Theorem logfac 15995

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 9565	. 2 |- ( N e. NN0 <-> ( N e. NN \/ N = 0 ) )
2		elnn1uz2 10007	. . . . 5 |- ( N e. NN <-> ( N = 1 \/ N e. ( ZZ>= ` 2 ) ) )
3		1zzd 9671	. . . . . . . . . . 11 |- ( N = 1 -> 1 e. ZZ )
4		fvi 5760	. . . . . . . . . . . . 13 |- ( k e. ( ZZ>= ` 1 ) -> ( _I ` k ) = k )
5		elnnuz 9959	. . . . . . . . . . . . . 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 9325	. . . . . . . . . . . 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 10899	. . . . . . . . . 10 |- ( N = 1 -> ( seq 1 ( x. , _I ) ` 1 ) = ( _I ` 1 ) )
12		1nn 9315	. . . . . . . . . . 11 |- 1 e. NN
13		fvi 5760	. . . . . . . . . . 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 5699	. . . . . . . 8 |- ( N = 1 -> ( log ` ( seq 1 ( x. , _I ) ` 1 ) ) = ( log ` 1 ) )
17		nnrp 10064	. . . . . . . . . . 11 |- ( k e. NN -> k e. RR+ )
18		relogcl 15963	. . . . . . . . . . 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 8305	. . . . . . . . . 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 10899	. . . . . . . 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 5700	. . . . . . 7 |- ( N = 1 -> ( log ` ( seq 1 ( x. , _I ) ` N ) ) = ( log ` ( seq 1 ( x. , _I ) ` 1 ) ) )
26		fveq2 5695	. . . . . . 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 10079	. . . . . . . . 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 5760	. . . . . . . . . 10 |- ( k e. ( ZZ>= ` 2 ) -> ( _I ` k ) = k )
31		eluz2nn 9966	. . . . . . . . . . 11 |- ( k e. ( ZZ>= ` 2 ) -> k e. NN )
32	31	nnrpd 10095	. . . . . . . . . 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 15970	. . . . . . . . 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 5760	. . . . . . . . . . 11 |- ( k e. _V -> ( _I ` k ) = k )
39	38	elv 2825	. . . . . . . . . 10 |- ( _I ` k ) = k
40	39	fveq2i 5698	. . . . . . . . 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 9317	. . . . . . . . . 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 10002	. . . . . . . . . 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 15989	. . . . . . . 8 |- ( ( N e. ( ZZ>= ` 2 ) /\ k e. ( ZZ>= ` 2 ) ) -> ( log ` k ) e. RR+ )
47		rpaddcl 10078	. . . . . . . . 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 10964	. . . . . . 7 |- ( N e. ( ZZ>= ` 2 ) -> ( log ` ( seq 2 ( x. , _I ) ` N ) ) = ( seq 2 ( + , log ) ` N ) )
50		remulcl 8307	. . . . . . . . . . . . 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 8354	. . . . . . . . . . . . . 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 8354	. . . . . . . . . . . . . 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 8354	. . . . . . . . . . . . . 14 |- ( ( k e. RR /\ n e. RR /\ m e. RR ) -> m e. CC )
58	53, 55, 57	mulassd 8349	. . . . . . . . . . . . 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 9421	. . . . . . . . . . . . . . 15 |- ( 1 + 1 ) = 2
61	60	fveq2i 5698	. . . . . . . . . . . . . 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 9671	. . . . . . . . . . . 12 |- ( N e. ( ZZ>= ` 2 ) -> 1 e. ZZ )
65	7	nnred 9317	. . . . . . . . . . . . 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 10927	. . . . . . . . . . 11 |- ( N e. ( ZZ>= ` 2 ) -> ( seq 1 ( x. , _I ) ` N ) = ( ( _I ` 1 ) x. ( seq ( 1 + 1 ) ( x. , _I ) ` N ) ) )
68	14	oveq1i 6095	. . . . . . . . . . 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 10887	. . . . . . . . . . . . . 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 5696	. . . . . . . . . . . 12 |- ( seq ( 1 + 1 ) ( x. , _I ) ` N ) = ( seq 2 ( x. , _I ) ` N )
73		eqid 2238	. . . . . . . . . . . . . . 15 |- ( ZZ>= ` 2 ) = ( ZZ>= ` 2 )
74		eluzel2 9926	. . . . . . . . . . . . . . 15 |- ( N e. ( ZZ>= ` 2 ) -> 2 e. ZZ )
75		eluzelz 9931	. . . . . . . . . . . . . . . . 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 9698	. . . . . . . . . . . . . . . 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 10901	. . . . . . . . . . . . . 14 |- ( N e. ( ZZ>= ` 2 ) -> seq 2 ( x. , _I ) : ( ZZ>= ` 2 ) --> ZZ )
81	80, 35	ffvelcdmd 5844	. . . . . . . . . . . . 13 |- ( N e. ( ZZ>= ` 2 ) -> ( seq 2 ( x. , _I ) ` N ) e. ZZ )
82	81	zcnd 9769	. . . . . . . . . . . 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 8344	. . . . . . . . . 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 5699	. . . . . . 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 8348	. . . . . . . . . . . 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 10095	. . . . . . . . . . . . 13 |- ( k e. ( ZZ>= ` 1 ) -> k e. RR+ )
92	91	relogcld 15983	. . . . . . . . . . . 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 10927	. . . . . . . . . 10 |- ( N e. ( ZZ>= ` 2 ) -> ( seq 1 ( + , log ) ` N ) = ( ( log ` 1 ) + ( seq ( 1 + 1 ) ( + , log ) ` N ) ) )
95		seqeq1 10887	. . . . . . . . . . . . 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 5696	. . . . . . . . . . 11 |- ( seq ( 1 + 1 ) ( + , log ) ` N ) = ( seq 2 ( + , log ) ` N )
98	97	oveq2i 6096	. . . . . . . . . 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 15967	. . . . . . . . . 10 |- ( log ` 1 ) = 0
101	100	oveq1i 6095	. . . . . . . . 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 10901	. . . . . . . . . . 11 |- ( N e. ( ZZ>= ` 2 ) -> seq 2 ( + , log ) : ( ZZ>= ` 2 ) --> RR+ )
104	103, 35	ffvelcdmd 5844	. . . . . . . . . 10 |- ( N e. ( ZZ>= ` 2 ) -> ( seq 2 ( + , log ) ` N ) e. RR+ )
105	104	rpcnd 10099	. . . . . . . . 9 |- ( N e. ( ZZ>= ` 2 ) -> ( seq 2 ( + , log ) ` N ) e. CC )
106	105	addlidd 8476	. . . . . . . 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 11165	. . . . 5 |- ( N e. NN -> ( ! ` N ) = ( seq 1 ( x. , _I ) ` N ) )
112	111	fveq2d 5699	. . . 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 9959	. . . . . 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 8354	. . . . 5 |- ( ( N e. NN /\ k e. ( ZZ>= ` 1 ) ) -> ( log ` k ) e. CC )
118	113, 115, 117	fsum3ser 12164	. . . 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 12155	. . . . 5 |- sum_ k e. (/) ( log ` k ) = 0
121	100, 120	eqtr4i 2262	. . . 4 |- ( log ` 1 ) = sum_ k e. (/) ( log ` k )
122		fveq2 5695	. . . . . 6 |- ( N = 0 -> ( ! ` N ) = ( ! ` 0 ) )
123		fac0 11166	. . . . . 6 |- ( ! ` 0 ) = 1
124	122, 123	eqtrdi 2287	. . . . 5 |- ( N = 0 -> ( ! ` N ) = 1 )
125	124	fveq2d 5699	. . . 4 |- ( N = 0 -> ( log ` ( ! ` N ) ) = ( log ` 1 ) )
126		oveq2 6093	. . . . . 6 |- ( N = 0 -> ( 1 ... N ) = ( 1 ... 0 ) )
127		fz10 10450	. . . . . 6 |- ( 1 ... 0 ) = (/)
128	126, 127	eqtrdi 2287	. . . . 5 |- ( N = 0 -> ( 1 ... N ) = (/) )
129	128	sumeq1d 12132	. . . 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 ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   \/ wo 720   /\ w3a 1009   = wceq 1402   e. wcel 2209   _Vcvv 2821   (/)c0 3520   class class class wbr 4130   _I cid 4433   ` cfv 5377  (class class class)co 6085   CCcc 8177   RRcr 8178   0cc0 8179   1c1 8180   + caddc 8182   x. cmul 8184   < clt 8360   NNcn 9304   2c2 9355   NN0cn0 9563   ZZcz 9644   ZZ>=cuz 9921   RR+crp 10054   ...cfz 10411   seqcseq 10884   !cfa 11163   sum_csu 12119   logclog 15957

