Theorem logsqvma 22917

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 11905	. . . 4 |- ( N e. NN -> ( 1 ... N ) e. Fin )
2		sgmss 22570	. . . 4 |- ( N e. NN -> { x e. NN | x || N } C_ ( 1 ... N ) )
3		ssfi 7637	. . . 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 661	. . 3 |- ( N e. NN -> { x e. NN | x || N } e. Fin )
5		fzfid 11905	. . . . 5 |- ( ( N e. NN /\ d e. { x e. NN | x || N } ) -> ( 1 ... d ) e. Fin )
6		elrabi 3214	. . . . . . 7 |- ( d e. { x e. NN | x || N } -> d e. NN )
7	6	adantl 466	. . . . . 6 |- ( ( N e. NN /\ d e. { x e. NN | x || N } ) -> d e. NN )
8		sgmss 22570	. . . . . 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 7637	. . . . 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 661	. . . 4 |- ( ( N e. NN /\ d e. { x e. NN | x || N } ) -> { x e. NN | x || d } e. Fin )
12		elrabi 3214	. . . . . . . . 9 |- ( u e. { x e. NN | x || d } -> u e. NN )
13	12	ad2antll 728	. . . . . . . 8 |- ( ( N e. NN /\ ( d e. { x e. NN | x || N } /\ u e. { x e. NN | x || d } ) ) -> u e. NN )
14		vmacl 22582	. . . . . . . 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 4396	. . . . . . . . . . . 12 |- ( x = u -> ( x || d <-> u || d ) )
17	16	elrab 3217	. . . . . . . . . . 11 |- ( u e. { x e. NN | x || d } <-> ( u e. NN /\ u || d ) )
18	17	simprbi 464	. . . . . . . . . 10 |- ( u e. { x e. NN | x || d } -> u || d )
19	18	ad2antll 728	. . . . . . . . 9 |- ( ( N e. NN /\ ( d e. { x e. NN | x || N } /\ u e. { x e. NN | x || d } ) ) -> u || d )
20	6	ad2antrl 727	. . . . . . . . . 10 |- ( ( N e. NN /\ ( d e. { x e. NN | x || N } /\ u e. { x e. NN | x || d } ) ) -> d e. NN )
21		nndivdvds 13652	. . . . . . . . . 10 |- ( ( d e. NN /\ u e. NN ) -> ( u || d <-> ( d / u ) e. NN ) )
22	20, 13, 21	syl2anc 661	. . . . . . . . 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 22582	. . . . . . . 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 9518	. . . . . 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 9516	. . . . 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 648	. . . 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 13321	. . 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 22582	. . . . . 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 11130	. . . . . 6 |- ( ( N e. NN /\ d e. { x e. NN | x || N } ) -> d e. RR+ )
33	32	relogcld 22198	. . . . 5 |- ( ( N e. NN /\ d e. { x e. NN | x || N } ) -> ( log ` d ) e. RR )
34	31, 33	remulcld 9518	. . . 4 |- ( ( N e. NN /\ d e. { x e. NN | x || N } ) -> ( (Λ ` d ) x. ( log ` d ) ) e. RR )
35	34	recnd 9516	. . 3 |- ( ( N e. NN /\ d e. { x e. NN | x || N } ) -> ( (Λ ` d ) x. ( log ` d ) ) e. CC )
36	4, 29, 35	fsumadd 13326	. 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 6200	. . . . . . 7 |- ( d = ( u x. k ) -> ( d / u ) = ( ( u x. k ) / u ) )
39	38	fveq2d 5796	. . . . . 6 |- ( d = ( u x. k ) -> (Λ ` ( d / u ) ) = (Λ ` ( ( u x. k ) / u ) ) )
40	39	oveq2d 6209	. . . . 5 |- ( d = ( u x. k ) -> ( (Λ ` u ) x. (Λ ` ( d / u ) ) ) = ( (Λ ` u ) x. (Λ ` ( ( u x. k ) / u ) ) ) )
41	37, 40, 27	fsumdvdscom 22651	. . . 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 3538	. . . . . . . . . . . . 13 |- { x e. NN | x || ( N / u ) } C_ NN
43		simpr 461	. . . . . . . . . . . . 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 3455	. . . . . . . . . . . 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 10442	. . . . . . . . . . 11 |- ( ( ( N e. NN /\ u e. { x e. NN | x || N } ) /\ k e. { x e. NN | x || ( N / u ) } ) -> k e. CC )
46		ssrab2 3538	. . . . . . . . . . . . . 14 |- { x e. NN | x || N } C_ NN
47		simpr 461	. . . . . . . . . . . . . 14 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> u e. { x e. NN | x || N } )
48	46, 47	sseldi 3455	. . . . . . . . . . . . 13 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> u e. NN )
