binomcxplemnn0 - Mathbox for Steve Rodriguez

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Theorem binomcxplemnn0 43412

Description: Lemma for binomcxp 43420. When 𝐶 is a nonnegative integer, the binomial's finite sum value by the standard binomial theorem binom 15782 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 13024	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℂ)
3		binomcxp.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℝ)
4	3	recnd 11248	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℂ)
5		binom 15782	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℕ₀) → ((𝐴 + 𝐵)↑𝐶) = Σ𝑘 ∈ (0...𝐶)((𝐶C𝑘) · ((𝐴↑(𝐶 − 𝑘)) · (𝐵↑𝑘))))
6	5	3expia 1119	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐶 ∈ ℕ₀ → ((𝐴 + 𝐵)↑𝐶) = Σ𝑘 ∈ (0...𝐶)((𝐶C𝑘) · ((𝐴↑(𝐶 − 𝑘)) · (𝐵↑𝑘)))))
7	2, 4, 6	syl2anc 582	. . . . . 6 ⊢ (𝜑 → (𝐶 ∈ ℕ₀ → ((𝐴 + 𝐵)↑𝐶) = Σ𝑘 ∈ (0...𝐶)((𝐶C𝑘) · ((𝐴↑(𝐶 − 𝑘)) · (𝐵↑𝑘)))))
8	7	imp 405	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ ℕ₀) → ((𝐴 + 𝐵)↑𝐶) = Σ𝑘 ∈ (0...𝐶)((𝐶C𝑘) · ((𝐴↑(𝐶 − 𝑘)) · (𝐵↑𝑘))))
9	2	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ ℕ₀) → 𝐴 ∈ ℂ)
10	4	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ ℕ₀) → 𝐵 ∈ ℂ)
11	9, 10	addcld 11239	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 ∈ ℕ₀) → (𝐴 + 𝐵) ∈ ℂ)
12		simpr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 ∈ ℕ₀) → 𝐶 ∈ ℕ₀)
13		cxpexp 26410	. . . . . 6 ⊢ (((𝐴 + 𝐵) ∈ ℂ ∧ 𝐶 ∈ ℕ₀) → ((𝐴 + 𝐵)↑_𝑐𝐶) = ((𝐴 + 𝐵)↑𝐶))
14	11, 12, 13	syl2anc 582	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ ℕ₀) → ((𝐴 + 𝐵)↑_𝑐𝐶) = ((𝐴 + 𝐵)↑𝐶))
15		elfznn0 13600	. . . . . . . 8 ⊢ (𝑘 ∈ (0...𝐶) → 𝑘 ∈ ℕ₀)
16		simplr 765	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → 𝐶 ∈ ℕ₀)
17		simpr 483	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → 𝑘 ∈ ℕ₀)
18	16, 17	bccbc 43408	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → (𝐶C_𝑐𝑘) = (𝐶C𝑘))
19	15, 18	sylan2 591	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝐶)) → (𝐶C_𝑐𝑘) = (𝐶C𝑘))
20	2	ad2antrr 722	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝐶)) → 𝐴 ∈ ℂ)
21		elfzle2 13511	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (0...𝐶) → 𝑘 ≤ 𝐶)
22	21	adantl 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝐶)) → 𝑘 ≤ 𝐶)
23		nn0sub 12528	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) → (𝑘 ≤ 𝐶 ↔ (𝐶 − 𝑘) ∈ ℕ₀))
24	23	ancoms 457	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) → (𝑘 ≤ 𝐶 ↔ (𝐶 − 𝑘) ∈ ℕ₀))
25	24	adantll 710	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → (𝑘 ≤ 𝐶 ↔ (𝐶 − 𝑘) ∈ ℕ₀))
26	15, 25	sylan2 591	. . . . . . . . . 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 26410	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (𝐶 − 𝑘) ∈ ℕ₀) → (𝐴↑_𝑐(𝐶 − 𝑘)) = (𝐴↑(𝐶 − 𝑘)))
29	20, 27, 28	syl2anc 582	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝐶)) → (𝐴↑_𝑐(𝐶 − 𝑘)) = (𝐴↑(𝐶 − 𝑘)))
30	29	oveq1d 7428	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝐶)) → ((𝐴↑_𝑐(𝐶 − 𝑘)) · (𝐵↑𝑘)) = ((𝐴↑(𝐶 − 𝑘)) · (𝐵↑𝑘)))
