bccolsum - Mathbox for Scott Fenton

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Theorem bccolsum 34647

Description: A column-sum rule for binomial coefficents. (Contributed by Scott Fenton, 24-Jun-2020.)

**Assertion**
Ref	Expression
bccolsum	⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) → Σ𝑘 ∈ (0...𝑁)(𝑘C𝐶) = ((𝑁 + 1)C(𝐶 + 1)))

Distinct variable groups: 𝑘,𝑁 𝐶,𝑘

Proof of Theorem bccolsum Dummy variables 𝑛 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 7412	. . . . . 6 ⊢ (𝑚 = 0 → (0...𝑚) = (0...0))
2	1	sumeq1d 15643	. . . . 5 ⊢ (𝑚 = 0 → Σ𝑘 ∈ (0...𝑚)(𝑘C𝐶) = Σ𝑘 ∈ (0...0)(𝑘C𝐶))
3		oveq1 7411	. . . . . . 7 ⊢ (𝑚 = 0 → (𝑚 + 1) = (0 + 1))
4		0p1e1 12330	. . . . . . 7 ⊢ (0 + 1) = 1
5	3, 4	eqtrdi 2789	. . . . . 6 ⊢ (𝑚 = 0 → (𝑚 + 1) = 1)
6	5	oveq1d 7419	. . . . 5 ⊢ (𝑚 = 0 → ((𝑚 + 1)C(𝐶 + 1)) = (1C(𝐶 + 1)))
7	2, 6	eqeq12d 2749	. . . 4 ⊢ (𝑚 = 0 → (Σ𝑘 ∈ (0...𝑚)(𝑘C𝐶) = ((𝑚 + 1)C(𝐶 + 1)) ↔ Σ𝑘 ∈ (0...0)(𝑘C𝐶) = (1C(𝐶 + 1))))
8	7	imbi2d 341	. . 3 ⊢ (𝑚 = 0 → ((𝐶 ∈ ℕ₀ → Σ𝑘 ∈ (0...𝑚)(𝑘C𝐶) = ((𝑚 + 1)C(𝐶 + 1))) ↔ (𝐶 ∈ ℕ₀ → Σ𝑘 ∈ (0...0)(𝑘C𝐶) = (1C(𝐶 + 1)))))
9		oveq2 7412	. . . . . 6 ⊢ (𝑚 = 𝑛 → (0...𝑚) = (0...𝑛))
10	9	sumeq1d 15643	. . . . 5 ⊢ (𝑚 = 𝑛 → Σ𝑘 ∈ (0...𝑚)(𝑘C𝐶) = Σ𝑘 ∈ (0...𝑛)(𝑘C𝐶))
11		oveq1 7411	. . . . . 6 ⊢ (𝑚 = 𝑛 → (𝑚 + 1) = (𝑛 + 1))
12	11	oveq1d 7419	. . . . 5 ⊢ (𝑚 = 𝑛 → ((𝑚 + 1)C(𝐶 + 1)) = ((𝑛 + 1)C(𝐶 + 1)))
13	10, 12	eqeq12d 2749	. . . 4 ⊢ (𝑚 = 𝑛 → (Σ𝑘 ∈ (0...𝑚)(𝑘C𝐶) = ((𝑚 + 1)C(𝐶 + 1)) ↔ Σ𝑘 ∈ (0...𝑛)(𝑘C𝐶) = ((𝑛 + 1)C(𝐶 + 1))))
14	13	imbi2d 341	. . 3 ⊢ (𝑚 = 𝑛 → ((𝐶 ∈ ℕ₀ → Σ𝑘 ∈ (0...𝑚)(𝑘C𝐶) = ((𝑚 + 1)C(𝐶 + 1))) ↔ (𝐶 ∈ ℕ₀ → Σ𝑘 ∈ (0...𝑛)(𝑘C𝐶) = ((𝑛 + 1)C(𝐶 + 1)))))
15		oveq2 7412	. . . . . 6 ⊢ (𝑚 = (𝑛 + 1) → (0...𝑚) = (0...(𝑛 + 1)))
16	15	sumeq1d 15643	. . . . 5 ⊢ (𝑚 = (𝑛 + 1) → Σ𝑘 ∈ (0...𝑚)(𝑘C𝐶) = Σ𝑘 ∈ (0...(𝑛 + 1))(𝑘C𝐶))
17		oveq1 7411	. . . . . 6 ⊢ (𝑚 = (𝑛 + 1) → (𝑚 + 1) = ((𝑛 + 1) + 1))
18	17	oveq1d 7419	. . . . 5 ⊢ (𝑚 = (𝑛 + 1) → ((𝑚 + 1)C(𝐶 + 1)) = (((𝑛 + 1) + 1)C(𝐶 + 1)))