49	48	nncnd 10442	. . . . . . . . . . . 12 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> u e. CC )
50	49	adantr 465	. . . . . . . . . . 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 10470	. . . . . . . . . . . 12 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> u =/= 0 )
52	51	adantr 465	. . . . . . . . . . 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 10216	. . . . . . . . . 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 5796	. . . . . . . . 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 13291	. . . . . . . 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 22647	. . . . . . . . . 10 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> ( N / u ) e. { x e. NN | x || N } )
57	46, 56	sseldi 3455	. . . . . . . . 9 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> ( N / u ) e. NN )
58		vmasum 22681	. . . . . . . . 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 11104	. . . . . . . . . 10 |- ( N e. NN -> N e. RR+ )
61	60	adantr 465	. . . . . . . . 9 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> N e. RR+ )
62	48	nnrpd 11130	. . . . . . . . 9 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> u e. RR+ )
63	61, 62	relogdivd 22201	. . . . . . . 8 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> ( log ` ( N / u ) ) = ( ( log ` N ) - ( log ` u ) ) )
64	55, 59, 63	3eqtrd 2496	. . . . . . 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 6209	. . . . . 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 11905	. . . . . . . 8 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> ( 1 ... ( N / u ) ) e. Fin )
67		sgmss 22570	. . . . . . . . 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 7637	. . . . . . . 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 661	. . . . . . 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 9516	. . . . . . 7 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> (Λ ` u ) e. CC )
73		vmacl 22582	. . . . . . . . . 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 9516	. . . . . . . 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 2539	. . . . . . 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 13362	. . . . . 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 22153	. . . . . . . . 9 |- ( N e. RR+ -> ( log ` N ) e. RR )
79	78	recnd 9516	. . . . . . . 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 22198	. . . . . . . 8 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> ( log ` u ) e. RR )
82	81	recnd 9516	. . . . . . 7 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> ( log ` u ) e. CC )
83	72, 80, 82	subdid 9904	. . . . . 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 2500	. . . . 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 13291	. . . 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 9510	. . . . . 6 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> ( (Λ ` u ) x. ( log ` N ) ) e. CC )
87	72, 82	mulcld 9510	. . . . . 6 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> ( (Λ ` u ) x. ( log ` u ) ) e. CC )
88	4, 86, 87	fsumsub 13366	. . . . 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 12115	. . . . . . 7 |- ( N e. NN -> ( ( log ` N ) ^ 2 ) = ( ( log ` N ) x. ( log ` N ) ) )
91		vmasum 22681	. . . . . . . 8 |- ( N e. NN -> sum_ u e. { x e. NN | x || N } (Λ ` u ) = ( log ` N ) )
92	91	oveq1d 6208	. . . . . . 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 13363	. . . . . . 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 2499	. . . . . 6 |- ( N e. NN -> sum_ u e. { x e. NN | x || N } ( (Λ ` u ) x. ( log ` N ) ) = ( ( log ` N ) ^ 2 ) )
95		fveq2 5792	. . . . . . . . 9 |- ( u = d -> (Λ ` u ) = (Λ ` d ) )
96		fveq2 5792	. . . . . . . . 9 |- ( u = d -> ( log ` u ) = ( log ` d ) )
