Theorem logsqvma 21189

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 11267	. . . 4 |- ( N e. NN -> ( 1 ... N ) e. Fin )
2		sgmss 20842	. . . 4 |- ( N e. NN -> { x e. NN | x || N } C_ ( 1 ... N ) )
3		ssfi 7288	. . . 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 643	. . 3 |- ( N e. NN -> { x e. NN | x || N } e. Fin )
5		fzfid 11267	. . . . 5 |- ( ( N e. NN /\ d e. { x e. NN | x || N } ) -> ( 1 ... d ) e. Fin )
6		elrabi 3050	. . . . . . 7 |- ( d e. { x e. NN | x || N } -> d e. NN )
7	6	adantl 453	. . . . . 6 |- ( ( N e. NN /\ d e. { x e. NN | x || N } ) -> d e. NN )
8		sgmss 20842	. . . . . 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 7288	. . . . 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 643	. . . 4 |- ( ( N e. NN /\ d e. { x e. NN | x || N } ) -> { x e. NN | x || d } e. Fin )
12		elrabi 3050	. . . . . . . . 9 |- ( u e. { x e. NN | x || d } -> u e. NN )
13	12	ad2antll 710	. . . . . . . 8 |- ( ( N e. NN /\ ( d e. { x e. NN | x || N } /\ u e. { x e. NN | x || d } ) ) -> u e. NN )
14		vmacl 20854	. . . . . . . 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 4175	. . . . . . . . . . . 12 |- ( x = u -> ( x || d <-> u || d ) )
17	16	elrab 3052	. . . . . . . . . . 11 |- ( u e. { x e. NN | x || d } <-> ( u e. NN /\ u || d ) )
18	17	simprbi 451	. . . . . . . . . 10 |- ( u e. { x e. NN | x || d } -> u || d )
19	18	ad2antll 710	. . . . . . . . 9 |- ( ( N e. NN /\ ( d e. { x e. NN | x || N } /\ u e. { x e. NN | x || d } ) ) -> u || d )
20	6	ad2antrl 709	. . . . . . . . . 10 |- ( ( N e. NN /\ ( d e. { x e. NN | x || N } /\ u e. { x e. NN | x || d } ) ) -> d e. NN )
21		nndivdvds 12813	. . . . . . . . . 10 |- ( ( d e. NN /\ u e. NN ) -> ( u || d <-> ( d / u ) e. NN ) )
22	20, 13, 21	syl2anc 643	. . . . . . . . 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 202	. . . . . . . 8 |- ( ( N e. NN /\ ( d e. { x e. NN | x || N } /\ u e. { x e. NN | x || d } ) ) -> ( d / u ) e. NN )
24		vmacl 20854	. . . . . . . 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 9072	. . . . . 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 9070	. . . . 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 630	. . . 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 12482	. . 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 20854	. . . . . 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 10603	. . . . . 6 |- ( ( N e. NN /\ d e. { x e. NN | x || N } ) -> d e. RR+ )
33	32	relogcld 20471	. . . . 5 |- ( ( N e. NN /\ d e. { x e. NN | x || N } ) -> ( log ` d ) e. RR )
34	31, 33	remulcld 9072	. . . 4 |- ( ( N e. NN /\ d e. { x e. NN | x || N } ) -> ( (Λ ` d ) x. ( log ` d ) ) e. RR )
35	34	recnd 9070	. . 3 |- ( ( N e. NN /\ d e. { x e. NN | x || N } ) -> ( (Λ ` d ) x. ( log ` d ) ) e. CC )
36	4, 29, 35	fsumadd 12487	. 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 20	. . . . 5 |- ( N e. NN -> N e. NN )
38		oveq1 6047	. . . . . . 7 |- ( d = ( u x. k ) -> ( d / u ) = ( ( u x. k ) / u ) )
39	38	fveq2d 5691	. . . . . 6 |- ( d = ( u x. k ) -> (Λ ` ( d / u ) ) = (Λ ` ( ( u x. k ) / u ) ) )
40	39	oveq2d 6056	. . . . 5 |- ( d = ( u x. k ) -> ( (Λ ` u ) x. (Λ ` ( d / u ) ) ) = ( (Λ ` u ) x. (Λ ` ( ( u x. k ) / u ) ) ) )
41	37, 40, 27	fsumdvdscom 20923	. . . 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 3388	. . . . . . . . . . . . 13 |- { x e. NN | x || ( N / u ) } C_ NN
43		simpr 448	. . . . . . . . . . . . 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 3306	. . . . . . . . . . . 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 9972	. . . . . . . . . . 11 |- ( ( ( N e. NN /\ u e. { x e. NN | x || N } ) /\ k e. { x e. NN | x || ( N / u ) } ) -> k e. CC )
46		ssrab2 3388	. . . . . . . . . . . . . 14 |- { x e. NN | x || N } C_ NN
47		simpr 448	. . . . . . . . . . . . . 14 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> u e. { x e. NN | x || N } )
48	46, 47	sseldi 3306	. . . . . . . . . . . . 13 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> u e. NN )
49	48	nncnd 9972	. . . . . . . . . . . 12 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> u e. CC )
50	49	adantr 452	. . . . . . . . . . 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 10000	. . . . . . . . . . . 12 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> u =/= 0 )
52	51	adantr 452	. . . . . . . . . . 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 9751	. . . . . . . . . 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 5691	. . . . . . . . 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 12452	. . . . . . . 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 20919	. . . . . . . . . 10 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> ( N / u ) e. { x e. NN | x || N } )
57	46, 56	sseldi 3306	. . . . . . . . 9 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> ( N / u ) e. NN )
