subfacval2 - Mathbox for Mario Carneiro

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Theorem subfacval2 32438

Description: A closed-form expression for the subfactorial. (Contributed by Mario Carneiro, 23-Jan-2015.)

**Hypotheses**
Ref	Expression
derang.d	⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)}))
subfac.n	⊢ 𝑆 = (𝑛 ∈ ℕ₀ ↦ (𝐷‘(1...𝑛)))

**Assertion**
Ref	Expression
subfacval2	⊢ (𝑁 ∈ ℕ₀ → (𝑆‘𝑁) = ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)((-1↑𝑘) / (!‘𝑘))))

Distinct variable groups: 𝑓,𝑛,𝑥,𝑦,𝑘,𝑁 𝐷,𝑛 𝑆,𝑛,𝑥,𝑦 Allowed substitution hints: 𝐷(𝑥,𝑦,𝑓,𝑘) 𝑆(𝑓,𝑘)

Proof of Theorem subfacval2 Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6673	. . . . . 6 ⊢ (𝑥 = 0 → (𝑆‘𝑥) = (𝑆‘0))
2		derang.d	. . . . . . 7 ⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)}))
3		subfac.n	. . . . . . 7 ⊢ 𝑆 = (𝑛 ∈ ℕ₀ ↦ (𝐷‘(1...𝑛)))
4	2, 3	subfac0 32428	. . . . . 6 ⊢ (𝑆‘0) = 1
5	1, 4	syl6eq 2875	. . . . 5 ⊢ (𝑥 = 0 → (𝑆‘𝑥) = 1)
6		fveq2 6673	. . . . . . 7 ⊢ (𝑥 = 0 → (!‘𝑥) = (!‘0))
7		fac0 13639	. . . . . . 7 ⊢ (!‘0) = 1
8	6, 7	syl6eq 2875	. . . . . 6 ⊢ (𝑥 = 0 → (!‘𝑥) = 1)
9		oveq2 7167	. . . . . . 7 ⊢ (𝑥 = 0 → (0...𝑥) = (0...0))
10	9	sumeq1d 15061	. . . . . 6 ⊢ (𝑥 = 0 → Σ𝑘 ∈ (0...𝑥)((-1↑𝑘) / (!‘𝑘)) = Σ𝑘 ∈ (0...0)((-1↑𝑘) / (!‘𝑘)))
11	8, 10	oveq12d 7177	. . . . 5 ⊢ (𝑥 = 0 → ((!‘𝑥) · Σ𝑘 ∈ (0...𝑥)((-1↑𝑘) / (!‘𝑘))) = (1 · Σ𝑘 ∈ (0...0)((-1↑𝑘) / (!‘𝑘))))
12	5, 11	eqeq12d 2840	. . . 4 ⊢ (𝑥 = 0 → ((𝑆‘𝑥) = ((!‘𝑥) · Σ𝑘 ∈ (0...𝑥)((-1↑𝑘) / (!‘𝑘))) ↔ 1 = (1 · Σ𝑘 ∈ (0...0)((-1↑𝑘) / (!‘𝑘)))))
13		fv0p1e1 11763	. . . . . 6 ⊢ (𝑥 = 0 → (𝑆‘(𝑥 + 1)) = (𝑆‘1))
14	2, 3	subfac1 32429	. . . . . 6 ⊢ (𝑆‘1) = 0
15	13, 14	syl6eq 2875	. . . . 5 ⊢ (𝑥 = 0 → (𝑆‘(𝑥 + 1)) = 0)
16		fv0p1e1 11763	. . . . . . 7 ⊢ (𝑥 = 0 → (!‘(𝑥 + 1)) = (!‘1))
17		fac1 13640	. . . . . . 7 ⊢ (!‘1) = 1
18	16, 17	syl6eq 2875	. . . . . 6 ⊢ (𝑥 = 0 → (!‘(𝑥 + 1)) = 1)
19		oveq1 7166	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝑥 + 1) = (0 + 1))
20		0p1e1 11762	. . . . . . . . 9 ⊢ (0 + 1) = 1
21	19, 20	syl6eq 2875	. . . . . . . 8 ⊢ (𝑥 = 0 → (𝑥 + 1) = 1)
22	21	oveq2d 7175	. . . . . . 7 ⊢ (𝑥 = 0 → (0...(𝑥 + 1)) = (0...1))
23	22	sumeq1d 15061	. . . . . 6 ⊢ (𝑥 = 0 → Σ𝑘 ∈ (0...(𝑥 + 1))((-1↑𝑘) / (!‘𝑘)) = Σ𝑘 ∈ (0...1)((-1↑𝑘) / (!‘𝑘)))
24	18, 23	oveq12d 7177	. . . . 5 ⊢ (𝑥 = 0 → ((!‘(𝑥 + 1)) · Σ𝑘 ∈ (0...(𝑥 + 1))((-1↑𝑘) / (!‘𝑘))) = (1 · Σ𝑘 ∈ (0...1)((-1↑𝑘) / (!‘𝑘))))
25	15, 24	eqeq12d 2840	. . . 4 ⊢ (𝑥 = 0 → ((𝑆‘(𝑥 + 1)) = ((!‘(𝑥 + 1)) · Σ𝑘 ∈ (0...(𝑥 + 1))((-1↑𝑘) / (!‘𝑘))) ↔ 0 = (1 · Σ𝑘 ∈ (0...1)((-1↑𝑘) / (!‘𝑘)))))
26	12, 25	anbi12d 632	. . 3 ⊢ (𝑥 = 0 → (((𝑆‘𝑥) = ((!‘𝑥) · Σ𝑘 ∈ (0...𝑥)((-1↑𝑘) / (!‘𝑘))) ∧ (𝑆‘(𝑥 + 1)) = ((!‘(𝑥 + 1)) · Σ𝑘 ∈ (0...(𝑥 + 1))((-1↑𝑘) / (!‘𝑘)))) ↔ (1 = (1 · Σ𝑘 ∈ (0...0)((-1↑𝑘) / (!‘𝑘))) ∧ 0 = (1 · Σ𝑘 ∈ (0...1)((-1↑𝑘) / (!‘𝑘))))))
27		fveq2 6673	. . . . 5 ⊢ (𝑥 = 𝑚 → (𝑆‘𝑥) = (𝑆‘𝑚))
28		fveq2 6673	. . . . . 6 ⊢ (𝑥 = 𝑚 → (!‘𝑥) = (!‘𝑚))
29		oveq2 7167	. . . . . . 7 ⊢ (𝑥 = 𝑚 → (0...𝑥) = (0...𝑚))
30	29	sumeq1d 15061	. . . . . 6 ⊢ (𝑥 = 𝑚 → Σ𝑘 ∈ (0...𝑥)((-1↑𝑘) / (!‘𝑘)) = Σ𝑘 ∈ (0...𝑚)((-1↑𝑘) / (!‘𝑘)))
31	28, 30	oveq12d 7177	. . . . 5 ⊢ (𝑥 = 𝑚 → ((!‘𝑥) · Σ𝑘 ∈ (0...𝑥)((-1↑𝑘) / (!‘𝑘))) = ((!‘𝑚) · Σ𝑘 ∈ (0...𝑚)((-1↑𝑘) / (!‘𝑘))))
32	27, 31	eqeq12d 2840	. . . 4 ⊢ (𝑥 = 𝑚 → ((𝑆‘𝑥) = ((!‘𝑥) · Σ𝑘 ∈ (0...𝑥)((-1↑𝑘) / (!‘𝑘))) ↔ (𝑆‘𝑚) = ((!‘𝑚) · Σ𝑘 ∈ (0...𝑚)((-1↑𝑘) / (!‘𝑘)))))
33		fvoveq1 7182	. . . . 5 ⊢ (𝑥 = 𝑚 → (𝑆‘(𝑥 + 1)) = (𝑆‘(𝑚 + 1)))
34		fvoveq1 7182	. . . . . 6 ⊢ (𝑥 = 𝑚 → (!‘(𝑥 + 1)) = (!‘(𝑚 + 1)))
35		oveq1 7166	. . . . . . . 8 ⊢ (𝑥 = 𝑚 → (𝑥 + 1) = (𝑚 + 1))
36	35	oveq2d 7175	. . . . . . 7 ⊢ (𝑥 = 𝑚 → (0...(𝑥 + 1)) = (0...(𝑚 + 1)))
37	36	sumeq1d 15061	. . . . . 6 ⊢ (𝑥 = 𝑚 → Σ𝑘 ∈ (0...(𝑥 + 1))((-1↑𝑘) / (!‘𝑘)) = Σ𝑘 ∈ (0...(𝑚 + 1))((-1↑𝑘) / (!‘𝑘)))
38	34, 37	oveq12d 7177	. . . . 5 ⊢ (𝑥 = 𝑚 → ((!‘(𝑥 + 1)) · Σ𝑘 ∈ (0...(𝑥 + 1))((-1↑𝑘) / (!‘𝑘))) = ((!‘(𝑚 + 1)) · Σ𝑘 ∈ (0...(𝑚 + 1))((-1↑𝑘) / (!‘𝑘))))
39	33, 38	eqeq12d 2840	. . . 4 ⊢ (𝑥 = 𝑚 → ((𝑆‘(𝑥 + 1)) = ((!‘(𝑥 + 1)) · Σ𝑘 ∈ (0...(𝑥 + 1))((-1↑𝑘) / (!‘𝑘))) ↔ (𝑆‘(𝑚 + 1)) = ((!‘(𝑚 + 1)) · Σ𝑘 ∈ (0...(𝑚 + 1))((-1↑𝑘) / (!‘𝑘)))))
