Theorem logfac 15918

Description: The logarithm of a factorial can be expressed as a finite sum of logs. (Contributed by Mario Carneiro, 17-Apr-2015.)

Assertion

Ref	Expression
logfac	|- ( N e. NN0 -> ( log ` ( ! ` N ) ) = sum_ k e. ( 1 ... N ) ( log ` k ) )

Distinct variable group:   k, N

Proof of Theorem logfac

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

Step	Hyp	Ref	Expression
1		elnn0 9544	. 2 |- ( N e. NN0 <-> ( N e. NN \/ N = 0 ) )
2		elnn1uz2 9986	. . . . 5 |- ( N e. NN <-> ( N = 1 \/ N e. ( ZZ>= ` 2 ) ) )
3		1zzd 9650	. . . . . . . . . . 11 |- ( N = 1 -> 1 e. ZZ )
4		fvi 5754	. . . . . . . . . . . . 13 |- ( k e. ( ZZ>= ` 1 ) -> ( _I ` k ) = k )
5		elnnuz 9938	. . . . . . . . . . . . . 14 |- ( k e. NN <-> k e. ( ZZ>= ` 1 ) )
6	5	biimpri 133	. . . . . . . . . . . . 13 |- ( k e. ( ZZ>= ` 1 ) -> k e. NN )
7	4, 6	eqeltrd 2315	. . . . . . . . . . . 12 |- ( k e. ( ZZ>= ` 1 ) -> ( _I ` k ) e. NN )
8	7	adantl 277	. . . . . . . . . . 11 |- ( ( N = 1 /\ k e. ( ZZ>= ` 1 ) ) -> ( _I ` k ) e. NN )
9		nnmulcl 9304	. . . . . . . . . . . 12 |- ( ( k e. NN /\ n e. NN ) -> ( k x. n ) e. NN )
10	9	adantl 277	. . . . . . . . . . 11 |- ( ( N = 1 /\ ( k e. NN /\ n e. NN ) ) -> ( k x. n ) e. NN )
11	3, 8, 10	seq3-1 10877	. . . . . . . . . 10 |- ( N = 1 -> ( seq 1 ( x. , _I ) ` 1 ) = ( _I ` 1 ) )
12		1nn 9294	. . . . . . . . . . 11 |- 1 e. NN
13		fvi 5754	. . . . . . . . . . 11 |- ( 1 e. NN -> ( _I ` 1 ) = 1 )
14	12, 13	ax-mp 5	. . . . . . . . . 10 |- ( _I ` 1 ) = 1
15	11, 14	eqtrdi 2287	. . . . . . . . 9 |- ( N = 1 -> ( seq 1 ( x. , _I ) ` 1 ) = 1 )
16	15	fveq2d 5694	. . . . . . . 8 |- ( N = 1 -> ( log ` ( seq 1 ( x. , _I ) ` 1 ) ) = ( log ` 1 ) )
17		nnrp 10043	. . . . . . . . . . 11 |- ( k e. NN -> k e. RR+ )
18		relogcl 15886	. . . . . . . . . . 11 |- ( k e. RR+ -> ( log ` k ) e. RR )
19	6, 17, 18	3syl 17	. . . . . . . . . 10 |- ( k e. ( ZZ>= ` 1 ) -> ( log ` k ) e. RR )
20	19	adantl 277	. . . . . . . . 9 |- ( ( N = 1 /\ k e. ( ZZ>= ` 1 ) ) -> ( log ` k ) e. RR )
21		readdcl 8295	. . . . . . . . . 10 |- ( ( k e. RR /\ n e. RR ) -> ( k + n ) e. RR )
22	21	adantl 277	. . . . . . . . 9 |- ( ( N = 1 /\ ( k e. RR /\ n e. RR ) ) -> ( k + n ) e. RR )