97	95, 96	oveq12d 6211	. . . . . . . 8 |- ( u = d -> ( (Λ ` u ) x. ( log ` u ) ) = ( (Λ ` d ) x. ( log ` d ) ) )
98	97	cbvsumv 13284	. . . . . . 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 6211	. . . . 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 2492	. . . 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 2496	. . 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 6208	. 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 12116	. . 3 |- ( N e. NN -> ( ( log ` N ) ^ 2 ) e. CC )
105	4, 35	fsumcl 13321	. . 3 |- ( N e. NN -> sum_ d e. { x e. NN | x || N } ( (Λ ` d ) x. ( log ` d ) ) e. CC )
106	104, 105	npcand 9827	. 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 2496	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 369   = wceq 1370   e. wcel 1758   =/= wne 2644   {crab 2799   C_ wss 3429   class class class wbr 4393   ` cfv 5519  (class class class)co 6193   Fincfn 7413   CCcc 9384   RRcr 9385   0cc0 9386   1c1 9387   + caddc 9389   x. cmul 9391   - cmin 9699   / cdiv 10097   NNcn 10426   2c2 10475   RR+crp 11095   ...cfz 11547   ^cexp 11975   sum_csu 13274   || cdivides 13646   logclog 22132  Λcvma 22555

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-inf2 7951  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463  ax-pre-sup 9464  ax-addf 9465  ax-mulf 9466

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-iin 4275  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-se 4781  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-isom 5528  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-of 6423  df-om 6580  df-1st 6680  df-2nd 6681  df-supp 6794  df-recs 6935  df-rdg 6969  df-1o 7023  df-2o 7024  df-oadd 7027  df-er 7204  df-map 7319  df-pm 7320  df-ixp 7367  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-fsupp 7725  df-fi 7765  df-sup 7795  df-oi 7828  df-card 8213  df-cda 8441  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-div 10098  df-nn 10427  df-2 10484  df-3 10485  df-4 10486  df-5 10487  df-6 10488  df-7 10489  df-8 10490  df-9 10491  df-10 10492  df-n0 10684  df-z 10751  df-dec 10860  df-uz 10966  df-q 11058  df-rp 11096  df-xneg 11193  df-xadd 11194  df-xmul 11195  df-ioo 11408  df-ioc 11409  df-ico 11410  df-icc 11411  df-fz 11548  df-fzo 11659  df-fl 11752  df-mod 11819  df-seq 11917  df-exp 11976  df-fac 12162  df-bc 12189  df-hash 12214  df-shft 12667  df-cj 12699  df-re 12700  df-im 12701  df-sqr 12835  df-abs 12836  df-limsup 13060  df-clim 13077  df-rlim 13078  df-sum 13275  df-ef 13464  df-sin 13466  df-cos 13467  df-pi 13469  df-dvds 13647  df-gcd 13802  df-prm 13875  df-pc 14015  df-struct 14287  df-ndx 14288  df-slot 14289  df-base 14290  df-sets 14291  df-ress 14292  df-plusg 14362  df-mulr 14363  df-starv 14364  df-sca 14365  df-vsca 14366  df-ip 14367  df-tset 14368  df-ple 14369  df-ds 14371  df-unif 14372  df-hom 14373  df-cco 14374  df-rest 14472  df-topn 14473  df-0g 14491  df-gsum 14492  df-topgen 14493  df-pt 14494  df-prds 14497  df-xrs 14551  df-qtop 14556  df-imas 14557  df-xps 14559  df-mre 14635  df-mrc 14636  df-acs 14638  df-mnd 15526  df-submnd 15576  df-mulg 15659  df-cntz 15946  df-cmn 16392  df-psmet 17927  df-xmet 17928  df-met 17929  df-bl 17930  df-mopn 17931  df-fbas 17932  df-fg 17933  df-cnfld 17937  df-top 18628  df-bases 18630  df-topon 18631  df-topsp 18632  df-cld 18748  df-ntr 18749  df-cls 18750  df-nei 18827  df-lp 18865  df-perf 18866  df-cn 18956  df-cnp 18957  df-haus 19044  df-tx 19260  df-hmeo 19453  df-fil 19544  df-fm 19636  df-flim 19637  df-flf 19638  df-xms 20020  df-ms 20021  df-tms 20022  df-cncf 20579  df-limc 21467  df-dv 21468  df-log 22134  df-vma 22561

This theorem is referenced by:  logsqvma2  22918