Theorem logsqvma2 25232

Description: The Möbius inverse of logsqvma 25231. Equation 10.4.8 of [Shapiro], p. 418. (Contributed by Mario Carneiro, 13-May-2016.)

Assertion

Ref	Expression
logsqvma2	|- ( N e. NN -> sum_ d e. { x e. NN | x || N } ( ( mmu ` d ) x. ( ( log ` ( N / d ) ) ^ 2 ) ) = ( sum_ d e. { x e. NN | x || N } ( (Λ ` d ) x. (Λ ` ( N / d ) ) ) + ( (Λ ` N ) x. ( log ` N ) ) ) )

Distinct variable group:   x, d, N

Proof of Theorem logsqvma2

Dummy variables i j k m n are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fzfid 12772	. . . . . . . . . 10 |- ( k e. NN -> ( 1 ... k ) e. Fin )
2		dvdsssfz1 15040	. . . . . . . . . 10 |- ( k e. NN -> { x e. NN | x || k } C_ ( 1 ... k ) )
3		ssfi 8180	. . . . . . . . . 10 |- ( ( ( 1 ... k ) e. Fin /\ { x e. NN | x || k } C_ ( 1 ... k ) ) -> { x e. NN | x || k } e. Fin )
4	1, 2, 3	syl2anc 693	. . . . . . . . 9 |- ( k e. NN -> { x e. NN | x || k } e. Fin )
5		ssrab2 3687	. . . . . . . . . . . 12 |- { x e. NN | x || k } C_ NN
6		simpr 477	. . . . . . . . . . . 12 |- ( ( k e. NN /\ d e. { x e. NN | x || k } ) -> d e. { x e. NN | x || k } )
7	5, 6	sseldi 3601	. . . . . . . . . . 11 |- ( ( k e. NN /\ d e. { x e. NN | x || k } ) -> d e. NN )
8		vmacl 24844	. . . . . . . . . . 11 |- ( d e. NN -> (Λ ` d ) e. RR )
9	7, 8	syl 17	. . . . . . . . . 10 |- ( ( k e. NN /\ d e. { x e. NN | x || k } ) -> (Λ ` d ) e. RR )
10		dvdsdivcl 15038	. . . . . . . . . . . 12 |- ( ( k e. NN /\ d e. { x e. NN | x || k } ) -> ( k / d ) e. { x e. NN | x || k } )
11	5, 10	sseldi 3601	. . . . . . . . . . 11 |- ( ( k e. NN /\ d e. { x e. NN | x || k } ) -> ( k / d ) e. NN )
12		vmacl 24844	. . . . . . . . . . 11 |- ( ( k / d ) e. NN -> (Λ ` ( k / d ) ) e. RR )
13	11, 12	syl 17	. . . . . . . . . 10 |- ( ( k e. NN /\ d e. { x e. NN | x || k } ) -> (Λ ` ( k / d ) ) e. RR )
14	9, 13	remulcld 10070	. . . . . . . . 9 |- ( ( k e. NN /\ d e. { x e. NN | x || k } ) -> ( (Λ ` d ) x. (Λ ` ( k / d ) ) ) e. RR )
15	4, 14	fsumrecl 14465	. . . . . . . 8 |- ( k e. NN -> sum_ d e. { x e. NN | x || k } ( (Λ ` d ) x. (Λ ` ( k / d ) ) ) e. RR )
16		vmacl 24844	. . . . . . . . 9 |- ( k e. NN -> (Λ ` k ) e. RR )
17		nnrp 11842	. . . . . . . . . 10 |- ( k e. NN -> k e. RR+ )
18	17	relogcld 24369	. . . . . . . . 9 |- ( k e. NN -> ( log ` k ) e. RR )