23	3, 20, 22	seq3-1 10877	. . . . . . . 8 |- ( N = 1 -> ( seq 1 ( + , log ) ` 1 ) = ( log ` 1 ) )
24	16, 23	eqtr4d 2274	. . . . . . 7 |- ( N = 1 -> ( log ` ( seq 1 ( x. , _I ) ` 1 ) ) = ( seq 1 ( + , log ) ` 1 ) )
25		2fveq3 5695	. . . . . . 7 |- ( N = 1 -> ( log ` ( seq 1 ( x. , _I ) ` N ) ) = ( log ` ( seq 1 ( x. , _I ) ` 1 ) ) )
26		fveq2 5690	. . . . . . 7 |- ( N = 1 -> ( seq 1 ( + , log ) ` N ) = ( seq 1 ( + , log ) ` 1 ) )
27	24, 25, 26	3eqtr4d 2281	. . . . . 6 |- ( N = 1 -> ( log ` ( seq 1 ( x. , _I ) ` N ) ) = ( seq 1 ( + , log ) ` N ) )
28		rpmulcl 10058	. . . . . . . . 9 |- ( ( k e. RR+ /\ n e. RR+ ) -> ( k x. n ) e. RR+ )
29	28	adantl 277	. . . . . . . 8 |- ( ( N e. ( ZZ>= ` 2 ) /\ ( k e. RR+ /\ n e. RR+ ) ) -> ( k x. n ) e. RR+ )
30		fvi 5754	. . . . . . . . . 10 |- ( k e. ( ZZ>= ` 2 ) -> ( _I ` k ) = k )
31		eluz2nn 9945	. . . . . . . . . . 11 |- ( k e. ( ZZ>= ` 2 ) -> k e. NN )
32	31	nnrpd 10074	. . . . . . . . . 10 |- ( k e. ( ZZ>= ` 2 ) -> k e. RR+ )
33	30, 32	eqeltrd 2315	. . . . . . . . 9 |- ( k e. ( ZZ>= ` 2 ) -> ( _I ` k ) e. RR+ )
34	33	adantl 277	. . . . . . . 8 |- ( ( N e. ( ZZ>= ` 2 ) /\ k e. ( ZZ>= ` 2 ) ) -> ( _I ` k ) e. RR+ )
35		id 19	. . . . . . . 8 |- ( N e. ( ZZ>= ` 2 ) -> N e. ( ZZ>= ` 2 ) )
36		relogmul 15893	. . . . . . . . 9 |- ( ( k e. RR+ /\ n e. RR+ ) -> ( log ` ( k x. n ) ) = ( ( log ` k ) + ( log ` n ) ) )
37	36	adantl 277	. . . . . . . 8 |- ( ( N e. ( ZZ>= ` 2 ) /\ ( k e. RR+ /\ n e. RR+ ) ) -> ( log ` ( k x. n ) ) = ( ( log ` k ) + ( log ` n ) ) )
38		fvi 5754	. . . . . . . . . . 11 |- ( k e. _V -> ( _I ` k ) = k )
39	38	elv 2825	. . . . . . . . . 10 |- ( _I ` k ) = k
40	39	fveq2i 5693	. . . . . . . . 9 |- ( log ` ( _I ` k ) ) = ( log ` k )
41	40	a1i 9	. . . . . . . 8 |- ( ( N e. ( ZZ>= ` 2 ) /\ k e. ( ZZ>= ` 2 ) ) -> ( log ` ( _I ` k ) ) = ( log ` k ) )
42	31	nnred 9296	. . . . . . . . . 10 |- ( k e. ( ZZ>= ` 2 ) -> k e. RR )
43	42	adantl 277	. . . . . . . . 9 |- ( ( N e. ( ZZ>= ` 2 ) /\ k e. ( ZZ>= ` 2 ) ) -> k e. RR )
44		eluz2gt1 9981	. . . . . . . . . 10 |- ( k e. ( ZZ>= ` 2 ) -> 1 < k )
45	44	adantl 277	. . . . . . . . 9 |- ( ( N e. ( ZZ>= ` 2 ) /\ k e. ( ZZ>= ` 2 ) ) -> 1 < k )
