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 ∈ ℕ0 → (log‘(!‘𝑁)) = Σ𝑘 ∈ (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 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
2		elnn1uz2 9986	. . . . 5 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 = 1 ∨ 𝑁 ∈ (ℤ≥‘2)))
3		1zzd 9650	. . . . . . . . . . 11 ⊢ (𝑁 = 1 → 1 ∈ ℤ)
4		fvi 5754	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (ℤ≥‘1) → ( I ‘𝑘) = 𝑘)
5		elnnuz 9938	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ ↔ 𝑘 ∈ (ℤ≥‘1))
6	5	biimpri 133	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (ℤ≥‘1) → 𝑘 ∈ ℕ)
7	4, 6	eqeltrd 2315	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (ℤ≥‘1) → ( I ‘𝑘) ∈ ℕ)
8	7	adantl 277	. . . . . . . . . . 11 ⊢ ((𝑁 = 1 ∧ 𝑘 ∈ (ℤ≥‘1)) → ( I ‘𝑘) ∈ ℕ)
9		nnmulcl 9304	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ ℕ ∧ 𝑛 ∈ ℕ) → (𝑘 · 𝑛) ∈ ℕ)
10	9	adantl 277	. . . . . . . . . . 11 ⊢ ((𝑁 = 1 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ ℕ)) → (𝑘 · 𝑛) ∈ ℕ)
11	3, 8, 10	seq3-1 10877	. . . . . . . . . 10 ⊢ (𝑁 = 1 → (seq1( · , I )‘1) = ( I ‘1))
12		1nn 9294	. . . . . . . . . . 11 ⊢ 1 ∈ ℕ
13		fvi 5754	. . . . . . . . . . 11 ⊢ (1 ∈ ℕ → ( I ‘1) = 1)
14	12, 13	ax-mp 5	. . . . . . . . . 10 ⊢ ( I ‘1) = 1
15	11, 14	eqtrdi 2287	. . . . . . . . 9 ⊢ (𝑁 = 1 → (seq1( · , I )‘1) = 1)
16	15	fveq2d 5694	. . . . . . . 8 ⊢ (𝑁 = 1 → (log‘(seq1( · , I )‘1)) = (log‘1))
17		nnrp 10043	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℝ+)
18		relogcl 15886	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℝ+ → (log‘𝑘) ∈ ℝ)
19	6, 17, 18	3syl 17	. . . . . . . . . 10 ⊢ (𝑘 ∈ (ℤ≥‘1) → (log‘𝑘) ∈ ℝ)
20	19	adantl 277	. . . . . . . . 9 ⊢ ((𝑁 = 1 ∧ 𝑘 ∈ (ℤ≥‘1)) → (log‘𝑘) ∈ ℝ)
21		readdcl 8295	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℝ ∧ 𝑛 ∈ ℝ) → (𝑘 + 𝑛) ∈ ℝ)
22	21	adantl 277	. . . . . . . . 9 ⊢ ((𝑁 = 1 ∧ (𝑘 ∈ ℝ ∧ 𝑛 ∈ ℝ)) → (𝑘 + 𝑛) ∈ ℝ)
23	3, 20, 22	seq3-1 10877	. . . . . . . 8 ⊢ (𝑁 = 1 → (seq1( + , log)‘1) = (log‘1))
24	16, 23	eqtr4d 2274	. . . . . . 7 ⊢ (𝑁 = 1 → (log‘(seq1( · , I )‘1)) = (seq1( + , log)‘1))
25		2fveq3 5695	. . . . . . 7 ⊢ (𝑁 = 1 → (log‘(seq1( · , I )‘𝑁)) = (log‘(seq1( · , I )‘1)))
26		fveq2 5690	. . . . . . 7 ⊢ (𝑁 = 1 → (seq1( + , log)‘𝑁) = (seq1( + , log)‘1))
27	24, 25, 26	3eqtr4d 2281	. . . . . 6 ⊢ (𝑁 = 1 → (log‘(seq1( · , I )‘𝑁)) = (seq1( + , log)‘𝑁))
28		rpmulcl 10058	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+) → (𝑘 · 𝑛) ∈ ℝ+)
29	28	adantl 277	. . . . . . . 8 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ (𝑘 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+)) → (𝑘 · 𝑛) ∈ ℝ+)