31	19, 30	oveq12d 7431	. . . . . 6 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝐶)) → ((𝐶C_𝑐𝑘) · ((𝐴↑_𝑐(𝐶 − 𝑘)) · (𝐵↑𝑘))) = ((𝐶C𝑘) · ((𝐴↑(𝐶 − 𝑘)) · (𝐵↑𝑘))))
32	31	sumeq2dv 15655	. . . . 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 2780	. . . 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 479	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ ℕ₀) → 𝐶 ∈ ℂ)
36	11, 35	cxpcld 26450	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ ℕ₀) → ((𝐴 + 𝐵)↑_𝑐𝐶) ∈ ℂ)
37	33, 36	eqeltrrd 2832	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ ℕ₀) → Σ𝑘 ∈ (0...𝐶)((𝐶C_𝑐𝑘) · ((𝐴↑_𝑐(𝐶 − 𝑘)) · (𝐵↑𝑘))) ∈ ℂ)
38	37	addridd 11420	. 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 12870	. . . 4 ⊢ ℕ₀ = (ℤ_≥‘0)
40		eqid 2730	. . . 4 ⊢ (ℤ_≥‘(𝐶 + 1)) = (ℤ_≥‘(𝐶 + 1))
41		1nn0 12494	. . . . . 6 ⊢ 1 ∈ ℕ₀
42	41	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ ℕ₀) → 1 ∈ ℕ₀)
43	12, 42	nn0addcld 12542	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ ℕ₀) → (𝐶 + 1) ∈ ℕ₀)
44		eqidd 2731	. . . . 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 483	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑗 = 𝑘) → 𝑗 = 𝑘)
46	45	oveq2d 7429	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑗 = 𝑘) → (𝐶C_𝑐𝑗) = (𝐶C_𝑐𝑘))
47	45	oveq2d 7429	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑗 = 𝑘) → (𝐶 − 𝑗) = (𝐶 − 𝑘))
48	47	oveq2d 7429	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑗 = 𝑘) → (𝐴↑_𝑐(𝐶 − 𝑗)) = (𝐴↑_𝑐(𝐶 − 𝑘)))
49	45	oveq2d 7429	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑗 = 𝑘) → (𝐵↑𝑗) = (𝐵↑𝑘))
50	48, 49	oveq12d 7431	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑗 = 𝑘) → ((𝐴↑_𝑐(𝐶 − 𝑗)) · (𝐵↑𝑗)) = ((𝐴↑_𝑐(𝐶 − 𝑘)) · (𝐵↑𝑘)))
51	46, 50	oveq12d 7431	. . . . 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 722	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → 𝐶 ∈ ℂ)
53	52, 17	bcccl 43402	. . . . . 6 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → (𝐶C_𝑐𝑘) ∈ ℂ)
54	2	ad2antrr 722	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → 𝐴 ∈ ℂ)
55	17	nn0cnd 12540	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → 𝑘 ∈ ℂ)
56	52, 55	subcld 11577	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → (𝐶 − 𝑘) ∈ ℂ)
57	54, 56	cxpcld 26450	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → (𝐴↑_𝑐(𝐶 − 𝑘)) ∈ ℂ)
58	4	ad2antrr 722	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → 𝐵 ∈ ℂ)
59	58, 17	expcld 14117	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → (𝐵↑𝑘) ∈ ℂ)
60	57, 59	mulcld 11240	. . . . . 6 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → ((𝐴↑_𝑐(𝐶 − 𝑘)) · (𝐵↑𝑘)) ∈ ℂ)
61	53, 60	mulcld 11240	. . . . 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 12518	. . . . . 6 ⊢ (𝐶 ∈ ℕ₀ → (𝐶 + 1) ∈ ℕ₀)
64	63	adantl 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ ℕ₀) → (𝐶 + 1) ∈ ℕ₀)
65		c0ex 11214	. . . . . . . . 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 11223	. . . . . . . 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 2831	. . . . 5 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → ((𝑗 ∈ ℕ₀ ↦ ((𝐶C_𝑐𝑗) · ((𝐴↑_𝑐(𝐶 − 𝑗)) · (𝐵↑𝑗))))‘𝑘) ∈ ℂ)
