bccolsum - Mathbox for Scott Fenton

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Theorem bccolsum 33233

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 7164	. . . . . 6 ⊢ (𝑚 = 0 → (0...𝑚) = (0...0))
2	1	sumeq1d 15119	. . . . 5 ⊢ (𝑚 = 0 → Σ𝑘 ∈ (0...𝑚)(𝑘C𝐶) = Σ𝑘 ∈ (0...0)(𝑘C𝐶))
3		oveq1 7163	. . . . . . 7 ⊢ (𝑚 = 0 → (𝑚 + 1) = (0 + 1))
4		0p1e1 11809	. . . . . . 7 ⊢ (0 + 1) = 1
5	3, 4	eqtrdi 2809	. . . . . 6 ⊢ (𝑚 = 0 → (𝑚 + 1) = 1)
6	5	oveq1d 7171	. . . . 5 ⊢ (𝑚 = 0 → ((𝑚 + 1)C(𝐶 + 1)) = (1C(𝐶 + 1)))
7	2, 6	eqeq12d 2774	. . . 4 ⊢ (𝑚 = 0 → (Σ𝑘 ∈ (0...𝑚)(𝑘C𝐶) = ((𝑚 + 1)C(𝐶 + 1)) ↔ Σ𝑘 ∈ (0...0)(𝑘C𝐶) = (1C(𝐶 + 1))))
8	7	imbi2d 344	. . 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 7164	. . . . . 6 ⊢ (𝑚 = 𝑛 → (0...𝑚) = (0...𝑛))
10	9	sumeq1d 15119	. . . . 5 ⊢ (𝑚 = 𝑛 → Σ𝑘 ∈ (0...𝑚)(𝑘C𝐶) = Σ𝑘 ∈ (0...𝑛)(𝑘C𝐶))
11		oveq1 7163	. . . . . 6 ⊢ (𝑚 = 𝑛 → (𝑚 + 1) = (𝑛 + 1))
12	11	oveq1d 7171	. . . . 5 ⊢ (𝑚 = 𝑛 → ((𝑚 + 1)C(𝐶 + 1)) = ((𝑛 + 1)C(𝐶 + 1)))
13	10, 12	eqeq12d 2774	. . . 4 ⊢ (𝑚 = 𝑛 → (Σ𝑘 ∈ (0...𝑚)(𝑘C𝐶) = ((𝑚 + 1)C(𝐶 + 1)) ↔ Σ𝑘 ∈ (0...𝑛)(𝑘C𝐶) = ((𝑛 + 1)C(𝐶 + 1))))
14	13	imbi2d 344	. . 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 7164	. . . . . 6 ⊢ (𝑚 = (𝑛 + 1) → (0...𝑚) = (0...(𝑛 + 1)))
16	15	sumeq1d 15119	. . . . 5 ⊢ (𝑚 = (𝑛 + 1) → Σ𝑘 ∈ (0...𝑚)(𝑘C𝐶) = Σ𝑘 ∈ (0...(𝑛 + 1))(𝑘C𝐶))
17		oveq1 7163	. . . . . 6 ⊢ (𝑚 = (𝑛 + 1) → (𝑚 + 1) = ((𝑛 + 1) + 1))
18	17	oveq1d 7171	. . . . 5 ⊢ (𝑚 = (𝑛 + 1) → ((𝑚 + 1)C(𝐶 + 1)) = (((𝑛 + 1) + 1)C(𝐶 + 1)))
19	16, 18	eqeq12d 2774	. . . 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 344	. . 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 7164	. . . . . 6 ⊢ (𝑚 = 𝑁 → (0...𝑚) = (0...𝑁))
22	21	sumeq1d 15119	. . . . 5 ⊢ (𝑚 = 𝑁 → Σ𝑘 ∈ (0...𝑚)(𝑘C𝐶) = Σ𝑘 ∈ (0...𝑁)(𝑘C𝐶))
23		oveq1 7163	. . . . . 6 ⊢ (𝑚 = 𝑁 → (𝑚 + 1) = (𝑁 + 1))
24	23	oveq1d 7171	. . . . 5 ⊢ (𝑚 = 𝑁 → ((𝑚 + 1)C(𝐶 + 1)) = ((𝑁 + 1)C(𝐶 + 1)))
25	22, 24	eqeq12d 2774	. . . 4 ⊢ (𝑚 = 𝑁 → (Σ𝑘 ∈ (0...𝑚)(𝑘C𝐶) = ((𝑚 + 1)C(𝐶 + 1)) ↔ Σ𝑘 ∈ (0...𝑁)(𝑘C𝐶) = ((𝑁 + 1)C(𝐶 + 1))))
26	25	imbi2d 344	. . 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 12044	. . . . 5 ⊢ 0 ∈ ℤ
28		0nn0 11962	. . . . . . 7 ⊢ 0 ∈ ℕ₀
29		nn0z 12057	. . . . . . 7 ⊢ (𝐶 ∈ ℕ₀ → 𝐶 ∈ ℤ)
30		bccl 13745	. . . . . . 7 ⊢ ((0 ∈ ℕ₀ ∧ 𝐶 ∈ ℤ) → (0C𝐶) ∈ ℕ₀)
31	28, 29, 30	sylancr 590	. . . . . 6 ⊢ (𝐶 ∈ ℕ₀ → (0C𝐶) ∈ ℕ₀)