19	16, 18	remulcld 10070	. . . . . . . 8 |- ( k e. NN -> ( (Λ ` k ) x. ( log ` k ) ) e. RR )
20	15, 19	readdcld 10069	. . . . . . 7 |- ( k e. NN -> ( sum_ d e. { x e. NN | x || k } ( (Λ ` d ) x. (Λ ` ( k / d ) ) ) + ( (Λ ` k ) x. ( log ` k ) ) ) e. RR )
21	20	recnd 10068	. . . . . 6 |- ( k e. NN -> ( sum_ d e. { x e. NN | x || k } ( (Λ ` d ) x. (Λ ` ( k / d ) ) ) + ( (Λ ` k ) x. ( log ` k ) ) ) e. CC )
22	21	adantl 482	. . . . 5 |- ( ( N e. NN /\ k e. NN ) -> ( sum_ d e. { x e. NN | x || k } ( (Λ ` d ) x. (Λ ` ( k / d ) ) ) + ( (Λ ` k ) x. ( log ` k ) ) ) e. CC )
23		eqid 2622	. . . . 5 |- ( k e. NN |-> ( sum_ d e. { x e. NN | x || k } ( (Λ ` d ) x. (Λ ` ( k / d ) ) ) + ( (Λ ` k ) x. ( log ` k ) ) ) ) = ( k e. NN |-> ( sum_ d e. { x e. NN | x || k } ( (Λ ` d ) x. (Λ ` ( k / d ) ) ) + ( (Λ ` k ) x. ( log ` k ) ) ) )
24	22, 23	fmptd 6385	. . . 4 |- ( N e. NN -> ( k e. NN |-> ( sum_ d e. { x e. NN | x || k } ( (Λ ` d ) x. (Λ ` ( k / d ) ) ) + ( (Λ ` k ) x. ( log ` k ) ) ) ) : NN --> CC )
25		ssrab2 3687	. . . . . . . . 9 |- { x e. NN | x || n } C_ NN
26		simpr 477	. . . . . . . . 9 |- ( ( ( N e. NN /\ n e. NN ) /\ m e. { x e. NN | x || n } ) -> m e. { x e. NN | x || n } )
27	25, 26	sseldi 3601	. . . . . . . 8 |- ( ( ( N e. NN /\ n e. NN ) /\ m e. { x e. NN | x || n } ) -> m e. NN )
28		breq2 4657	. . . . . . . . . . . 12 |- ( k = m -> ( x || k <-> x || m ) )
29	28	rabbidv 3189	. . . . . . . . . . 11 |- ( k = m -> { x e. NN | x || k } = { x e. NN | x || m } )
30		oveq1 6657	. . . . . . . . . . . . . 14 |- ( k = m -> ( k / d ) = ( m / d ) )
31	30	fveq2d 6195	. . . . . . . . . . . . 13 |- ( k = m -> (Λ ` ( k / d ) ) = (Λ ` ( m / d ) ) )
32	31	oveq2d 6666	. . . . . . . . . . . 12 |- ( k = m -> ( (Λ ` d ) x. (Λ ` ( k / d ) ) ) = ( (Λ ` d ) x. (Λ ` ( m / d ) ) ) )
33	32	adantr 481	. . . . . . . . . . 11 |- ( ( k = m /\ d e. { x e. NN | x || k } ) -> ( (Λ ` d ) x. (Λ ` ( k / d ) ) ) = ( (Λ ` d ) x. (Λ ` ( m / d ) ) ) )
34	29, 33	sumeq12dv 14437	. . . . . . . . . 10 |- ( k = m -> sum_ d e. { x e. NN | x || k } ( (Λ ` d ) x. (Λ ` ( k / d ) ) ) = sum_ d e. { x e. NN | x || m } ( (Λ ` d ) x. (Λ ` ( m / d ) ) ) )
35		fveq2 6191	. . . . . . . . . . 11 |- ( k = m -> (Λ ` k ) = (Λ ` m ) )
36		fveq2 6191	. . . . . . . . . . 11 |- ( k = m -> ( log ` k ) = ( log ` m ) )