58		vmasum 20953	. . . . . . . . 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 10577	. . . . . . . . . 10 |- ( N e. NN -> N e. RR+ )
61	60	adantr 452	. . . . . . . . 9 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> N e. RR+ )
62	48	nnrpd 10603	. . . . . . . . 9 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> u e. RR+ )
63	61, 62	relogdivd 20474	. . . . . . . 8 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> ( log ` ( N / u ) ) = ( ( log ` N ) - ( log ` u ) ) )
64	55, 59, 63	3eqtrd 2440	. . . . . . 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 6056	. . . . . 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 11267	. . . . . . . 8 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> ( 1 ... ( N / u ) ) e. Fin )
67		sgmss 20842	. . . . . . . . 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 7288	. . . . . . . 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 643	. . . . . . 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 9070	. . . . . . 7 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> (Λ ` u ) e. CC )
73		vmacl 20854	. . . . . . . . . 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 9070	. . . . . . . 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 2478	. . . . . . 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 12522	. . . . . 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 20426	. . . . . . . . 9 |- ( N e. RR+ -> ( log ` N ) e. RR )
79	78	recnd 9070	. . . . . . . 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 20471	. . . . . . . 8 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> ( log ` u ) e. RR )
82	81	recnd 9070	. . . . . . 7 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> ( log ` u ) e. CC )
83	72, 80, 82	subdid 9445	. . . . . 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 2444	. . . . 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 12452	. . . 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 9064	. . . . . 6 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> ( (Λ ` u ) x. ( log ` N ) ) e. CC )
87	72, 82	mulcld 9064	. . . . . 6 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> ( (Λ ` u ) x. ( log ` u ) ) e. CC )
88	4, 86, 87	fsumsub 12526	. . . . 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 11475	. . . . . . 7 |- ( N e. NN -> ( ( log ` N ) ^ 2 ) = ( ( log ` N ) x. ( log ` N ) ) )
91		vmasum 20953	. . . . . . . 8 |- ( N e. NN -> sum_ u e. { x e. NN | x || N } (Λ ` u ) = ( log ` N ) )
92	91	oveq1d 6055	. . . . . . 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 12523	. . . . . . 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 2443	. . . . . 6 |- ( N e. NN -> sum_ u e. { x e. NN | x || N } ( (Λ ` u ) x. ( log ` N ) ) = ( ( log ` N ) ^ 2 ) )
95		fveq2 5687	. . . . . . . . 9 |- ( u = d -> (Λ ` u ) = (Λ ` d ) )
96		fveq2 5687	. . . . . . . . 9 |- ( u = d -> ( log ` u ) = ( log ` d ) )
97	95, 96	oveq12d 6058	. . . . . . . 8 |- ( u = d -> ( (Λ ` u ) x. ( log ` u ) ) = ( (Λ ` d ) x. ( log ` d ) ) )
98	97	cbvsumv 12445	. . . . . . 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 6058	. . . . 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 2436	. . . 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 2440	. . 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 6055	. 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 11476	. . 3 |- ( N e. NN -> ( ( log ` N ) ^ 2 ) e. CC )
105	4, 35	fsumcl 12482	. . 3 |- ( N e. NN -> sum_ d e. { x e. NN | x || N } ( (Λ ` d ) x. ( log ` d ) ) e. CC )
106	104, 105	npcand 9371	. 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 2440	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 177   /\ wa 359   = wceq 1649   e. wcel 1721   =/= wne 2567   {crab 2670   C_ wss 3280   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   Fincfn 7068   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947   + caddc 8949   x. cmul 8951   - cmin 9247   / cdiv 9633   NNcn 9956   2c2 10005   RR+crp 10568   ...cfz 10999   ^cexp 11337   sum_csu 12434   || cdivides 12807   logclog 20405  Λcvma 20827

This theorem is referenced by:  logsqvma2  21190

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-ioc 10877  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-fac 11522  df-bc 11549  df-hash 11574  df-shft 11837  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-limsup 12220  df-clim 12237  df-rlim 12238  df-sum 12435  df-ef 12625  df-sin 12627  df-cos 12628  df-pi 12630  df-dvds 12808  df-gcd 12962  df-prm 13035  df-pc 13166  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-fbas 16654  df-fg 16655  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-lp 17155  df-perf 17156  df-cn 17245  df-cnp 17246  df-haus 17333  df-tx 17547  df-hmeo 17740  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-xms 18303  df-ms 18304  df-tms 18305  df-cncf 18861  df-limc 19706  df-dv 19707  df-log 20407  df-vma 20833