32	31	nn0cnd 12009	. . . . 5 ⊢ (𝐶 ∈ ℕ₀ → (0C𝐶) ∈ ℂ)
33		oveq1 7163	. . . . . 6 ⊢ (𝑘 = 0 → (𝑘C𝐶) = (0C𝐶))
34	33	fsum1 15163	. . . . 5 ⊢ ((0 ∈ ℤ ∧ (0C𝐶) ∈ ℂ) → Σ𝑘 ∈ (0...0)(𝑘C𝐶) = (0C𝐶))
35	27, 32, 34	sylancr 590	. . . 4 ⊢ (𝐶 ∈ ℕ₀ → Σ𝑘 ∈ (0...0)(𝑘C𝐶) = (0C𝐶))
36		elnn0 11949	. . . . 5 ⊢ (𝐶 ∈ ℕ₀ ↔ (𝐶 ∈ ℕ ∨ 𝐶 = 0))
37		1red 10693	. . . . . . . . . . 11 ⊢ (𝐶 ∈ ℕ → 1 ∈ ℝ)
38		nnrp 12454	. . . . . . . . . . 11 ⊢ (𝐶 ∈ ℕ → 𝐶 ∈ ℝ⁺)
39	37, 38	ltaddrp2d 12519	. . . . . . . . . 10 ⊢ (𝐶 ∈ ℕ → 1 < (𝐶 + 1))
40		peano2nn 11699	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ ℕ → (𝐶 + 1) ∈ ℕ)
41	40	nnred 11702	. . . . . . . . . . 11 ⊢ (𝐶 ∈ ℕ → (𝐶 + 1) ∈ ℝ)
42	37, 41	ltnled 10838	. . . . . . . . . 10 ⊢ (𝐶 ∈ ℕ → (1 < (𝐶 + 1) ↔ ¬ (𝐶 + 1) ≤ 1))
43	39, 42	mpbid 235	. . . . . . . . 9 ⊢ (𝐶 ∈ ℕ → ¬ (𝐶 + 1) ≤ 1)
44		elfzle2 12973	. . . . . . . . 9 ⊢ ((𝐶 + 1) ∈ (0...1) → (𝐶 + 1) ≤ 1)
45	43, 44	nsyl 142	. . . . . . . 8 ⊢ (𝐶 ∈ ℕ → ¬ (𝐶 + 1) ∈ (0...1))
46	45	iffalsed 4434	. . . . . . 7 ⊢ (𝐶 ∈ ℕ → if((𝐶 + 1) ∈ (0...1), ((!‘1) / ((!‘(1 − (𝐶 + 1))) · (!‘(𝐶 + 1)))), 0) = 0)
47		1nn0 11963	. . . . . . . 8 ⊢ 1 ∈ ℕ₀
48	40	nnzd 12138	. . . . . . . 8 ⊢ (𝐶 ∈ ℕ → (𝐶 + 1) ∈ ℤ)
49		bcval 13727	. . . . . . . 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 590	. . . . . . 7 ⊢ (𝐶 ∈ ℕ → (1C(𝐶 + 1)) = if((𝐶 + 1) ∈ (0...1), ((!‘1) / ((!‘(1 − (𝐶 + 1))) · (!‘(𝐶 + 1)))), 0))
51		bc0k 13734	. . . . . . 7 ⊢ (𝐶 ∈ ℕ → (0C𝐶) = 0)
52	46, 50, 51	3eqtr4rd 2804	. . . . . 6 ⊢ (𝐶 ∈ ℕ → (0C𝐶) = (1C(𝐶 + 1)))
53		bcnn 13735	. . . . . . . . 9 ⊢ (0 ∈ ℕ₀ → (0C0) = 1)
54	28, 53	ax-mp 5	. . . . . . . 8 ⊢ (0C0) = 1
55		bcnn 13735	. . . . . . . . 9 ⊢ (1 ∈ ℕ₀ → (1C1) = 1)
56	47, 55	ax-mp 5	. . . . . . . 8 ⊢ (1C1) = 1
57	54, 56	eqtr4i 2784	. . . . . . 7 ⊢ (0C0) = (1C1)
58		oveq2 7164	. . . . . . 7 ⊢ (𝐶 = 0 → (0C𝐶) = (0C0))
59		oveq1 7163	. . . . . . . . 9 ⊢ (𝐶 = 0 → (𝐶 + 1) = (0 + 1))
60	59, 4	eqtrdi 2809	. . . . . . . 8 ⊢ (𝐶 = 0 → (𝐶 + 1) = 1)
61	60	oveq2d 7172	. . . . . . 7 ⊢ (𝐶 = 0 → (1C(𝐶 + 1)) = (1C1))
62	57, 58, 61	3eqtr4a 2819	. . . . . 6 ⊢ (𝐶 = 0 → (0C𝐶) = (1C(𝐶 + 1)))
63	52, 62	jaoi 854	. . . . 5 ⊢ ((𝐶 ∈ ℕ ∨ 𝐶 = 0) → (0C𝐶) = (1C(𝐶 + 1)))
64	36, 63	sylbi 220	. . . 4 ⊢ (𝐶 ∈ ℕ₀ → (0C𝐶) = (1C(𝐶 + 1)))
65	35, 64	eqtrd 2793	. . 3 ⊢ (𝐶 ∈ ℕ₀ → Σ𝑘 ∈ (0...0)(𝑘C𝐶) = (1C(𝐶 + 1)))
66		elnn0uz 12336	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ₀ ↔ 𝑛 ∈ (ℤ_≥‘0))