73		climrel 15442	. . . . . . 7 ⊢ Rel ⇝
74	39	xpeq1i 5703	. . . . . . . . 9 ⊢ (ℕ₀ × {0}) = ((ℤ_≥‘0) × {0})
75		seqeq3 13977	. . . . . . . . 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 12575	. . . . . . . . 9 ⊢ 0 ∈ ℤ
78		serclim0 15527	. . . . . . . . 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 688	. . . . . 6 ⊢ seq0( + , (ℕ₀ × {0})) ∈ dom ⇝
83	82	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ ℕ₀) → seq0( + , (ℕ₀ × {0})) ∈ dom ⇝ )
84		eluznn0 12907	. . . . . . . . . . . 12 ⊢ (((𝐶 + 1) ∈ ℕ₀ ∧ 𝑘 ∈ (ℤ_≥‘(𝐶 + 1))) → 𝑘 ∈ ℕ₀)
85	64, 84	sylan 578	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (ℤ_≥‘(𝐶 + 1))) → 𝑘 ∈ ℕ₀)
86	85, 62	syldan 589	. . . . . . . . . 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 12576	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (ℤ_≥‘(𝐶 + 1))) → 0 ∈ ℤ)
88	85	nn0zd 12590	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (ℤ_≥‘(𝐶 + 1))) → 𝑘 ∈ ℤ)
89		1zzd 12599	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (ℤ_≥‘(𝐶 + 1))) → 1 ∈ ℤ)
90	88, 89	zsubcld 12677	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (ℤ_≥‘(𝐶 + 1))) → (𝑘 − 1) ∈ ℤ)
91	12	nn0zd 12590	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐶 ∈ ℕ₀) → 𝐶 ∈ ℤ)
92	91	adantr 479	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (ℤ_≥‘(𝐶 + 1))) → 𝐶 ∈ ℤ)
93	12	nn0ge0d 12541	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐶 ∈ ℕ₀) → 0 ≤ 𝐶)
94	93	adantr 479	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (ℤ_≥‘(𝐶 + 1))) → 0 ≤ 𝐶)
95		eluzle 12841	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (ℤ_≥‘(𝐶 + 1)) → (𝐶 + 1) ≤ 𝑘)
96	95	adantl 480	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (ℤ_≥‘(𝐶 + 1))) → (𝐶 + 1) ≤ 𝑘)
97	92	zred 12672	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (ℤ_≥‘(𝐶 + 1))) → 𝐶 ∈ ℝ)
98		1red 11221	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (ℤ_≥‘(𝐶 + 1))) → 1 ∈ ℝ)
99	85	nn0red 12539	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (ℤ_≥‘(𝐶 + 1))) → 𝑘 ∈ ℝ)
100		leaddsub 11696	. . . . . . . . . . . . . . . 16 ⊢ ((𝐶 ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝑘 ∈ ℝ) → ((𝐶 + 1) ≤ 𝑘 ↔ 𝐶 ≤ (𝑘 − 1)))
101	97, 98, 99, 100	syl3anc 1369	. . . . . . . . . . . . . . 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 13498	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (ℤ_≥‘(𝐶 + 1))) → 𝐶 ∈ (0...(𝑘 − 1)))
104	34	ad2antrr 722	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (ℤ_≥‘(𝐶 + 1))) → 𝐶 ∈ ℂ)
105	104, 85	bcc0 43403	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (ℤ_≥‘(𝐶 + 1))) → ((𝐶C_𝑐𝑘) = 0 ↔ 𝐶 ∈ (0...(𝑘 − 1))))
106	103, 105	mpbird 256	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (ℤ_≥‘(𝐶 + 1))) → (𝐶C_𝑐𝑘) = 0)
107	106	oveq1d 7428	. . . . . . . . . . 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 722	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (ℤ_≥‘(𝐶 + 1))) → 𝐴 ∈ ℂ)
109		eluzelcn 12840	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (ℤ_≥‘(𝐶 + 1)) → 𝑘 ∈ ℂ)
