binomcxplemnn0 - Mathbox for Steve Rodriguez

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Theorem binomcxplemnn0 43108

Description: Lemma for binomcxp 43116. When 𝐶 is a nonnegative integer, the binomial's finite sum value by the standard binomial theorem binom 15776 equals this generalized infinite sum: the generalized binomial coefficient and exponentiation operators give exactly the same values in the standard index set (0...𝐶), and when the index set is widened beyond 𝐶 the additional values are just zeroes. (Contributed by Steve Rodriguez, 22-Apr-2020.)

**Hypotheses**
Ref	Expression
binomcxp.a	⊢ (𝜑 → 𝐴 ∈ ℝ⁺)
binomcxp.b	⊢ (𝜑 → 𝐵 ∈ ℝ)
binomcxp.lt	⊢ (𝜑 → (abs‘𝐵) < (abs‘𝐴))
binomcxp.c	⊢ (𝜑 → 𝐶 ∈ ℂ)

**Assertion**
Ref	Expression
binomcxplemnn0	⊢ ((𝜑 ∧ 𝐶 ∈ ℕ₀) → ((𝐴 + 𝐵)↑_𝑐𝐶) = Σ𝑘 ∈ ℕ₀ ((𝐶C_𝑐𝑘) · ((𝐴↑_𝑐(𝐶 − 𝑘)) · (𝐵↑𝑘))))

Distinct variable groups: 𝜑,𝑘 𝐴,𝑘 𝐵,𝑘 𝐶,𝑘

Proof of Theorem binomcxplemnn0 Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		binomcxp.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ⁺)
2	1	rpcnd 13018	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℂ)
3		binomcxp.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℝ)
4	3	recnd 11242	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℂ)
5		binom 15776	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℕ₀) → ((𝐴 + 𝐵)↑𝐶) = Σ𝑘 ∈ (0...𝐶)((𝐶C𝑘) · ((𝐴↑(𝐶 − 𝑘)) · (𝐵↑𝑘))))
6	5	3expia 1122	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐶 ∈ ℕ₀ → ((𝐴 + 𝐵)↑𝐶) = Σ𝑘 ∈ (0...𝐶)((𝐶C𝑘) · ((𝐴↑(𝐶 − 𝑘)) · (𝐵↑𝑘)))))
7	2, 4, 6	syl2anc 585	. . . . . 6 ⊢ (𝜑 → (𝐶 ∈ ℕ₀ → ((𝐴 + 𝐵)↑𝐶) = Σ𝑘 ∈ (0...𝐶)((𝐶C𝑘) · ((𝐴↑(𝐶 − 𝑘)) · (𝐵↑𝑘)))))
8	7	imp 408	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ ℕ₀) → ((𝐴 + 𝐵)↑𝐶) = Σ𝑘 ∈ (0...𝐶)((𝐶C𝑘) · ((𝐴↑(𝐶 − 𝑘)) · (𝐵↑𝑘))))
9	2	adantr 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ ℕ₀) → 𝐴 ∈ ℂ)
10	4	adantr 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ ℕ₀) → 𝐵 ∈ ℂ)
11	9, 10	addcld 11233	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 ∈ ℕ₀) → (𝐴 + 𝐵) ∈ ℂ)
12		simpr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 ∈ ℕ₀) → 𝐶 ∈ ℕ₀)
13		cxpexp 26176	. . . . . 6 ⊢ (((𝐴 + 𝐵) ∈ ℂ ∧ 𝐶 ∈ ℕ₀) → ((𝐴 + 𝐵)↑_𝑐𝐶) = ((𝐴 + 𝐵)↑𝐶))
14	11, 12, 13	syl2anc 585	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ ℕ₀) → ((𝐴 + 𝐵)↑_𝑐𝐶) = ((𝐴 + 𝐵)↑𝐶))
15		elfznn0 13594	. . . . . . . 8 ⊢ (𝑘 ∈ (0...𝐶) → 𝑘 ∈ ℕ₀)
