bccolsum - Mathbox for Scott Fenton

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Theorem bccolsum 33705

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 7283	. . . . . 6 ⊢ (𝑚 = 0 → (0...𝑚) = (0...0))
2	1	sumeq1d 15413	. . . . 5 ⊢ (𝑚 = 0 → Σ𝑘 ∈ (0...𝑚)(𝑘C𝐶) = Σ𝑘 ∈ (0...0)(𝑘C𝐶))
3		oveq1 7282	. . . . . . 7 ⊢ (𝑚 = 0 → (𝑚 + 1) = (0 + 1))
4		0p1e1 12095	. . . . . . 7 ⊢ (0 + 1) = 1
5	3, 4	eqtrdi 2794	. . . . . 6 ⊢ (𝑚 = 0 → (𝑚 + 1) = 1)
6	5	oveq1d 7290	. . . . 5 ⊢ (𝑚 = 0 → ((𝑚 + 1)C(𝐶 + 1)) = (1C(𝐶 + 1)))
7	2, 6	eqeq12d 2754	. . . 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 7283	. . . . . 6 ⊢ (𝑚 = 𝑛 → (0...𝑚) = (0...𝑛))
10	9	sumeq1d 15413	. . . . 5 ⊢ (𝑚 = 𝑛 → Σ𝑘 ∈ (0...𝑚)(𝑘C𝐶) = Σ𝑘 ∈ (0...𝑛)(𝑘C𝐶))
11		oveq1 7282	. . . . . 6 ⊢ (𝑚 = 𝑛 → (𝑚 + 1) = (𝑛 + 1))
12	11	oveq1d 7290	. . . . 5 ⊢ (𝑚 = 𝑛 → ((𝑚 + 1)C(𝐶 + 1)) = ((𝑛 + 1)C(𝐶 + 1)))
13	10, 12	eqeq12d 2754	. . . 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 7283	. . . . . 6 ⊢ (𝑚 = (𝑛 + 1) → (0...𝑚) = (0...(𝑛 + 1)))
16	15	sumeq1d 15413	. . . . 5 ⊢ (𝑚 = (𝑛 + 1) → Σ𝑘 ∈ (0...𝑚)(𝑘C𝐶) = Σ𝑘 ∈ (0...(𝑛 + 1))(𝑘C𝐶))
17		oveq1 7282	. . . . . 6 ⊢ (𝑚 = (𝑛 + 1) → (𝑚 + 1) = ((𝑛 + 1) + 1))
18	17	oveq1d 7290	. . . . 5 ⊢ (𝑚 = (𝑛 + 1) → ((𝑚 + 1)C(𝐶 + 1)) = (((𝑛 + 1) + 1)C(𝐶 + 1)))
19	16, 18	eqeq12d 2754	. . . 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 7283	. . . . . 6 ⊢ (𝑚 = 𝑁 → (0...𝑚) = (0...𝑁))
22	21	sumeq1d 15413	. . . . 5 ⊢ (𝑚 = 𝑁 → Σ𝑘 ∈ (0...𝑚)(𝑘C𝐶) = Σ𝑘 ∈ (0...𝑁)(𝑘C𝐶))
23		oveq1 7282	. . . . . 6 ⊢ (𝑚 = 𝑁 → (𝑚 + 1) = (𝑁 + 1))
24	23	oveq1d 7290	. . . . 5 ⊢ (𝑚 = 𝑁 → ((𝑚 + 1)C(𝐶 + 1)) = ((𝑁 + 1)C(𝐶 + 1)))
25	22, 24	eqeq12d 2754	. . . 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 12330	. . . . 5 ⊢ 0 ∈ ℤ
28		0nn0 12248	. . . . . . 7 ⊢ 0 ∈ ℕ₀
29		nn0z 12343	. . . . . . 7 ⊢ (𝐶 ∈ ℕ₀ → 𝐶 ∈ ℤ)
30		bccl 14036	. . . . . . 7 ⊢ ((0 ∈ ℕ₀ ∧ 𝐶 ∈ ℤ) → (0C𝐶) ∈ ℕ₀)
31	28, 29, 30	sylancr 587	. . . . . 6 ⊢ (𝐶 ∈ ℕ₀ → (0C𝐶) ∈ ℕ₀)
32	31	nn0cnd 12295	. . . . 5 ⊢ (𝐶 ∈ ℕ₀ → (0C𝐶) ∈ ℂ)
33		oveq1 7282	. . . . . 6 ⊢ (𝑘 = 0 → (𝑘C𝐶) = (0C𝐶))
34	33	fsum1 15459	. . . . 5 ⊢ ((0 ∈ ℤ ∧ (0C𝐶) ∈ ℂ) → Σ𝑘 ∈ (0...0)(𝑘C𝐶) = (0C𝐶))
35	27, 32, 34	sylancr 587	. . . 4 ⊢ (𝐶 ∈ ℕ₀ → Σ𝑘 ∈ (0...0)(𝑘C𝐶) = (0C𝐶))
36		elnn0 12235	. . . . 5 ⊢ (𝐶 ∈ ℕ₀ ↔ (𝐶 ∈ ℕ ∨ 𝐶 = 0))
37		1red 10976	. . . . . . . . . . 11 ⊢ (𝐶 ∈ ℕ → 1 ∈ ℝ)
38		nnrp 12741	. . . . . . . . . . 11 ⊢ (𝐶 ∈ ℕ → 𝐶 ∈ ℝ⁺)
39	37, 38	ltaddrp2d 12806	. . . . . . . . . 10 ⊢ (𝐶 ∈ ℕ → 1 < (𝐶 + 1))
40		peano2nn 11985	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ ℕ → (𝐶 + 1) ∈ ℕ)