110	109	adantl 480	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (ℤ_≥‘(𝐶 + 1))) → 𝑘 ∈ ℂ)
111	104, 110	subcld 11577	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (ℤ_≥‘(𝐶 + 1))) → (𝐶 − 𝑘) ∈ ℂ)
112	108, 111	cxpcld 26450	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (ℤ_≥‘(𝐶 + 1))) → (𝐴↑_𝑐(𝐶 − 𝑘)) ∈ ℂ)
113	4	ad2antrr 722	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (ℤ_≥‘(𝐶 + 1))) → 𝐵 ∈ ℂ)
114	113, 85	expcld 14117	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (ℤ_≥‘(𝐶 + 1))) → (𝐵↑𝑘) ∈ ℂ)
115	112, 114	mulcld 11240	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (ℤ_≥‘(𝐶 + 1))) → ((𝐴↑_𝑐(𝐶 − 𝑘)) · (𝐵↑𝑘)) ∈ ℂ)
116	115	mul02d 11418	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (ℤ_≥‘(𝐶 + 1))) → (0 · ((𝐴↑_𝑐(𝐶 − 𝑘)) · (𝐵↑𝑘))) = 0)
117	107, 116	eqtrd 2770	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (ℤ_≥‘(𝐶 + 1))) → ((𝐶C_𝑐𝑘) · ((𝐴↑_𝑐(𝐶 − 𝑘)) · (𝐵↑𝑘))) = 0)
118	86, 117	eqtrd 2770	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (ℤ_≥‘(𝐶 + 1))) → ((𝑗 ∈ ℕ₀ ↦ ((𝐶C_𝑐𝑗) · ((𝐴↑_𝑐(𝐶 − 𝑗)) · (𝐵↑𝑗))))‘𝑘) = 0)
119	118	abs00bd 15244	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (ℤ_≥‘(𝐶 + 1))) → (abs‘((𝑗 ∈ ℕ₀ ↦ ((𝐶C_𝑐𝑗) · ((𝐴↑_𝑐(𝐶 − 𝑗)) · (𝐵↑𝑗))))‘𝑘)) = 0)
120		0re 11222	. . . . . . . 8 ⊢ 0 ∈ ℝ
121	119, 120	eqeltrdi 2839	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (ℤ_≥‘(𝐶 + 1))) → (abs‘((𝑗 ∈ ℕ₀ ↦ ((𝐶C_𝑐𝑗) · ((𝐴↑_𝑐(𝐶 − 𝑗)) · (𝐵↑𝑗))))‘𝑘)) ∈ ℝ)
122		eqle 11322	. . . . . . 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 582	. . . . . 6 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (ℤ_≥‘(𝐶 + 1))) → (abs‘((𝑗 ∈ ℕ₀ ↦ ((𝐶C_𝑐𝑗) · ((𝐴↑_𝑐(𝐶 − 𝑗)) · (𝐵↑𝑗))))‘𝑘)) ≤ 0)
124	71	recnd 11248	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → ((ℕ₀ × {0})‘𝑘) ∈ ℂ)
125	85, 124	syldan 589	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 ∈ ℕ₀) ∧ 𝑘 ∈ (ℤ_≥‘(𝐶 + 1))) → ((ℕ₀ × {0})‘𝑘) ∈ ℂ)
126	125	mul02d 11418	. . . . . 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 15770	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ ℕ₀) → seq0( + , (𝑗 ∈ ℕ₀ ↦ ((𝐶C_𝑐𝑗) · ((𝐴↑_𝑐(𝐶 − 𝑗)) · (𝐵↑𝑗))))) ∈ dom ⇝ )
129	39, 40, 43, 62, 61, 128	isumsplit 15792	. . 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 11215	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ ℕ₀) → 1 ∈ ℂ)
131	35, 130	pncand 11578	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 ∈ ℕ₀) → ((𝐶 + 1) − 1) = 𝐶)
132	131	oveq2d 7429	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ ℕ₀) → (0...((𝐶 + 1) − 1)) = (0...𝐶))
133	132	sumeq1d 15653	. . . 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 7428	. . 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 15655	. . . . 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 861	. . . . . 6 ⊢ ((ℤ_≥‘(𝐶 + 1)) ⊆ (ℤ_≥‘(𝐶 + 1)) ∨ (ℤ_≥‘(𝐶 + 1)) ∈ Fin)
138		sumz 15674	. . . . . 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 2786	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ ℕ₀) → Σ𝑘 ∈ (ℤ_≥‘(𝐶 + 1))((𝐶C_𝑐𝑘) · ((𝐴↑_𝑐(𝐶 − 𝑘)) · (𝐵↑𝑘))) = 0)
