bccolsum - Mathbox for Scott Fenton

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Theorem bccolsum 35719

Description: A column-sum rule for binomial coefficients. (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 7439	. . . . . 6 ⊢ (𝑚 = 0 → (0...𝑚) = (0...0))
2	1	sumeq1d 15733	. . . . 5 ⊢ (𝑚 = 0 → Σ𝑘 ∈ (0...𝑚)(𝑘C𝐶) = Σ𝑘 ∈ (0...0)(𝑘C𝐶))
3		oveq1 7438	. . . . . . 7 ⊢ (𝑚 = 0 → (𝑚 + 1) = (0 + 1))
4		0p1e1 12386	. . . . . . 7 ⊢ (0 + 1) = 1
5	3, 4	eqtrdi 2791	. . . . . 6 ⊢ (𝑚 = 0 → (𝑚 + 1) = 1)
6	5	oveq1d 7446	. . . . 5 ⊢ (𝑚 = 0 → ((𝑚 + 1)C(𝐶 + 1)) = (1C(𝐶 + 1)))
7	2, 6	eqeq12d 2751	. . . 4 ⊢ (𝑚 = 0 → (Σ𝑘 ∈ (0...𝑚)(𝑘C𝐶) = ((𝑚 + 1)C(𝐶 + 1)) ↔ Σ𝑘 ∈ (0...0)(𝑘C𝐶) = (1C(𝐶 + 1))))
8	7	imbi2d 340	. . 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 7439	. . . . . 6 ⊢ (𝑚 = 𝑛 → (0...𝑚) = (0...𝑛))
10	9	sumeq1d 15733	. . . . 5 ⊢ (𝑚 = 𝑛 → Σ𝑘 ∈ (0...𝑚)(𝑘C𝐶) = Σ𝑘 ∈ (0...𝑛)(𝑘C𝐶))
11		oveq1 7438	. . . . . 6 ⊢ (𝑚 = 𝑛 → (𝑚 + 1) = (𝑛 + 1))
12	11	oveq1d 7446	. . . . 5 ⊢ (𝑚 = 𝑛 → ((𝑚 + 1)C(𝐶 + 1)) = ((𝑛 + 1)C(𝐶 + 1)))
13	10, 12	eqeq12d 2751	. . . 4 ⊢ (𝑚 = 𝑛 → (Σ𝑘 ∈ (0...𝑚)(𝑘C𝐶) = ((𝑚 + 1)C(𝐶 + 1)) ↔ Σ𝑘 ∈ (0...𝑛)(𝑘C𝐶) = ((𝑛 + 1)C(𝐶 + 1))))
14	13	imbi2d 340	. . 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 7439	. . . . . 6 ⊢ (𝑚 = (𝑛 + 1) → (0...𝑚) = (0...(𝑛 + 1)))
16	15	sumeq1d 15733	. . . . 5 ⊢ (𝑚 = (𝑛 + 1) → Σ𝑘 ∈ (0...𝑚)(𝑘C𝐶) = Σ𝑘 ∈ (0...(𝑛 + 1))(𝑘C𝐶))
17		oveq1 7438	. . . . . 6 ⊢ (𝑚 = (𝑛 + 1) → (𝑚 + 1) = ((𝑛 + 1) + 1))
18	17	oveq1d 7446	. . . . 5 ⊢ (𝑚 = (𝑛 + 1) → ((𝑚 + 1)C(𝐶 + 1)) = (((𝑛 + 1) + 1)C(𝐶 + 1)))
19	16, 18	eqeq12d 2751	. . . 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 340	. . 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 7439	. . . . . 6 ⊢ (𝑚 = 𝑁 → (0...𝑚) = (0...𝑁))
22	21	sumeq1d 15733	. . . . 5 ⊢ (𝑚 = 𝑁 → Σ𝑘 ∈ (0...𝑚)(𝑘C𝐶) = Σ𝑘 ∈ (0...𝑁)(𝑘C𝐶))
23		oveq1 7438	. . . . . 6 ⊢ (𝑚 = 𝑁 → (𝑚 + 1) = (𝑁 + 1))
24	23	oveq1d 7446	. . . . 5 ⊢ (𝑚 = 𝑁 → ((𝑚 + 1)C(𝐶 + 1)) = ((𝑁 + 1)C(𝐶 + 1)))
25	22, 24	eqeq12d 2751	. . . 4 ⊢ (𝑚 = 𝑁 → (Σ𝑘 ∈ (0...𝑚)(𝑘C𝐶) = ((𝑚 + 1)C(𝐶 + 1)) ↔ Σ𝑘 ∈ (0...𝑁)(𝑘C𝐶) = ((𝑁 + 1)C(𝐶 + 1))))
26	25	imbi2d 340	. . 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 12622	. . . . 5 ⊢ 0 ∈ ℤ
28		0nn0 12539	. . . . . . 7 ⊢ 0 ∈ ℕ₀
29		nn0z 12636	. . . . . . 7 ⊢ (𝐶 ∈ ℕ₀ → 𝐶 ∈ ℤ)
30		bccl 14358	. . . . . . 7 ⊢ ((0 ∈ ℕ₀ ∧ 𝐶 ∈ ℤ) → (0C𝐶) ∈ ℕ₀)
31	28, 29, 30	sylancr 587	. . . . . 6 ⊢ (𝐶 ∈ ℕ₀ → (0C𝐶) ∈ ℕ₀)
32	31	nn0cnd 12587	. . . . 5 ⊢ (𝐶 ∈ ℕ₀ → (0C𝐶) ∈ ℂ)
33		oveq1 7438	. . . . . 6 ⊢ (𝑘 = 0 → (𝑘C𝐶) = (0C𝐶))