40	32, 39	anbi12d 632	. . 3 ⊢ (𝑥 = 𝑚 → (((𝑆‘𝑥) = ((!‘𝑥) · Σ𝑘 ∈ (0...𝑥)((-1↑𝑘) / (!‘𝑘))) ∧ (𝑆‘(𝑥 + 1)) = ((!‘(𝑥 + 1)) · Σ𝑘 ∈ (0...(𝑥 + 1))((-1↑𝑘) / (!‘𝑘)))) ↔ ((𝑆‘𝑚) = ((!‘𝑚) · Σ𝑘 ∈ (0...𝑚)((-1↑𝑘) / (!‘𝑘))) ∧ (𝑆‘(𝑚 + 1)) = ((!‘(𝑚 + 1)) · Σ𝑘 ∈ (0...(𝑚 + 1))((-1↑𝑘) / (!‘𝑘))))))
41		fveq2 6673	. . . . 5 ⊢ (𝑥 = (𝑚 + 1) → (𝑆‘𝑥) = (𝑆‘(𝑚 + 1)))
42		fveq2 6673	. . . . . 6 ⊢ (𝑥 = (𝑚 + 1) → (!‘𝑥) = (!‘(𝑚 + 1)))
43		oveq2 7167	. . . . . . 7 ⊢ (𝑥 = (𝑚 + 1) → (0...𝑥) = (0...(𝑚 + 1)))
44	43	sumeq1d 15061	. . . . . 6 ⊢ (𝑥 = (𝑚 + 1) → Σ𝑘 ∈ (0...𝑥)((-1↑𝑘) / (!‘𝑘)) = Σ𝑘 ∈ (0...(𝑚 + 1))((-1↑𝑘) / (!‘𝑘)))
45	42, 44	oveq12d 7177	. . . . 5 ⊢ (𝑥 = (𝑚 + 1) → ((!‘𝑥) · Σ𝑘 ∈ (0...𝑥)((-1↑𝑘) / (!‘𝑘))) = ((!‘(𝑚 + 1)) · Σ𝑘 ∈ (0...(𝑚 + 1))((-1↑𝑘) / (!‘𝑘))))
46	41, 45	eqeq12d 2840	. . . 4 ⊢ (𝑥 = (𝑚 + 1) → ((𝑆‘𝑥) = ((!‘𝑥) · Σ𝑘 ∈ (0...𝑥)((-1↑𝑘) / (!‘𝑘))) ↔ (𝑆‘(𝑚 + 1)) = ((!‘(𝑚 + 1)) · Σ𝑘 ∈ (0...(𝑚 + 1))((-1↑𝑘) / (!‘𝑘)))))
47		fvoveq1 7182	. . . . 5 ⊢ (𝑥 = (𝑚 + 1) → (𝑆‘(𝑥 + 1)) = (𝑆‘((𝑚 + 1) + 1)))
48		fvoveq1 7182	. . . . . 6 ⊢ (𝑥 = (𝑚 + 1) → (!‘(𝑥 + 1)) = (!‘((𝑚 + 1) + 1)))
49		oveq1 7166	. . . . . . . 8 ⊢ (𝑥 = (𝑚 + 1) → (𝑥 + 1) = ((𝑚 + 1) + 1))
50	49	oveq2d 7175	. . . . . . 7 ⊢ (𝑥 = (𝑚 + 1) → (0...(𝑥 + 1)) = (0...((𝑚 + 1) + 1)))
51	50	sumeq1d 15061	. . . . . 6 ⊢ (𝑥 = (𝑚 + 1) → Σ𝑘 ∈ (0...(𝑥 + 1))((-1↑𝑘) / (!‘𝑘)) = Σ𝑘 ∈ (0...((𝑚 + 1) + 1))((-1↑𝑘) / (!‘𝑘)))
52	48, 51	oveq12d 7177	. . . . 5 ⊢ (𝑥 = (𝑚 + 1) → ((!‘(𝑥 + 1)) · Σ𝑘 ∈ (0...(𝑥 + 1))((-1↑𝑘) / (!‘𝑘))) = ((!‘((𝑚 + 1) + 1)) · Σ𝑘 ∈ (0...((𝑚 + 1) + 1))((-1↑𝑘) / (!‘𝑘))))
53	47, 52	eqeq12d 2840	. . . 4 ⊢ (𝑥 = (𝑚 + 1) → ((𝑆‘(𝑥 + 1)) = ((!‘(𝑥 + 1)) · Σ𝑘 ∈ (0...(𝑥 + 1))((-1↑𝑘) / (!‘𝑘))) ↔ (𝑆‘((𝑚 + 1) + 1)) = ((!‘((𝑚 + 1) + 1)) · Σ𝑘 ∈ (0...((𝑚 + 1) + 1))((-1↑𝑘) / (!‘𝑘)))))
54	46, 53	anbi12d 632	. . 3 ⊢ (𝑥 = (𝑚 + 1) → (((𝑆‘𝑥) = ((!‘𝑥) · Σ𝑘 ∈ (0...𝑥)((-1↑𝑘) / (!‘𝑘))) ∧ (𝑆‘(𝑥 + 1)) = ((!‘(𝑥 + 1)) · Σ𝑘 ∈ (0...(𝑥 + 1))((-1↑𝑘) / (!‘𝑘)))) ↔ ((𝑆‘(𝑚 + 1)) = ((!‘(𝑚 + 1)) · Σ𝑘 ∈ (0...(𝑚 + 1))((-1↑𝑘) / (!‘𝑘))) ∧ (𝑆‘((𝑚 + 1) + 1)) = ((!‘((𝑚 + 1) + 1)) · Σ𝑘 ∈ (0...((𝑚 + 1) + 1))((-1↑𝑘) / (!‘𝑘))))))
55		fveq2 6673	. . . . 5 ⊢ (𝑥 = 𝑁 → (𝑆‘𝑥) = (𝑆‘𝑁))
56		fveq2 6673	. . . . . 6 ⊢ (𝑥 = 𝑁 → (!‘𝑥) = (!‘𝑁))
57		oveq2 7167	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (0...𝑥) = (0...𝑁))
58	57	sumeq1d 15061	. . . . . 6 ⊢ (𝑥 = 𝑁 → Σ𝑘 ∈ (0...𝑥)((-1↑𝑘) / (!‘𝑘)) = Σ𝑘 ∈ (0...𝑁)((-1↑𝑘) / (!‘𝑘)))
59	56, 58	oveq12d 7177	. . . . 5 ⊢ (𝑥 = 𝑁 → ((!‘𝑥) · Σ𝑘 ∈ (0...𝑥)((-1↑𝑘) / (!‘𝑘))) = ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)((-1↑𝑘) / (!‘𝑘))))
60	55, 59	eqeq12d 2840	. . . 4 ⊢ (𝑥 = 𝑁 → ((𝑆‘𝑥) = ((!‘𝑥) · Σ𝑘 ∈ (0...𝑥)((-1↑𝑘) / (!‘𝑘))) ↔ (𝑆‘𝑁) = ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)((-1↑𝑘) / (!‘𝑘)))))
61		fvoveq1 7182	. . . . 5 ⊢ (𝑥 = 𝑁 → (𝑆‘(𝑥 + 1)) = (𝑆‘(𝑁 + 1)))
62		fvoveq1 7182	. . . . . 6 ⊢ (𝑥 = 𝑁 → (!‘(𝑥 + 1)) = (!‘(𝑁 + 1)))
63		oveq1 7166	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (𝑥 + 1) = (𝑁 + 1))
64	63	oveq2d 7175	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (0...(𝑥 + 1)) = (0...(𝑁 + 1)))
65	64	sumeq1d 15061	. . . . . 6 ⊢ (𝑥 = 𝑁 → Σ𝑘 ∈ (0...(𝑥 + 1))((-1↑𝑘) / (!‘𝑘)) = Σ𝑘 ∈ (0...(𝑁 + 1))((-1↑𝑘) / (!‘𝑘)))
66	62, 65	oveq12d 7177	. . . . 5 ⊢ (𝑥 = 𝑁 → ((!‘(𝑥 + 1)) · Σ𝑘 ∈ (0...(𝑥 + 1))((-1↑𝑘) / (!‘𝑘))) = ((!‘(𝑁 + 1)) · Σ𝑘 ∈ (0...(𝑁 + 1))((-1↑𝑘) / (!‘𝑘))))
67	61, 66	eqeq12d 2840	. . . 4 ⊢ (𝑥 = 𝑁 → ((𝑆‘(𝑥 + 1)) = ((!‘(𝑥 + 1)) · Σ𝑘 ∈ (0...(𝑥 + 1))((-1↑𝑘) / (!‘𝑘))) ↔ (𝑆‘(𝑁 + 1)) = ((!‘(𝑁 + 1)) · Σ𝑘 ∈ (0...(𝑁 + 1))((-1↑𝑘) / (!‘𝑘)))))
68	60, 67	anbi12d 632	. . 3 ⊢ (𝑥 = 𝑁 → (((𝑆‘𝑥) = ((!‘𝑥) · Σ𝑘 ∈ (0...𝑥)((-1↑𝑘) / (!‘𝑘))) ∧ (𝑆‘(𝑥 + 1)) = ((!‘(𝑥 + 1)) · Σ𝑘 ∈ (0...(𝑥 + 1))((-1↑𝑘) / (!‘𝑘)))) ↔ ((𝑆‘𝑁) = ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)((-1↑𝑘) / (!‘𝑘))) ∧ (𝑆‘(𝑁 + 1)) = ((!‘(𝑁 + 1)) · Σ𝑘 ∈ (0...(𝑁 + 1))((-1↑𝑘) / (!‘𝑘))))))
69		0z 11995	. . . . . . 7 ⊢ 0 ∈ ℤ
70		ax-1cn 10598	. . . . . . 7 ⊢ 1 ∈ ℂ
71		oveq2 7167	. . . . . . . . . . 11 ⊢ (𝑘 = 0 → (-1↑𝑘) = (-1↑0))
72		neg1cn 11754	. . . . . . . . . . . 12 ⊢ -1 ∈ ℂ
73		exp0 13436	. . . . . . . . . . . 12 ⊢ (-1 ∈ ℂ → (-1↑0) = 1)
74	72, 73	ax-mp 5	. . . . . . . . . . 11 ⊢ (-1↑0) = 1
75	71, 74	syl6eq 2875	. . . . . . . . . 10 ⊢ (𝑘 = 0 → (-1↑𝑘) = 1)
76		fveq2 6673	. . . . . . . . . . 11 ⊢ (𝑘 = 0 → (!‘𝑘) = (!‘0))
77	76, 7	syl6eq 2875	. . . . . . . . . 10 ⊢ (𝑘 = 0 → (!‘𝑘) = 1)