16		simplr 768	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → 𝐶 ∈ ℕ₀)
17		simpr 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → 𝑘 ∈ ℕ₀)
18	16, 17	bccbc 43104	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → (𝐶C_𝑐𝑘) = (𝐶C𝑘))
19	15, 18	sylan2 594	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝐶)) → (𝐶C_𝑐𝑘) = (𝐶C𝑘))
20	2	ad2antrr 725	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝐶)) → 𝐴 ∈ ℂ)
21		elfzle2 13505	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (0...𝐶) → 𝑘 ≤ 𝐶)
22	21	adantl 483	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝐶)) → 𝑘 ≤ 𝐶)
23		nn0sub 12522	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) → (𝑘 ≤ 𝐶 ↔ (𝐶 − 𝑘) ∈ ℕ₀))
24	23	ancoms 460	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) → (𝑘 ≤ 𝐶 ↔ (𝐶 − 𝑘) ∈ ℕ₀))
25	24	adantll 713	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → (𝑘 ≤ 𝐶 ↔ (𝐶 − 𝑘) ∈ ℕ₀))
26	15, 25	sylan2 594	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝐶)) → (𝑘 ≤ 𝐶 ↔ (𝐶 − 𝑘) ∈ ℕ₀))
27	22, 26	mpbid 231	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝐶)) → (𝐶 − 𝑘) ∈ ℕ₀)
28		cxpexp 26176	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (𝐶 − 𝑘) ∈ ℕ₀) → (𝐴↑_𝑐(𝐶 − 𝑘)) = (𝐴↑(𝐶 − 𝑘)))
29	20, 27, 28	syl2anc 585	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝐶)) → (𝐴↑_𝑐(𝐶 − 𝑘)) = (𝐴↑(𝐶 − 𝑘)))
30	29	oveq1d 7424	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝐶)) → ((𝐴↑_𝑐(𝐶 − 𝑘)) · (𝐵↑𝑘)) = ((𝐴↑(𝐶 − 𝑘)) · (𝐵↑𝑘)))
31	19, 30	oveq12d 7427	. . . . . 6 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝐶)) → ((𝐶C_𝑐𝑘) · ((𝐴↑_𝑐(𝐶 − 𝑘)) · (𝐵↑𝑘))) = ((𝐶C𝑘) · ((𝐴↑(𝐶 − 𝑘)) · (𝐵↑𝑘))))
32	31	sumeq2dv 15649	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ ℕ₀) → Σ𝑘 ∈ (0...𝐶)((𝐶C_𝑐𝑘) · ((𝐴↑_𝑐(𝐶 − 𝑘)) · (𝐵↑𝑘))) = Σ𝑘 ∈ (0...𝐶)((𝐶C𝑘) · ((𝐴↑(𝐶 − 𝑘)) · (𝐵↑𝑘))))
33	8, 14, 32	3eqtr4d 2783	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ ℕ₀) → ((𝐴 + 𝐵)↑_𝑐𝐶) = Σ𝑘 ∈ (0...𝐶)((𝐶C_𝑐𝑘) · ((𝐴↑_𝑐(𝐶 − 𝑘)) · (𝐵↑𝑘))))
34		binomcxp.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ℂ)
35	34	adantr 482	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ ℕ₀) → 𝐶 ∈ ℂ)
36	11, 35	cxpcld 26216	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ ℕ₀) → ((𝐴 + 𝐵)↑_𝑐𝐶) ∈ ℂ)
37	33, 36	eqeltrrd 2835	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ ℕ₀) → Σ𝑘 ∈ (0...𝐶)((𝐶C_𝑐𝑘) · ((𝐴↑_𝑐(𝐶 − 𝑘)) · (𝐵↑𝑘))) ∈ ℂ)
