binomcxplemdvbinom - Mathbox for Steve Rodriguez

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Theorem binomcxplemdvbinom 43713

Description: Lemma for binomcxp 43717. By the power and chain rules, calculate the derivative of ((1 + 𝑏)↑_𝑐-𝐶), with respect to 𝑏 in the disk of convergence 𝐷. We later multiply the derivative in the later binomcxplemdvsum 43715 by this derivative to show that ((1 + 𝑏)↑_𝑐𝐶) (with a nonnegated 𝐶) and the later sum, since both at 𝑏 = 0 equal one, are the same. (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)
binomcxplem.f	⊢ 𝐹 = (𝑗 ∈ ℕ₀ ↦ (𝐶C_𝑐𝑗))
binomcxplem.s	⊢ 𝑆 = (𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ₀ ↦ ((𝐹‘𝑘) · (𝑏↑𝑘))))
binomcxplem.r	⊢ 𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑆‘𝑟)) ∈ dom ⇝ }, ℝ^*, < )
binomcxplem.e	⊢ 𝐸 = (𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ ↦ ((𝑘 · (𝐹‘𝑘)) · (𝑏↑(𝑘 − 1)))))
binomcxplem.d	⊢ 𝐷 = (^◡abs “ (0[,)𝑅))

**Assertion**
Ref	Expression
binomcxplemdvbinom	⊢ ((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) → (ℂ D (𝑏 ∈ 𝐷 ↦ ((1 + 𝑏)↑_𝑐-𝐶))) = (𝑏 ∈ 𝐷 ↦ (-𝐶 · ((1 + 𝑏)↑_𝑐(-𝐶 − 1)))))

Distinct variable groups: 𝑗,𝑘,𝜑 𝑘,𝑏,𝐶 𝐶,𝑗 𝐹,𝑏,𝑘 𝑆,𝑟 𝑟,𝑏 Allowed substitution hints: 𝜑(𝑟,𝑏) 𝐴(𝑗,𝑘,𝑟,𝑏) 𝐵(𝑗,𝑘,𝑟,𝑏) 𝐶(𝑟) 𝐷(𝑗,𝑘,𝑟,𝑏) 𝑅(𝑗,𝑘,𝑟,𝑏) 𝑆(𝑗,𝑘,𝑏) 𝐸(𝑗,𝑘,𝑟,𝑏) 𝐹(𝑗,𝑟)

Proof of Theorem binomcxplemdvbinom Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		binomcxplem.d	. . . . 5 ⊢ 𝐷 = (^◡abs “ (0[,)𝑅))
2		nfcv 2898	. . . . . 6 ⊢ Ⅎ𝑏^◡abs
3		nfcv 2898	. . . . . . 7 ⊢ Ⅎ𝑏0
4		nfcv 2898	. . . . . . 7 ⊢ Ⅎ𝑏[,)
5		binomcxplem.r	. . . . . . . 8 ⊢ 𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑆‘𝑟)) ∈ dom ⇝ }, ℝ^*, < )
6		nfcv 2898	. . . . . . . . . . . 12 ⊢ Ⅎ𝑏 +
7		binomcxplem.s	. . . . . . . . . . . . . 14 ⊢ 𝑆 = (𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ₀ ↦ ((𝐹‘𝑘) · (𝑏↑𝑘))))
8		nfmpt1 5250	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑏(𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ₀ ↦ ((𝐹‘𝑘) · (𝑏↑𝑘))))
9	7, 8	nfcxfr 2896	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑏𝑆
10		nfcv 2898	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑏𝑟
11	9, 10	nffv 6901	. . . . . . . . . . . 12 ⊢ Ⅎ𝑏(𝑆‘𝑟)
12	3, 6, 11	nfseq 14000	. . . . . . . . . . 11 ⊢ Ⅎ𝑏seq0( + , (𝑆‘𝑟))
13	12	nfel1 2914	. . . . . . . . . 10 ⊢ Ⅎ𝑏seq0( + , (𝑆‘𝑟)) ∈ dom ⇝
14		nfcv 2898	. . . . . . . . . 10 ⊢ Ⅎ𝑏ℝ