78	75, 77	oveq12d 7177	. . . . . . . . 9 ⊢ (𝑘 = 0 → ((-1↑𝑘) / (!‘𝑘)) = (1 / 1))
79	70	div1i 11371	. . . . . . . . 9 ⊢ (1 / 1) = 1
80	78, 79	syl6eq 2875	. . . . . . . 8 ⊢ (𝑘 = 0 → ((-1↑𝑘) / (!‘𝑘)) = 1)
81	80	fsum1 15105	. . . . . . 7 ⊢ ((0 ∈ ℤ ∧ 1 ∈ ℂ) → Σ𝑘 ∈ (0...0)((-1↑𝑘) / (!‘𝑘)) = 1)
82	69, 70, 81	mp2an 690	. . . . . 6 ⊢ Σ𝑘 ∈ (0...0)((-1↑𝑘) / (!‘𝑘)) = 1
83	82	oveq2i 7170	. . . . 5 ⊢ (1 · Σ𝑘 ∈ (0...0)((-1↑𝑘) / (!‘𝑘))) = (1 · 1)
84		1t1e1 11802	. . . . 5 ⊢ (1 · 1) = 1
85	83, 84	eqtr2i 2848	. . . 4 ⊢ 1 = (1 · Σ𝑘 ∈ (0...0)((-1↑𝑘) / (!‘𝑘)))
86		nn0uz 12283	. . . . . . . . 9 ⊢ ℕ₀ = (ℤ_≥‘0)
87		1e0p1 12143	. . . . . . . . 9 ⊢ 1 = (0 + 1)
88		oveq2 7167	. . . . . . . . . . . 12 ⊢ (𝑘 = 1 → (-1↑𝑘) = (-1↑1))
89		exp1 13438	. . . . . . . . . . . . 13 ⊢ (-1 ∈ ℂ → (-1↑1) = -1)
90	72, 89	ax-mp 5	. . . . . . . . . . . 12 ⊢ (-1↑1) = -1
91	88, 90	syl6eq 2875	. . . . . . . . . . 11 ⊢ (𝑘 = 1 → (-1↑𝑘) = -1)
92		fveq2 6673	. . . . . . . . . . . 12 ⊢ (𝑘 = 1 → (!‘𝑘) = (!‘1))
93	92, 17	syl6eq 2875	. . . . . . . . . . 11 ⊢ (𝑘 = 1 → (!‘𝑘) = 1)
94	91, 93	oveq12d 7177	. . . . . . . . . 10 ⊢ (𝑘 = 1 → ((-1↑𝑘) / (!‘𝑘)) = (-1 / 1))
95	72	div1i 11371	. . . . . . . . . 10 ⊢ (-1 / 1) = -1
96	94, 95	syl6eq 2875	. . . . . . . . 9 ⊢ (𝑘 = 1 → ((-1↑𝑘) / (!‘𝑘)) = -1)
97		neg1rr 11755	. . . . . . . . . . . . 13 ⊢ -1 ∈ ℝ
98		reexpcl 13449	. . . . . . . . . . . . 13 ⊢ ((-1 ∈ ℝ ∧ 𝑘 ∈ ℕ₀) → (-1↑𝑘) ∈ ℝ)
99	97, 98	mpan 688	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ₀ → (-1↑𝑘) ∈ ℝ)
100		faccl 13646	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ₀ → (!‘𝑘) ∈ ℕ)
101	99, 100	nndivred 11694	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ₀ → ((-1↑𝑘) / (!‘𝑘)) ∈ ℝ)
102	101	recnd 10672	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ₀ → ((-1↑𝑘) / (!‘𝑘)) ∈ ℂ)
103	102	adantl 484	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → ((-1↑𝑘) / (!‘𝑘)) ∈ ℂ)
104		0nn0 11915	. . . . . . . . . . 11 ⊢ 0 ∈ ℕ₀
105	104, 82	pm3.2i 473	. . . . . . . . . 10 ⊢ (0 ∈ ℕ₀ ∧ Σ𝑘 ∈ (0...0)((-1↑𝑘) / (!‘𝑘)) = 1)
106	105	a1i 11	. . . . . . . . 9 ⊢ (⊤ → (0 ∈ ℕ₀ ∧ Σ𝑘 ∈ (0...0)((-1↑𝑘) / (!‘𝑘)) = 1))
107		1pneg1e0 11759	. . . . . . . . . 10 ⊢ (1 + -1) = 0
108	107	a1i 11	. . . . . . . . 9 ⊢ (⊤ → (1 + -1) = 0)
109	86, 87, 96, 103, 106, 108	fsump1i 15127	. . . . . . . 8 ⊢ (⊤ → (1 ∈ ℕ₀ ∧ Σ𝑘 ∈ (0...1)((-1↑𝑘) / (!‘𝑘)) = 0))
110	109	mptru 1543	. . . . . . 7 ⊢ (1 ∈ ℕ₀ ∧ Σ𝑘 ∈ (0...1)((-1↑𝑘) / (!‘𝑘)) = 0)
111	110	simpri 488	. . . . . 6 ⊢ Σ𝑘 ∈ (0...1)((-1↑𝑘) / (!‘𝑘)) = 0
112	111	oveq2i 7170	. . . . 5 ⊢ (1 · Σ𝑘 ∈ (0...1)((-1↑𝑘) / (!‘𝑘))) = (1 · 0)
113	70	mul01i 10833	. . . . 5 ⊢ (1 · 0) = 0
114	112, 113	eqtr2i 2848	. . . 4 ⊢ 0 = (1 · Σ𝑘 ∈ (0...1)((-1↑𝑘) / (!‘𝑘)))
115	85, 114	pm3.2i 473	. . 3 ⊢ (1 = (1 · Σ𝑘 ∈ (0...0)((-1↑𝑘) / (!‘𝑘))) ∧ 0 = (1 · Σ𝑘 ∈ (0...1)((-1↑𝑘) / (!‘𝑘))))
116		simpr 487	. . . . 5 ⊢ (((𝑆‘𝑚) = ((!‘𝑚) · Σ𝑘 ∈ (0...𝑚)((-1↑𝑘) / (!‘𝑘))) ∧ (𝑆‘(𝑚 + 1)) = ((!‘(𝑚 + 1)) · Σ𝑘 ∈ (0...(𝑚 + 1))((-1↑𝑘) / (!‘𝑘)))) → (𝑆‘(𝑚 + 1)) = ((!‘(𝑚 + 1)) · Σ𝑘 ∈ (0...(𝑚 + 1))((-1↑𝑘) / (!‘𝑘))))
117	116	a1i 11	. . . 4 ⊢ (𝑚 ∈ ℕ₀ → (((𝑆‘𝑚) = ((!‘𝑚) · Σ𝑘 ∈ (0...𝑚)((-1↑𝑘) / (!‘𝑘))) ∧ (𝑆‘(𝑚 + 1)) = ((!‘(𝑚 + 1)) · Σ𝑘 ∈ (0...(𝑚 + 1))((-1↑𝑘) / (!‘𝑘)))) → (𝑆‘(𝑚 + 1)) = ((!‘(𝑚 + 1)) · Σ𝑘 ∈ (0...(𝑚 + 1))((-1↑𝑘) / (!‘𝑘)))))
118		oveq12 7168	. . . . . . 7 ⊢ (((𝑆‘(𝑚 + 1)) = ((!‘(𝑚 + 1)) · Σ𝑘 ∈ (0...(𝑚 + 1))((-1↑𝑘) / (!‘𝑘))) ∧ (𝑆‘𝑚) = ((!‘𝑚) · Σ𝑘 ∈ (0...𝑚)((-1↑𝑘) / (!‘𝑘)))) → ((𝑆‘(𝑚 + 1)) + (𝑆‘𝑚)) = (((!‘(𝑚 + 1)) · Σ𝑘 ∈ (0...(𝑚 + 1))((-1↑𝑘) / (!‘𝑘))) + ((!‘𝑚) · Σ𝑘 ∈ (0...𝑚)((-1↑𝑘) / (!‘𝑘)))))
119	118	ancoms 461	. . . . . 6 ⊢ (((𝑆‘𝑚) = ((!‘𝑚) · Σ𝑘 ∈ (0...𝑚)((-1↑𝑘) / (!‘𝑘))) ∧ (𝑆‘(𝑚 + 1)) = ((!‘(𝑚 + 1)) · Σ𝑘 ∈ (0...(𝑚 + 1))((-1↑𝑘) / (!‘𝑘)))) → ((𝑆‘(𝑚 + 1)) + (𝑆‘𝑚)) = (((!‘(𝑚 + 1)) · Σ𝑘 ∈ (0...(𝑚 + 1))((-1↑𝑘) / (!‘𝑘))) + ((!‘𝑚) · Σ𝑘 ∈ (0...𝑚)((-1↑𝑘) / (!‘𝑘)))))
120	119	oveq2d 7175	. . . . 5 ⊢ (((𝑆‘𝑚) = ((!‘𝑚) · Σ𝑘 ∈ (0...𝑚)((-1↑𝑘) / (!‘𝑘))) ∧ (𝑆‘(𝑚 + 1)) = ((!‘(𝑚 + 1)) · Σ𝑘 ∈ (0...(𝑚 + 1))((-1↑𝑘) / (!‘𝑘)))) → ((𝑚 + 1) · ((𝑆‘(𝑚 + 1)) + (𝑆‘𝑚))) = ((𝑚 + 1) · (((!‘(𝑚 + 1)) · Σ𝑘 ∈ (0...(𝑚 + 1))((-1↑𝑘) / (!‘𝑘))) + ((!‘𝑚) · Σ𝑘 ∈ (0...𝑚)((-1↑𝑘) / (!‘𝑘))))))
121		nn0p1nn 11939	. . . . . . . 8 ⊢ (𝑚 ∈ ℕ₀ → (𝑚 + 1) ∈ ℕ)
