Theorem logsqvma 22675

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 11778	. . . 4 |- ( N e. NN -> ( 1 ... N ) e. Fin )
2		sgmss 22328	. . . 4 |- ( N e. NN -> { x e. NN | x || N } C_ ( 1 ... N ) )
3		ssfi 7521	. . . 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 654	. . 3 |- ( N e. NN -> { x e. NN | x || N } e. Fin )
5		fzfid 11778	. . . . 5 |- ( ( N e. NN /\ d e. { x e. NN | x || N } ) -> ( 1 ... d ) e. Fin )
6		elrabi 3103	. . . . . . 7 |- ( d e. { x e. NN | x || N } -> d e. NN )
7	6	adantl 463	. . . . . 6 |- ( ( N e. NN /\ d e. { x e. NN | x || N } ) -> d e. NN )
8		sgmss 22328	. . . . . 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 7521	. . . . 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 654	. . . 4 |- ( ( N e. NN /\ d e. { x e. NN | x || N } ) -> { x e. NN | x || d } e. Fin )
12		elrabi 3103	. . . . . . . . 9 |- ( u e. { x e. NN | x || d } -> u e. NN )
13	12	ad2antll 721	. . . . . . . 8 |- ( ( N e. NN /\ ( d e. { x e. NN | x || N } /\ u e. { x e. NN | x || d } ) ) -> u e. NN )
14		vmacl 22340	. . . . . . . 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 4283	. . . . . . . . . . . 12 |- ( x = u -> ( x || d <-> u || d ) )
17	16	elrab 3106	. . . . . . . . . . 11 |- ( u e. { x e. NN | x || d } <-> ( u e. NN /\ u || d ) )
18	17	simprbi 461	. . . . . . . . . 10 |- ( u e. { x e. NN | x || d } -> u || d )
19	18	ad2antll 721	. . . . . . . . 9 |- ( ( N e. NN /\ ( d e. { x e. NN | x || N } /\ u e. { x e. NN | x || d } ) ) -> u || d )
20	6	ad2antrl 720	. . . . . . . . . 10 |- ( ( N e. NN /\ ( d e. { x e. NN | x || N } /\ u e. { x e. NN | x || d } ) ) -> d e. NN )
21		nndivdvds 13523	. . . . . . . . . 10 |- ( ( d e. NN /\ u e. NN ) -> ( u || d <-> ( d / u ) e. NN ) )
22	20, 13, 21	syl2anc 654	. . . . . . . . 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 22340	. . . . . . . 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 9401	. . . . . 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 9399	. . . . 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 641	. . . 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 13193	. . 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 22340	. . . . . 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 11013	. . . . . 6 |- ( ( N e. NN /\ d e. { x e. NN | x || N } ) -> d e. RR+ )
33	32	relogcld 21956	. . . . 5 |- ( ( N e. NN /\ d e. { x e. NN | x || N } ) -> ( log ` d ) e. RR )
34	31, 33	remulcld 9401	. . . 4 |- ( ( N e. NN /\ d e. { x e. NN | x || N } ) -> ( (Λ ` d ) x. ( log ` d ) ) e. RR )
35	34	recnd 9399	. . 3 |- ( ( N e. NN /\ d e. { x e. NN | x || N } ) -> ( (Λ ` d ) x. ( log ` d ) ) e. CC )
36	4, 29, 35	fsumadd 13198	. 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 6087	. . . . . . 7 |- ( d = ( u x. k ) -> ( d / u ) = ( ( u x. k ) / u ) )
39	38	fveq2d 5683	. . . . . 6 |- ( d = ( u x. k ) -> (Λ ` ( d / u ) ) = (Λ ` ( ( u x. k ) / u ) ) )
40	39	oveq2d 6096	. . . . 5 |- ( d = ( u x. k ) -> ( (Λ ` u ) x. (Λ ` ( d / u ) ) ) = ( (Λ ` u ) x. (Λ ` ( ( u x. k ) / u ) ) ) )
41	37, 40, 27	fsumdvdscom 22409	. . . 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 3425	. . . . . . . . . . . . 13 |- { x e. NN | x || ( N / u ) } C_ NN
43		simpr 458	. . . . . . . . . . . . 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 3342	. . . . . . . . . . . 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 10325	. . . . . . . . . . 11 |- ( ( ( N e. NN /\ u e. { x e. NN | x || N } ) /\ k e. { x e. NN | x || ( N / u ) } ) -> k e. CC )
46		ssrab2 3425	. . . . . . . . . . . . . 14 |- { x e. NN | x || N } C_ NN
47		simpr 458	. . . . . . . . . . . . . 14 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> u e. { x e. NN | x || N } )
48	46, 47	sseldi 3342	. . . . . . . . . . . . 13 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> u e. NN )