30		fvi 5754	. . . . . . . . . 10 ⊢ (𝑘 ∈ (ℤ≥‘2) → ( I ‘𝑘) = 𝑘)
31		eluz2nn 9945	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (ℤ≥‘2) → 𝑘 ∈ ℕ)
32	31	nnrpd 10074	. . . . . . . . . 10 ⊢ (𝑘 ∈ (ℤ≥‘2) → 𝑘 ∈ ℝ+)
33	30, 32	eqeltrd 2315	. . . . . . . . 9 ⊢ (𝑘 ∈ (ℤ≥‘2) → ( I ‘𝑘) ∈ ℝ+)
34	33	adantl 277	. . . . . . . 8 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑘 ∈ (ℤ≥‘2)) → ( I ‘𝑘) ∈ ℝ+)
35		id 19	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ≥‘2) → 𝑁 ∈ (ℤ≥‘2))
36		relogmul 15893	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+) → (log‘(𝑘 · 𝑛)) = ((log‘𝑘) + (log‘𝑛)))
37	36	adantl 277	. . . . . . . 8 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ (𝑘 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+)) → (log‘(𝑘 · 𝑛)) = ((log‘𝑘) + (log‘𝑛)))
38		fvi 5754	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 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 ∈ (ℤ≥‘2) ∧ 𝑘 ∈ (ℤ≥‘2)) → (log‘( I ‘𝑘)) = (log‘𝑘))
42	31	nnred 9296	. . . . . . . . . 10 ⊢ (𝑘 ∈ (ℤ≥‘2) → 𝑘 ∈ ℝ)
43	42	adantl 277	. . . . . . . . 9 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑘 ∈ (ℤ≥‘2)) → 𝑘 ∈ ℝ)
44		eluz2gt1 9981	. . . . . . . . . 10 ⊢ (𝑘 ∈ (ℤ≥‘2) → 1 < 𝑘)
45	44	adantl 277	. . . . . . . . 9 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑘 ∈ (ℤ≥‘2)) → 1 < 𝑘)
46	43, 45	rplogcld 15912	. . . . . . . 8 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑘 ∈ (ℤ≥‘2)) → (log‘𝑘) ∈ ℝ+)
47		rpaddcl 10057	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+) → (𝑘 + 𝑛) ∈ ℝ+)
48	47	adantl 277	. . . . . . . 8 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ (𝑘 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+)) → (𝑘 + 𝑛) ∈ ℝ+)
49	29, 34, 35, 37, 41, 46, 48	seq3homo 10942	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘2) → (log‘(seq2( · , I )‘𝑁)) = (seq2( + , log)‘𝑁))
50		remulcl 8297	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℝ ∧ 𝑛 ∈ ℝ) → (𝑘 · 𝑛) ∈ ℝ)
51	50	adantl 277	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ (𝑘 ∈ ℝ ∧ 𝑛 ∈ ℝ)) → (𝑘 · 𝑛) ∈ ℝ)
52		simp1 1028	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ℝ ∧ 𝑛 ∈ ℝ ∧ 𝑚 ∈ ℝ) → 𝑘 ∈ ℝ)
53	52	recnd 8344	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ ℝ ∧ 𝑛 ∈ ℝ ∧ 𝑚 ∈ ℝ) → 𝑘 ∈ ℂ)
54		simp2 1029	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ℝ ∧ 𝑛 ∈ ℝ ∧ 𝑚 ∈ ℝ) → 𝑛 ∈ ℝ)
55	54	recnd 8344	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ ℝ ∧ 𝑛 ∈ ℝ ∧ 𝑚 ∈ ℝ) → 𝑛 ∈ ℂ)