19	16, 18	eqeq12d 2749	. . . 4 ⊢ (𝑚 = (𝑛 + 1) → (Σ𝑘 ∈ (0...𝑚)(𝑘C𝐶) = ((𝑚 + 1)C(𝐶 + 1)) ↔ Σ𝑘 ∈ (0...(𝑛 + 1))(𝑘C𝐶) = (((𝑛 + 1) + 1)C(𝐶 + 1))))
20	19	imbi2d 341	. . 3 ⊢ (𝑚 = (𝑛 + 1) → ((𝐶 ∈ ℕ₀ → Σ𝑘 ∈ (0...𝑚)(𝑘C𝐶) = ((𝑚 + 1)C(𝐶 + 1))) ↔ (𝐶 ∈ ℕ₀ → Σ𝑘 ∈ (0...(𝑛 + 1))(𝑘C𝐶) = (((𝑛 + 1) + 1)C(𝐶 + 1)))))
21		oveq2 7412	. . . . . 6 ⊢ (𝑚 = 𝑁 → (0...𝑚) = (0...𝑁))
22	21	sumeq1d 15643	. . . . 5 ⊢ (𝑚 = 𝑁 → Σ𝑘 ∈ (0...𝑚)(𝑘C𝐶) = Σ𝑘 ∈ (0...𝑁)(𝑘C𝐶))
23		oveq1 7411	. . . . . 6 ⊢ (𝑚 = 𝑁 → (𝑚 + 1) = (𝑁 + 1))
24	23	oveq1d 7419	. . . . 5 ⊢ (𝑚 = 𝑁 → ((𝑚 + 1)C(𝐶 + 1)) = ((𝑁 + 1)C(𝐶 + 1)))
25	22, 24	eqeq12d 2749	. . . 4 ⊢ (𝑚 = 𝑁 → (Σ𝑘 ∈ (0...𝑚)(𝑘C𝐶) = ((𝑚 + 1)C(𝐶 + 1)) ↔ Σ𝑘 ∈ (0...𝑁)(𝑘C𝐶) = ((𝑁 + 1)C(𝐶 + 1))))
26	25	imbi2d 341	. . 3 ⊢ (𝑚 = 𝑁 → ((𝐶 ∈ ℕ₀ → Σ𝑘 ∈ (0...𝑚)(𝑘C𝐶) = ((𝑚 + 1)C(𝐶 + 1))) ↔ (𝐶 ∈ ℕ₀ → Σ𝑘 ∈ (0...𝑁)(𝑘C𝐶) = ((𝑁 + 1)C(𝐶 + 1)))))
27		0z 12565	. . . . 5 ⊢ 0 ∈ ℤ
28		0nn0 12483	. . . . . . 7 ⊢ 0 ∈ ℕ₀
29		nn0z 12579	. . . . . . 7 ⊢ (𝐶 ∈ ℕ₀ → 𝐶 ∈ ℤ)
30		bccl 14278	. . . . . . 7 ⊢ ((0 ∈ ℕ₀ ∧ 𝐶 ∈ ℤ) → (0C𝐶) ∈ ℕ₀)
31	28, 29, 30	sylancr 588	. . . . . 6 ⊢ (𝐶 ∈ ℕ₀ → (0C𝐶) ∈ ℕ₀)
32	31	nn0cnd 12530	. . . . 5 ⊢ (𝐶 ∈ ℕ₀ → (0C𝐶) ∈ ℂ)
33		oveq1 7411	. . . . . 6 ⊢ (𝑘 = 0 → (𝑘C𝐶) = (0C𝐶))
34	33	fsum1 15689	. . . . 5 ⊢ ((0 ∈ ℤ ∧ (0C𝐶) ∈ ℂ) → Σ𝑘 ∈ (0...0)(𝑘C𝐶) = (0C𝐶))
35	27, 32, 34	sylancr 588	. . . 4 ⊢ (𝐶 ∈ ℕ₀ → Σ𝑘 ∈ (0...0)(𝑘C𝐶) = (0C𝐶))
36		elnn0 12470	. . . . 5 ⊢ (𝐶 ∈ ℕ₀ ↔ (𝐶 ∈ ℕ ∨ 𝐶 = 0))
37		1red 11211	. . . . . . . . . . 11 ⊢ (𝐶 ∈ ℕ → 1 ∈ ℝ)
38		nnrp 12981	. . . . . . . . . . 11 ⊢ (𝐶 ∈ ℕ → 𝐶 ∈ ℝ⁺)
39	37, 38	ltaddrp2d 13046	. . . . . . . . . 10 ⊢ (𝐶 ∈ ℕ → 1 < (𝐶 + 1))
40		peano2nn 12220	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ ℕ → (𝐶 + 1) ∈ ℕ)
41	40	nnred 12223	. . . . . . . . . . 11 ⊢ (𝐶 ∈ ℕ → (𝐶 + 1) ∈ ℝ)
42	37, 41	ltnled 11357	. . . . . . . . . 10 ⊢ (𝐶 ∈ ℕ → (1 < (𝐶 + 1) ↔ ¬ (𝐶 + 1) ≤ 1))
43	39, 42	mpbid 231	. . . . . . . . 9 ⊢ (𝐶 ∈ ℕ → ¬ (𝐶 + 1) ≤ 1)
44		elfzle2 13501	. . . . . . . . 9 ⊢ ((𝐶 + 1) ∈ (0...1) → (𝐶 + 1) ≤ 1)
45	43, 44	nsyl 140	. . . . . . . 8 ⊢ (𝐶 ∈ ℕ → ¬ (𝐶 + 1) ∈ (0...1))