49	48	nncnd 10325	. . . . . . . . . . . 12 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> u e. CC )
50	49	adantr 462	. . . . . . . . . . 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 10353	. . . . . . . . . . . 12 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> u =/= 0 )
52	51	adantr 462	. . . . . . . . . . 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 10099	. . . . . . . . . 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 5683	. . . . . . . . 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 13163	. . . . . . . 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 22405	. . . . . . . . . 10 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> ( N / u ) e. { x e. NN | x || N } )
57	46, 56	sseldi 3342	. . . . . . . . 9 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> ( N / u ) e. NN )
58		vmasum 22439	. . . . . . . . 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 10987	. . . . . . . . . 10 |- ( N e. NN -> N e. RR+ )
61	60	adantr 462	. . . . . . . . 9 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> N e. RR+ )
62	48	nnrpd 11013	. . . . . . . . 9 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> u e. RR+ )
63	61, 62	relogdivd 21959	. . . . . . . 8 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> ( log ` ( N / u ) ) = ( ( log ` N ) - ( log ` u ) ) )
64	55, 59, 63	3eqtrd 2469	. . . . . . 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 6096	. . . . . 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 11778	. . . . . . . 8 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> ( 1 ... ( N / u ) ) e. Fin )
67		sgmss 22328	. . . . . . . . 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 7521	. . . . . . . 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 654	. . . . . . 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 9399	. . . . . . 7 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> (Λ ` u ) e. CC )
73		vmacl 22340	. . . . . . . . . 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 9399	. . . . . . . 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 2507	. . . . . . 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 13233	. . . . . 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 21911	. . . . . . . . 9 |- ( N e. RR+ -> ( log ` N ) e. RR )
79	78	recnd 9399	. . . . . . . 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 21956	. . . . . . . 8 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> ( log ` u ) e. RR )
82	81	recnd 9399	. . . . . . 7 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> ( log ` u ) e. CC )
83	72, 80, 82	subdid 9787	. . . . . 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 2473	. . . . 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 13163	. . . 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 9393	. . . . . 6 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> ( (Λ ` u ) x. ( log ` N ) ) e. CC )
87	72, 82	mulcld 9393	. . . . . 6 |- ( ( N e. NN /\ u e. { x e. NN | x || N } ) -> ( (Λ ` u ) x. ( log ` u ) ) e. CC )
88	4, 86, 87	fsumsub 13237	. . . . 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 11988	. . . . . . 7 |- ( N e. NN -> ( ( log ` N ) ^ 2 ) = ( ( log ` N ) x. ( log ` N ) ) )
91		vmasum 22439	. . . . . . . 8 |- ( N e. NN -> sum_ u e. { x e. NN | x || N } (Λ ` u ) = ( log ` N ) )
92	91	oveq1d 6095	. . . . . . 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 13234	. . . . . . 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 2472	. . . . . 6 |- ( N e. NN -> sum_ u e. { x e. NN | x || N } ( (Λ ` u ) x. ( log ` N ) ) = ( ( log ` N ) ^ 2 ) )
95		fveq2 5679	. . . . . . . . 9 |- ( u = d -> (Λ ` u ) = (Λ ` d ) )
96		fveq2 5679	. . . . . . . . 9 |- ( u = d -> ( log ` u ) = ( log ` d ) )
97	95, 96	oveq12d 6098	. . . . . . . 8 |- ( u = d -> ( (Λ ` u ) x. ( log ` u ) ) = ( (Λ ` d ) x. ( log ` d ) ) )