141	140	oveq2d 7429	. . 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 2774	. 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 2781	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 394 ∨ wo 843 = wceq 1539 ∈ wcel 2104 ⊆ 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 7413 Fincfn 8943 ℂcc 11112 ℝcr 11113 0cc0 11114 1c1 11115 + caddc 11117 · cmul 11119 < clt 11254 ≤ cle 11255 − cmin 11450 ℕ₀cn0 12478 ℤcz 12564 ℤ_≥cuz 12828 ℝ⁺crp 12980 ...cfz 13490 seqcseq 13972 ↑cexp 14033 Ccbc 14268 abscabs 15187 ⇝ cli 15434 Σcsu 15638 ↑_𝑐ccxp 26298 C_𝑐cbcc 43399

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7729 ax-inf2 9640 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 ax-pre-sup 11192 ax-addf 11193 ax-mulf 11194

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 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 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-of 7674 df-om 7860 df-1st 7979 df-2nd 7980 df-supp 8151 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-2o 8471 df-er 8707 df-map 8826 df-pm 8827 df-ixp 8896 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-fsupp 9366 df-fi 9410 df-sup 9441 df-inf 9442 df-oi 9509 df-card 9938 df-pnf 11256 df-mnf 11257 df-xr 11258 df-ltxr 11259 df-le 11260 df-sub 11452 df-neg 11453 df-div 11878 df-nn 12219 df-2 12281 df-3 12282 df-4 12283 df-5 12284 df-6 12285 df-7 12286 df-8 12287 df-9 12288 df-n0 12479 df-z 12565 df-dec 12684 df-uz 12829 df-q 12939 df-rp 12981 df-xneg 13098 df-xadd 13099 df-xmul 13100 df-ioo 13334 df-ioc 13335 df-ico 13336 df-icc 13337 df-fz 13491 df-fzo 13634 df-fl 13763 df-mod 13841 df-seq 13973 df-exp 14034 df-fac 14240 df-bc 14269 df-hash 14297 df-shft 15020 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-limsup 15421 df-clim 15438 df-rlim 15439 df-sum 15639 df-prod 15856 df-fallfac 15957 df-ef 16017 df-sin 16019 df-cos 16020 df-pi 16022 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17151 df-ress 17180 df-plusg 17216 df-mulr 17217 df-starv 17218 df-sca 17219 df-vsca 17220 df-ip 17221 df-tset 17222 df-ple 17223 df-ds 17225 df-unif 17226 df-hom 17227 df-cco 17228 df-rest 17374 df-topn 17375 df-0g 17393 df-gsum 17394 df-topgen 17395 df-pt 17396 df-prds 17399 df-xrs 17454 df-qtop 17459 df-imas 17460 df-xps 17462 df-mre 17536 df-mrc 17537 df-acs 17539 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-submnd 18708 df-mulg 18989 df-cntz 19224 df-cmn 19693 df-psmet 21138 df-xmet 21139 df-met 21140 df-bl 21141 df-mopn 21142 df-fbas 21143 df-fg 21144 df-cnfld 21147 df-top 22618 df-topon 22635 df-topsp 22657 df-bases 22671 df-cld 22745 df-ntr 22746 df-cls 22747 df-nei 22824 df-lp 22862 df-perf 22863 df-cn 22953 df-cnp 22954 df-haus 23041 df-tx 23288 df-hmeo 23481 df-fil 23572 df-fm 23664 df-flim 23665 df-flf 23666 df-xms 24048 df-ms 24049 df-tms 24050 df-cncf 24620 df-limc 25617 df-dv 25618 df-log 26299 df-cxp 26300 df-bcc 43400

This theorem is referenced by: binomcxp 43420

W3C validator