38	37	addridd 11414	. 2 ⊢ ((𝜑 ∧ 𝐶 ∈ ℕ₀) → (Σ𝑘 ∈ (0...𝐶)((𝐶C_𝑐𝑘) · ((𝐴↑_𝑐(𝐶 − 𝑘)) · (𝐵↑𝑘))) + 0) = Σ𝑘 ∈ (0...𝐶)((𝐶C_𝑐𝑘) · ((𝐴↑_𝑐(𝐶 − 𝑘)) · (𝐵↑𝑘))))
39		nn0uz 12864	. . . 4 ⊢ ℕ₀ = (ℤ_≥‘0)
40		eqid 2733	. . . 4 ⊢ (ℤ_≥‘(𝐶 + 1)) = (ℤ_≥‘(𝐶 + 1))
41		1nn0 12488	. . . . . 6 ⊢ 1 ∈ ℕ₀
42	41	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ ℕ₀) → 1 ∈ ℕ₀)
43	12, 42	nn0addcld 12536	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ ℕ₀) → (𝐶 + 1) ∈ ℕ₀)
44		eqidd 2734	. . . . 5 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → (𝑗 ∈ ℕ₀ ↦ ((𝐶C_𝑐𝑗) · ((𝐴↑_𝑐(𝐶 − 𝑗)) · (𝐵↑𝑗)))) = (𝑗 ∈ ℕ₀ ↦ ((𝐶C_𝑐𝑗) · ((𝐴↑_𝑐(𝐶 − 𝑗)) · (𝐵↑𝑗)))))
45		simpr 486	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑗 = 𝑘) → 𝑗 = 𝑘)
46	45	oveq2d 7425	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑗 = 𝑘) → (𝐶C_𝑐𝑗) = (𝐶C_𝑐𝑘))
47	45	oveq2d 7425	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑗 = 𝑘) → (𝐶 − 𝑗) = (𝐶 − 𝑘))
48	47	oveq2d 7425	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑗 = 𝑘) → (𝐴↑_𝑐(𝐶 − 𝑗)) = (𝐴↑_𝑐(𝐶 − 𝑘)))
49	45	oveq2d 7425	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑗 = 𝑘) → (𝐵↑𝑗) = (𝐵↑𝑘))
50	48, 49	oveq12d 7427	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑗 = 𝑘) → ((𝐴↑_𝑐(𝐶 − 𝑗)) · (𝐵↑𝑗)) = ((𝐴↑_𝑐(𝐶 − 𝑘)) · (𝐵↑𝑘)))
51	46, 50	oveq12d 7427	. . . . 5 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑗 = 𝑘) → ((𝐶C_𝑐𝑗) · ((𝐴↑_𝑐(𝐶 − 𝑗)) · (𝐵↑𝑗))) = ((𝐶C_𝑐𝑘) · ((𝐴↑_𝑐(𝐶 − 𝑘)) · (𝐵↑𝑘))))
52	34	ad2antrr 725	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → 𝐶 ∈ ℂ)
53	52, 17	bcccl 43098	. . . . . 6 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → (𝐶C_𝑐𝑘) ∈ ℂ)
54	2	ad2antrr 725	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → 𝐴 ∈ ℂ)
55	17	nn0cnd 12534	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → 𝑘 ∈ ℂ)
56	52, 55	subcld 11571	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → (𝐶 − 𝑘) ∈ ℂ)
57	54, 56	cxpcld 26216	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → (𝐴↑_𝑐(𝐶 − 𝑘)) ∈ ℂ)
58	4	ad2antrr 725	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → 𝐵 ∈ ℂ)
59	58, 17	expcld 14111	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → (𝐵↑𝑘) ∈ ℂ)
60	57, 59	mulcld 11234	. . . . . 6 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → ((𝐴↑_𝑐(𝐶 − 𝑘)) · (𝐵↑𝑘)) ∈ ℂ)
61	53, 60	mulcld 11234	. . . . 5 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → ((𝐶C_𝑐𝑘) · ((𝐴↑_𝑐(𝐶 − 𝑘)) · (𝐵↑𝑘))) ∈ ℂ)