34	33	fsum1 15780	. . . . 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 12526	. . . . 5 ⊢ (𝐶 ∈ ℕ₀ ↔ (𝐶 ∈ ℕ ∨ 𝐶 = 0))
37		1red 11260	. . . . . . . . . . 11 ⊢ (𝐶 ∈ ℕ → 1 ∈ ℝ)
38		nnrp 13044	. . . . . . . . . . 11 ⊢ (𝐶 ∈ ℕ → 𝐶 ∈ ℝ⁺)
39	37, 38	ltaddrp2d 13109	. . . . . . . . . 10 ⊢ (𝐶 ∈ ℕ → 1 < (𝐶 + 1))
40		peano2nn 12276	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ ℕ → (𝐶 + 1) ∈ ℕ)
41	40	nnred 12279	. . . . . . . . . . 11 ⊢ (𝐶 ∈ ℕ → (𝐶 + 1) ∈ ℝ)
42	37, 41	ltnled 11406	. . . . . . . . . 10 ⊢ (𝐶 ∈ ℕ → (1 < (𝐶 + 1) ↔ ¬ (𝐶 + 1) ≤ 1))
43	39, 42	mpbid 232	. . . . . . . . 9 ⊢ (𝐶 ∈ ℕ → ¬ (𝐶 + 1) ≤ 1)
44		elfzle2 13565	. . . . . . . . 9 ⊢ ((𝐶 + 1) ∈ (0...1) → (𝐶 + 1) ≤ 1)
45	43, 44	nsyl 140	. . . . . . . 8 ⊢ (𝐶 ∈ ℕ → ¬ (𝐶 + 1) ∈ (0...1))
46	45	iffalsed 4542	. . . . . . 7 ⊢ (𝐶 ∈ ℕ → if((𝐶 + 1) ∈ (0...1), ((!‘1) / ((!‘(1 − (𝐶 + 1))) · (!‘(𝐶 + 1)))), 0) = 0)
47		1nn0 12540	. . . . . . . 8 ⊢ 1 ∈ ℕ₀
48	40	nnzd 12638	. . . . . . . 8 ⊢ (𝐶 ∈ ℕ → (𝐶 + 1) ∈ ℤ)
49		bcval 14340	. . . . . . . 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 14347	. . . . . . 7 ⊢ (𝐶 ∈ ℕ → (0C𝐶) = 0)
52	46, 50, 51	3eqtr4rd 2786	. . . . . 6 ⊢ (𝐶 ∈ ℕ → (0C𝐶) = (1C(𝐶 + 1)))
53		bcnn 14348	. . . . . . . . 9 ⊢ (0 ∈ ℕ₀ → (0C0) = 1)
54	28, 53	ax-mp 5	. . . . . . . 8 ⊢ (0C0) = 1
55		bcnn 14348	. . . . . . . . 9 ⊢ (1 ∈ ℕ₀ → (1C1) = 1)
56	47, 55	ax-mp 5	. . . . . . . 8 ⊢ (1C1) = 1
57	54, 56	eqtr4i 2766	. . . . . . 7 ⊢ (0C0) = (1C1)
58		oveq2 7439	. . . . . . 7 ⊢ (𝐶 = 0 → (0C𝐶) = (0C0))
59		oveq1 7438	. . . . . . . . 9 ⊢ (𝐶 = 0 → (𝐶 + 1) = (0 + 1))
60	59, 4	eqtrdi 2791	. . . . . . . 8 ⊢ (𝐶 = 0 → (𝐶 + 1) = 1)
61	60	oveq2d 7447	. . . . . . 7 ⊢ (𝐶 = 0 → (1C(𝐶 + 1)) = (1C1))
62	57, 58, 61	3eqtr4a 2801	. . . . . 6 ⊢ (𝐶 = 0 → (0C𝐶) = (1C(𝐶 + 1)))
63	52, 62	jaoi 857	. . . . 5 ⊢ ((𝐶 ∈ ℕ ∨ 𝐶 = 0) → (0C𝐶) = (1C(𝐶 + 1)))
64	36, 63	sylbi 217	. . . 4 ⊢ (𝐶 ∈ ℕ₀ → (0C𝐶) = (1C(𝐶 + 1)))
65	35, 64	eqtrd 2775	. . 3 ⊢ (𝐶 ∈ ℕ₀ → Σ𝑘 ∈ (0...0)(𝑘C𝐶) = (1C(𝐶 + 1)))
66		elnn0uz 12921	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ₀ ↔ 𝑛 ∈ (ℤ_≥‘0))
67	66	biimpi 216	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ₀ → 𝑛 ∈ (ℤ_≥‘0))
68	67	adantr 480	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) → 𝑛 ∈ (ℤ_≥‘0))
69		elfznn0 13657	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (0...(𝑛 + 1)) → 𝑘 ∈ ℕ₀)
70	69	adantl 481	. . . . . . . . . 10 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (0...(𝑛 + 1))) → 𝑘 ∈ ℕ₀)
71		simplr 769	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (0...(𝑛 + 1))) → 𝐶 ∈ ℕ₀)