56		simp3 1030	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ℝ ∧ 𝑛 ∈ ℝ ∧ 𝑚 ∈ ℝ) → 𝑚 ∈ ℝ)
57	56	recnd 8344	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ ℝ ∧ 𝑛 ∈ ℝ ∧ 𝑚 ∈ ℝ) → 𝑚 ∈ ℂ)
58	53, 55, 57	mulassd 8339	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℝ ∧ 𝑛 ∈ ℝ ∧ 𝑚 ∈ ℝ) → ((𝑘 · 𝑛) · 𝑚) = (𝑘 · (𝑛 · 𝑚)))
59	58	adantl 277	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ (𝑘 ∈ ℝ ∧ 𝑛 ∈ ℝ ∧ 𝑚 ∈ ℝ)) → ((𝑘 · 𝑛) · 𝑚) = (𝑘 · (𝑛 · 𝑚)))
60		1p1e2 9400	. . . . . . . . . . . . . . 15 ⊢ (1 + 1) = 2
61	60	fveq2i 5693	. . . . . . . . . . . . . 14 ⊢ (ℤ≥‘(1 + 1)) = (ℤ≥‘2)
62	61	eleq2i 2305	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ (ℤ≥‘(1 + 1)) ↔ 𝑁 ∈ (ℤ≥‘2))
63	62	biimpri 133	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (ℤ≥‘2) → 𝑁 ∈ (ℤ≥‘(1 + 1)))
64		1zzd 9650	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (ℤ≥‘2) → 1 ∈ ℤ)
65	7	nnred 9296	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (ℤ≥‘1) → ( I ‘𝑘) ∈ ℝ)
66	65	adantl 277	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑘 ∈ (ℤ≥‘1)) → ( I ‘𝑘) ∈ ℝ)
67	51, 59, 63, 64, 66	seq3-1p 10905	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ≥‘2) → (seq1( · , I )‘𝑁) = (( I ‘1) · (seq(1 + 1)( · , I )‘𝑁)))
68	14	oveq1i 6085	. . . . . . . . . . 11 ⊢ (( I ‘1) · (seq(1 + 1)( · , I )‘𝑁)) = (1 · (seq(1 + 1)( · , I )‘𝑁))
69	67, 68	eqtrdi 2287	. . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ≥‘2) → (seq1( · , I )‘𝑁) = (1 · (seq(1 + 1)( · , I )‘𝑁)))
70		seqeq1 10865	. . . . . . . . . . . . . 14 ⊢ ((1 + 1) = 2 → seq(1 + 1)( · , I ) = seq2( · , I ))
71	60, 70	ax-mp 5	. . . . . . . . . . . . 13 ⊢ seq(1 + 1)( · , I ) = seq2( · , I )
72	71	fveq1i 5691	. . . . . . . . . . . 12 ⊢ (seq(1 + 1)( · , I )‘𝑁) = (seq2( · , I )‘𝑁)
73		eqid 2238	. . . . . . . . . . . . . . 15 ⊢ (ℤ≥‘2) = (ℤ≥‘2)
74		eluzel2 9905	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ (ℤ≥‘2) → 2 ∈ ℤ)
75		eluzelz 9910	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (ℤ≥‘2) → 𝑘 ∈ ℤ)
76	30, 75	eqeltrd 2315	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (ℤ≥‘2) → ( I ‘𝑘) ∈ ℤ)
77	76	adantl 277	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑘 ∈ (ℤ≥‘2)) → ( I ‘𝑘) ∈ ℤ)
78		zmulcl 9677	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑘 · 𝑛) ∈ ℤ)
79	78	adantl 277	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ (𝑘 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → (𝑘 · 𝑛) ∈ ℤ)
80	73, 74, 77, 79	seqf 10879	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (ℤ≥‘2) → seq2( · , I ):(ℤ≥‘2)⟶ℤ)