15	13, 14	nfrabw 3463	. . . . . . . . 9 ⊢ Ⅎ𝑏{𝑟 ∈ ℝ ∣ seq0( + , (𝑆‘𝑟)) ∈ dom ⇝ }
16		nfcv 2898	. . . . . . . . 9 ⊢ Ⅎ𝑏ℝ^*
17		nfcv 2898	. . . . . . . . 9 ⊢ Ⅎ𝑏 <
18	15, 16, 17	nfsup 9466	. . . . . . . 8 ⊢ Ⅎ𝑏sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑆‘𝑟)) ∈ dom ⇝ }, ℝ^*, < )
19	5, 18	nfcxfr 2896	. . . . . . 7 ⊢ Ⅎ𝑏𝑅
20	3, 4, 19	nfov 7444	. . . . . 6 ⊢ Ⅎ𝑏(0[,)𝑅)
21	2, 20	nfima 6065	. . . . 5 ⊢ Ⅎ𝑏(^◡abs “ (0[,)𝑅))
22	1, 21	nfcxfr 2896	. . . 4 ⊢ Ⅎ𝑏𝐷
23		nfcv 2898	. . . 4 ⊢ Ⅎ𝑦𝐷
24		nfcv 2898	. . . 4 ⊢ Ⅎ𝑦((1 + 𝑏)↑_𝑐-𝐶)
25		nfcv 2898	. . . 4 ⊢ Ⅎ𝑏((1 + 𝑦)↑_𝑐-𝐶)
26		oveq2 7422	. . . . 5 ⊢ (𝑏 = 𝑦 → (1 + 𝑏) = (1 + 𝑦))
27	26	oveq1d 7429	. . . 4 ⊢ (𝑏 = 𝑦 → ((1 + 𝑏)↑_𝑐-𝐶) = ((1 + 𝑦)↑_𝑐-𝐶))
28	22, 23, 24, 25, 27	cbvmptf 5251	. . 3 ⊢ (𝑏 ∈ 𝐷 ↦ ((1 + 𝑏)↑_𝑐-𝐶)) = (𝑦 ∈ 𝐷 ↦ ((1 + 𝑦)↑_𝑐-𝐶))
29	28	oveq2i 7425	. 2 ⊢ (ℂ D (𝑏 ∈ 𝐷 ↦ ((1 + 𝑏)↑_𝑐-𝐶))) = (ℂ D (𝑦 ∈ 𝐷 ↦ ((1 + 𝑦)↑_𝑐-𝐶)))
30		cnelprrecn 11223	. . . . 5 ⊢ ℂ ∈ {ℝ, ℂ}
31	30	a1i 11	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) → ℂ ∈ {ℝ, ℂ})
32		1cnd 11231	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑦 ∈ 𝐷) → 1 ∈ ℂ)
33		cnvimass 6079	. . . . . . . . . 10 ⊢ (^◡abs “ (0[,)𝑅)) ⊆ dom abs
34	1, 33	eqsstri 4012	. . . . . . . . 9 ⊢ 𝐷 ⊆ dom abs
35		absf 15308	. . . . . . . . . 10 ⊢ abs:ℂ⟶ℝ
36	35	fdmi 6728	. . . . . . . . 9 ⊢ dom abs = ℂ
37	34, 36	sseqtri 4014	. . . . . . . 8 ⊢ 𝐷 ⊆ ℂ
38	37	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) → 𝐷 ⊆ ℂ)
39	38	sselda 3978	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑦 ∈ 𝐷) → 𝑦 ∈ ℂ)
40	32, 39	addcld 11255	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑦 ∈ 𝐷) → (1 + 𝑦) ∈ ℂ)
41		simpr 484	. . . . . . 7 ⊢ ((((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑦 ∈ 𝐷) ∧ (1 + 𝑦) ∈ ℝ) → (1 + 𝑦) ∈ ℝ)
42		1cnd 11231	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑦 ∈ 𝐷) ∧ (1 + 𝑦) ∈ ℝ) → 1 ∈ ℂ)
43	39	adantr 480	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑦 ∈ 𝐷) ∧ (1 + 𝑦) ∈ ℝ) → 𝑦 ∈ ℂ)
44	42, 43	pncan2d 11595	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑦 ∈ 𝐷) ∧ (1 + 𝑦) ∈ ℝ) → ((1 + 𝑦) − 1) = 𝑦)
45		1red 11237	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑦 ∈ 𝐷) ∧ (1 + 𝑦) ∈ ℝ) → 1 ∈ ℝ)
46	41, 45	resubcld 11664	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑦 ∈ 𝐷) ∧ (1 + 𝑦) ∈ ℝ) → ((1 + 𝑦) − 1) ∈ ℝ)