46	45	iffalsed 4538	. . . . . . 7 ⊢ (𝐶 ∈ ℕ → if((𝐶 + 1) ∈ (0...1), ((!‘1) / ((!‘(1 − (𝐶 + 1))) · (!‘(𝐶 + 1)))), 0) = 0)
47		1nn0 12484	. . . . . . . 8 ⊢ 1 ∈ ℕ₀
48	40	nnzd 12581	. . . . . . . 8 ⊢ (𝐶 ∈ ℕ → (𝐶 + 1) ∈ ℤ)
49		bcval 14260	. . . . . . . 8 ⊢ ((1 ∈ ℕ₀ ∧ (𝐶 + 1) ∈ ℤ) → (1C(𝐶 + 1)) = if((𝐶 + 1) ∈ (0...1), ((!‘1) / ((!‘(1 − (𝐶 + 1))) · (!‘(𝐶 + 1)))), 0))
50	47, 48, 49	sylancr 588	. . . . . . 7 ⊢ (𝐶 ∈ ℕ → (1C(𝐶 + 1)) = if((𝐶 + 1) ∈ (0...1), ((!‘1) / ((!‘(1 − (𝐶 + 1))) · (!‘(𝐶 + 1)))), 0))
51		bc0k 14267	. . . . . . 7 ⊢ (𝐶 ∈ ℕ → (0C𝐶) = 0)
52	46, 50, 51	3eqtr4rd 2784	. . . . . 6 ⊢ (𝐶 ∈ ℕ → (0C𝐶) = (1C(𝐶 + 1)))
53		bcnn 14268	. . . . . . . . 9 ⊢ (0 ∈ ℕ₀ → (0C0) = 1)
54	28, 53	ax-mp 5	. . . . . . . 8 ⊢ (0C0) = 1
55		bcnn 14268	. . . . . . . . 9 ⊢ (1 ∈ ℕ₀ → (1C1) = 1)
56	47, 55	ax-mp 5	. . . . . . . 8 ⊢ (1C1) = 1
57	54, 56	eqtr4i 2764	. . . . . . 7 ⊢ (0C0) = (1C1)
58		oveq2 7412	. . . . . . 7 ⊢ (𝐶 = 0 → (0C𝐶) = (0C0))
59		oveq1 7411	. . . . . . . . 9 ⊢ (𝐶 = 0 → (𝐶 + 1) = (0 + 1))
60	59, 4	eqtrdi 2789	. . . . . . . 8 ⊢ (𝐶 = 0 → (𝐶 + 1) = 1)
61	60	oveq2d 7420	. . . . . . 7 ⊢ (𝐶 = 0 → (1C(𝐶 + 1)) = (1C1))
62	57, 58, 61	3eqtr4a 2799	. . . . . 6 ⊢ (𝐶 = 0 → (0C𝐶) = (1C(𝐶 + 1)))
63	52, 62	jaoi 856	. . . . 5 ⊢ ((𝐶 ∈ ℕ ∨ 𝐶 = 0) → (0C𝐶) = (1C(𝐶 + 1)))
64	36, 63	sylbi 216	. . . 4 ⊢ (𝐶 ∈ ℕ₀ → (0C𝐶) = (1C(𝐶 + 1)))
65	35, 64	eqtrd 2773	. . 3 ⊢ (𝐶 ∈ ℕ₀ → Σ𝑘 ∈ (0...0)(𝑘C𝐶) = (1C(𝐶 + 1)))
66		elnn0uz 12863	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ₀ ↔ 𝑛 ∈ (ℤ_≥‘0))
67	66	biimpi 215	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ₀ → 𝑛 ∈ (ℤ_≥‘0))
68	67	adantr 482	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) → 𝑛 ∈ (ℤ_≥‘0))
69		elfznn0 13590	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (0...(𝑛 + 1)) → 𝑘 ∈ ℕ₀)
70	69	adantl 483	. . . . . . . . . 10 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (0...(𝑛 + 1))) → 𝑘 ∈ ℕ₀)
71		simplr 768	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (0...(𝑛 + 1))) → 𝐶 ∈ ℕ₀)