122	2, 3	subfacp1 32437	. . . . . . . 8 ⊢ ((𝑚 + 1) ∈ ℕ → (𝑆‘((𝑚 + 1) + 1)) = ((𝑚 + 1) · ((𝑆‘(𝑚 + 1)) + (𝑆‘((𝑚 + 1) − 1)))))
123	121, 122	syl 17	. . . . . . 7 ⊢ (𝑚 ∈ ℕ₀ → (𝑆‘((𝑚 + 1) + 1)) = ((𝑚 + 1) · ((𝑆‘(𝑚 + 1)) + (𝑆‘((𝑚 + 1) − 1)))))
124		nn0cn 11910	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕ₀ → 𝑚 ∈ ℂ)
125		pncan 10895	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑚 + 1) − 1) = 𝑚)
126	124, 70, 125	sylancl 588	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ₀ → ((𝑚 + 1) − 1) = 𝑚)
127	126	fveq2d 6677	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ₀ → (𝑆‘((𝑚 + 1) − 1)) = (𝑆‘𝑚))
128	127	oveq2d 7175	. . . . . . . 8 ⊢ (𝑚 ∈ ℕ₀ → ((𝑆‘(𝑚 + 1)) + (𝑆‘((𝑚 + 1) − 1))) = ((𝑆‘(𝑚 + 1)) + (𝑆‘𝑚)))
129	128	oveq2d 7175	. . . . . . 7 ⊢ (𝑚 ∈ ℕ₀ → ((𝑚 + 1) · ((𝑆‘(𝑚 + 1)) + (𝑆‘((𝑚 + 1) − 1)))) = ((𝑚 + 1) · ((𝑆‘(𝑚 + 1)) + (𝑆‘𝑚))))
130	123, 129	eqtrd 2859	. . . . . 6 ⊢ (𝑚 ∈ ℕ₀ → (𝑆‘((𝑚 + 1) + 1)) = ((𝑚 + 1) · ((𝑆‘(𝑚 + 1)) + (𝑆‘𝑚))))
131		peano2nn0 11940	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ₀ → (𝑚 + 1) ∈ ℕ₀)
132		peano2nn0 11940	. . . . . . . . . . . 12 ⊢ ((𝑚 + 1) ∈ ℕ₀ → ((𝑚 + 1) + 1) ∈ ℕ₀)
133	131, 132	syl 17	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕ₀ → ((𝑚 + 1) + 1) ∈ ℕ₀)
134		faccl 13646	. . . . . . . . . . 11 ⊢ (((𝑚 + 1) + 1) ∈ ℕ₀ → (!‘((𝑚 + 1) + 1)) ∈ ℕ)
135	133, 134	syl 17	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ₀ → (!‘((𝑚 + 1) + 1)) ∈ ℕ)
136	135	nncnd 11657	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ₀ → (!‘((𝑚 + 1) + 1)) ∈ ℂ)
137		fzfid 13344	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ₀ → (0...(𝑚 + 1)) ∈ Fin)
138		elfznn0 13003	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (0...(𝑚 + 1)) → 𝑘 ∈ ℕ₀)
139	138	adantl 484	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ ℕ₀ ∧ 𝑘 ∈ (0...(𝑚 + 1))) → 𝑘 ∈ ℕ₀)
140	139, 102	syl 17	. . . . . . . . . 10 ⊢ ((𝑚 ∈ ℕ₀ ∧ 𝑘 ∈ (0...(𝑚 + 1))) → ((-1↑𝑘) / (!‘𝑘)) ∈ ℂ)
141	137, 140	fsumcl 15093	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ₀ → Σ𝑘 ∈ (0...(𝑚 + 1))((-1↑𝑘) / (!‘𝑘)) ∈ ℂ)
142		expcl 13450	. . . . . . . . . . 11 ⊢ ((-1 ∈ ℂ ∧ ((𝑚 + 1) + 1) ∈ ℕ₀) → (-1↑((𝑚 + 1) + 1)) ∈ ℂ)
143	72, 133, 142	sylancr 589	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ₀ → (-1↑((𝑚 + 1) + 1)) ∈ ℂ)
144	135	nnne0d 11690	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ₀ → (!‘((𝑚 + 1) + 1)) ≠ 0)
145	143, 136, 144	divcld 11419	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ₀ → ((-1↑((𝑚 + 1) + 1)) / (!‘((𝑚 + 1) + 1))) ∈ ℂ)
146	136, 141, 145	adddid 10668	. . . . . . . 8 ⊢ (𝑚 ∈ ℕ₀ → ((!‘((𝑚 + 1) + 1)) · (Σ𝑘 ∈ (0...(𝑚 + 1))((-1↑𝑘) / (!‘𝑘)) + ((-1↑((𝑚 + 1) + 1)) / (!‘((𝑚 + 1) + 1))))) = (((!‘((𝑚 + 1) + 1)) · Σ𝑘 ∈ (0...(𝑚 + 1))((-1↑𝑘) / (!‘𝑘))) + ((!‘((𝑚 + 1) + 1)) · ((-1↑((𝑚 + 1) + 1)) / (!‘((𝑚 + 1) + 1))))))
147		id 22	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ₀ → 𝑚 ∈ ℕ₀)
148	147, 86	eleqtrdi 2926	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ₀ → 𝑚 ∈ (ℤ_≥‘0))
149		oveq2 7167	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝑚 + 1) → (-1↑𝑘) = (-1↑(𝑚 + 1)))
150		fveq2 6673	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝑚 + 1) → (!‘𝑘) = (!‘(𝑚 + 1)))
151	149, 150	oveq12d 7177	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑚 + 1) → ((-1↑𝑘) / (!‘𝑘)) = ((-1↑(𝑚 + 1)) / (!‘(𝑚 + 1))))
152	148, 140, 151	fsump1 15114	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕ₀ → Σ𝑘 ∈ (0...(𝑚 + 1))((-1↑𝑘) / (!‘𝑘)) = (Σ𝑘 ∈ (0...𝑚)((-1↑𝑘) / (!‘𝑘)) + ((-1↑(𝑚 + 1)) / (!‘(𝑚 + 1)))))
153	152	oveq2d 7175	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ₀ → ((!‘((𝑚 + 1) + 1)) · Σ𝑘 ∈ (0...(𝑚 + 1))((-1↑𝑘) / (!‘𝑘))) = ((!‘((𝑚 + 1) + 1)) · (Σ𝑘 ∈ (0...𝑚)((-1↑𝑘) / (!‘𝑘)) + ((-1↑(𝑚 + 1)) / (!‘(𝑚 + 1))))))
154		fzfid 13344	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ₀ → (0...𝑚) ∈ Fin)
155		elfznn0 13003	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (0...𝑚) → 𝑘 ∈ ℕ₀)
156	155	adantl 484	. . . . . . . . . . . . 13 ⊢ ((𝑚 ∈ ℕ₀ ∧ 𝑘 ∈ (0...𝑚)) → 𝑘 ∈ ℕ₀)
157	156, 102	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑚 ∈ ℕ₀ ∧ 𝑘 ∈ (0...𝑚)) → ((-1↑𝑘) / (!‘𝑘)) ∈ ℂ)
158	154, 157	fsumcl 15093	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕ₀ → Σ𝑘 ∈ (0...𝑚)((-1↑𝑘) / (!‘𝑘)) ∈ ℂ)