47	44, 46	eqeltrrd 2829	. . . . . . . 8 ⊢ ((((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑦 ∈ 𝐷) ∧ (1 + 𝑦) ∈ ℝ) → 𝑦 ∈ ℝ)
48		1pneg1e0 12353	. . . . . . . . 9 ⊢ (1 + -1) = 0
49		1red 11237	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑦 ∈ 𝐷) ∧ 𝑦 ∈ ℝ) → 1 ∈ ℝ)
50	49	renegcld 11663	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑦 ∈ 𝐷) ∧ 𝑦 ∈ ℝ) → -1 ∈ ℝ)
51		simpr 484	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑦 ∈ 𝐷) ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ)
52		ffn 6716	. . . . . . . . . . . . . . . . . . . 20 ⊢ (abs:ℂ⟶ℝ → abs Fn ℂ)
53		elpreima 7061	. . . . . . . . . . . . . . . . . . . 20 ⊢ (abs Fn ℂ → (𝑦 ∈ (^◡abs “ (0[,)𝑅)) ↔ (𝑦 ∈ ℂ ∧ (abs‘𝑦) ∈ (0[,)𝑅))))
54	35, 52, 53	mp2b 10	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ (^◡abs “ (0[,)𝑅)) ↔ (𝑦 ∈ ℂ ∧ (abs‘𝑦) ∈ (0[,)𝑅)))
55	54	simprbi 496	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ (^◡abs “ (0[,)𝑅)) → (abs‘𝑦) ∈ (0[,)𝑅))
56	55, 1	eleq2s 2846	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ 𝐷 → (abs‘𝑦) ∈ (0[,)𝑅))
57		0re 11238	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ ℝ
58		ssrab2 4073	. . . . . . . . . . . . . . . . . . . . 21 ⊢ {𝑟 ∈ ℝ ∣ seq0( + , (𝑆‘𝑟)) ∈ dom ⇝ } ⊆ ℝ
59		ressxr 11280	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ℝ ⊆ ℝ^*
60	58, 59	sstri 3987	. . . . . . . . . . . . . . . . . . . 20 ⊢ {𝑟 ∈ ℝ ∣ seq0( + , (𝑆‘𝑟)) ∈ dom ⇝ } ⊆ ℝ^*
61		supxrcl 13318	. . . . . . . . . . . . . . . . . . . 20 ⊢ ({𝑟 ∈ ℝ ∣ seq0( + , (𝑆‘𝑟)) ∈ dom ⇝ } ⊆ ℝ^* → sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑆‘𝑟)) ∈ dom ⇝ }, ℝ^, < ) ∈ ℝ^)
62	60, 61	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑆‘𝑟)) ∈ dom ⇝ }, ℝ^, < ) ∈ ℝ^
63	5, 62	eqeltri 2824	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑅 ∈ ℝ^*
64		elico2 13412	. . . . . . . . . . . . . . . . . 18 ⊢ ((0 ∈ ℝ ∧ 𝑅 ∈ ℝ^*) → ((abs‘𝑦) ∈ (0[,)𝑅) ↔ ((abs‘𝑦) ∈ ℝ ∧ 0 ≤ (abs‘𝑦) ∧ (abs‘𝑦) < 𝑅)))
65	57, 63, 64	mp2an 691	. . . . . . . . . . . . . . . . 17 ⊢ ((abs‘𝑦) ∈ (0[,)𝑅) ↔ ((abs‘𝑦) ∈ ℝ ∧ 0 ≤ (abs‘𝑦) ∧ (abs‘𝑦) < 𝑅))
66	56, 65	sylib 217	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ 𝐷 → ((abs‘𝑦) ∈ ℝ ∧ 0 ≤ (abs‘𝑦) ∧ (abs‘𝑦) < 𝑅))
67	66	simp3d 1142	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ 𝐷 → (abs‘𝑦) < 𝑅)
68	67	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑦 ∈ 𝐷) → (abs‘𝑦) < 𝑅)
69		binomcxp.a	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐴 ∈ ℝ⁺)