46	43, 45	rplogcld 15912	. . . . . . . 8 |- ( ( N e. ( ZZ>= ` 2 ) /\ k e. ( ZZ>= ` 2 ) ) -> ( log ` k ) e. RR+ )
47		rpaddcl 10057	. . . . . . . . 9 |- ( ( k e. RR+ /\ n e. RR+ ) -> ( k + n ) e. RR+ )
48	47	adantl 277	. . . . . . . 8 |- ( ( N e. ( ZZ>= ` 2 ) /\ ( k e. RR+ /\ n e. RR+ ) ) -> ( k + n ) e. RR+ )
49	29, 34, 35, 37, 41, 46, 48	seq3homo 10942	. . . . . . 7 |- ( N e. ( ZZ>= ` 2 ) -> ( log ` ( seq 2 ( x. , _I ) ` N ) ) = ( seq 2 ( + , log ) ` N ) )
50		remulcl 8297	. . . . . . . . . . . . 13 |- ( ( k e. RR /\ n e. RR ) -> ( k x. n ) e. RR )
51	50	adantl 277	. . . . . . . . . . . 12 |- ( ( N e. ( ZZ>= ` 2 ) /\ ( k e. RR /\ n e. RR ) ) -> ( k x. n ) e. RR )
52		simp1 1028	. . . . . . . . . . . . . . 15 |- ( ( k e. RR /\ n e. RR /\ m e. RR ) -> k e. RR )
53	52	recnd 8344	. . . . . . . . . . . . . 14 |- ( ( k e. RR /\ n e. RR /\ m e. RR ) -> k e. CC )
54		simp2 1029	. . . . . . . . . . . . . . 15 |- ( ( k e. RR /\ n e. RR /\ m e. RR ) -> n e. RR )
55	54	recnd 8344	. . . . . . . . . . . . . 14 |- ( ( k e. RR /\ n e. RR /\ m e. RR ) -> n e. CC )
56		simp3 1030	. . . . . . . . . . . . . . 15 |- ( ( k e. RR /\ n e. RR /\ m e. RR ) -> m e. RR )
57	56	recnd 8344	. . . . . . . . . . . . . 14 |- ( ( k e. RR /\ n e. RR /\ m e. RR ) -> m e. CC )
58	53, 55, 57	mulassd 8339	. . . . . . . . . . . . 13 |- ( ( k e. RR /\ n e. RR /\ m e. RR ) -> ( ( k x. n ) x. m ) = ( k x. ( n x. m ) ) )
59	58	adantl 277	. . . . . . . . . . . 12 |- ( ( N e. ( ZZ>= ` 2 ) /\ ( k e. RR /\ n e. RR /\ m e. RR ) ) -> ( ( k x. n ) x. m ) = ( k x. ( n x. m ) ) )
60		1p1e2 9400	. . . . . . . . . . . . . . 15 |- ( 1 + 1 ) = 2
61	60	fveq2i 5693	. . . . . . . . . . . . . 14 |- ( ZZ>= ` ( 1 + 1 ) ) = ( ZZ>= ` 2 )
62	61	eleq2i 2305	. . . . . . . . . . . . 13 |- ( N e. ( ZZ>= ` ( 1 + 1 ) ) <-> N e. ( ZZ>= ` 2 ) )
63	62	biimpri 133	. . . . . . . . . . . 12 |- ( N e. ( ZZ>= ` 2 ) -> N e. ( ZZ>= ` ( 1 + 1 ) ) )
64		1zzd 9650	. . . . . . . . . . . 12 |- ( N e. ( ZZ>= ` 2 ) -> 1 e. ZZ )
65	7	nnred 9296	. . . . . . . . . . . . 13 |- ( k e. ( ZZ>= ` 1 ) -> ( _I ` k ) e. RR )
66	65	adantl 277	. . . . . . . . . . . 12 |- ( ( N e. ( ZZ>= ` 2 ) /\ k e. ( ZZ>= ` 1 ) ) -> ( _I ` k ) e. RR )