This proof depends on 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 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299  ax-pre-suploc 8300  ax-addf 8301  ax-mulf 8302

This proof 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 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-disj 4107  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-isom 5386  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-of 6302  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-frec 6662  df-1o 6687  df-oadd 6691  df-er 6807  df-map 6924  df-pm 6925  df-en 7023  df-dom 7024  df-fin 7025  df-sup 7324  df-inf 7325  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-n0 9564  df-z 9645  df-uz 9922  df-q 10020  df-rp 10055  df-xneg 10174  df-xadd 10175  df-ioo 10294  df-ico 10296  df-icc 10297  df-fz 10412  df-fzo 10550  df-seqfrec 10885  df-exp 10976  df-fac 11164  df-bc 11186  df-ihash 11215  df-shft 11580  df-cj 11607  df-re 11608  df-im 11609  df-rsqrt 11764  df-abs 11765  df-clim 12045  df-sumdc 12120  df-ef 12415  df-e 12416  df-rest 13595  df-topgen 13614  df-psmet 14880  df-xmet 14881  df-met 14882  df-bl 14883  df-mopn 14884  df-top 15099  df-topon 15112  df-bases 15144  df-ntr 15197  df-cn 15289  df-cnp 15290  df-tx 15354  df-cncf 15672  df-limced 15757  df-dvap 15758  df-relog 15959

This theorem is used by:  birthdaylem2  16088