70		binomcxp.b	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐵 ∈ ℝ)
71		binomcxp.lt	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (abs‘𝐵) < (abs‘𝐴))
72		binomcxp.c	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐶 ∈ ℂ)
73		binomcxplem.f	. . . . . . . . . . . . . . . 16 ⊢ 𝐹 = (𝑗 ∈ ℕ₀ ↦ (𝐶C_𝑐𝑗))
74	69, 70, 71, 72, 73, 7, 5	binomcxplemradcnv 43712	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) → 𝑅 = 1)
75	74	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑦 ∈ 𝐷) → 𝑅 = 1)
76	68, 75	breqtrd 5168	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑦 ∈ 𝐷) → (abs‘𝑦) < 1)
77	76	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑦 ∈ 𝐷) ∧ 𝑦 ∈ ℝ) → (abs‘𝑦) < 1)
78	51, 49	absltd 15400	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑦 ∈ 𝐷) ∧ 𝑦 ∈ ℝ) → ((abs‘𝑦) < 1 ↔ (-1 < 𝑦 ∧ 𝑦 < 1)))
79	77, 78	mpbid 231	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑦 ∈ 𝐷) ∧ 𝑦 ∈ ℝ) → (-1 < 𝑦 ∧ 𝑦 < 1))
80	79	simpld 494	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑦 ∈ 𝐷) ∧ 𝑦 ∈ ℝ) → -1 < 𝑦)
81	50, 51, 49, 80	ltadd2dd 11395	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑦 ∈ 𝐷) ∧ 𝑦 ∈ ℝ) → (1 + -1) < (1 + 𝑦))
82	48, 81	eqbrtrrid 5178	. . . . . . . 8 ⊢ ((((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑦 ∈ 𝐷) ∧ 𝑦 ∈ ℝ) → 0 < (1 + 𝑦))
83	47, 82	syldan 590	. . . . . . 7 ⊢ ((((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑦 ∈ 𝐷) ∧ (1 + 𝑦) ∈ ℝ) → 0 < (1 + 𝑦))
84	41, 83	elrpd 13037	. . . . . 6 ⊢ ((((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑦 ∈ 𝐷) ∧ (1 + 𝑦) ∈ ℝ) → (1 + 𝑦) ∈ ℝ⁺)
85	84	ex 412	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑦 ∈ 𝐷) → ((1 + 𝑦) ∈ ℝ → (1 + 𝑦) ∈ ℝ⁺))
86		eqid 2727	. . . . . 6 ⊢ (ℂ ∖ (-∞(,]0)) = (ℂ ∖ (-∞(,]0))
87	86	ellogdm 26560	. . . . 5 ⊢ ((1 + 𝑦) ∈ (ℂ ∖ (-∞(,]0)) ↔ ((1 + 𝑦) ∈ ℂ ∧ ((1 + 𝑦) ∈ ℝ → (1 + 𝑦) ∈ ℝ⁺)))
88	40, 85, 87	sylanbrc 582	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑦 ∈ 𝐷) → (1 + 𝑦) ∈ (ℂ ∖ (-∞(,]0)))
89		eldifi 4122	. . . . . 6 ⊢ (𝑥 ∈ (ℂ ∖ (-∞(,]0)) → 𝑥 ∈ ℂ)
90	89	adantl 481	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑥 ∈ (ℂ ∖ (-∞(,]0))) → 𝑥 ∈ ℂ)
91	72	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) → 𝐶 ∈ ℂ)
92	91	negcld 11580	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) → -𝐶 ∈ ℂ)
93	92	adantr 480	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑥 ∈ (ℂ ∖ (-∞(,]0))) → -𝐶 ∈ ℂ)