37	35, 36	oveq12d 6668	. . . . . . . . . 10 |- ( k = m -> ( (Λ ` k ) x. ( log ` k ) ) = ( (Λ ` m ) x. ( log ` m ) ) )
38	34, 37	oveq12d 6668	. . . . . . . . 9 |- ( k = m -> ( sum_ d e. { x e. NN | x || k } ( (Λ ` d ) x. (Λ ` ( k / d ) ) ) + ( (Λ ` k ) x. ( log ` k ) ) ) = ( sum_ d e. { x e. NN | x || m } ( (Λ ` d ) x. (Λ ` ( m / d ) ) ) + ( (Λ ` m ) x. ( log ` m ) ) ) )
39		ovex 6678	. . . . . . . . 9 |- ( sum_ d e. { x e. NN | x || k } ( (Λ ` d ) x. (Λ ` ( k / d ) ) ) + ( (Λ ` k ) x. ( log ` k ) ) ) e. _V
40	38, 23, 39	fvmpt3i 6287	. . . . . . . 8 |- ( m e. NN -> ( ( k e. NN |-> ( sum_ d e. { x e. NN | x || k } ( (Λ ` d ) x. (Λ ` ( k / d ) ) ) + ( (Λ ` k ) x. ( log ` k ) ) ) ) ` m ) = ( sum_ d e. { x e. NN | x || m } ( (Λ ` d ) x. (Λ ` ( m / d ) ) ) + ( (Λ ` m ) x. ( log ` m ) ) ) )
41	27, 40	syl 17	. . . . . . 7 |- ( ( ( N e. NN /\ n e. NN ) /\ m e. { x e. NN | x || n } ) -> ( ( k e. NN |-> ( sum_ d e. { x e. NN | x || k } ( (Λ ` d ) x. (Λ ` ( k / d ) ) ) + ( (Λ ` k ) x. ( log ` k ) ) ) ) ` m ) = ( sum_ d e. { x e. NN | x || m } ( (Λ ` d ) x. (Λ ` ( m / d ) ) ) + ( (Λ ` m ) x. ( log ` m ) ) ) )
42	41	sumeq2dv 14433	. . . . . 6 |- ( ( N e. NN /\ n e. NN ) -> sum_ m e. { x e. NN | x || n } ( ( k e. NN |-> ( sum_ d e. { x e. NN | x || k } ( (Λ ` d ) x. (Λ ` ( k / d ) ) ) + ( (Λ ` k ) x. ( log ` k ) ) ) ) ` m ) = sum_ m e. { x e. NN | x || n } ( sum_ d e. { x e. NN | x || m } ( (Λ ` d ) x. (Λ ` ( m / d ) ) ) + ( (Λ ` m ) x. ( log ` m ) ) ) )
43		logsqvma 25231	. . . . . . 7 |- ( n e. NN -> sum_ m e. { x e. NN | x || n } ( sum_ d e. { x e. NN | x || m } ( (Λ ` d ) x. (Λ ` ( m / d ) ) ) + ( (Λ ` m ) x. ( log ` m ) ) ) = ( ( log ` n ) ^ 2 ) )
44	43	adantl 482	. . . . . 6 |- ( ( N e. NN /\ n e. NN ) -> sum_ m e. { x e. NN | x || n } ( sum_ d e. { x e. NN | x || m } ( (Λ ` d ) x. (Λ ` ( m / d ) ) ) + ( (Λ ` m ) x. ( log ` m ) ) ) = ( ( log ` n ) ^ 2 ) )
45	42, 44	eqtr2d 2657	. . . . 5 |- ( ( N e. NN /\ n e. NN ) -> ( ( log ` n ) ^ 2 ) = sum_ m e. { x e. NN | x || n } ( ( k e. NN |-> ( sum_ d e. { x e. NN | x || k } ( (Λ ` d ) x. (Λ ` ( k / d ) ) ) + ( (Λ ` k ) x. ( log ` k ) ) ) ) ` m ) )
46	45	mpteq2dva 4744	. . . 4 |- ( N e. NN -> ( n e. NN |-> ( ( log ` n ) ^ 2 ) ) = ( n e. NN |-> sum_ m e. { x e. NN | x || n } ( ( k e. NN |-> ( sum_ d e. { x e. NN | x || k } ( (Λ ` d ) x. (Λ ` ( k / d ) ) ) + ( (Λ ` k ) x. ( log ` k ) ) ) ) ` m ) ) )