67	51, 59, 63, 64, 66	seq3-1p 10905	. . . . . . . . . . 11 |- ( N e. ( ZZ>= ` 2 ) -> ( seq 1 ( x. , _I ) ` N ) = ( ( _I ` 1 ) x. ( seq ( 1 + 1 ) ( x. , _I ) ` N ) ) )
68	14	oveq1i 6085	. . . . . . . . . . 11 |- ( ( _I ` 1 ) x. ( seq ( 1 + 1 ) ( x. , _I ) ` N ) ) = ( 1 x. ( seq ( 1 + 1 ) ( x. , _I ) ` N ) )
69	67, 68	eqtrdi 2287	. . . . . . . . . 10 |- ( N e. ( ZZ>= ` 2 ) -> ( seq 1 ( x. , _I ) ` N ) = ( 1 x. ( seq ( 1 + 1 ) ( x. , _I ) ` N ) ) )
70		seqeq1 10865	. . . . . . . . . . . . . 14 |- ( ( 1 + 1 ) = 2 -> seq ( 1 + 1 ) ( x. , _I ) = seq 2 ( x. , _I ) )
71	60, 70	ax-mp 5	. . . . . . . . . . . . 13 |- seq ( 1 + 1 ) ( x. , _I ) = seq 2 ( x. , _I )
72	71	fveq1i 5691	. . . . . . . . . . . 12 |- ( seq ( 1 + 1 ) ( x. , _I ) ` N ) = ( seq 2 ( x. , _I ) ` N )
73		eqid 2238	. . . . . . . . . . . . . . 15 |- ( ZZ>= ` 2 ) = ( ZZ>= ` 2 )
74		eluzel2 9905	. . . . . . . . . . . . . . 15 |- ( N e. ( ZZ>= ` 2 ) -> 2 e. ZZ )
75		eluzelz 9910	. . . . . . . . . . . . . . . . 17 |- ( k e. ( ZZ>= ` 2 ) -> k e. ZZ )
76	30, 75	eqeltrd 2315	. . . . . . . . . . . . . . . 16 |- ( k e. ( ZZ>= ` 2 ) -> ( _I ` k ) e. ZZ )
77	76	adantl 277	. . . . . . . . . . . . . . 15 |- ( ( N e. ( ZZ>= ` 2 ) /\ k e. ( ZZ>= ` 2 ) ) -> ( _I ` k ) e. ZZ )
78		zmulcl 9677	. . . . . . . . . . . . . . . 16 |- ( ( k e. ZZ /\ n e. ZZ ) -> ( k x. n ) e. ZZ )
79	78	adantl 277	. . . . . . . . . . . . . . 15 |- ( ( N e. ( ZZ>= ` 2 ) /\ ( k e. ZZ /\ n e. ZZ ) ) -> ( k x. n ) e. ZZ )
80	73, 74, 77, 79	seqf 10879	. . . . . . . . . . . . . 14 |- ( N e. ( ZZ>= ` 2 ) -> seq 2 ( x. , _I ) : ( ZZ>= ` 2 ) --> ZZ )
81	80, 35	ffvelcdmd 5835	. . . . . . . . . . . . 13 |- ( N e. ( ZZ>= ` 2 ) -> ( seq 2 ( x. , _I ) ` N ) e. ZZ )
82	81	zcnd 9748	. . . . . . . . . . . 12 |- ( N e. ( ZZ>= ` 2 ) -> ( seq 2 ( x. , _I ) ` N ) e. CC )
83	72, 82	eqeltrid 2325	. . . . . . . . . . 11 |- ( N e. ( ZZ>= ` 2 ) -> ( seq ( 1 + 1 ) ( x. , _I ) ` N ) e. CC )
84	83	mullidd 8334	. . . . . . . . . 10 |- ( N e. ( ZZ>= ` 2 ) -> ( 1 x. ( seq ( 1 + 1 ) ( x. , _I ) ` N ) ) = ( seq ( 1 + 1 ) ( x. , _I ) ` N ) )
85	69, 84	eqtrd 2271	. . . . . . . . 9 |- ( N e. ( ZZ>= ` 2 ) -> ( seq 1 ( x. , _I ) ` N ) = ( seq ( 1 + 1 ) ( x. , _I ) ` N ) )