72	71	nn0zd 12580	. . . . . . . . . 10 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (0...(𝑛 + 1))) → 𝐶 ∈ ℤ)
73		bccl 14278	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℕ₀ ∧ 𝐶 ∈ ℤ) → (𝑘C𝐶) ∈ ℕ₀)
74	70, 72, 73	syl2anc 585	. . . . . . . . 9 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (0...(𝑛 + 1))) → (𝑘C𝐶) ∈ ℕ₀)
75	74	nn0cnd 12530	. . . . . . . 8 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (0...(𝑛 + 1))) → (𝑘C𝐶) ∈ ℂ)
76		oveq1 7411	. . . . . . . 8 ⊢ (𝑘 = (𝑛 + 1) → (𝑘C𝐶) = ((𝑛 + 1)C𝐶))
77	68, 75, 76	fsump1 15698	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) → Σ𝑘 ∈ (0...(𝑛 + 1))(𝑘C𝐶) = (Σ𝑘 ∈ (0...𝑛)(𝑘C𝐶) + ((𝑛 + 1)C𝐶)))
78	77	adantr 482	. . . . . 6 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) ∧ Σ𝑘 ∈ (0...𝑛)(𝑘C𝐶) = ((𝑛 + 1)C(𝐶 + 1))) → Σ𝑘 ∈ (0...(𝑛 + 1))(𝑘C𝐶) = (Σ𝑘 ∈ (0...𝑛)(𝑘C𝐶) + ((𝑛 + 1)C𝐶)))
79		id 22	. . . . . . 7 ⊢ (Σ𝑘 ∈ (0...𝑛)(𝑘C𝐶) = ((𝑛 + 1)C(𝐶 + 1)) → Σ𝑘 ∈ (0...𝑛)(𝑘C𝐶) = ((𝑛 + 1)C(𝐶 + 1)))
80		nn0cn 12478	. . . . . . . . . . 11 ⊢ (𝐶 ∈ ℕ₀ → 𝐶 ∈ ℂ)
81	80	adantl 483	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) → 𝐶 ∈ ℂ)
82		1cnd 11205	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) → 1 ∈ ℂ)
83	81, 82	pncand 11568	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) → ((𝐶 + 1) − 1) = 𝐶)
84	83	oveq2d 7420	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) → ((𝑛 + 1)C((𝐶 + 1) − 1)) = ((𝑛 + 1)C𝐶))
85	84	eqcomd 2739	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) → ((𝑛 + 1)C𝐶) = ((𝑛 + 1)C((𝐶 + 1) − 1)))
86	79, 85	oveqan12rd 7424	. . . . . 6 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) ∧ Σ𝑘 ∈ (0...𝑛)(𝑘C𝐶) = ((𝑛 + 1)C(𝐶 + 1))) → (Σ𝑘 ∈ (0...𝑛)(𝑘C𝐶) + ((𝑛 + 1)C𝐶)) = (((𝑛 + 1)C(𝐶 + 1)) + ((𝑛 + 1)C((𝐶 + 1) − 1))))
87		peano2nn0 12508	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → (𝑛 + 1) ∈ ℕ₀)
88		peano2nn0 12508	. . . . . . . . 9 ⊢ (𝐶 ∈ ℕ₀ → (𝐶 + 1) ∈ ℕ₀)
89	88	nn0zd 12580	. . . . . . . 8 ⊢ (𝐶 ∈ ℕ₀ → (𝐶 + 1) ∈ ℤ)