98	97	cbvsumv 13156	. . . . . . 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 6098	. . . . 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 2465	. . . 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 2469	. . 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 6095	. 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 11989	. . 3 |- ( N e. NN -> ( ( log ` N ) ^ 2 ) e. CC )
105	4, 35	fsumcl 13193	. . 3 |- ( N e. NN -> sum_ d e. { x e. NN | x || N } ( (Λ ` d ) x. ( log ` d ) ) e. CC )
106	104, 105	npcand 9710	. 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 2469	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 1362   e. wcel 1755   =/= wne 2596   {crab 2709   C_ wss 3316   class class class wbr 4280   ` cfv 5406  (class class class)co 6080   Fincfn 7298   CCcc 9267   RRcr 9268   0cc0 9269   1c1 9270   + caddc 9272   x. cmul 9274   - cmin 9582   / cdiv 9980   NNcn 10309   2c2 10358   RR+crp 10978   ...cfz 11423   ^cexp 11848   sum_csu 13146   || cdivides 13517   logclog 21890  Λcvma 22313

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-inf2 7835  ax-cnex 9325  ax-resscn 9326  ax-1cn 9327  ax-icn 9328  ax-addcl 9329  ax-addrcl 9330  ax-mulcl 9331  ax-mulrcl 9332  ax-mulcom 9333  ax-addass 9334  ax-mulass 9335  ax-distr 9336  ax-i2m1 9337  ax-1ne0 9338  ax-1rid 9339  ax-rnegex 9340  ax-rrecex 9341  ax-cnre 9342  ax-pre-lttri 9343  ax-pre-lttrn 9344  ax-pre-ltadd 9345  ax-pre-mulgt0 9346  ax-pre-sup 9347  ax-addf 9348  ax-mulf 9349

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-fal 1368  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-iin 4162  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-se 4667  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-of 6309  df-om 6466  df-1st 6566  df-2nd 6567  df-supp 6680  df-recs 6818  df-rdg 6852  df-1o 6908  df-2o 6909  df-oadd 6912  df-er 7089  df-map 7204  df-pm 7205  df-ixp 7252  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-fsupp 7609  df-fi 7649  df-sup 7679  df-oi 7712  df-card 8097  df-cda 8325  df-pnf 9407  df-mnf 9408  df-xr 9409  df-ltxr 9410  df-le 9411  df-sub 9584  df-neg 9585  df-div 9981  df-nn 10310  df-2 10367  df-3 10368  df-4 10369  df-5 10370  df-6 10371  df-7 10372  df-8 10373  df-9 10374  df-10 10375  df-n0 10567  df-z 10634  df-dec 10743  df-uz 10849  df-q 10941  df-rp 10979  df-xneg 11076  df-xadd 11077  df-xmul 11078  df-ioo 11291  df-ioc 11292  df-ico 11293  df-icc 11294  df-fz 11424  df-fzo 11532  df-fl 11625  df-mod 11692  df-seq 11790  df-exp 11849  df-fac 12035  df-bc 12062  df-hash 12087  df-shft 12539  df-cj 12571  df-re 12572  df-im 12573  df-sqr 12707  df-abs 12708  df-limsup 12932  df-clim 12949  df-rlim 12950  df-sum 13147  df-ef 13335  df-sin 13337  df-cos 13338  df-pi 13340  df-dvds 13518  df-gcd 13673  df-prm 13746  df-pc 13886  df-struct 14158  df-ndx 14159  df-slot 14160  df-base 14161  df-sets 14162  df-ress 14163  df-plusg 14233  df-mulr 14234  df-starv 14235  df-sca 14236  df-vsca 14237  df-ip 14238  df-tset 14239  df-ple 14240  df-ds 14242  df-unif 14243  df-hom 14244  df-cco 14245  df-rest 14343  df-topn 14344  df-0g 14362  df-gsum 14363  df-topgen 14364  df-pt 14365  df-prds 14368  df-xrs 14422  df-qtop 14427  df-imas 14428  df-xps 14430  df-mre 14506  df-mrc 14507  df-acs 14509  df-mnd 15397  df-submnd 15447  df-mulg 15527  df-cntz 15814  df-cmn 16258  df-psmet 17652  df-xmet 17653  df-met 17654  df-bl 17655  df-mopn 17656  df-fbas 17657  df-fg 17658  df-cnfld 17662  df-top 18344  df-bases 18346  df-topon 18347  df-topsp 18348  df-cld 18464  df-ntr 18465  df-cls 18466  df-nei 18543  df-lp 18581  df-perf 18582  df-cn 18672  df-cnp 18673  df-haus 18760  df-tx 18976  df-hmeo 19169  df-fil 19260  df-fm 19352  df-flim 19353  df-flf 19354  df-xms 19736  df-ms 19737  df-tms 19738  df-cncf 20295  df-limc 21182  df-dv 21183  df-log 21892  df-vma 22319

This theorem is referenced by:  logsqvma2  22676