94	90, 93	cxpcld 26629	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑥 ∈ (ℂ ∖ (-∞(,]0))) → (𝑥↑_𝑐-𝐶) ∈ ℂ)
95		ovexd 7449	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑥 ∈ (ℂ ∖ (-∞(,]0))) → (-𝐶 · (𝑥↑_𝑐(-𝐶 − 1))) ∈ V)
96		1cnd 11231	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑥 ∈ ℂ) → 1 ∈ ℂ)
97		simpr 484	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑥 ∈ ℂ) → 𝑥 ∈ ℂ)
98	96, 97	addcld 11255	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑥 ∈ ℂ) → (1 + 𝑥) ∈ ℂ)
99		c0ex 11230	. . . . . . . . 9 ⊢ 0 ∈ V
100	99	a1i 11	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑥 ∈ ℂ) → 0 ∈ V)
101		1cnd 11231	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) → 1 ∈ ℂ)
102	31, 101	dvmptc 25877	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) → (ℂ D (𝑥 ∈ ℂ ↦ 1)) = (𝑥 ∈ ℂ ↦ 0))
103	31	dvmptid 25876	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) → (ℂ D (𝑥 ∈ ℂ ↦ 𝑥)) = (𝑥 ∈ ℂ ↦ 1))
104	31, 96, 100, 102, 97, 96, 103	dvmptadd 25879	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) → (ℂ D (𝑥 ∈ ℂ ↦ (1 + 𝑥))) = (𝑥 ∈ ℂ ↦ (0 + 1)))
105		0p1e1 12356	. . . . . . . 8 ⊢ (0 + 1) = 1
106	105	mpteq2i 5247	. . . . . . 7 ⊢ (𝑥 ∈ ℂ ↦ (0 + 1)) = (𝑥 ∈ ℂ ↦ 1)
107	104, 106	eqtrdi 2783	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) → (ℂ D (𝑥 ∈ ℂ ↦ (1 + 𝑥))) = (𝑥 ∈ ℂ ↦ 1))
108		fvex 6904	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) ∈ V
109		cnfldtps 24681	. . . . . . . . . 10 ⊢ ℂ_fld ∈ TopSp
110		cnfldbas 21270	. . . . . . . . . . 11 ⊢ ℂ = (Base‘ℂ_fld)
111		eqid 2727	. . . . . . . . . . 11 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
112	110, 111	tpsuni 22825	. . . . . . . . . 10 ⊢ (ℂ_fld ∈ TopSp → ℂ = ∪ (TopOpen‘ℂ_fld))
113	109, 112	ax-mp 5	. . . . . . . . 9 ⊢ ℂ = ∪ (TopOpen‘ℂ_fld)
114	113	restid 17406	. . . . . . . 8 ⊢ ((TopOpen‘ℂ_fld) ∈ V → ((TopOpen‘ℂ_fld) ↾_t ℂ) = (TopOpen‘ℂ_fld))
115	108, 114	ax-mp 5	. . . . . . 7 ⊢ ((TopOpen‘ℂ_fld) ↾_t ℂ) = (TopOpen‘ℂ_fld)
116	115	eqcomi 2736	. . . . . 6 ⊢ (TopOpen‘ℂ_fld) = ((TopOpen‘ℂ_fld) ↾_t ℂ)
117	111	cnfldtop 24687	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) ∈ Top
118		eqid 2727	. . . . . . . . . . . 12 ⊢ (abs ∘ − ) = (abs ∘ − )
119	118	cnbl0 24677	. . . . . . . . . . 11 ⊢ (𝑅 ∈ ℝ^* → (^◡abs “ (0[,)𝑅)) = (0(ball‘(abs ∘ − ))𝑅))
120	63, 119	ax-mp 5	. . . . . . . . . 10 ⊢ (^◡abs “ (0[,)𝑅)) = (0(ball‘(abs ∘ − ))𝑅)