81	80, 35	ffvelcdmd 5835	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ (ℤ≥‘2) → (seq2( · , I )‘𝑁) ∈ ℤ)
82	81	zcnd 9748	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (ℤ≥‘2) → (seq2( · , I )‘𝑁) ∈ ℂ)
83	72, 82	eqeltrid 2325	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ≥‘2) → (seq(1 + 1)( · , I )‘𝑁) ∈ ℂ)
84	83	mullidd 8334	. . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ≥‘2) → (1 · (seq(1 + 1)( · , I )‘𝑁)) = (seq(1 + 1)( · , I )‘𝑁))
85	69, 84	eqtrd 2271	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ≥‘2) → (seq1( · , I )‘𝑁) = (seq(1 + 1)( · , I )‘𝑁))
86	85, 72	eqtrdi 2287	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ≥‘2) → (seq1( · , I )‘𝑁) = (seq2( · , I )‘𝑁))
87	86	fveq2d 5694	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘2) → (log‘(seq1( · , I )‘𝑁)) = (log‘(seq2( · , I )‘𝑁)))
88	21	adantl 277	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ (𝑘 ∈ ℝ ∧ 𝑛 ∈ ℝ)) → (𝑘 + 𝑛) ∈ ℝ)
89	53, 55, 57	addassd 8338	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ ℝ ∧ 𝑛 ∈ ℝ ∧ 𝑚 ∈ ℝ) → ((𝑘 + 𝑛) + 𝑚) = (𝑘 + (𝑛 + 𝑚)))
90	89	adantl 277	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ (𝑘 ∈ ℝ ∧ 𝑛 ∈ ℝ ∧ 𝑚 ∈ ℝ)) → ((𝑘 + 𝑛) + 𝑚) = (𝑘 + (𝑛 + 𝑚)))
91	6	nnrpd 10074	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (ℤ≥‘1) → 𝑘 ∈ ℝ+)
92	91	relogcld 15906	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (ℤ≥‘1) → (log‘𝑘) ∈ ℝ)
93	92	adantl 277	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑘 ∈ (ℤ≥‘1)) → (log‘𝑘) ∈ ℝ)
94	88, 90, 63, 64, 93	seq3-1p 10905	. . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ≥‘2) → (seq1( + , log)‘𝑁) = ((log‘1) + (seq(1 + 1)( + , log)‘𝑁)))
95		seqeq1 10865	. . . . . . . . . . . . 13 ⊢ ((1 + 1) = 2 → seq(1 + 1)( + , log) = seq2( + , log))
96	60, 95	ax-mp 5	. . . . . . . . . . . 12 ⊢ seq(1 + 1)( + , log) = seq2( + , log)
97	96	fveq1i 5691	. . . . . . . . . . 11 ⊢ (seq(1 + 1)( + , log)‘𝑁) = (seq2( + , log)‘𝑁)
98	97	oveq2i 6086	. . . . . . . . . 10 ⊢ ((log‘1) + (seq(1 + 1)( + , log)‘𝑁)) = ((log‘1) + (seq2( + , log)‘𝑁))
99	94, 98	eqtrdi 2287	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ≥‘2) → (seq1( + , log)‘𝑁) = ((log‘1) + (seq2( + , log)‘𝑁)))
100		log1 15890	. . . . . . . . . 10 ⊢ (log‘1) = 0
101	100	oveq1i 6085	. . . . . . . . 9 ⊢ ((log‘1) + (seq2( + , log)‘𝑁)) = (0 + (seq2( + , log)‘𝑁))
102	99, 101	eqtrdi 2287	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ≥‘2) → (seq1( + , log)‘𝑁) = (0 + (seq2( + , log)‘𝑁)))
103	73, 74, 46, 48	seqf 10879	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ≥‘2) → seq2( + , log):(ℤ≥‘2)⟶ℝ+)
