. RR )
72	71	recnd 9666	. . . . . . 7 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> (Λ ` u ) e. CC )
73		vmacl 24038	. . . . . . . . . 10 |- ( k e. NN -> (Λ ` k ) e. RR )
74	44, 73	syl 17	. . . . . . . . 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 9666	. . . . . . . 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 2528	. . . . . . 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 13838	. . . . . 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 23518	. . . . . . . . 9 |- ( N e. RR+ -> ( log ` N ) e. RR )
79	78	recnd 9666	. . . . . . . 8 |- ( N e. RR+ -> ( log ` N ) e. CC )
80	61, 79	syl 17	. . . . . . 7 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> ( log ` N ) e. CC )
81	62	relogcld 23565	. . . . . . . 8 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> ( log ` u ) e. RR )
82	81	recnd 9666	. . . . . . 7 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> ( log ` u ) e. CC )
83	72, 80, 82	subdid 10071	. . . . . 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 2492	. . . . 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 13762	. . . 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 9660	. . . . . 6 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> ( (Λ ` u ) x. ( log ` N ) ) e. CC )
87	72, 82	mulcld 9660	. . . . . 6 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> ( (Λ ` u ) x. ( log ` u ) ) e. CC )
88	4, 86, 87	fsumsub 13842	. . . . 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 17	. . . . . . . 8 |- ( N e. NN -> ( log ` N ) e. CC )
90	89	sqvald 12410	. . . . . . 7 |- ( N e. NN -> ( ( log ` N ) ^ 2 ) = ( ( log ` N ) x. ( log ` N ) ) )
91		vmasum 24137	. . . . . . . 8 |- ( N e. NN -> sum_ u e. { x e. NN | x || N } (Λ ` u ) = ( log ` N ) )
92	91	oveq1d 6303	. . . . . . 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 13839	. . . . . . 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 2491	. . . . . 6 |- ( N e. NN -> sum_ u e. { x e. NN | x || N } ( (Λ ` u ) x. ( log ` N ) ) = ( ( log ` N ) ^ 2 ) )
95		fveq2 5863	. . . . . . . . 9 |- ( u = d -> (Λ ` u ) = (Λ ` d ) )
96		fveq2 5863	. . . . . . . . 9 |- ( u = d -> ( log ` u ) = ( log ` d ) )
97	95, 96	oveq12d 6306	. . . . . . . 8 |- ( u = d -> ( (Λ ` u ) x. ( log ` u ) ) = ( (Λ ` d ) x. ( log ` d ) ) )
98	97	cbvsumv 13755	. . . . . . 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 6306	. . . . 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 2484	. . . 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 2488	. . 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 6303	. 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 12411	. . 3 |- ( N e. NN -> ( ( log ` N ) ^ 2 ) e. CC )
105	4, 35	fsumcl 13792	. . 3 |- ( N e. NN -> sum_ d e. { x e. NN | x || N } ( (Λ ` d ) x. ( log ` d ) ) e. CC )
106	104, 105	npcand 9987	. 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 2488	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 188   /\ wa 371   = wceq 1443   e. wcel 1886   =/= wne 2621   {crab 2740   C_ wss 3403   class class class wbr 4401   ` cfv 5581  (class class class)co 6288   Fincfn 7566   CCcc 9534   RRcr 9535   0cc0 9536   1c1 9537   + caddc 9539   x. cmul 9541   - cmin 9857   / cdiv 10266   NNcn 10606   2c2 10656   RR+crp 11299   ...cfz 11781   ^cexp 12269   sum_csu 13745   || cdvds 14298   logclog 23497  Λcvma 24011

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-inf2 8143  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613  ax-pre-sup 9614  ax-addf 9615  ax-mulf 9616

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-fal 1449  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-iin 4280  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-se 4793  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-of 6528  df-om 6690  df-1st 6790  df-2nd 6791  df-supp 6912  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-2o 7180  df-oadd 7183  df-er 7360  df-map 7471  df-pm 7472  df-ixp 7520  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-fsupp 7881  df-fi 7922  df-sup 7953  df-inf 7954  df-oi 8022  df-card 8370  df-cda 8595  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-div 10267  df-nn 10607  df-2 10665  df-3 10666  df-4 10667  df-5 10668  df-6 10669  df-7 10670  df-8 10671  df-9 10672  df-10 10673  df-n0 10867  df-z 10935  df-dec 11049  df-uz 11157  df-q 11262  df-rp 11300  df-xneg 11406  df-xadd 11407  df-xmul 11408  df-ioo 11636  df-ioc 11637  df-ico 11638  df-icc 11639  df-fz 11782  df-fzo 11913  df-fl 12025  df-mod 12094  df-seq 12211  df-exp 12270  df-fac 12457  df-bc 12485  df-hash 12513  df-shft 13123  df-cj 13155  df-re 13156  df-im 13157  df-sqrt 13291  df-abs 13292  df-limsup 13519  df-clim 13545  df-rlim 13546  df-sum 13746  df-ef 14114  df-sin 14116  df-cos 14117  df-pi 14119  df-dvds 14299  df-gcd 14462  df-prm 14616  df-pc 14780  df-struct 15116  df-ndx 15117  df-slot 15118  df-base 15119  df-sets 15120  df-ress 15121  df-plusg 15196  df-mulr 15197  df-starv 15198  df-sca 15199  df-vsca 15200  df-ip 15201  df-tset 15202  df-ple 15203  df-ds 15205  df-unif 15206  df-hom 15207  df-cco 15208  df-rest 15314  df-topn 15315  df-0g 15333  df-gsum 15334  df-topgen 15335  df-pt 15336  df-prds 15339  df-xrs 15393  df-qtop 15399  df-imas 15400  df-xps 15403  df-mre 15485  df-mrc 15486  df-acs 15488  df-mgm 16481  df-sgrp 16520  df-mnd 16530  df-submnd 16576  df-mulg 16669  df-cntz 16964  df-cmn 17425  df-psmet 18955  df-xmet 18956  df-met 18957  df-bl 18958  df-mopn 18959  df-fbas 18960  df-fg 18961  df-cnfld 18964  df-top 19914  df-bases 19915  df-topon 19916  df-topsp 19917  df-cld 20027  df-ntr 20028  df-cls 20029  df-nei 20107  df-lp 20145  df-perf 20146  df-cn 20236  df-cnp 20237  df-haus 20324  df-tx 20570  df-hmeo 20763  df-fil 20854  df-fm 20946  df-flim 20947  df-flf 20948  df-xms 21328  df-ms 21329  df-tms 21330  df-cncf 21903  df-limc 22814  df-dv 22815  df-log 23499  df-vma 24017

This theorem is referenced by:  logsqvma2  24374