62	44, 51, 17, 61	fvmptd 7006	. . . 4 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → ((𝑗 ∈ ℕ₀ ↦ ((𝐶C_𝑐𝑗) · ((𝐴↑_𝑐(𝐶 − 𝑗)) · (𝐵↑𝑗))))‘𝑘) = ((𝐶C_𝑐𝑘) · ((𝐴↑_𝑐(𝐶 − 𝑘)) · (𝐵↑𝑘))))
63		peano2nn0 12512	. . . . . 6 ⊢ (𝐶 ∈ ℕ₀ → (𝐶 + 1) ∈ ℕ₀)
64	63	adantl 483	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ ℕ₀) → (𝐶 + 1) ∈ ℕ₀)
65		c0ex 11208	. . . . . . . . 9 ⊢ 0 ∈ V
66	65	fconst 6778	. . . . . . . 8 ⊢ (ℕ₀ × {0}):ℕ₀⟶{0}
67	66	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ ℕ₀) → (ℕ₀ × {0}):ℕ₀⟶{0})
68		0red 11217	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ ℕ₀) → 0 ∈ ℝ)
69	68	snssd 4813	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ ℕ₀) → {0} ⊆ ℝ)
70	67, 69	fssd 6736	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 ∈ ℕ₀) → (ℕ₀ × {0}):ℕ₀⟶ℝ)
71	70	ffvelcdmda 7087	. . . . 5 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → ((ℕ₀ × {0})‘𝑘) ∈ ℝ)
72	62, 61	eqeltrd 2834	. . . . 5 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → ((𝑗 ∈ ℕ₀ ↦ ((𝐶C_𝑐𝑗) · ((𝐴↑_𝑐(𝐶 − 𝑗)) · (𝐵↑𝑗))))‘𝑘) ∈ ℂ)
73		climrel 15436	. . . . . . 7 ⊢ Rel ⇝
74	39	xpeq1i 5703	. . . . . . . . 9 ⊢ (ℕ₀ × {0}) = ((ℤ_≥‘0) × {0})
75		seqeq3 13971	. . . . . . . . 9 ⊢ ((ℕ₀ × {0}) = ((ℤ_≥‘0) × {0}) → seq0( + , (ℕ₀ × {0})) = seq0( + , ((ℤ_≥‘0) × {0})))
76	74, 75	ax-mp 5	. . . . . . . 8 ⊢ seq0( + , (ℕ₀ × {0})) = seq0( + , ((ℤ_≥‘0) × {0}))
77		0z 12569	. . . . . . . . 9 ⊢ 0 ∈ ℤ
78		serclim0 15521	. . . . . . . . 9 ⊢ (0 ∈ ℤ → seq0( + , ((ℤ_≥‘0) × {0})) ⇝ 0)
79	77, 78	ax-mp 5	. . . . . . . 8 ⊢ seq0( + , ((ℤ_≥‘0) × {0})) ⇝ 0
80	76, 79	eqbrtri 5170	. . . . . . 7 ⊢ seq0( + , (ℕ₀ × {0})) ⇝ 0
81		releldm 5944	. . . . . . 7 ⊢ ((Rel ⇝ ∧ seq0( + , (ℕ₀ × {0})) ⇝ 0) → seq0( + , (ℕ₀ × {0})) ∈ dom ⇝ )
82	73, 80, 81	mp2an 691	. . . . . 6 ⊢ seq0( + , (ℕ₀ × {0})) ∈ dom ⇝
83	82	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ ℕ₀) → seq0( + , (ℕ₀ × {0})) ∈ dom ⇝ )
84		eluznn0 12901	. . . . . . . . . . . 12 ⊢ (((𝐶 + 1) ∈ ℕ₀ ∧ 𝑘 ∈ (ℤ_≥‘(𝐶 + 1))) → 𝑘 ∈ ℕ₀)
85	64, 84	sylan 581	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (ℤ_≥‘(𝐶 + 1))) → 𝑘 ∈ ℕ₀)
86	85, 62	syldan 592	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (ℤ_≥‘(𝐶 + 1))) → ((𝑗 ∈ ℕ₀ ↦ ((𝐶C_𝑐𝑗) · ((𝐴↑_𝑐(𝐶 − 𝑗)) · (𝐵↑𝑗))))‘𝑘) = ((𝐶C_𝑐𝑘) · ((𝐴↑_𝑐(𝐶 − 𝑘)) · (𝐵↑𝑘))))
87		0zd 12570	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (ℤ_≥‘(𝐶 + 1))) → 0 ∈ ℤ)
88	85	nn0zd 12584	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (ℤ_≥‘(𝐶 + 1))) → 𝑘 ∈ ℤ)