47	24, 46	muinv 24919	. . 3 |- ( N e. NN -> ( k e. NN |-> ( sum_ d e. { x e. NN | x || k } ( (Λ ` d ) x. (Λ ` ( k / d ) ) ) + ( (Λ ` k ) x. ( log ` k ) ) ) ) = ( i e. NN |-> sum_ j e. { x e. NN | x || i } ( ( mmu ` j ) x. ( ( n e. NN |-> ( ( log ` n ) ^ 2 ) ) ` ( i / j ) ) ) ) )
48	47	fveq1d 6193	. 2 |- ( N e. NN -> ( ( k e. NN |-> ( sum_ d e. { x e. NN | x || k } ( (Λ ` d ) x. (Λ ` ( k / d ) ) ) + ( (Λ ` k ) x. ( log ` k ) ) ) ) ` N ) = ( ( i e. NN |-> sum_ j e. { x e. NN | x || i } ( ( mmu ` j ) x. ( ( n e. NN |-> ( ( log ` n ) ^ 2 ) ) ` ( i / j ) ) ) ) ` N ) )
49		breq2 4657	. . . . . 6 |- ( k = N -> ( x || k <-> x || N ) )
50	49	rabbidv 3189	. . . . 5 |- ( k = N -> { x e. NN | x || k } = { x e. NN | x || N } )
51		oveq1 6657	. . . . . . . 8 |- ( k = N -> ( k / d ) = ( N / d ) )
52	51	fveq2d 6195	. . . . . . 7 |- ( k = N -> (Λ ` ( k / d ) ) = (Λ ` ( N / d ) ) )
53	52	oveq2d 6666	. . . . . 6 |- ( k = N -> ( (Λ ` d ) x. (Λ ` ( k / d ) ) ) = ( (Λ ` d ) x. (Λ ` ( N / d ) ) ) )
54	53	adantr 481	. . . . 5 |- ( ( k = N /\ d e. { x e. NN | x || k } ) -> ( (Λ ` d ) x. (Λ ` ( k / d ) ) ) = ( (Λ ` d ) x. (Λ ` ( N / d ) ) ) )
55	50, 54	sumeq12dv 14437	. . . 4 |- ( k = N -> sum_ d e. { x e. NN | x || k } ( (Λ ` d ) x. (Λ ` ( k / d ) ) ) = sum_ d e. { x e. NN | x || N } ( (Λ ` d ) x. (Λ ` ( N / d ) ) ) )
56		fveq2 6191	. . . . 5 |- ( k = N -> (Λ ` k ) = (Λ ` N ) )
57		fveq2 6191	. . . . 5 |- ( k = N -> ( log ` k ) = ( log ` N ) )
58	56, 57	oveq12d 6668	. . . 4 |- ( k = N -> ( (Λ ` k ) x. ( log ` k ) ) = ( (Λ ` N ) x. ( log ` N ) ) )
59	55, 58	oveq12d 6668	. . 3 |- ( k = N -> ( sum_ d e. { x e. NN | x || k } ( (Λ ` d ) x. (Λ ` ( k / d ) ) ) + ( (Λ ` k ) x. ( log ` k ) ) ) = ( sum_ d e. { x e. NN | x || N } ( (Λ ` d ) x. (Λ ` ( N / d ) ) ) + ( (Λ ` N ) x. ( log ` N ) ) ) )
60	59, 23, 39	fvmpt3i 6287	. 2 |- ( N e. NN -> ( ( k e. NN |-> ( sum_ d e. { x e. NN | x || k } ( (Λ ` d ) x. (Λ ` ( k / d ) ) ) + ( (Λ ` k ) x. ( log ` k ) ) ) ) ` N ) = ( sum_ d e. { x e. NN | x || N } ( (Λ ` d ) x. (Λ ` ( N / d ) ) ) + ( (Λ ` N ) x. ( log ` N ) ) ) )
61		fveq2 6191	. . . . . 6 |- ( j = d -> ( mmu ` j ) = ( mmu ` d ) )
62		oveq2 6658	. . . . . . . 8 |- ( j = d -> ( i / j ) = ( i / d ) )
63	62	fveq2d 6195	. . . . . . 7 |- ( j = d -> ( log ` ( i / j ) ) = ( log ` ( i / d ) ) )
