bccolsum - Mathbox for Scott Fenton

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Theorem bccolsum 31625

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 6658	. . . . . 6 ⊢ (𝑚 = 0 → (0...𝑚) = (0...0))
2	1	sumeq1d 14431	. . . . 5 ⊢ (𝑚 = 0 → Σ𝑘 ∈ (0...𝑚)(𝑘C𝐶) = Σ𝑘 ∈ (0...0)(𝑘C𝐶))
3		oveq1 6657	. . . . . . 7 ⊢ (𝑚 = 0 → (𝑚 + 1) = (0 + 1))
4		0p1e1 11132	. . . . . . 7 ⊢ (0 + 1) = 1
5	3, 4	syl6eq 2672	. . . . . 6 ⊢ (𝑚 = 0 → (𝑚 + 1) = 1)
6	5	oveq1d 6665	. . . . 5 ⊢ (𝑚 = 0 → ((𝑚 + 1)C(𝐶 + 1)) = (1C(𝐶 + 1)))
7	2, 6	eqeq12d 2637	. . . 4 ⊢ (𝑚 = 0 → (Σ𝑘 ∈ (0...𝑚)(𝑘C𝐶) = ((𝑚 + 1)C(𝐶 + 1)) ↔ Σ𝑘 ∈ (0...0)(𝑘C𝐶) = (1C(𝐶 + 1))))
8	7	imbi2d 330	. . 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 6658	. . . . . 6 ⊢ (𝑚 = 𝑛 → (0...𝑚) = (0...𝑛))
10	9	sumeq1d 14431	. . . . 5 ⊢ (𝑚 = 𝑛 → Σ𝑘 ∈ (0...𝑚)(𝑘C𝐶) = Σ𝑘 ∈ (0...𝑛)(𝑘C𝐶))
11		oveq1 6657	. . . . . 6 ⊢ (𝑚 = 𝑛 → (𝑚 + 1) = (𝑛 + 1))
12	11	oveq1d 6665	. . . . 5 ⊢ (𝑚 = 𝑛 → ((𝑚 + 1)C(𝐶 + 1)) = ((𝑛 + 1)C(𝐶 + 1)))
13	10, 12	eqeq12d 2637	. . . 4 ⊢ (𝑚 = 𝑛 → (Σ𝑘 ∈ (0...𝑚)(𝑘C𝐶) = ((𝑚 + 1)C(𝐶 + 1)) ↔ Σ𝑘 ∈ (0...𝑛)(𝑘C𝐶) = ((𝑛 + 1)C(𝐶 + 1))))
14	13	imbi2d 330	. . 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 6658	. . . . . 6 ⊢ (𝑚 = (𝑛 + 1) → (0...𝑚) = (0...(𝑛 + 1)))
16	15	sumeq1d 14431	. . . . 5 ⊢ (𝑚 = (𝑛 + 1) → Σ𝑘 ∈ (0...𝑚)(𝑘C𝐶) = Σ𝑘 ∈ (0...(𝑛 + 1))(𝑘C𝐶))
17		oveq1 6657	. . . . . 6 ⊢ (𝑚 = (𝑛 + 1) → (𝑚 + 1) = ((𝑛 + 1) + 1))
18	17	oveq1d 6665	. . . . 5 ⊢ (𝑚 = (𝑛 + 1) → ((𝑚 + 1)C(𝐶 + 1)) = (((𝑛 + 1) + 1)C(𝐶 + 1)))
19	16, 18	eqeq12d 2637	. . . 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 330	. . 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 6658	. . . . . 6 ⊢ (𝑚 = 𝑁 → (0...𝑚) = (0...𝑁))
22	21	sumeq1d 14431	. . . . 5 ⊢ (𝑚 = 𝑁 → Σ𝑘 ∈ (0...𝑚)(𝑘C𝐶) = Σ𝑘 ∈ (0...𝑁)(𝑘C𝐶))
23		oveq1 6657	. . . . . 6 ⊢ (𝑚 = 𝑁 → (𝑚 + 1) = (𝑁 + 1))
24	23	oveq1d 6665	. . . . 5 ⊢ (𝑚 = 𝑁 → ((𝑚 + 1)C(𝐶 + 1)) = ((𝑁 + 1)C(𝐶 + 1)))
25	22, 24	eqeq12d 2637	. . . 4 ⊢ (𝑚 = 𝑁 → (Σ𝑘 ∈ (0...𝑚)(𝑘C𝐶) = ((𝑚 + 1)C(𝐶 + 1)) ↔ Σ𝑘 ∈ (0...𝑁)(𝑘C𝐶) = ((𝑁 + 1)C(𝐶 + 1))))
26	25	imbi2d 330	. . 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 11388	. . . . 5 ⊢ 0 ∈ ℤ
28		0nn0 11307	. . . . . . 7 ⊢ 0 ∈ ℕ₀
29		nn0z 11400	. . . . . . 7 ⊢ (𝐶 ∈ ℕ₀ → 𝐶 ∈ ℤ)
30		bccl 13109	. . . . . . 7 ⊢ ((0 ∈ ℕ₀ ∧ 𝐶 ∈ ℤ) → (0C𝐶) ∈ ℕ₀)
31	28, 29, 30	sylancr 695	. . . . . 6 ⊢ (𝐶 ∈ ℕ₀ → (0C𝐶) ∈ ℕ₀)
32	31	nn0cnd 11353	. . . . 5 ⊢ (𝐶 ∈ ℕ₀ → (0C𝐶) ∈ ℂ)
33		oveq1 6657	. . . . . 6 ⊢ (𝑘 = 0 → (𝑘C𝐶) = (0C𝐶))
34	33	fsum1 14476	. . . . 5 ⊢ ((0 ∈ ℤ ∧ (0C𝐶) ∈ ℂ) → Σ𝑘 ∈ (0...0)(𝑘C𝐶) = (0C𝐶))