89		1zzd 12593	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (ℤ_≥‘(𝐶 + 1))) → 1 ∈ ℤ)
90	88, 89	zsubcld 12671	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (ℤ_≥‘(𝐶 + 1))) → (𝑘 − 1) ∈ ℤ)
91	12	nn0zd 12584	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐶 ∈ ℕ₀) → 𝐶 ∈ ℤ)
92	91	adantr 482	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (ℤ_≥‘(𝐶 + 1))) → 𝐶 ∈ ℤ)
93	12	nn0ge0d 12535	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐶 ∈ ℕ₀) → 0 ≤ 𝐶)
94	93	adantr 482	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (ℤ_≥‘(𝐶 + 1))) → 0 ≤ 𝐶)
95		eluzle 12835	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (ℤ_≥‘(𝐶 + 1)) → (𝐶 + 1) ≤ 𝑘)
96	95	adantl 483	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (ℤ_≥‘(𝐶 + 1))) → (𝐶 + 1) ≤ 𝑘)
97	92	zred 12666	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (ℤ_≥‘(𝐶 + 1))) → 𝐶 ∈ ℝ)
98		1red 11215	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (ℤ_≥‘(𝐶 + 1))) → 1 ∈ ℝ)
99	85	nn0red 12533	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (ℤ_≥‘(𝐶 + 1))) → 𝑘 ∈ ℝ)
100		leaddsub 11690	. . . . . . . . . . . . . . . 16 ⊢ ((𝐶 ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝑘 ∈ ℝ) → ((𝐶 + 1) ≤ 𝑘 ↔ 𝐶 ≤ (𝑘 − 1)))
101	97, 98, 99, 100	syl3anc 1372	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (ℤ_≥‘(𝐶 + 1))) → ((𝐶 + 1) ≤ 𝑘 ↔ 𝐶 ≤ (𝑘 − 1)))
102	96, 101	mpbid 231	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (ℤ_≥‘(𝐶 + 1))) → 𝐶 ≤ (𝑘 − 1))
103	87, 90, 92, 94, 102	elfzd 13492	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (ℤ_≥‘(𝐶 + 1))) → 𝐶 ∈ (0...(𝑘 − 1)))
104	34	ad2antrr 725	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (ℤ_≥‘(𝐶 + 1))) → 𝐶 ∈ ℂ)
105	104, 85	bcc0 43099	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (ℤ_≥‘(𝐶 + 1))) → ((𝐶C_𝑐𝑘) = 0 ↔ 𝐶 ∈ (0...(𝑘 − 1))))
106	103, 105	mpbird 257	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (ℤ_≥‘(𝐶 + 1))) → (𝐶C_𝑐𝑘) = 0)
107	106	oveq1d 7424	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (ℤ_≥‘(𝐶 + 1))) → ((𝐶C_𝑐𝑘) · ((𝐴↑_𝑐(𝐶 − 𝑘)) · (𝐵↑𝑘))) = (0 · ((𝐴↑_𝑐(𝐶 − 𝑘)) · (𝐵↑𝑘))))
108	2	ad2antrr 725	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (ℤ_≥‘(𝐶 + 1))) → 𝐴 ∈ ℂ)
109		eluzelcn 12834	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (ℤ_≥‘(𝐶 + 1)) → 𝑘 ∈ ℂ)
110	109	adantl 483	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (ℤ_≥‘(𝐶 + 1))) → 𝑘 ∈ ℂ)