121	1, 120	eqtri 2755	. . . . . . . . 9 ⊢ 𝐷 = (0(ball‘(abs ∘ − ))𝑅)
122		cnxmet 24676	. . . . . . . . . 10 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
123		0cn 11228	. . . . . . . . . 10 ⊢ 0 ∈ ℂ
124	111	cnfldtopn 24685	. . . . . . . . . . 11 ⊢ (TopOpen‘ℂ_fld) = (MetOpen‘(abs ∘ − ))
125	124	blopn 24396	. . . . . . . . . 10 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ 𝑅 ∈ ℝ^*) → (0(ball‘(abs ∘ − ))𝑅) ∈ (TopOpen‘ℂ_fld))
126	122, 123, 63, 125	mp3an 1458	. . . . . . . . 9 ⊢ (0(ball‘(abs ∘ − ))𝑅) ∈ (TopOpen‘ℂ_fld)
127	121, 126	eqeltri 2824	. . . . . . . 8 ⊢ 𝐷 ∈ (TopOpen‘ℂ_fld)
128		isopn3i 22973	. . . . . . . 8 ⊢ (((TopOpen‘ℂ_fld) ∈ Top ∧ 𝐷 ∈ (TopOpen‘ℂ_fld)) → ((int‘(TopOpen‘ℂ_fld))‘𝐷) = 𝐷)
129	117, 127, 128	mp2an 691	. . . . . . 7 ⊢ ((int‘(TopOpen‘ℂ_fld))‘𝐷) = 𝐷
130	129	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) → ((int‘(TopOpen‘ℂ_fld))‘𝐷) = 𝐷)
131	31, 98, 96, 107, 38, 116, 111, 130	dvmptres2 25881	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) → (ℂ D (𝑥 ∈ 𝐷 ↦ (1 + 𝑥))) = (𝑥 ∈ 𝐷 ↦ 1))
132		oveq2 7422	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (1 + 𝑥) = (1 + 𝑦))
133	132	cbvmptv 5255	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 ↦ (1 + 𝑥)) = (𝑦 ∈ 𝐷 ↦ (1 + 𝑦))
134	133	oveq2i 7425	. . . . 5 ⊢ (ℂ D (𝑥 ∈ 𝐷 ↦ (1 + 𝑥))) = (ℂ D (𝑦 ∈ 𝐷 ↦ (1 + 𝑦)))
135		eqidd 2728	. . . . . 6 ⊢ (𝑥 = 𝑦 → 1 = 1)
136	135	cbvmptv 5255	. . . . 5 ⊢ (𝑥 ∈ 𝐷 ↦ 1) = (𝑦 ∈ 𝐷 ↦ 1)
137	131, 134, 136	3eqtr3g 2790	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) → (ℂ D (𝑦 ∈ 𝐷 ↦ (1 + 𝑦))) = (𝑦 ∈ 𝐷 ↦ 1))
138	86	dvcncxp1 26664	. . . . 5 ⊢ (-𝐶 ∈ ℂ → (ℂ D (𝑥 ∈ (ℂ ∖ (-∞(,]0)) ↦ (𝑥↑_𝑐-𝐶))) = (𝑥 ∈ (ℂ ∖ (-∞(,]0)) ↦ (-𝐶 · (𝑥↑_𝑐(-𝐶 − 1)))))
139	92, 138	syl 17	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) → (ℂ D (𝑥 ∈ (ℂ ∖ (-∞(,]0)) ↦ (𝑥↑_𝑐-𝐶))) = (𝑥 ∈ (ℂ ∖ (-∞(,]0)) ↦ (-𝐶 · (𝑥↑_𝑐(-𝐶 − 1)))))
140		oveq1 7421	. . . 4 ⊢ (𝑥 = (1 + 𝑦) → (𝑥↑_𝑐-𝐶) = ((1 + 𝑦)↑_𝑐-𝐶))
141		oveq1 7421	. . . . 5 ⊢ (𝑥 = (1 + 𝑦) → (𝑥↑_𝑐(-𝐶 − 1)) = ((1 + 𝑦)↑_𝑐(-𝐶 − 1)))
142	141	oveq2d 7430	. . . 4 ⊢ (𝑥 = (1 + 𝑦) → (-𝐶 · (𝑥↑_𝑐(-𝐶 − 1))) = (-𝐶 · ((1 + 𝑦)↑_𝑐(-𝐶 − 1))))
143	31, 31, 88, 32, 94, 95, 137, 139, 140, 142	dvmptco 25891	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) → (ℂ D (𝑦 ∈ 𝐷 ↦ ((1 + 𝑦)↑_𝑐-𝐶))) = (𝑦 ∈ 𝐷 ↦ ((-𝐶 · ((1 + 𝑦)↑_𝑐(-𝐶 − 1))) · 1)))