67	66	biimpi 219	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ₀ → 𝑛 ∈ (ℤ_≥‘0))
68	67	adantr 484	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) → 𝑛 ∈ (ℤ_≥‘0))
69		elfznn0 13062	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (0...(𝑛 + 1)) → 𝑘 ∈ ℕ₀)
70	69	adantl 485	. . . . . . . . . 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 12137	. . . . . . . . . 10 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (0...(𝑛 + 1))) → 𝐶 ∈ ℤ)
73		bccl 13745	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℕ₀ ∧ 𝐶 ∈ ℤ) → (𝑘C𝐶) ∈ ℕ₀)
74	70, 72, 73	syl2anc 587	. . . . . . . . 9 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (0...(𝑛 + 1))) → (𝑘C𝐶) ∈ ℕ₀)
75	74	nn0cnd 12009	. . . . . . . 8 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (0...(𝑛 + 1))) → (𝑘C𝐶) ∈ ℂ)
76		oveq1 7163	. . . . . . . 8 ⊢ (𝑘 = (𝑛 + 1) → (𝑘C𝐶) = ((𝑛 + 1)C𝐶))
77	68, 75, 76	fsump1 15172	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) → Σ𝑘 ∈ (0...(𝑛 + 1))(𝑘C𝐶) = (Σ𝑘 ∈ (0...𝑛)(𝑘C𝐶) + ((𝑛 + 1)C𝐶)))
78	77	adantr 484	. . . . . 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 11957	. . . . . . . . . . 11 ⊢ (𝐶 ∈ ℕ₀ → 𝐶 ∈ ℂ)
81	80	adantl 485	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) → 𝐶 ∈ ℂ)
82		1cnd 10687	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) → 1 ∈ ℂ)
83	81, 82	pncand 11049	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) → ((𝐶 + 1) − 1) = 𝐶)
84	83	oveq2d 7172	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) → ((𝑛 + 1)C((𝐶 + 1) − 1)) = ((𝑛 + 1)C𝐶))
85	84	eqcomd 2764	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) → ((𝑛 + 1)C𝐶) = ((𝑛 + 1)C((𝐶 + 1) − 1)))
86	79, 85	oveqan12rd 7176	. . . . . 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 11987	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → (𝑛 + 1) ∈ ℕ₀)
88		peano2nn0 11987	. . . . . . . . 9 ⊢ (𝐶 ∈ ℕ₀ → (𝐶 + 1) ∈ ℕ₀)
89	88	nn0zd 12137	. . . . . . . 8 ⊢ (𝐶 ∈ ℕ₀ → (𝐶 + 1) ∈ ℤ)
90		bcpasc 13744	. . . . . . . 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 598	. . . . . . 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 484	. . . . . 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 2797	. . . . 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 423	. . . 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 12129	. 2 ⊢ (𝑁 ∈ ℕ₀ → (𝐶 ∈ ℕ₀ → Σ𝑘 ∈ (0...𝑁)(𝑘C𝐶) = ((𝑁 + 1)C(𝐶 + 1))))