111	104, 110	subcld 11571	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (ℤ_≥‘(𝐶 + 1))) → (𝐶 − 𝑘) ∈ ℂ)
112	108, 111	cxpcld 26216	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (ℤ_≥‘(𝐶 + 1))) → (𝐴↑_𝑐(𝐶 − 𝑘)) ∈ ℂ)
113	4	ad2antrr 725	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (ℤ_≥‘(𝐶 + 1))) → 𝐵 ∈ ℂ)
114	113, 85	expcld 14111	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (ℤ_≥‘(𝐶 + 1))) → (𝐵↑𝑘) ∈ ℂ)
115	112, 114	mulcld 11234	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (ℤ_≥‘(𝐶 + 1))) → ((𝐴↑_𝑐(𝐶 − 𝑘)) · (𝐵↑𝑘)) ∈ ℂ)
116	115	mul02d 11412	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (ℤ_≥‘(𝐶 + 1))) → (0 · ((𝐴↑_𝑐(𝐶 − 𝑘)) · (𝐵↑𝑘))) = 0)
117	107, 116	eqtrd 2773	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (ℤ_≥‘(𝐶 + 1))) → ((𝐶C_𝑐𝑘) · ((𝐴↑_𝑐(𝐶 − 𝑘)) · (𝐵↑𝑘))) = 0)
118	86, 117	eqtrd 2773	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (ℤ_≥‘(𝐶 + 1))) → ((𝑗 ∈ ℕ₀ ↦ ((𝐶C_𝑐𝑗) · ((𝐴↑_𝑐(𝐶 − 𝑗)) · (𝐵↑𝑗))))‘𝑘) = 0)
119	118	abs00bd 15238	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (ℤ_≥‘(𝐶 + 1))) → (abs‘((𝑗 ∈ ℕ₀ ↦ ((𝐶C_𝑐𝑗) · ((𝐴↑_𝑐(𝐶 − 𝑗)) · (𝐵↑𝑗))))‘𝑘)) = 0)
120		0re 11216	. . . . . . . 8 ⊢ 0 ∈ ℝ
121	119, 120	eqeltrdi 2842	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (ℤ_≥‘(𝐶 + 1))) → (abs‘((𝑗 ∈ ℕ₀ ↦ ((𝐶C_𝑐𝑗) · ((𝐴↑_𝑐(𝐶 − 𝑗)) · (𝐵↑𝑗))))‘𝑘)) ∈ ℝ)
122		eqle 11316	. . . . . . 7 ⊢ (((abs‘((𝑗 ∈ ℕ₀ ↦ ((𝐶C_𝑐𝑗) · ((𝐴↑_𝑐(𝐶 − 𝑗)) · (𝐵↑𝑗))))‘𝑘)) ∈ ℝ ∧ (abs‘((𝑗 ∈ ℕ₀ ↦ ((𝐶C_𝑐𝑗) · ((𝐴↑_𝑐(𝐶 − 𝑗)) · (𝐵↑𝑗))))‘𝑘)) = 0) → (abs‘((𝑗 ∈ ℕ₀ ↦ ((𝐶C_𝑐𝑗) · ((𝐴↑_𝑐(𝐶 − 𝑗)) · (𝐵↑𝑗))))‘𝑘)) ≤ 0)
123	121, 119, 122	syl2anc 585	. . . . . 6 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (ℤ_≥‘(𝐶 + 1))) → (abs‘((𝑗 ∈ ℕ₀ ↦ ((𝐶C_𝑐𝑗) · ((𝐴↑_𝑐(𝐶 − 𝑗)) · (𝐵↑𝑗))))‘𝑘)) ≤ 0)
124	71	recnd 11242	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → ((ℕ₀ × {0})‘𝑘) ∈ ℂ)
125	85, 124	syldan 592	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (ℤ_≥‘(𝐶 + 1))) → ((ℕ₀ × {0})‘𝑘) ∈ ℂ)
126	125	mul02d 11412	. . . . . 6 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (ℤ_≥‘(𝐶 + 1))) → (0 · ((ℕ₀ × {0})‘𝑘)) = 0)
127	123, 126	breqtrrd 5177	. . . . 5 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (ℤ_≥‘(𝐶 + 1))) → (abs‘((𝑗 ∈ ℕ₀ ↦ ((𝐶C_𝑐𝑗) · ((𝐴↑_𝑐(𝐶 − 𝑗)) · (𝐵↑𝑗))))‘𝑘)) ≤ (0 · ((ℕ₀ × {0})‘𝑘)))
128	39, 64, 71, 72, 83, 68, 127	cvgcmpce 15764	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ ℕ₀) → seq0( + , (𝑗 ∈ ℕ₀ ↦ ((𝐶C_𝑐𝑗) · ((𝐴↑_𝑐(𝐶 − 𝑗)) · (𝐵↑𝑗))))) ∈ dom ⇝ )