41	40	nnred 11988	. . . . . . . . . . 11 ⊢ (𝐶 ∈ ℕ → (𝐶 + 1) ∈ ℝ)
42	37, 41	ltnled 11122	. . . . . . . . . 10 ⊢ (𝐶 ∈ ℕ → (1 < (𝐶 + 1) ↔ ¬ (𝐶 + 1) ≤ 1))
43	39, 42	mpbid 231	. . . . . . . . 9 ⊢ (𝐶 ∈ ℕ → ¬ (𝐶 + 1) ≤ 1)
44		elfzle2 13260	. . . . . . . . 9 ⊢ ((𝐶 + 1) ∈ (0...1) → (𝐶 + 1) ≤ 1)
45	43, 44	nsyl 140	. . . . . . . 8 ⊢ (𝐶 ∈ ℕ → ¬ (𝐶 + 1) ∈ (0...1))
46	45	iffalsed 4470	. . . . . . 7 ⊢ (𝐶 ∈ ℕ → if((𝐶 + 1) ∈ (0...1), ((!‘1) / ((!‘(1 − (𝐶 + 1))) · (!‘(𝐶 + 1)))), 0) = 0)
47		1nn0 12249	. . . . . . . 8 ⊢ 1 ∈ ℕ₀
48	40	nnzd 12425	. . . . . . . 8 ⊢ (𝐶 ∈ ℕ → (𝐶 + 1) ∈ ℤ)
49		bcval 14018	. . . . . . . 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 587	. . . . . . 7 ⊢ (𝐶 ∈ ℕ → (1C(𝐶 + 1)) = if((𝐶 + 1) ∈ (0...1), ((!‘1) / ((!‘(1 − (𝐶 + 1))) · (!‘(𝐶 + 1)))), 0))
51		bc0k 14025	. . . . . . 7 ⊢ (𝐶 ∈ ℕ → (0C𝐶) = 0)
52	46, 50, 51	3eqtr4rd 2789	. . . . . 6 ⊢ (𝐶 ∈ ℕ → (0C𝐶) = (1C(𝐶 + 1)))
53		bcnn 14026	. . . . . . . . 9 ⊢ (0 ∈ ℕ₀ → (0C0) = 1)
54	28, 53	ax-mp 5	. . . . . . . 8 ⊢ (0C0) = 1
55		bcnn 14026	. . . . . . . . 9 ⊢ (1 ∈ ℕ₀ → (1C1) = 1)
56	47, 55	ax-mp 5	. . . . . . . 8 ⊢ (1C1) = 1
57	54, 56	eqtr4i 2769	. . . . . . 7 ⊢ (0C0) = (1C1)
58		oveq2 7283	. . . . . . 7 ⊢ (𝐶 = 0 → (0C𝐶) = (0C0))
59		oveq1 7282	. . . . . . . . 9 ⊢ (𝐶 = 0 → (𝐶 + 1) = (0 + 1))
60	59, 4	eqtrdi 2794	. . . . . . . 8 ⊢ (𝐶 = 0 → (𝐶 + 1) = 1)
61	60	oveq2d 7291	. . . . . . 7 ⊢ (𝐶 = 0 → (1C(𝐶 + 1)) = (1C1))
62	57, 58, 61	3eqtr4a 2804	. . . . . 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 216	. . . 4 ⊢ (𝐶 ∈ ℕ₀ → (0C𝐶) = (1C(𝐶 + 1)))
65	35, 64	eqtrd 2778	. . 3 ⊢ (𝐶 ∈ ℕ₀ → Σ𝑘 ∈ (0...0)(𝑘C𝐶) = (1C(𝐶 + 1)))
66		elnn0uz 12623	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ₀ ↔ 𝑛 ∈ (ℤ_≥‘0))
67	66	biimpi 215	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ₀ → 𝑛 ∈ (ℤ_≥‘0))
68	67	adantr 481	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) → 𝑛 ∈ (ℤ_≥‘0))
69		elfznn0 13349	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (0...(𝑛 + 1)) → 𝑘 ∈ ℕ₀)
70	69	adantl 482	. . . . . . . . . 10 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (0...(𝑛 + 1))) → 𝑘 ∈ ℕ₀)
71		simplr 766	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (0...(𝑛 + 1))) → 𝐶 ∈ ℕ₀)
72	71	nn0zd 12424	. . . . . . . . . 10 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (0...(𝑛 + 1))) → 𝐶 ∈ ℤ)
73		bccl 14036	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℕ₀ ∧ 𝐶 ∈ ℤ) → (𝑘C𝐶) ∈ ℕ₀)
74	70, 72, 73	syl2anc 584	. . . . . . . . 9 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (0...(𝑛 + 1))) → (𝑘C𝐶) ∈ ℕ₀)