97	96	imp 410	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 399 ∨ wo 844 = wceq 1538 ∈ wcel 2111 ifcif 4423 class class class wbr 5036 ‘cfv 6340 (class class class)co 7156 ℂcc 10586 0cc0 10588 1c1 10589 + caddc 10591 · cmul 10593 < clt 10726 ≤ cle 10727 − cmin 10921 / cdiv 11348 ℕcn 11687 ℕ₀cn0 11947 ℤcz 12033 ℤ_≥cuz 12295 ...cfz 12952 !cfa 13696 Ccbc 13725 Σcsu 15103

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-rep 5160 ax-sep 5173 ax-nul 5180 ax-pow 5238 ax-pr 5302 ax-un 7465 ax-inf2 9150 ax-cnex 10644 ax-resscn 10645 ax-1cn 10646 ax-icn 10647 ax-addcl 10648 ax-addrcl 10649 ax-mulcl 10650 ax-mulrcl 10651 ax-mulcom 10652 ax-addass 10653 ax-mulass 10654 ax-distr 10655 ax-i2m1 10656 ax-1ne0 10657 ax-1rid 10658 ax-rnegex 10659 ax-rrecex 10660 ax-cnre 10661 ax-pre-lttri 10662 ax-pre-lttrn 10663 ax-pre-ltadd 10664 ax-pre-mulgt0 10665 ax-pre-sup 10666

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-nel 3056 df-ral 3075 df-rex 3076 df-reu 3077 df-rmo 3078 df-rab 3079 df-v 3411 df-sbc 3699 df-csb 3808 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-pss 3879 df-nul 4228 df-if 4424 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4802 df-int 4842 df-iun 4888 df-br 5037 df-opab 5099 df-mpt 5117 df-tr 5143 df-id 5434 df-eprel 5439 df-po 5447 df-so 5448 df-fr 5487 df-se 5488 df-we 5489 df-xp 5534 df-rel 5535 df-cnv 5536 df-co 5537 df-dm 5538 df-rn 5539 df-res 5540 df-ima 5541 df-pred 6131 df-ord 6177 df-on 6178 df-lim 6179 df-suc 6180 df-iota 6299 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-isom 6349 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7586 df-1st 7699 df-2nd 7700 df-wrecs 7963 df-recs 8024 df-rdg 8062 df-1o 8118 df-er 8305 df-en 8541 df-dom 8542 df-sdom 8543 df-fin 8544 df-sup 8952 df-oi 9020 df-card 9414 df-pnf 10728 df-mnf 10729 df-xr 10730 df-ltxr 10731 df-le 10732 df-sub 10923 df-neg 10924 df-div 11349 df-nn 11688 df-2 11750 df-3 11751 df-n0 11948 df-z 12034 df-uz 12296 df-rp 12444 df-fz 12953 df-fzo 13096 df-seq 13432 df-exp 13493 df-fac 13697 df-bc 13726 df-hash 13754 df-cj 14519 df-re 14520 df-im 14521 df-sqrt 14655 df-abs 14656 df-clim 14906 df-sum 15104

This theorem is referenced by: (None)

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