129	39, 40, 43, 62, 61, 128	isumsplit 15786	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ ℕ₀) → Σ𝑘 ∈ ℕ₀ ((𝐶C_𝑐𝑘) · ((𝐴↑_𝑐(𝐶 − 𝑘)) · (𝐵↑𝑘))) = (Σ𝑘 ∈ (0...((𝐶 + 1) − 1))((𝐶C_𝑐𝑘) · ((𝐴↑_𝑐(𝐶 − 𝑘)) · (𝐵↑𝑘))) + Σ𝑘 ∈ (ℤ_≥‘(𝐶 + 1))((𝐶C_𝑐𝑘) · ((𝐴↑_𝑐(𝐶 − 𝑘)) · (𝐵↑𝑘)))))
130		1cnd 11209	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ ℕ₀) → 1 ∈ ℂ)
131	35, 130	pncand 11572	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 ∈ ℕ₀) → ((𝐶 + 1) − 1) = 𝐶)
132	131	oveq2d 7425	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ ℕ₀) → (0...((𝐶 + 1) − 1)) = (0...𝐶))
133	132	sumeq1d 15647	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ ℕ₀) → Σ𝑘 ∈ (0...((𝐶 + 1) − 1))((𝐶C_𝑐𝑘) · ((𝐴↑_𝑐(𝐶 − 𝑘)) · (𝐵↑𝑘))) = Σ𝑘 ∈ (0...𝐶)((𝐶C_𝑐𝑘) · ((𝐴↑_𝑐(𝐶 − 𝑘)) · (𝐵↑𝑘))))
134	133	oveq1d 7424	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ ℕ₀) → (Σ𝑘 ∈ (0...((𝐶 + 1) − 1))((𝐶C_𝑐𝑘) · ((𝐴↑_𝑐(𝐶 − 𝑘)) · (𝐵↑𝑘))) + Σ𝑘 ∈ (ℤ_≥‘(𝐶 + 1))((𝐶C_𝑐𝑘) · ((𝐴↑_𝑐(𝐶 − 𝑘)) · (𝐵↑𝑘)))) = (Σ𝑘 ∈ (0...𝐶)((𝐶C_𝑐𝑘) · ((𝐴↑_𝑐(𝐶 − 𝑘)) · (𝐵↑𝑘))) + Σ𝑘 ∈ (ℤ_≥‘(𝐶 + 1))((𝐶C_𝑐𝑘) · ((𝐴↑_𝑐(𝐶 − 𝑘)) · (𝐵↑𝑘)))))
135	117	sumeq2dv 15649	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ ℕ₀) → Σ𝑘 ∈ (ℤ_≥‘(𝐶 + 1))((𝐶C_𝑐𝑘) · ((𝐴↑_𝑐(𝐶 − 𝑘)) · (𝐵↑𝑘))) = Σ𝑘 ∈ (ℤ_≥‘(𝐶 + 1))0)
136		ssid 4005	. . . . . . 7 ⊢ (ℤ_≥‘(𝐶 + 1)) ⊆ (ℤ_≥‘(𝐶 + 1))
137	136	orci 864	. . . . . 6 ⊢ ((ℤ_≥‘(𝐶 + 1)) ⊆ (ℤ_≥‘(𝐶 + 1)) ∨ (ℤ_≥‘(𝐶 + 1)) ∈ Fin)
138		sumz 15668	. . . . . 6 ⊢ (((ℤ_≥‘(𝐶 + 1)) ⊆ (ℤ_≥‘(𝐶 + 1)) ∨ (ℤ_≥‘(𝐶 + 1)) ∈ Fin) → Σ𝑘 ∈ (ℤ_≥‘(𝐶 + 1))0 = 0)
139	137, 138	ax-mp 5	. . . . 5 ⊢ Σ𝑘 ∈ (ℤ_≥‘(𝐶 + 1))0 = 0
140	135, 139	eqtrdi 2789	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ ℕ₀) → Σ𝑘 ∈ (ℤ_≥‘(𝐶 + 1))((𝐶C_𝑐𝑘) · ((𝐴↑_𝑐(𝐶 − 𝑘)) · (𝐵↑𝑘))) = 0)
141	140	oveq2d 7425	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ ℕ₀) → (Σ𝑘 ∈ (0...𝐶)((𝐶C_𝑐𝑘) · ((𝐴↑_𝑐(𝐶 − 𝑘)) · (𝐵↑𝑘))) + Σ𝑘 ∈ (ℤ_≥‘(𝐶 + 1))((𝐶C_𝑐𝑘) · ((𝐴↑_𝑐(𝐶 − 𝑘)) · (𝐵↑𝑘)))) = (Σ𝑘 ∈ (0...𝐶)((𝐶C_𝑐𝑘) · ((𝐴↑_𝑐(𝐶 − 𝑘)) · (𝐵↑𝑘))) + 0))
142	129, 134, 141	3eqtrd 2777	. 2 ⊢ ((𝜑 ∧ 𝐶 ∈ ℕ₀) → Σ𝑘 ∈ ℕ₀ ((𝐶C_𝑐𝑘) · ((𝐴↑_𝑐(𝐶 − 𝑘)) · (𝐵↑𝑘))) = (Σ𝑘 ∈ (0...𝐶)((𝐶C_𝑐𝑘) · ((𝐴↑_𝑐(𝐶 − 𝑘)) · (𝐵↑𝑘))) + 0))