159		expcl 13450	. . . . . . . . . . . . 13 ⊢ ((-1 ∈ ℂ ∧ (𝑚 + 1) ∈ ℕ₀) → (-1↑(𝑚 + 1)) ∈ ℂ)
160	72, 131, 159	sylancr 589	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ₀ → (-1↑(𝑚 + 1)) ∈ ℂ)
161		faccl 13646	. . . . . . . . . . . . . 14 ⊢ ((𝑚 + 1) ∈ ℕ₀ → (!‘(𝑚 + 1)) ∈ ℕ)
162	131, 161	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ₀ → (!‘(𝑚 + 1)) ∈ ℕ)
163	162	nncnd 11657	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ₀ → (!‘(𝑚 + 1)) ∈ ℂ)
164	162	nnne0d 11690	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ₀ → (!‘(𝑚 + 1)) ≠ 0)
165	160, 163, 164	divcld 11419	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕ₀ → ((-1↑(𝑚 + 1)) / (!‘(𝑚 + 1))) ∈ ℂ)
166	136, 158, 165	adddid 10668	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ₀ → ((!‘((𝑚 + 1) + 1)) · (Σ𝑘 ∈ (0...𝑚)((-1↑𝑘) / (!‘𝑘)) + ((-1↑(𝑚 + 1)) / (!‘(𝑚 + 1))))) = (((!‘((𝑚 + 1) + 1)) · Σ𝑘 ∈ (0...𝑚)((-1↑𝑘) / (!‘𝑘))) + ((!‘((𝑚 + 1) + 1)) · ((-1↑(𝑚 + 1)) / (!‘(𝑚 + 1))))))
167		facp1 13641	. . . . . . . . . . . . . 14 ⊢ ((𝑚 + 1) ∈ ℕ₀ → (!‘((𝑚 + 1) + 1)) = ((!‘(𝑚 + 1)) · ((𝑚 + 1) + 1)))
168	131, 167	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ₀ → (!‘((𝑚 + 1) + 1)) = ((!‘(𝑚 + 1)) · ((𝑚 + 1) + 1)))
169		facp1 13641	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ ℕ₀ → (!‘(𝑚 + 1)) = ((!‘𝑚) · (𝑚 + 1)))
170		faccl 13646	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ ℕ₀ → (!‘𝑚) ∈ ℕ)
171	170	nncnd 11657	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ ℕ₀ → (!‘𝑚) ∈ ℂ)
172	121	nncnd 11657	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ ℕ₀ → (𝑚 + 1) ∈ ℂ)
173	171, 172	mulcomd 10665	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ ℕ₀ → ((!‘𝑚) · (𝑚 + 1)) = ((𝑚 + 1) · (!‘𝑚)))
174	169, 173	eqtrd 2859	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℕ₀ → (!‘(𝑚 + 1)) = ((𝑚 + 1) · (!‘𝑚)))
175	174	oveq1d 7174	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ₀ → ((!‘(𝑚 + 1)) · ((𝑚 + 1) + 1)) = (((𝑚 + 1) · (!‘𝑚)) · ((𝑚 + 1) + 1)))
176	133	nn0cnd 11960	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℕ₀ → ((𝑚 + 1) + 1) ∈ ℂ)
177	172, 171, 176	mulassd 10667	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ₀ → (((𝑚 + 1) · (!‘𝑚)) · ((𝑚 + 1) + 1)) = ((𝑚 + 1) · ((!‘𝑚) · ((𝑚 + 1) + 1))))
178	168, 175, 177	3eqtrd 2863	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ₀ → (!‘((𝑚 + 1) + 1)) = ((𝑚 + 1) · ((!‘𝑚) · ((𝑚 + 1) + 1))))
179	178	oveq1d 7174	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕ₀ → ((!‘((𝑚 + 1) + 1)) · Σ𝑘 ∈ (0...𝑚)((-1↑𝑘) / (!‘𝑘))) = (((𝑚 + 1) · ((!‘𝑚) · ((𝑚 + 1) + 1))) · Σ𝑘 ∈ (0...𝑚)((-1↑𝑘) / (!‘𝑘))))
180	136, 160, 163, 164	div12d 11455	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ₀ → ((!‘((𝑚 + 1) + 1)) · ((-1↑(𝑚 + 1)) / (!‘(𝑚 + 1)))) = ((-1↑(𝑚 + 1)) · ((!‘((𝑚 + 1) + 1)) / (!‘(𝑚 + 1)))))
181	168	oveq1d 7174	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℕ₀ → ((!‘((𝑚 + 1) + 1)) / (!‘(𝑚 + 1))) = (((!‘(𝑚 + 1)) · ((𝑚 + 1) + 1)) / (!‘(𝑚 + 1))))
182	176, 163, 164	divcan3d 11424	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℕ₀ → (((!‘(𝑚 + 1)) · ((𝑚 + 1) + 1)) / (!‘(𝑚 + 1))) = ((𝑚 + 1) + 1))
183	181, 182	eqtrd 2859	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ₀ → ((!‘((𝑚 + 1) + 1)) / (!‘(𝑚 + 1))) = ((𝑚 + 1) + 1))
184	183	oveq2d 7175	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ₀ → ((-1↑(𝑚 + 1)) · ((!‘((𝑚 + 1) + 1)) / (!‘(𝑚 + 1)))) = ((-1↑(𝑚 + 1)) · ((𝑚 + 1) + 1)))
185	180, 184	eqtrd 2859	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕ₀ → ((!‘((𝑚 + 1) + 1)) · ((-1↑(𝑚 + 1)) / (!‘(𝑚 + 1)))) = ((-1↑(𝑚 + 1)) · ((𝑚 + 1) + 1)))
186	179, 185	oveq12d 7177	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ₀ → (((!‘((𝑚 + 1) + 1)) · Σ𝑘 ∈ (0...𝑚)((-1↑𝑘) / (!‘𝑘))) + ((!‘((𝑚 + 1) + 1)) · ((-1↑(𝑚 + 1)) / (!‘(𝑚 + 1))))) = ((((𝑚 + 1) · ((!‘𝑚) · ((𝑚 + 1) + 1))) · Σ𝑘 ∈ (0...𝑚)((-1↑𝑘) / (!‘𝑘))) + ((-1↑(𝑚 + 1)) · ((𝑚 + 1) + 1))))
187	153, 166, 186	3eqtrd 2863	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ₀ → ((!‘((𝑚 + 1) + 1)) · Σ𝑘 ∈ (0...(𝑚 + 1))((-1↑𝑘) / (!‘𝑘))) = ((((𝑚 + 1) · ((!‘𝑚) · ((𝑚 + 1) + 1))) · Σ𝑘 ∈ (0...𝑚)((-1↑𝑘) / (!‘𝑘))) + ((-1↑(𝑚 + 1)) · ((𝑚 + 1) + 1))))