35	27, 32, 34	sylancr 695	. . . 4 ⊢ (𝐶 ∈ ℕ₀ → Σ𝑘 ∈ (0...0)(𝑘C𝐶) = (0C𝐶))
36		elnn0 11294	. . . . 5 ⊢ (𝐶 ∈ ℕ₀ ↔ (𝐶 ∈ ℕ ∨ 𝐶 = 0))
37		1red 10055	. . . . . . . . . . 11 ⊢ (𝐶 ∈ ℕ → 1 ∈ ℝ)
38		nnrp 11842	. . . . . . . . . . 11 ⊢ (𝐶 ∈ ℕ → 𝐶 ∈ ℝ⁺)
39	37, 38	ltaddrp2d 11906	. . . . . . . . . 10 ⊢ (𝐶 ∈ ℕ → 1 < (𝐶 + 1))
40		peano2nn 11032	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ ℕ → (𝐶 + 1) ∈ ℕ)
41	40	nnred 11035	. . . . . . . . . . 11 ⊢ (𝐶 ∈ ℕ → (𝐶 + 1) ∈ ℝ)
42	37, 41	ltnled 10184	. . . . . . . . . 10 ⊢ (𝐶 ∈ ℕ → (1 < (𝐶 + 1) ↔ ¬ (𝐶 + 1) ≤ 1))
43	39, 42	mpbid 222	. . . . . . . . 9 ⊢ (𝐶 ∈ ℕ → ¬ (𝐶 + 1) ≤ 1)
44		elfzle2 12345	. . . . . . . . 9 ⊢ ((𝐶 + 1) ∈ (0...1) → (𝐶 + 1) ≤ 1)
45	43, 44	nsyl 135	. . . . . . . 8 ⊢ (𝐶 ∈ ℕ → ¬ (𝐶 + 1) ∈ (0...1))
46	45	iffalsed 4097	. . . . . . 7 ⊢ (𝐶 ∈ ℕ → if((𝐶 + 1) ∈ (0...1), ((!‘1) / ((!‘(1 − (𝐶 + 1))) · (!‘(𝐶 + 1)))), 0) = 0)
47		1nn0 11308	. . . . . . . 8 ⊢ 1 ∈ ℕ₀
48	40	nnzd 11481	. . . . . . . 8 ⊢ (𝐶 ∈ ℕ → (𝐶 + 1) ∈ ℤ)
49		bcval 13091	. . . . . . . 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 695	. . . . . . 7 ⊢ (𝐶 ∈ ℕ → (1C(𝐶 + 1)) = if((𝐶 + 1) ∈ (0...1), ((!‘1) / ((!‘(1 − (𝐶 + 1))) · (!‘(𝐶 + 1)))), 0))
51		bc0k 13098	. . . . . . 7 ⊢ (𝐶 ∈ ℕ → (0C𝐶) = 0)
52	46, 50, 51	3eqtr4rd 2667	. . . . . 6 ⊢ (𝐶 ∈ ℕ → (0C𝐶) = (1C(𝐶 + 1)))
53		bcnn 13099	. . . . . . . . 9 ⊢ (0 ∈ ℕ₀ → (0C0) = 1)
54	28, 53	ax-mp 5	. . . . . . . 8 ⊢ (0C0) = 1
55		bcnn 13099	. . . . . . . . 9 ⊢ (1 ∈ ℕ₀ → (1C1) = 1)
56	47, 55	ax-mp 5	. . . . . . . 8 ⊢ (1C1) = 1
57	54, 56	eqtr4i 2647	. . . . . . 7 ⊢ (0C0) = (1C1)
58		oveq2 6658	. . . . . . 7 ⊢ (𝐶 = 0 → (0C𝐶) = (0C0))
59		oveq1 6657	. . . . . . . . 9 ⊢ (𝐶 = 0 → (𝐶 + 1) = (0 + 1))
60	59, 4	syl6eq 2672	. . . . . . . 8 ⊢ (𝐶 = 0 → (𝐶 + 1) = 1)
61	60	oveq2d 6666	. . . . . . 7 ⊢ (𝐶 = 0 → (1C(𝐶 + 1)) = (1C1))
62	57, 58, 61	3eqtr4a 2682	. . . . . 6 ⊢ (𝐶 = 0 → (0C𝐶) = (1C(𝐶 + 1)))
63	52, 62	jaoi 394	. . . . 5 ⊢ ((𝐶 ∈ ℕ ∨ 𝐶 = 0) → (0C𝐶) = (1C(𝐶 + 1)))
64	36, 63	sylbi 207	. . . 4 ⊢ (𝐶 ∈ ℕ₀ → (0C𝐶) = (1C(𝐶 + 1)))
65	35, 64	eqtrd 2656	. . 3 ⊢ (𝐶 ∈ ℕ₀ → Σ𝑘 ∈ (0...0)(𝑘C𝐶) = (1C(𝐶 + 1)))
66		elnn0uz 11725	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ₀ ↔ 𝑛 ∈ (ℤ_≥‘0))
67	66	biimpi 206	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ₀ → 𝑛 ∈ (ℤ_≥‘0))
68	67	adantr 481	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) → 𝑛 ∈ (ℤ_≥‘0))
69		elfznn0 12433	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (0...(𝑛 + 1)) → 𝑘 ∈ ℕ₀)
70	69	adantl 482	. . . . . . . . . 10 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (0...(𝑛 + 1))) → 𝑘 ∈ ℕ₀)