90		bcpasc 14277	. . . . . . . 8 ⊢ (((𝑛 + 1) ∈ ℕ₀ ∧ (𝐶 + 1) ∈ ℤ) → (((𝑛 + 1)C(𝐶 + 1)) + ((𝑛 + 1)C((𝐶 + 1) − 1))) = (((𝑛 + 1) + 1)C(𝐶 + 1)))
91	87, 89, 90	syl2an 597	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) → (((𝑛 + 1)C(𝐶 + 1)) + ((𝑛 + 1)C((𝐶 + 1) − 1))) = (((𝑛 + 1) + 1)C(𝐶 + 1)))
92	91	adantr 482	. . . . . 6 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) ∧ Σ𝑘 ∈ (0...𝑛)(𝑘C𝐶) = ((𝑛 + 1)C(𝐶 + 1))) → (((𝑛 + 1)C(𝐶 + 1)) + ((𝑛 + 1)C((𝐶 + 1) − 1))) = (((𝑛 + 1) + 1)C(𝐶 + 1)))
93	78, 86, 92	3eqtrd 2777	. . . . 5 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) ∧ Σ𝑘 ∈ (0...𝑛)(𝑘C𝐶) = ((𝑛 + 1)C(𝐶 + 1))) → Σ𝑘 ∈ (0...(𝑛 + 1))(𝑘C𝐶) = (((𝑛 + 1) + 1)C(𝐶 + 1)))
94	93	exp31 421	. . . 4 ⊢ (𝑛 ∈ ℕ₀ → (𝐶 ∈ ℕ₀ → (Σ𝑘 ∈ (0...𝑛)(𝑘C𝐶) = ((𝑛 + 1)C(𝐶 + 1)) → Σ𝑘 ∈ (0...(𝑛 + 1))(𝑘C𝐶) = (((𝑛 + 1) + 1)C(𝐶 + 1)))))
95	94	a2d 29	. . 3 ⊢ (𝑛 ∈ ℕ₀ → ((𝐶 ∈ ℕ₀ → Σ𝑘 ∈ (0...𝑛)(𝑘C𝐶) = ((𝑛 + 1)C(𝐶 + 1))) → (𝐶 ∈ ℕ₀ → Σ𝑘 ∈ (0...(𝑛 + 1))(𝑘C𝐶) = (((𝑛 + 1) + 1)C(𝐶 + 1)))))
96	8, 14, 20, 26, 65, 95	nn0ind 12653	. 2 ⊢ (𝑁 ∈ ℕ₀ → (𝐶 ∈ ℕ₀ → Σ𝑘 ∈ (0...𝑁)(𝑘C𝐶) = ((𝑁 + 1)C(𝐶 + 1))))
97	96	imp 408	1 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) → Σ𝑘 ∈ (0...𝑁)(𝑘C𝐶) = ((𝑁 + 1)C(𝐶 + 1)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 ∨ wo 846 = wceq 1542 ∈ wcel 2107 ifcif 4527 class class class wbr 5147 ‘cfv 6540 (class class class)co 7404 ℂcc 11104 0cc0 11106 1c1 11107 + caddc 11109 · cmul 11111 < clt 11244 ≤ cle 11245 − cmin 11440 / cdiv 11867 ℕcn 12208 ℕ₀cn0 12468 ℤcz 12554 ℤ_≥cuz 12818 ...cfz 13480 !cfa 14229 Ccbc 14258 Σcsu 15628

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7851 df-1st 7970 df-2nd 7971 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-oi 9501 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-n0 12469 df-z 12555 df-uz 12819 df-rp 12971 df-fz 13481 df-fzo 13624 df-seq 13963 df-exp 14024 df-fac 14230 df-bc 14259 df-hash 14287 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-clim 15428 df-sum 15629

This theorem is referenced by: (None)

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