188	143, 136, 144	divcan2d 11421	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ₀ → ((!‘((𝑚 + 1) + 1)) · ((-1↑((𝑚 + 1) + 1)) / (!‘((𝑚 + 1) + 1)))) = (-1↑((𝑚 + 1) + 1)))
189	187, 188	oveq12d 7177	. . . . . . . 8 ⊢ (𝑚 ∈ ℕ₀ → (((!‘((𝑚 + 1) + 1)) · Σ𝑘 ∈ (0...(𝑚 + 1))((-1↑𝑘) / (!‘𝑘))) + ((!‘((𝑚 + 1) + 1)) · ((-1↑((𝑚 + 1) + 1)) / (!‘((𝑚 + 1) + 1))))) = (((((𝑚 + 1) · ((!‘𝑚) · ((𝑚 + 1) + 1))) · Σ𝑘 ∈ (0...𝑚)((-1↑𝑘) / (!‘𝑘))) + ((-1↑(𝑚 + 1)) · ((𝑚 + 1) + 1))) + (-1↑((𝑚 + 1) + 1))))
190	171, 176	mulcld 10664	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕ₀ → ((!‘𝑚) · ((𝑚 + 1) + 1)) ∈ ℂ)
191	172, 190, 158	mulassd 10667	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ₀ → (((𝑚 + 1) · ((!‘𝑚) · ((𝑚 + 1) + 1))) · Σ𝑘 ∈ (0...𝑚)((-1↑𝑘) / (!‘𝑘))) = ((𝑚 + 1) · (((!‘𝑚) · ((𝑚 + 1) + 1)) · Σ𝑘 ∈ (0...𝑚)((-1↑𝑘) / (!‘𝑘)))))
192	72	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ₀ → -1 ∈ ℂ)
193	160, 176, 192	adddid 10668	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ₀ → ((-1↑(𝑚 + 1)) · (((𝑚 + 1) + 1) + -1)) = (((-1↑(𝑚 + 1)) · ((𝑚 + 1) + 1)) + ((-1↑(𝑚 + 1)) · -1)))
194		negsub 10937	. . . . . . . . . . . . . . 15 ⊢ ((((𝑚 + 1) + 1) ∈ ℂ ∧ 1 ∈ ℂ) → (((𝑚 + 1) + 1) + -1) = (((𝑚 + 1) + 1) − 1))
195	176, 70, 194	sylancl 588	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℕ₀ → (((𝑚 + 1) + 1) + -1) = (((𝑚 + 1) + 1) − 1))
196		pncan 10895	. . . . . . . . . . . . . . 15 ⊢ (((𝑚 + 1) ∈ ℂ ∧ 1 ∈ ℂ) → (((𝑚 + 1) + 1) − 1) = (𝑚 + 1))
197	172, 70, 196	sylancl 588	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℕ₀ → (((𝑚 + 1) + 1) − 1) = (𝑚 + 1))
198	195, 197	eqtrd 2859	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ₀ → (((𝑚 + 1) + 1) + -1) = (𝑚 + 1))
199	198	oveq2d 7175	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ₀ → ((-1↑(𝑚 + 1)) · (((𝑚 + 1) + 1) + -1)) = ((-1↑(𝑚 + 1)) · (𝑚 + 1)))
200	193, 199	eqtr3d 2861	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕ₀ → (((-1↑(𝑚 + 1)) · ((𝑚 + 1) + 1)) + ((-1↑(𝑚 + 1)) · -1)) = ((-1↑(𝑚 + 1)) · (𝑚 + 1)))
201		expp1 13439	. . . . . . . . . . . . 13 ⊢ ((-1 ∈ ℂ ∧ (𝑚 + 1) ∈ ℕ₀) → (-1↑((𝑚 + 1) + 1)) = ((-1↑(𝑚 + 1)) · -1))
202	72, 131, 201	sylancr 589	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ₀ → (-1↑((𝑚 + 1) + 1)) = ((-1↑(𝑚 + 1)) · -1))
203	202	oveq2d 7175	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕ₀ → (((-1↑(𝑚 + 1)) · ((𝑚 + 1) + 1)) + (-1↑((𝑚 + 1) + 1))) = (((-1↑(𝑚 + 1)) · ((𝑚 + 1) + 1)) + ((-1↑(𝑚 + 1)) · -1)))
204	172, 160	mulcomd 10665	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕ₀ → ((𝑚 + 1) · (-1↑(𝑚 + 1))) = ((-1↑(𝑚 + 1)) · (𝑚 + 1)))
205	200, 203, 204	3eqtr4d 2869	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ₀ → (((-1↑(𝑚 + 1)) · ((𝑚 + 1) + 1)) + (-1↑((𝑚 + 1) + 1))) = ((𝑚 + 1) · (-1↑(𝑚 + 1))))
206	191, 205	oveq12d 7177	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ₀ → ((((𝑚 + 1) · ((!‘𝑚) · ((𝑚 + 1) + 1))) · Σ𝑘 ∈ (0...𝑚)((-1↑𝑘) / (!‘𝑘))) + (((-1↑(𝑚 + 1)) · ((𝑚 + 1) + 1)) + (-1↑((𝑚 + 1) + 1)))) = (((𝑚 + 1) · (((!‘𝑚) · ((𝑚 + 1) + 1)) · Σ𝑘 ∈ (0...𝑚)((-1↑𝑘) / (!‘𝑘)))) + ((𝑚 + 1) · (-1↑(𝑚 + 1)))))
207	172, 190	mulcld 10664	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕ₀ → ((𝑚 + 1) · ((!‘𝑚) · ((𝑚 + 1) + 1))) ∈ ℂ)
208	207, 158	mulcld 10664	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ₀ → (((𝑚 + 1) · ((!‘𝑚) · ((𝑚 + 1) + 1))) · Σ𝑘 ∈ (0...𝑚)((-1↑𝑘) / (!‘𝑘))) ∈ ℂ)
209	160, 176	mulcld 10664	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ₀ → ((-1↑(𝑚 + 1)) · ((𝑚 + 1) + 1)) ∈ ℂ)
210	208, 209, 143	addassd 10666	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ₀ → (((((𝑚 + 1) · ((!‘𝑚) · ((𝑚 + 1) + 1))) · Σ𝑘 ∈ (0...𝑚)((-1↑𝑘) / (!‘𝑘))) + ((-1↑(𝑚 + 1)) · ((𝑚 + 1) + 1))) + (-1↑((𝑚 + 1) + 1))) = ((((𝑚 + 1) · ((!‘𝑚) · ((𝑚 + 1) + 1))) · Σ𝑘 ∈ (0...𝑚)((-1↑𝑘) / (!‘𝑘))) + (((-1↑(𝑚 + 1)) · ((𝑚 + 1) + 1)) + (-1↑((𝑚 + 1) + 1)))))
211	190, 158	mulcld 10664	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ₀ → (((!‘𝑚) · ((𝑚 + 1) + 1)) · Σ𝑘 ∈ (0...𝑚)((-1↑𝑘) / (!‘𝑘))) ∈ ℂ)
212	172, 211, 160	adddid 10668	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ₀ → ((𝑚 + 1) · ((((!‘𝑚) · ((𝑚 + 1) + 1)) · Σ𝑘 ∈ (0...𝑚)((-1↑𝑘) / (!‘𝑘))) + (-1↑(𝑚 + 1)))) = (((𝑚 + 1) · (((!‘𝑚) · ((𝑚 + 1) + 1)) · Σ𝑘 ∈ (0...𝑚)((-1↑𝑘) / (!‘𝑘)))) + ((𝑚 + 1) · (-1↑(𝑚 + 1)))))