72	71	nn0zd 12637	. . . . . . . . . 10 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (0...(𝑛 + 1))) → 𝐶 ∈ ℤ)
73		bccl 14358	. . . . . . . . . 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 12587	. . . . . . . 8 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (0...(𝑛 + 1))) → (𝑘C𝐶) ∈ ℂ)
76		oveq1 7438	. . . . . . . 8 ⊢ (𝑘 = (𝑛 + 1) → (𝑘C𝐶) = ((𝑛 + 1)C𝐶))
77	68, 75, 76	fsump1 15789	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) → Σ𝑘 ∈ (0...(𝑛 + 1))(𝑘C𝐶) = (Σ𝑘 ∈ (0...𝑛)(𝑘C𝐶) + ((𝑛 + 1)C𝐶)))
78	77	adantr 480	. . . . . 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 12534	. . . . . . . . . . 11 ⊢ (𝐶 ∈ ℕ₀ → 𝐶 ∈ ℂ)
81	80	adantl 481	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) → 𝐶 ∈ ℂ)
82		1cnd 11254	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) → 1 ∈ ℂ)
83	81, 82	pncand 11619	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) → ((𝐶 + 1) − 1) = 𝐶)
84	83	oveq2d 7447	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) → ((𝑛 + 1)C((𝐶 + 1) − 1)) = ((𝑛 + 1)C𝐶))
85	84	eqcomd 2741	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) → ((𝑛 + 1)C𝐶) = ((𝑛 + 1)C((𝐶 + 1) − 1)))
86	79, 85	oveqan12rd 7451	. . . . . 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 12564	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → (𝑛 + 1) ∈ ℕ₀)
88		peano2nn0 12564	. . . . . . . . 9 ⊢ (𝐶 ∈ ℕ₀ → (𝐶 + 1) ∈ ℕ₀)
89	88	nn0zd 12637	. . . . . . . 8 ⊢ (𝐶 ∈ ℕ₀ → (𝐶 + 1) ∈ ℤ)
90		bcpasc 14357	. . . . . . . 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 480	. . . . . 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 2779	. . . . 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 419	. . . 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 12711	. 2 ⊢ (𝑁 ∈ ℕ₀ → (𝐶 ∈ ℕ₀ → Σ𝑘 ∈ (0...𝑁)(𝑘C𝐶) = ((𝑁 + 1)C(𝐶 + 1))))
97	96	imp 406	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 395 ∨ wo 847 = wceq 1537 ∈ wcel 2106 ifcif 4531 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 0cc0 11153 1c1 11154 + caddc 11156 · cmul 11158 < clt 11293 ≤ cle 11294 − cmin 11490 / cdiv 11918 ℕcn 12264 ℕ₀cn0 12524 ℤcz 12611 ℤ_≥cuz 12876 ...cfz 13544 !cfa 14309 Ccbc 14338 Σcsu 15719

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-fz 13545 df-fzo 13692 df-seq 14040 df-exp 14100 df-fac 14310 df-bc 14339 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-clim 15521 df-sum 15720

This theorem is referenced by: (None)

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