86	85, 72	eqtrdi 2287	. . . . . . . 8 |- ( N e. ( ZZ>= ` 2 ) -> ( seq 1 ( x. , _I ) ` N ) = ( seq 2 ( x. , _I ) ` N ) )
87	86	fveq2d 5694	. . . . . . 7 |- ( N e. ( ZZ>= ` 2 ) -> ( log ` ( seq 1 ( x. , _I ) ` N ) ) = ( log ` ( seq 2 ( x. , _I ) ` N ) ) )
88	21	adantl 277	. . . . . . . . . . 11 |- ( ( N e. ( ZZ>= ` 2 ) /\ ( k e. RR /\ n e. RR ) ) -> ( k + n ) e. RR )
89	53, 55, 57	addassd 8338	. . . . . . . . . . . 12 |- ( ( k e. RR /\ n e. RR /\ m e. RR ) -> ( ( k + n ) + m ) = ( k + ( n + m ) ) )
90	89	adantl 277	. . . . . . . . . . 11 |- ( ( N e. ( ZZ>= ` 2 ) /\ ( k e. RR /\ n e. RR /\ m e. RR ) ) -> ( ( k + n ) + m ) = ( k + ( n + m ) ) )
91	6	nnrpd 10074	. . . . . . . . . . . . 13 |- ( k e. ( ZZ>= ` 1 ) -> k e. RR+ )
92	91	relogcld 15906	. . . . . . . . . . . 12 |- ( k e. ( ZZ>= ` 1 ) -> ( log ` k ) e. RR )
93	92	adantl 277	. . . . . . . . . . 11 |- ( ( N e. ( ZZ>= ` 2 ) /\ k e. ( ZZ>= ` 1 ) ) -> ( log ` k ) e. RR )
94	88, 90, 63, 64, 93	seq3-1p 10905	. . . . . . . . . 10 |- ( N e. ( ZZ>= ` 2 ) -> ( seq 1 ( + , log ) ` N ) = ( ( log ` 1 ) + ( seq ( 1 + 1 ) ( + , log ) ` N ) ) )
95		seqeq1 10865	. . . . . . . . . . . . 13 |- ( ( 1 + 1 ) = 2 -> seq ( 1 + 1 ) ( + , log ) = seq 2 ( + , log ) )
96	60, 95	ax-mp 5	. . . . . . . . . . . 12 |- seq ( 1 + 1 ) ( + , log ) = seq 2 ( + , log )
97	96	fveq1i 5691	. . . . . . . . . . 11 |- ( seq ( 1 + 1 ) ( + , log ) ` N ) = ( seq 2 ( + , log ) ` N )
98	97	oveq2i 6086	. . . . . . . . . 10 |- ( ( log ` 1 ) + ( seq ( 1 + 1 ) ( + , log ) ` N ) ) = ( ( log ` 1 ) + ( seq 2 ( + , log ) ` N ) )
99	94, 98	eqtrdi 2287	. . . . . . . . 9 |- ( N e. ( ZZ>= ` 2 ) -> ( seq 1 ( + , log ) ` N ) = ( ( log ` 1 ) + ( seq 2 ( + , log ) ` N ) ) )
100		log1 15890	. . . . . . . . . 10 |- ( log ` 1 ) = 0
101	100	oveq1i 6085	. . . . . . . . 9 |- ( ( log ` 1 ) + ( seq 2 ( + , log ) ` N ) ) = ( 0 + ( seq 2 ( + , log ) ` N ) )
102	99, 101	eqtrdi 2287	. . . . . . . 8 |- ( N e. ( ZZ>= ` 2 ) -> ( seq 1 ( + , log ) ` N ) = ( 0 + ( seq 2 ( + , log ) ` N ) ) )
103	73, 74, 46, 48	seqf 10879	. . . . . . . . . . 11 |- ( N e. ( ZZ>= ` 2 ) -> seq 2 ( + , log ) : ( ZZ>= ` 2 ) --> RR+ )