64	63	oveq1d 6665	. . . . . 6 |- ( j = d -> ( ( log ` ( i / j ) ) ^ 2 ) = ( ( log ` ( i / d ) ) ^ 2 ) )
65	61, 64	oveq12d 6668	. . . . 5 |- ( j = d -> ( ( mmu ` j ) x. ( ( log ` ( i / j ) ) ^ 2 ) ) = ( ( mmu ` d ) x. ( ( log ` ( i / d ) ) ^ 2 ) ) )
66	65	cbvsumv 14426	. . . 4 |- sum_ j e. { x e. NN | x || i } ( ( mmu ` j ) x. ( ( log ` ( i / j ) ) ^ 2 ) ) = sum_ d e. { x e. NN | x || i } ( ( mmu ` d ) x. ( ( log ` ( i / d ) ) ^ 2 ) )
67		breq2 4657	. . . . . 6 |- ( i = N -> ( x || i <-> x || N ) )
68	67	rabbidv 3189	. . . . 5 |- ( i = N -> { x e. NN | x || i } = { x e. NN | x || N } )
69		oveq1 6657	. . . . . . . . 9 |- ( i = N -> ( i / d ) = ( N / d ) )
70	69	fveq2d 6195	. . . . . . . 8 |- ( i = N -> ( log ` ( i / d ) ) = ( log ` ( N / d ) ) )
71	70	oveq1d 6665	. . . . . . 7 |- ( i = N -> ( ( log ` ( i / d ) ) ^ 2 ) = ( ( log ` ( N / d ) ) ^ 2 ) )
72	71	oveq2d 6666	. . . . . 6 |- ( i = N -> ( ( mmu ` d ) x. ( ( log ` ( i / d ) ) ^ 2 ) ) = ( ( mmu ` d ) x. ( ( log ` ( N / d ) ) ^ 2 ) ) )
73	72	adantr 481	. . . . 5 |- ( ( i = N /\ d e. { x e. NN | x || i } ) -> ( ( mmu ` d ) x. ( ( log ` ( i / d ) ) ^ 2 ) ) = ( ( mmu ` d ) x. ( ( log ` ( N / d ) ) ^ 2 ) ) )
74	68, 73	sumeq12dv 14437	. . . 4 |- ( i = N -> sum_ d e. { x e. NN | x || i } ( ( mmu ` d ) x. ( ( log ` ( i / d ) ) ^ 2 ) ) = sum_ d e. { x e. NN | x || N } ( ( mmu ` d ) x. ( ( log ` ( N / d ) ) ^ 2 ) ) )
75	66, 74	syl5eq 2668	. . 3 |- ( i = N -> sum_ j e. { x e. NN | x || i } ( ( mmu ` j ) x. ( ( log ` ( i / j ) ) ^ 2 ) ) = sum_ d e. { x e. NN | x || N } ( ( mmu ` d ) x. ( ( log ` ( N / d ) ) ^ 2 ) ) )
76		ssrab2 3687	. . . . . . . 8 |- { x e. NN | x || i } C_ NN
77		dvdsdivcl 15038	. . . . . . . 8 |- ( ( i e. NN /\ j e. { x e. NN | x || i } ) -> ( i / j ) e. { x e. NN | x || i } )
78	76, 77	sseldi 3601	. . . . . . 7 |- ( ( i e. NN /\ j e. { x e. NN | x || i } ) -> ( i / j ) e. NN )
79		fveq2 6191	. . . . . . . . 9 |- ( n = ( i / j ) -> ( log ` n ) = ( log ` ( i / j ) ) )
80	79	oveq1d 6665	. . . . . . . 8 |- ( n = ( i / j ) -> ( ( log ` n ) ^ 2 ) = ( ( log ` ( i / j ) ) ^ 2 ) )
81		eqid 2622	. . . . . . . 8 |- ( n e. NN |-> ( ( log ` n ) ^ 2 ) ) = ( n e. NN |-> ( ( log ` n ) ^ 2 ) )
82		ovex 6678	. . . . . . . 8 |- ( ( log ` n ) ^ 2 ) e. _V