144	91	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑦 ∈ 𝐷) → 𝐶 ∈ ℂ)
145	144	negcld 11580	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑦 ∈ 𝐷) → -𝐶 ∈ ℂ)
146	145, 32	subcld 11593	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑦 ∈ 𝐷) → (-𝐶 − 1) ∈ ℂ)
147	40, 146	cxpcld 26629	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑦 ∈ 𝐷) → ((1 + 𝑦)↑_𝑐(-𝐶 − 1)) ∈ ℂ)
148	145, 147	mulcld 11256	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑦 ∈ 𝐷) → (-𝐶 · ((1 + 𝑦)↑_𝑐(-𝐶 − 1))) ∈ ℂ)
149	148	mulridd 11253	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑦 ∈ 𝐷) → ((-𝐶 · ((1 + 𝑦)↑_𝑐(-𝐶 − 1))) · 1) = (-𝐶 · ((1 + 𝑦)↑_𝑐(-𝐶 − 1))))
150	149	mpteq2dva 5242	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) → (𝑦 ∈ 𝐷 ↦ ((-𝐶 · ((1 + 𝑦)↑_𝑐(-𝐶 − 1))) · 1)) = (𝑦 ∈ 𝐷 ↦ (-𝐶 · ((1 + 𝑦)↑_𝑐(-𝐶 − 1)))))
151		nfcv 2898	. . . . 5 ⊢ Ⅎ𝑏(-𝐶 · ((1 + 𝑦)↑_𝑐(-𝐶 − 1)))
152		nfcv 2898	. . . . 5 ⊢ Ⅎ𝑦(-𝐶 · ((1 + 𝑏)↑_𝑐(-𝐶 − 1)))
153		oveq2 7422	. . . . . . 7 ⊢ (𝑦 = 𝑏 → (1 + 𝑦) = (1 + 𝑏))
154	153	oveq1d 7429	. . . . . 6 ⊢ (𝑦 = 𝑏 → ((1 + 𝑦)↑_𝑐(-𝐶 − 1)) = ((1 + 𝑏)↑_𝑐(-𝐶 − 1)))
155	154	oveq2d 7430	. . . . 5 ⊢ (𝑦 = 𝑏 → (-𝐶 · ((1 + 𝑦)↑_𝑐(-𝐶 − 1))) = (-𝐶 · ((1 + 𝑏)↑_𝑐(-𝐶 − 1))))
156	23, 22, 151, 152, 155	cbvmptf 5251	. . . 4 ⊢ (𝑦 ∈ 𝐷 ↦ (-𝐶 · ((1 + 𝑦)↑_𝑐(-𝐶 − 1)))) = (𝑏 ∈ 𝐷 ↦ (-𝐶 · ((1 + 𝑏)↑_𝑐(-𝐶 − 1))))
157	156	a1i 11	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) → (𝑦 ∈ 𝐷 ↦ (-𝐶 · ((1 + 𝑦)↑_𝑐(-𝐶 − 1)))) = (𝑏 ∈ 𝐷 ↦ (-𝐶 · ((1 + 𝑏)↑_𝑐(-𝐶 − 1)))))
158	143, 150, 157	3eqtrd 2771	. 2 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) → (ℂ D (𝑦 ∈ 𝐷 ↦ ((1 + 𝑦)↑_𝑐-𝐶))) = (𝑏 ∈ 𝐷 ↦ (-𝐶 · ((1 + 𝑏)↑_𝑐(-𝐶 − 1)))))
159	29, 158	eqtrid 2779	1 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) → (ℂ D (𝑏 ∈ 𝐷 ↦ ((1 + 𝑏)↑_𝑐-𝐶))) = (𝑏 ∈ 𝐷 ↦ (-𝐶 · ((1 + 𝑏)↑_𝑐(-𝐶 − 1)))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 {crab 3427 Vcvv 3469 ∖ cdif 3941 ⊆ wss 3944 {cpr 4626 ∪ cuni 4903 class class class wbr 5142 ↦ cmpt 5225 ^◡ccnv 5671 dom cdm 5672 “ cima 5675 ∘ ccom 5676 Fn wfn 6537 ⟶wf 6538 ‘cfv 6542 (class class class)co 7414 supcsup 9455 ℂcc 11128 ℝcr 11129 0cc0 11130 1c1 11131 + caddc 11133 · cmul 11135 -∞cmnf 11268 ℝ^*cxr 11269 < clt 11270 ≤ cle 11271 − cmin 11466 -cneg 11467 ℕcn 12234 ℕ₀cn0 12494 ℝ⁺crp 12998 (,]cioc 13349 [,)cico 13350 seqcseq 13990 ↑cexp 14050 abscabs 15205 ⇝ cli 15452 ↾_t crest 17393 TopOpenctopn 17394 ∞Metcxmet 21251 ballcbl 21253 ℂ_fldccnfld 21266 Topctop 22782 TopSpctps 22821 intcnt 22908 D cdv 25779 ↑_𝑐ccxp 26476 C_𝑐cbcc 43696