71		simplr 792	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (0...(𝑛 + 1))) → 𝐶 ∈ ℕ₀)
72	71	nn0zd 11480	. . . . . . . . . 10 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (0...(𝑛 + 1))) → 𝐶 ∈ ℤ)
73		bccl 13109	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℕ₀ ∧ 𝐶 ∈ ℤ) → (𝑘C𝐶) ∈ ℕ₀)
74	70, 72, 73	syl2anc 693	. . . . . . . . 9 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (0...(𝑛 + 1))) → (𝑘C𝐶) ∈ ℕ₀)
75	74	nn0cnd 11353	. . . . . . . 8 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (0...(𝑛 + 1))) → (𝑘C𝐶) ∈ ℂ)
76		oveq1 6657	. . . . . . . 8 ⊢ (𝑘 = (𝑛 + 1) → (𝑘C𝐶) = ((𝑛 + 1)C𝐶))
77	68, 75, 76	fsump1 14487	. . . . . . 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 11302	. . . . . . . . . . 11 ⊢ (𝐶 ∈ ℕ₀ → 𝐶 ∈ ℂ)
81	80	adantl 482	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) → 𝐶 ∈ ℂ)
82		1cnd 10056	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) → 1 ∈ ℂ)
83	81, 82	pncand 10393	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) → ((𝐶 + 1) − 1) = 𝐶)
84	83	oveq2d 6666	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) → ((𝑛 + 1)C((𝐶 + 1) − 1)) = ((𝑛 + 1)C𝐶))
85	84	eqcomd 2628	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) → ((𝑛 + 1)C𝐶) = ((𝑛 + 1)C((𝐶 + 1) − 1)))
86	79, 85	oveqan12rd 6670	. . . . . 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 11333	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → (𝑛 + 1) ∈ ℕ₀)
88		peano2nn0 11333	. . . . . . . . 9 ⊢ (𝐶 ∈ ℕ₀ → (𝐶 + 1) ∈ ℕ₀)
89	88	nn0zd 11480	. . . . . . . 8 ⊢ (𝐶 ∈ ℕ₀ → (𝐶 + 1) ∈ ℤ)
90		bcpasc 13108	. . . . . . . 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 494	. . . . . . 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 2660	. . . . 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 630	. . . 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 11472	. 2 ⊢ (𝑁 ∈ ℕ₀ → (𝐶 ∈ ℕ₀ → Σ𝑘 ∈ (0...𝑁)(𝑘C𝐶) = ((𝑁 + 1)C(𝐶 + 1))))
97	96	imp 445	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 ∨ wo 383 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ifcif 4086 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 < clt 10074 ≤ cle 10075 − cmin 10266 / cdiv 10684 ℕcn 11020 ℕ₀cn0 11292 ℤcz 11377 ℤ_≥cuz 11687 ...cfz 12326 !cfa 13060 Ccbc 13089 Σcsu 14416

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-fz 12327 df-fzo 12466 df-seq 12802 df-exp 12861 df-fac 13061 df-bc 13090 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-sum 14417

This theorem is referenced by: (None)

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