213	206, 210, 212	3eqtr4d 2869	. . . . . . . 8 ⊢ (𝑚 ∈ ℕ₀ → (((((𝑚 + 1) · ((!‘𝑚) · ((𝑚 + 1) + 1))) · Σ𝑘 ∈ (0...𝑚)((-1↑𝑘) / (!‘𝑘))) + ((-1↑(𝑚 + 1)) · ((𝑚 + 1) + 1))) + (-1↑((𝑚 + 1) + 1))) = ((𝑚 + 1) · ((((!‘𝑚) · ((𝑚 + 1) + 1)) · Σ𝑘 ∈ (0...𝑚)((-1↑𝑘) / (!‘𝑘))) + (-1↑(𝑚 + 1)))))
214	146, 189, 213	3eqtrd 2863	. . . . . . 7 ⊢ (𝑚 ∈ ℕ₀ → ((!‘((𝑚 + 1) + 1)) · (Σ𝑘 ∈ (0...(𝑚 + 1))((-1↑𝑘) / (!‘𝑘)) + ((-1↑((𝑚 + 1) + 1)) / (!‘((𝑚 + 1) + 1))))) = ((𝑚 + 1) · ((((!‘𝑚) · ((𝑚 + 1) + 1)) · Σ𝑘 ∈ (0...𝑚)((-1↑𝑘) / (!‘𝑘))) + (-1↑(𝑚 + 1)))))
215	131, 86	eleqtrdi 2926	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ₀ → (𝑚 + 1) ∈ (ℤ_≥‘0))
216		elfznn0 13003	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (0...((𝑚 + 1) + 1)) → 𝑘 ∈ ℕ₀)
217	216	adantl 484	. . . . . . . . . 10 ⊢ ((𝑚 ∈ ℕ₀ ∧ 𝑘 ∈ (0...((𝑚 + 1) + 1))) → 𝑘 ∈ ℕ₀)
218	217, 102	syl 17	. . . . . . . . 9 ⊢ ((𝑚 ∈ ℕ₀ ∧ 𝑘 ∈ (0...((𝑚 + 1) + 1))) → ((-1↑𝑘) / (!‘𝑘)) ∈ ℂ)
219		oveq2 7167	. . . . . . . . . 10 ⊢ (𝑘 = ((𝑚 + 1) + 1) → (-1↑𝑘) = (-1↑((𝑚 + 1) + 1)))
220		fveq2 6673	. . . . . . . . . 10 ⊢ (𝑘 = ((𝑚 + 1) + 1) → (!‘𝑘) = (!‘((𝑚 + 1) + 1)))
221	219, 220	oveq12d 7177	. . . . . . . . 9 ⊢ (𝑘 = ((𝑚 + 1) + 1) → ((-1↑𝑘) / (!‘𝑘)) = ((-1↑((𝑚 + 1) + 1)) / (!‘((𝑚 + 1) + 1))))
222	215, 218, 221	fsump1 15114	. . . . . . . 8 ⊢ (𝑚 ∈ ℕ₀ → Σ𝑘 ∈ (0...((𝑚 + 1) + 1))((-1↑𝑘) / (!‘𝑘)) = (Σ𝑘 ∈ (0...(𝑚 + 1))((-1↑𝑘) / (!‘𝑘)) + ((-1↑((𝑚 + 1) + 1)) / (!‘((𝑚 + 1) + 1)))))
223	222	oveq2d 7175	. . . . . . 7 ⊢ (𝑚 ∈ ℕ₀ → ((!‘((𝑚 + 1) + 1)) · Σ𝑘 ∈ (0...((𝑚 + 1) + 1))((-1↑𝑘) / (!‘𝑘))) = ((!‘((𝑚 + 1) + 1)) · (Σ𝑘 ∈ (0...(𝑚 + 1))((-1↑𝑘) / (!‘𝑘)) + ((-1↑((𝑚 + 1) + 1)) / (!‘((𝑚 + 1) + 1))))))
224	163, 158	mulcld 10664	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ₀ → ((!‘(𝑚 + 1)) · Σ𝑘 ∈ (0...𝑚)((-1↑𝑘) / (!‘𝑘))) ∈ ℂ)
225	171, 158	mulcld 10664	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ₀ → ((!‘𝑚) · Σ𝑘 ∈ (0...𝑚)((-1↑𝑘) / (!‘𝑘))) ∈ ℂ)
226	224, 160, 225	add32d 10870	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ₀ → ((((!‘(𝑚 + 1)) · Σ𝑘 ∈ (0...𝑚)((-1↑𝑘) / (!‘𝑘))) + (-1↑(𝑚 + 1))) + ((!‘𝑚) · Σ𝑘 ∈ (0...𝑚)((-1↑𝑘) / (!‘𝑘)))) = ((((!‘(𝑚 + 1)) · Σ𝑘 ∈ (0...𝑚)((-1↑𝑘) / (!‘𝑘))) + ((!‘𝑚) · Σ𝑘 ∈ (0...𝑚)((-1↑𝑘) / (!‘𝑘)))) + (-1↑(𝑚 + 1))))
227	152	oveq2d 7175	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕ₀ → ((!‘(𝑚 + 1)) · Σ𝑘 ∈ (0...(𝑚 + 1))((-1↑𝑘) / (!‘𝑘))) = ((!‘(𝑚 + 1)) · (Σ𝑘 ∈ (0...𝑚)((-1↑𝑘) / (!‘𝑘)) + ((-1↑(𝑚 + 1)) / (!‘(𝑚 + 1))))))
228	163, 158, 165	adddid 10668	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕ₀ → ((!‘(𝑚 + 1)) · (Σ𝑘 ∈ (0...𝑚)((-1↑𝑘) / (!‘𝑘)) + ((-1↑(𝑚 + 1)) / (!‘(𝑚 + 1))))) = (((!‘(𝑚 + 1)) · Σ𝑘 ∈ (0...𝑚)((-1↑𝑘) / (!‘𝑘))) + ((!‘(𝑚 + 1)) · ((-1↑(𝑚 + 1)) / (!‘(𝑚 + 1))))))
229	160, 163, 164	divcan2d 11421	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ₀ → ((!‘(𝑚 + 1)) · ((-1↑(𝑚 + 1)) / (!‘(𝑚 + 1)))) = (-1↑(𝑚 + 1)))
230	229	oveq2d 7175	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕ₀ → (((!‘(𝑚 + 1)) · Σ𝑘 ∈ (0...𝑚)((-1↑𝑘) / (!‘𝑘))) + ((!‘(𝑚 + 1)) · ((-1↑(𝑚 + 1)) / (!‘(𝑚 + 1))))) = (((!‘(𝑚 + 1)) · Σ𝑘 ∈ (0...𝑚)((-1↑𝑘) / (!‘𝑘))) + (-1↑(𝑚 + 1))))
231	227, 228, 230	3eqtrd 2863	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ₀ → ((!‘(𝑚 + 1)) · Σ𝑘 ∈ (0...(𝑚 + 1))((-1↑𝑘) / (!‘𝑘))) = (((!‘(𝑚 + 1)) · Σ𝑘 ∈ (0...𝑚)((-1↑𝑘) / (!‘𝑘))) + (-1↑(𝑚 + 1))))
232	231	oveq1d 7174	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ₀ → (((!‘(𝑚 + 1)) · Σ𝑘 ∈ (0...(𝑚 + 1))((-1↑𝑘) / (!‘𝑘))) + ((!‘𝑚) · Σ𝑘 ∈ (0...𝑚)((-1↑𝑘) / (!‘𝑘)))) = ((((!‘(𝑚 + 1)) · Σ𝑘 ∈ (0...𝑚)((-1↑𝑘) / (!‘𝑘))) + (-1↑(𝑚 + 1))) + ((!‘𝑚) · Σ𝑘 ∈ (0...𝑚)((-1↑𝑘) / (!‘𝑘)))))
233	70	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℕ₀ → 1 ∈ ℂ)
234	171, 172, 233	adddid 10668	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ₀ → ((!‘𝑚) · ((𝑚 + 1) + 1)) = (((!‘𝑚) · (𝑚 + 1)) + ((!‘𝑚) · 1)))
235	169	eqcomd 2830	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℕ₀ → ((!‘𝑚) · (𝑚 + 1)) = (!‘(𝑚 + 1)))
236	171	mulid1d 10661	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℕ₀ → ((!‘𝑚) · 1) = (!‘𝑚))