75	74	nn0cnd 12295	. . . . . . . 8 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (0...(𝑛 + 1))) → (𝑘C𝐶) ∈ ℂ)
76		oveq1 7282	. . . . . . . 8 ⊢ (𝑘 = (𝑛 + 1) → (𝑘C𝐶) = ((𝑛 + 1)C𝐶))
77	68, 75, 76	fsump1 15468	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) → Σ𝑘 ∈ (0...(𝑛 + 1))(𝑘C𝐶) = (Σ𝑘 ∈ (0...𝑛)(𝑘C𝐶) + ((𝑛 + 1)C𝐶)))
78	77	adantr 481	. . . . . 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 12243	. . . . . . . . . . 11 ⊢ (𝐶 ∈ ℕ₀ → 𝐶 ∈ ℂ)
81	80	adantl 482	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) → 𝐶 ∈ ℂ)
82		1cnd 10970	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) → 1 ∈ ℂ)
83	81, 82	pncand 11333	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) → ((𝐶 + 1) − 1) = 𝐶)
84	83	oveq2d 7291	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) → ((𝑛 + 1)C((𝐶 + 1) − 1)) = ((𝑛 + 1)C𝐶))
85	84	eqcomd 2744	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) → ((𝑛 + 1)C𝐶) = ((𝑛 + 1)C((𝐶 + 1) − 1)))
86	79, 85	oveqan12rd 7295	. . . . . 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 12273	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → (𝑛 + 1) ∈ ℕ₀)
88		peano2nn0 12273	. . . . . . . . 9 ⊢ (𝐶 ∈ ℕ₀ → (𝐶 + 1) ∈ ℕ₀)
89	88	nn0zd 12424	. . . . . . . 8 ⊢ (𝐶 ∈ ℕ₀ → (𝐶 + 1) ∈ ℤ)
90		bcpasc 14035	. . . . . . . 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 596	. . . . . . 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 481	. . . . . 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 2782	. . . . 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 420	. . . 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 12415	. 2 ⊢ (𝑁 ∈ ℕ₀ → (𝐶 ∈ ℕ₀ → Σ𝑘 ∈ (0...𝑁)(𝑘C𝐶) = ((𝑁 + 1)C(𝐶 + 1))))
97	96	imp 407	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 396 ∨ wo 844 = wceq 1539 ∈ wcel 2106 ifcif 4459 class class class wbr 5074 ‘cfv 6433 (class class class)co 7275 ℂcc 10869 0cc0 10871 1c1 10872 + caddc 10874 · cmul 10876 < clt 11009 ≤ cle 11010 − cmin 11205 / cdiv 11632 ℕcn 11973 ℕ₀cn0 12233 ℤcz 12319 ℤ_≥cuz 12582 ...cfz 13239 !cfa 13987 Ccbc 14016 Σcsu 15397

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-sup 9201 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-n0 12234 df-z 12320 df-uz 12583 df-rp 12731 df-fz 13240 df-fzo 13383 df-seq 13722 df-exp 13783 df-fac 13988 df-bc 14017 df-hash 14045 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-clim 15197 df-sum 15398

This theorem is referenced by: (None)

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