83	80, 81, 82	fvmpt3i 6287	. . . . . . 7 |- ( ( i / j ) e. NN -> ( ( n e. NN |-> ( ( log ` n ) ^ 2 ) ) ` ( i / j ) ) = ( ( log ` ( i / j ) ) ^ 2 ) )
84	78, 83	syl 17	. . . . . 6 |- ( ( i e. NN /\ j e. { x e. NN | x || i } ) -> ( ( n e. NN |-> ( ( log ` n ) ^ 2 ) ) ` ( i / j ) ) = ( ( log ` ( i / j ) ) ^ 2 ) )
85	84	oveq2d 6666	. . . . 5 |- ( ( i e. NN /\ j e. { x e. NN | x || i } ) -> ( ( mmu ` j ) x. ( ( n e. NN |-> ( ( log ` n ) ^ 2 ) ) ` ( i / j ) ) ) = ( ( mmu ` j ) x. ( ( log ` ( i / j ) ) ^ 2 ) ) )
86	85	sumeq2dv 14433	. . . 4 |- ( i e. NN -> sum_ j e. { x e. NN | x || i } ( ( mmu ` j ) x. ( ( n e. NN |-> ( ( log ` n ) ^ 2 ) ) ` ( i / j ) ) ) = sum_ j e. { x e. NN | x || i } ( ( mmu ` j ) x. ( ( log ` ( i / j ) ) ^ 2 ) ) )
87	86	mpteq2ia 4740	. . 3 |- ( i e. NN |-> sum_ j e. { x e. NN | x || i } ( ( mmu ` j ) x. ( ( n e. NN |-> ( ( log ` n ) ^ 2 ) ) ` ( i / j ) ) ) ) = ( i e. NN |-> sum_ j e. { x e. NN | x || i } ( ( mmu ` j ) x. ( ( log ` ( i / j ) ) ^ 2 ) ) )
88		sumex 14418	. . 3 |- sum_ j e. { x e. NN | x || i } ( ( mmu ` j ) x. ( ( log ` ( i / j ) ) ^ 2 ) ) e. _V
89	75, 87, 88	fvmpt3i 6287	. 2 |- ( N e. NN -> ( ( i e. NN |-> sum_ j e. { x e. NN | x || i } ( ( mmu ` j ) x. ( ( n e. NN |-> ( ( log ` n ) ^ 2 ) ) ` ( i / j ) ) ) ) ` N ) = sum_ d e. { x e. NN | x || N } ( ( mmu ` d ) x. ( ( log ` ( N / d ) ) ^ 2 ) ) )
90	48, 60, 89	3eqtr3rd 2665	1 |- ( N e. NN -> sum_ d e. { x e. NN | x || N } ( ( mmu ` d ) x. ( ( log ` ( N / d ) ) ^ 2 ) ) = ( sum_ d e. { x e. NN | x || N } ( (Λ ` d ) x. (Λ ` ( N / d ) ) ) + ( (Λ ` N ) x. ( log ` N ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   C_ wss 3574   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   Fincfn 7955   CCcc 9934   RRcr 9935   1c1 9937   + caddc 9939   x. cmul 9941   / cdiv 10684   NNcn 11020   2c2 11070   ...cfz 12326   ^cexp 12860   sum_csu 14416   || cdvds 14983   logclog 24301  Λcvma 24818   mmucmu 24821

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-disj 4621  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-xnn0 11364  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ef 14798  df-sin 14800  df-cos 14801  df-pi 14803  df-dvds 14984  df-gcd 15217  df-prm 15386  df-pc 15542  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-limc 23630  df-dv 23631  df-log 24303  df-vma 24824  df-mu 24827

This theorem is referenced by:  selberg  25237