104	103, 35	ffvelcdmd 5835	. . . . . . . . . 10 |- ( N e. ( ZZ>= ` 2 ) -> ( seq 2 ( + , log ) ` N ) e. RR+ )
105	104	rpcnd 10078	. . . . . . . . 9 |- ( N e. ( ZZ>= ` 2 ) -> ( seq 2 ( + , log ) ` N ) e. CC )
106	105	addlidd 8466	. . . . . . . 8 |- ( N e. ( ZZ>= ` 2 ) -> ( 0 + ( seq 2 ( + , log ) ` N ) ) = ( seq 2 ( + , log ) ` N ) )
107	102, 106	eqtrd 2271	. . . . . . 7 |- ( N e. ( ZZ>= ` 2 ) -> ( seq 1 ( + , log ) ` N ) = ( seq 2 ( + , log ) ` N ) )
108	49, 87, 107	3eqtr4d 2281	. . . . . 6 |- ( N e. ( ZZ>= ` 2 ) -> ( log ` ( seq 1 ( x. , _I ) ` N ) ) = ( seq 1 ( + , log ) ` N ) )
109	27, 108	jaoi 728	. . . . 5 |- ( ( N = 1 \/ N e. ( ZZ>= ` 2 ) ) -> ( log ` ( seq 1 ( x. , _I ) ` N ) ) = ( seq 1 ( + , log ) ` N ) )
110	2, 109	sylbi 121	. . . 4 |- ( N e. NN -> ( log ` ( seq 1 ( x. , _I ) ` N ) ) = ( seq 1 ( + , log ) ` N ) )
111		facnn 11143	. . . . 5 |- ( N e. NN -> ( ! ` N ) = ( seq 1 ( x. , _I ) ` N ) )
112	111	fveq2d 5694	. . . 4 |- ( N e. NN -> ( log ` ( ! ` N ) ) = ( log ` ( seq 1 ( x. , _I ) ` N ) ) )
113		eqidd 2239	. . . . 5 |- ( ( N e. NN /\ k e. ( ZZ>= ` 1 ) ) -> ( log ` k ) = ( log ` k ) )
114		elnnuz 9938	. . . . . 6 |- ( N e. NN <-> N e. ( ZZ>= ` 1 ) )
115	114	biimpi 120	. . . . 5 |- ( N e. NN -> N e. ( ZZ>= ` 1 ) )
116	19	adantl 277	. . . . . 6 |- ( ( N e. NN /\ k e. ( ZZ>= ` 1 ) ) -> ( log ` k ) e. RR )
117	116	recnd 8344	. . . . 5 |- ( ( N e. NN /\ k e. ( ZZ>= ` 1 ) ) -> ( log ` k ) e. CC )
118	113, 115, 117	fsum3ser 12142	. . . 4 |- ( N e. NN -> sum_ k e. ( 1 ... N ) ( log ` k ) = ( seq 1 ( + , log ) ` N ) )
119	110, 112, 118	3eqtr4d 2281	. . 3 |- ( N e. NN -> ( log ` ( ! ` N ) ) = sum_ k e. ( 1 ... N ) ( log ` k ) )
120		sum0 12133	. . . . 5 |- sum_ k e. (/) ( log ` k ) = 0
121	100, 120	eqtr4i 2262	. . . 4 |- ( log ` 1 ) = sum_ k e. (/) ( log ` k )
122		fveq2 5690	. . . . . 6 |- ( N = 0 -> ( ! ` N ) = ( ! ` 0 ) )
123		fac0 11144	. . . . . 6 |- ( ! ` 0 ) = 1
124	122, 123	eqtrdi 2287	. . . . 5 |- ( N = 0 -> ( ! ` N ) = 1 )
125	124	fveq2d 5694	. . . 4 |- ( N = 0 -> ( log ` ( ! ` N ) ) = ( log ` 1 ) )
126		oveq2 6083	. . . . . 6 |- ( N = 0 -> ( 1 ... N ) = ( 1 ... 0 ) )
127		fz10 10429	. . . . . 6 |- ( 1 ... 0 ) = (/)
128	126, 127	eqtrdi 2287	. . . . 5 |- ( N = 0 -> ( 1 ... N ) = (/) )