237	235, 236	oveq12d 7177	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ₀ → (((!‘𝑚) · (𝑚 + 1)) + ((!‘𝑚) · 1)) = ((!‘(𝑚 + 1)) + (!‘𝑚)))
238	234, 237	eqtrd 2859	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ₀ → ((!‘𝑚) · ((𝑚 + 1) + 1)) = ((!‘(𝑚 + 1)) + (!‘𝑚)))
239	238	oveq1d 7174	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕ₀ → (((!‘𝑚) · ((𝑚 + 1) + 1)) · Σ𝑘 ∈ (0...𝑚)((-1↑𝑘) / (!‘𝑘))) = (((!‘(𝑚 + 1)) + (!‘𝑚)) · Σ𝑘 ∈ (0...𝑚)((-1↑𝑘) / (!‘𝑘))))
240	163, 171, 158	adddird 10669	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕ₀ → (((!‘(𝑚 + 1)) + (!‘𝑚)) · Σ𝑘 ∈ (0...𝑚)((-1↑𝑘) / (!‘𝑘))) = (((!‘(𝑚 + 1)) · Σ𝑘 ∈ (0...𝑚)((-1↑𝑘) / (!‘𝑘))) + ((!‘𝑚) · Σ𝑘 ∈ (0...𝑚)((-1↑𝑘) / (!‘𝑘)))))
241	239, 240	eqtrd 2859	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ₀ → (((!‘𝑚) · ((𝑚 + 1) + 1)) · Σ𝑘 ∈ (0...𝑚)((-1↑𝑘) / (!‘𝑘))) = (((!‘(𝑚 + 1)) · Σ𝑘 ∈ (0...𝑚)((-1↑𝑘) / (!‘𝑘))) + ((!‘𝑚) · Σ𝑘 ∈ (0...𝑚)((-1↑𝑘) / (!‘𝑘)))))
242	241	oveq1d 7174	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ₀ → ((((!‘𝑚) · ((𝑚 + 1) + 1)) · Σ𝑘 ∈ (0...𝑚)((-1↑𝑘) / (!‘𝑘))) + (-1↑(𝑚 + 1))) = ((((!‘(𝑚 + 1)) · Σ𝑘 ∈ (0...𝑚)((-1↑𝑘) / (!‘𝑘))) + ((!‘𝑚) · Σ𝑘 ∈ (0...𝑚)((-1↑𝑘) / (!‘𝑘)))) + (-1↑(𝑚 + 1))))
243	226, 232, 242	3eqtr4d 2869	. . . . . . . 8 ⊢ (𝑚 ∈ ℕ₀ → (((!‘(𝑚 + 1)) · Σ𝑘 ∈ (0...(𝑚 + 1))((-1↑𝑘) / (!‘𝑘))) + ((!‘𝑚) · Σ𝑘 ∈ (0...𝑚)((-1↑𝑘) / (!‘𝑘)))) = ((((!‘𝑚) · ((𝑚 + 1) + 1)) · Σ𝑘 ∈ (0...𝑚)((-1↑𝑘) / (!‘𝑘))) + (-1↑(𝑚 + 1))))
244	243	oveq2d 7175	. . . . . . 7 ⊢ (𝑚 ∈ ℕ₀ → ((𝑚 + 1) · (((!‘(𝑚 + 1)) · Σ𝑘 ∈ (0...(𝑚 + 1))((-1↑𝑘) / (!‘𝑘))) + ((!‘𝑚) · Σ𝑘 ∈ (0...𝑚)((-1↑𝑘) / (!‘𝑘))))) = ((𝑚 + 1) · ((((!‘𝑚) · ((𝑚 + 1) + 1)) · Σ𝑘 ∈ (0...𝑚)((-1↑𝑘) / (!‘𝑘))) + (-1↑(𝑚 + 1)))))
245	214, 223, 244	3eqtr4d 2869	. . . . . 6 ⊢ (𝑚 ∈ ℕ₀ → ((!‘((𝑚 + 1) + 1)) · Σ𝑘 ∈ (0...((𝑚 + 1) + 1))((-1↑𝑘) / (!‘𝑘))) = ((𝑚 + 1) · (((!‘(𝑚 + 1)) · Σ𝑘 ∈ (0...(𝑚 + 1))((-1↑𝑘) / (!‘𝑘))) + ((!‘𝑚) · Σ𝑘 ∈ (0...𝑚)((-1↑𝑘) / (!‘𝑘))))))
246	130, 245	eqeq12d 2840	. . . . 5 ⊢ (𝑚 ∈ ℕ₀ → ((𝑆‘((𝑚 + 1) + 1)) = ((!‘((𝑚 + 1) + 1)) · Σ𝑘 ∈ (0...((𝑚 + 1) + 1))((-1↑𝑘) / (!‘𝑘))) ↔ ((𝑚 + 1) · ((𝑆‘(𝑚 + 1)) + (𝑆‘𝑚))) = ((𝑚 + 1) · (((!‘(𝑚 + 1)) · Σ𝑘 ∈ (0...(𝑚 + 1))((-1↑𝑘) / (!‘𝑘))) + ((!‘𝑚) · Σ𝑘 ∈ (0...𝑚)((-1↑𝑘) / (!‘𝑘)))))))
247	120, 246	syl5ibr 248	. . . 4 ⊢ (𝑚 ∈ ℕ₀ → (((𝑆‘𝑚) = ((!‘𝑚) · Σ𝑘 ∈ (0...𝑚)((-1↑𝑘) / (!‘𝑘))) ∧ (𝑆‘(𝑚 + 1)) = ((!‘(𝑚 + 1)) · Σ𝑘 ∈ (0...(𝑚 + 1))((-1↑𝑘) / (!‘𝑘)))) → (𝑆‘((𝑚 + 1) + 1)) = ((!‘((𝑚 + 1) + 1)) · Σ𝑘 ∈ (0...((𝑚 + 1) + 1))((-1↑𝑘) / (!‘𝑘)))))
248	117, 247	jcad 515	. . 3 ⊢ (𝑚 ∈ ℕ₀ → (((𝑆‘𝑚) = ((!‘𝑚) · Σ𝑘 ∈ (0...𝑚)((-1↑𝑘) / (!‘𝑘))) ∧ (𝑆‘(𝑚 + 1)) = ((!‘(𝑚 + 1)) · Σ𝑘 ∈ (0...(𝑚 + 1))((-1↑𝑘) / (!‘𝑘)))) → ((𝑆‘(𝑚 + 1)) = ((!‘(𝑚 + 1)) · Σ𝑘 ∈ (0...(𝑚 + 1))((-1↑𝑘) / (!‘𝑘))) ∧ (𝑆‘((𝑚 + 1) + 1)) = ((!‘((𝑚 + 1) + 1)) · Σ𝑘 ∈ (0...((𝑚 + 1) + 1))((-1↑𝑘) / (!‘𝑘))))))
249	26, 40, 54, 68, 115, 248	nn0ind 12080	. 2 ⊢ (𝑁 ∈ ℕ₀ → ((𝑆‘𝑁) = ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)((-1↑𝑘) / (!‘𝑘))) ∧ (𝑆‘(𝑁 + 1)) = ((!‘(𝑁 + 1)) · Σ𝑘 ∈ (0...(𝑁 + 1))((-1↑𝑘) / (!‘𝑘)))))
250	249	simpld 497	1 ⊢ (𝑁 ∈ ℕ₀ → (𝑆‘𝑁) = ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)((-1↑𝑘) / (!‘𝑘))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ⊤wtru 1537 ∈ wcel 2113 {cab 2802 ≠ wne 3019 ∀wral 3141 ↦ cmpt 5149 –1-1-onto→wf1o 6357 ‘cfv 6358 (class class class)co 7159 Fincfn 8512 ℂcc 10538 ℝcr 10539 0cc0 10540 1c1 10541 + caddc 10543 · cmul 10545 − cmin 10873 -cneg 10874 / cdiv 11300 ℕcn 11641 ℕ₀cn0 11900 ℤcz 11984 ℤ_≥cuz 12246 ...cfz 12895 ↑cexp 13432 !cfa 13636 ♯chash 13693 Σcsu 15045

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-inf2 9107 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-se 5518 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-2o 8106 df-oadd 8109 df-er 8292 df-map 8411 df-pm 8412 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-sup 8909 df-oi 8977 df-dju 9333 df-card 9371 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 df-2 11703 df-3 11704 df-n0 11901 df-xnn0 11971 df-z 11985 df-uz 12247 df-rp 12393 df-fz 12896 df-fzo 13037 df-seq 13373 df-exp 13433 df-fac 13637 df-hash 13694 df-cj 14461 df-re 14462 df-im 14463 df-sqrt 14597 df-abs 14598 df-clim 14848 df-sum 15046

This theorem is referenced by: subfaclim 32439

W3C validator