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-inf2 9656 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 ax-pre-sup 11208 ax-addf 11209

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-om 7865 df-1st 7987 df-2nd 7988 df-supp 8160 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-er 8718 df-map 8838 df-pm 8839 df-ixp 8908 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-fsupp 9378 df-fi 9426 df-sup 9457 df-inf 9458 df-oi 9525 df-card 9954 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-div 11894 df-nn 12235 df-2 12297 df-3 12298 df-4 12299 df-5 12300 df-6 12301 df-7 12302 df-8 12303 df-9 12304 df-n0 12495 df-z 12581 df-dec 12700 df-uz 12845 df-q 12955 df-rp 12999 df-xneg 13116 df-xadd 13117 df-xmul 13118 df-ioo 13352 df-ioc 13353 df-ico 13354 df-icc 13355 df-fz 13509 df-fzo 13652 df-fl 13781 df-mod 13859 df-seq 13991 df-exp 14051 df-fac 14257 df-bc 14286 df-hash 14314 df-shft 15038 df-cj 15070 df-re 15071 df-im 15072 df-sqrt 15206 df-abs 15207 df-limsup 15439 df-clim 15456 df-rlim 15457 df-sum 15657 df-prod 15874 df-fallfac 15975 df-ef 16035 df-sin 16037 df-cos 16038 df-tan 16039 df-pi 16040 df-struct 17107 df-sets 17124 df-slot 17142 df-ndx 17154 df-base 17172 df-ress 17201 df-plusg 17237 df-mulr 17238 df-starv 17239 df-sca 17240 df-vsca 17241 df-ip 17242 df-tset 17243 df-ple 17244 df-ds 17246 df-unif 17247 df-hom 17248 df-cco 17249 df-rest 17395 df-topn 17396 df-0g 17414 df-gsum 17415 df-topgen 17416 df-pt 17417 df-prds 17420 df-xrs 17475 df-qtop 17480 df-imas 17481 df-xps 17483 df-mre 17557 df-mrc 17558 df-acs 17560 df-mgm 18591 df-sgrp 18670 df-mnd 18686 df-submnd 18732 df-mulg 19015 df-cntz 19259 df-cmn 19728 df-psmet 21258 df-xmet 21259 df-met 21260 df-bl 21261 df-mopn 21262 df-fbas 21263 df-fg 21264 df-cnfld 21267 df-top 22783 df-topon 22800 df-topsp 22822 df-bases 22836 df-cld 22910 df-ntr 22911 df-cls 22912 df-nei 22989 df-lp 23027 df-perf 23028 df-cn 23118 df-cnp 23119 df-haus 23206 df-cmp 23278 df-tx 23453 df-hmeo 23646 df-fil 23737 df-fm 23829 df-flim 23830 df-flf 23831 df-xms 24213 df-ms 24214 df-tms 24215 df-cncf 24785 df-limc 25782 df-dv 25783 df-log 26477 df-cxp 26478 df-bcc 43697

This theorem is referenced by: binomcxplemnotnn0 43716

W3C validator