104	103, 35	ffvelcdmd 5835	. . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ≥‘2) → (seq2( + , log)‘𝑁) ∈ ℝ+)
105	104	rpcnd 10078	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ≥‘2) → (seq2( + , log)‘𝑁) ∈ ℂ)
106	105	addlidd 8466	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ≥‘2) → (0 + (seq2( + , log)‘𝑁)) = (seq2( + , log)‘𝑁))
107	102, 106	eqtrd 2271	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘2) → (seq1( + , log)‘𝑁) = (seq2( + , log)‘𝑁))
108	49, 87, 107	3eqtr4d 2281	. . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘2) → (log‘(seq1( · , I )‘𝑁)) = (seq1( + , log)‘𝑁))
109	27, 108	jaoi 728	. . . . 5 ⊢ ((𝑁 = 1 ∨ 𝑁 ∈ (ℤ≥‘2)) → (log‘(seq1( · , I )‘𝑁)) = (seq1( + , log)‘𝑁))
110	2, 109	sylbi 121	. . . 4 ⊢ (𝑁 ∈ ℕ → (log‘(seq1( · , I )‘𝑁)) = (seq1( + , log)‘𝑁))
111		facnn 11143	. . . . 5 ⊢ (𝑁 ∈ ℕ → (!‘𝑁) = (seq1( · , I )‘𝑁))
112	111	fveq2d 5694	. . . 4 ⊢ (𝑁 ∈ ℕ → (log‘(!‘𝑁)) = (log‘(seq1( · , I )‘𝑁)))
113		eqidd 2239	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (ℤ≥‘1)) → (log‘𝑘) = (log‘𝑘))
114		elnnuz 9938	. . . . . 6 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (ℤ≥‘1))
115	114	biimpi 120	. . . . 5 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (ℤ≥‘1))
116	19	adantl 277	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (ℤ≥‘1)) → (log‘𝑘) ∈ ℝ)
117	116	recnd 8344	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (ℤ≥‘1)) → (log‘𝑘) ∈ ℂ)
118	113, 115, 117	fsum3ser 12142	. . . 4 ⊢ (𝑁 ∈ ℕ → Σ𝑘 ∈ (1...𝑁)(log‘𝑘) = (seq1( + , log)‘𝑁))
119	110, 112, 118	3eqtr4d 2281	. . 3 ⊢ (𝑁 ∈ ℕ → (log‘(!‘𝑁)) = Σ𝑘 ∈ (1...𝑁)(log‘𝑘))
120		sum0 12133	. . . . 5 ⊢ Σ𝑘 ∈ ∅ (log‘𝑘) = 0
121	100, 120	eqtr4i 2262	. . . 4 ⊢ (log‘1) = Σ𝑘 ∈ ∅ (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 → Σ𝑘 ∈ (1...𝑁)(log‘𝑘) = Σ𝑘 ∈ ∅ (log‘𝑘))
130	121, 125, 129	3eqtr4a 2297	. . 3 ⊢ (𝑁 = 0 → (log‘(!‘𝑁)) = Σ𝑘 ∈ (1...𝑁)(log‘𝑘))
131	119, 130	jaoi 728	. 2 ⊢ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → (log‘(!‘𝑁)) = Σ𝑘 ∈ (1...𝑁)(log‘𝑘))
132	1, 131	sylbi 121	1 ⊢ (𝑁 ∈ ℕ0 → (log‘(!‘𝑁)) = Σ𝑘 ∈ (1...𝑁)(log‘𝑘))

Syntax hints:   → wi 4   ∧ wa 104   ∨ wo 720   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209  Vcvv 2821  ∅c0 3520   class class class wbr 4125   I cid 4428  ‘cfv 5372  (class class class)co 6075  ℂcc 8167  ℝcr 8168  0cc0 8169  1c1 8170   + caddc 8172   · cmul 8174   < clt 8350  ℕcn 9283  2c2 9334  ℕ₀cn0 9542  ℤcz 9623  ℤ_≥cuz 9900  ℝ⁺crp 10033  ...cfz 10390  seqcseq 10862  !cfa 11141  Σcsu 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)