129	128	sumeq1d 12110	. . . 4 |- ( N = 0 -> sum_ k e. ( 1 ... N ) ( log ` k ) = sum_ k e. (/) ( log ` k ) )
130	121, 125, 129	3eqtr4a 2297	. . 3 |- ( N = 0 -> ( log ` ( ! ` N ) ) = sum_ k e. ( 1 ... N ) ( log ` k ) )
131	119, 130	jaoi 728	. 2 |- ( ( N e. NN \/ N = 0 ) -> ( log ` ( ! ` N ) ) = sum_ k e. ( 1 ... N ) ( log ` k ) )
132	1, 131	sylbi 121	1 |- ( N e. NN0 -> ( log ` ( ! ` N ) ) = sum_ k e. ( 1 ... N ) ( log ` k ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 720   /\ w3a 1009   = wceq 1402   e. wcel 2209   _Vcvv 2821   (/)c0 3520   class class class wbr 4125   _I cid 4428   ` cfv 5372  (class class class)co 6075   CCcc 8167   RRcr 8168   0cc0 8169   1c1 8170   + caddc 8172   x. cmul 8174   < clt 8350   NNcn 9283   2c2 9334   NN0cn0 9542   ZZcz 9623   ZZ>=cuz 9900   RR+crp 10033   ...cfz 10390   seqcseq 10862   !cfa 11141   sum_csu 12097   logclog 15880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287  ax-arch 8288  ax-caucvg 8289  ax-pre-suploc 8290  ax-addf 8291  ax-mulf 8292

This theorem depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-disj 4102  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-isom 5381  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-of 6292  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-frec 6652  df-1o 6677  df-oadd 6681  df-er 6797  df-map 6914  df-pm 6915  df-en 7013  df-dom 7014  df-fin 7015  df-sup 7314  df-inf 7315  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-n0 9543  df-z 9624  df-uz 9901  df-q 9999  df-rp 10034  df-xneg 10153  df-xadd 10154  df-ioo 10273  df-ico 10275  df-icc 10276  df-fz 10391  df-fzo 10528  df-seqfrec 10863  df-exp 10954  df-fac 11142  df-bc 11164  df-ihash 11193  df-shft 11558  df-cj 11585  df-re 11586  df-im 11587  df-rsqrt 11742  df-abs 11743  df-clim 12023  df-sumdc 12098  df-ef 12393  df-e 12394  df-rest 13572  df-topgen 13591  df-psmet 14852  df-xmet 14853  df-met 14854  df-bl 14855  df-mopn 14856  df-top 15022  df-topon 15035  df-bases 15067  df-ntr 15120  df-cn 15212  df-cnp 15213  df-tx 15277  df-cncf 15595  df-limced 15680  df-dvap 15681  df-relog 15882

This theorem is referenced by: (None)