143	38, 142, 33	3eqtr4rd 2784	1 ⊢ ((𝜑 ∧ 𝐶 ∈ ℕ₀) → ((𝐴 + 𝐵)↑_𝑐𝐶) = Σ𝑘 ∈ ℕ₀ ((𝐶C_𝑐𝑘) · ((𝐴↑_𝑐(𝐶 − 𝑘)) · (𝐵↑𝑘))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∨ wo 846 = wceq 1542 ∈ wcel 2107 ⊆ wss 3949 {csn 4629 class class class wbr 5149 ↦ cmpt 5232 × cxp 5675 dom cdm 5677 Rel wrel 5682 ⟶wf 6540 ‘cfv 6544 (class class class)co 7409 Fincfn 8939 ℂcc 11108 ℝcr 11109 0cc0 11110 1c1 11111 + caddc 11113 · cmul 11115 < clt 11248 ≤ cle 11249 − cmin 11444 ℕ₀cn0 12472 ℤcz 12558 ℤ_≥cuz 12822 ℝ⁺crp 12974 ...cfz 13484 seqcseq 13966 ↑cexp 14027 Ccbc 14262 abscabs 15181 ⇝ cli 15428 Σcsu 15632 ↑_𝑐ccxp 26064 C_𝑐cbcc 43095

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 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188 ax-addf 11189 ax-mulf 11190

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 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7670 df-om 7856 df-1st 7975 df-2nd 7976 df-supp 8147 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-er 8703 df-map 8822 df-pm 8823 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9362 df-fi 9406 df-sup 9437 df-inf 9438 df-oi 9505 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-z 12559 df-dec 12678 df-uz 12823 df-q 12933 df-rp 12975 df-xneg 13092 df-xadd 13093 df-xmul 13094 df-ioo 13328 df-ioc 13329 df-ico 13330 df-icc 13331 df-fz 13485 df-fzo 13628 df-fl 13757 df-mod 13835 df-seq 13967 df-exp 14028 df-fac 14234 df-bc 14263 df-hash 14291 df-shft 15014 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-limsup 15415 df-clim 15432 df-rlim 15433 df-sum 15633 df-prod 15850 df-fallfac 15951 df-ef 16011 df-sin 16013 df-cos 16014 df-pi 16016 df-struct 17080 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-mulr 17211 df-starv 17212 df-sca 17213 df-vsca 17214 df-ip 17215 df-tset 17216 df-ple 17217 df-ds 17219 df-unif 17220 df-hom 17221 df-cco 17222 df-rest 17368 df-topn 17369 df-0g 17387 df-gsum 17388 df-topgen 17389 df-pt 17390 df-prds 17393 df-xrs 17448 df-qtop 17453 df-imas 17454 df-xps 17456 df-mre 17530 df-mrc 17531 df-acs 17533 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-submnd 18672 df-mulg 18951 df-cntz 19181 df-cmn 19650 df-psmet 20936 df-xmet 20937 df-met 20938 df-bl 20939 df-mopn 20940 df-fbas 20941 df-fg 20942 df-cnfld 20945 df-top 22396 df-topon 22413 df-topsp 22435 df-bases 22449 df-cld 22523 df-ntr 22524 df-cls 22525 df-nei 22602 df-lp 22640 df-perf 22641 df-cn 22731 df-cnp 22732 df-haus 22819 df-tx 23066 df-hmeo 23259 df-fil 23350 df-fm 23442 df-flim 23443 df-flf 23444 df-xms 23826 df-ms 23827 df-tms 23828 df-cncf 24394 df-limc 25383 df-dv 25384 df-log 26065 df-cxp 26066 df-bcc 43096

This theorem is referenced by: binomcxp 43116

W3C validator