binomcxplemdvbinom - Mathbox for Steve Rodriguez

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Theorem binomcxplemdvbinom 43855

Description: Lemma for binomcxp 43859. 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 43857 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 2892	. . . . . 6 ⊢ Ⅎ𝑏^◡abs
3		nfcv 2892	. . . . . . 7 ⊢ Ⅎ𝑏0
4		nfcv 2892	. . . . . . 7 ⊢ Ⅎ𝑏[,)
5		binomcxplem.r	. . . . . . . 8 ⊢ 𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑆‘𝑟)) ∈ dom ⇝ }, ℝ^*, < )
6		nfcv 2892	. . . . . . . . . . . 12 ⊢ Ⅎ𝑏 +
7		binomcxplem.s	. . . . . . . . . . . . . 14 ⊢ 𝑆 = (𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ₀ ↦ ((𝐹‘𝑘) · (𝑏↑𝑘))))
8		nfmpt1 5256	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑏(𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ₀ ↦ ((𝐹‘𝑘) · (𝑏↑𝑘))))
9	7, 8	nfcxfr 2890	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑏𝑆
10		nfcv 2892	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑏𝑟
11	9, 10	nffv 6904	. . . . . . . . . . . 12 ⊢ Ⅎ𝑏(𝑆‘𝑟)
12	3, 6, 11	nfseq 14008	. . . . . . . . . . 11 ⊢ Ⅎ𝑏seq0( + , (𝑆‘𝑟))
13	12	nfel1 2909	. . . . . . . . . 10 ⊢ Ⅎ𝑏seq0( + , (𝑆‘𝑟)) ∈ dom ⇝
14		nfcv 2892	. . . . . . . . . 10 ⊢ Ⅎ𝑏ℝ
15	13, 14	nfrabw 3457	. . . . . . . . 9 ⊢ Ⅎ𝑏{𝑟 ∈ ℝ ∣ seq0( + , (𝑆‘𝑟)) ∈ dom ⇝ }
16		nfcv 2892	. . . . . . . . 9 ⊢ Ⅎ𝑏ℝ^*
17		nfcv 2892	. . . . . . . . 9 ⊢ Ⅎ𝑏 <
18	15, 16, 17	nfsup 9474	. . . . . . . 8 ⊢ Ⅎ𝑏sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑆‘𝑟)) ∈ dom ⇝ }, ℝ^*, < )
19	5, 18	nfcxfr 2890	. . . . . . 7 ⊢ Ⅎ𝑏𝑅
20	3, 4, 19	nfov 7447	. . . . . 6 ⊢ Ⅎ𝑏(0[,)𝑅)
21	2, 20	nfima 6071	. . . . 5 ⊢ Ⅎ𝑏(^◡abs “ (0[,)𝑅))
22	1, 21	nfcxfr 2890	. . . 4 ⊢ Ⅎ𝑏𝐷
23		nfcv 2892	. . . 4 ⊢ Ⅎ𝑦𝐷
24		nfcv 2892	. . . 4 ⊢ Ⅎ𝑦((1 + 𝑏)↑_𝑐-𝐶)
25		nfcv 2892	. . . 4 ⊢ Ⅎ𝑏((1 + 𝑦)↑_𝑐-𝐶)
26		oveq2 7425	. . . . 5 ⊢ (𝑏 = 𝑦 → (1 + 𝑏) = (1 + 𝑦))
27	26	oveq1d 7432	. . . 4 ⊢ (𝑏 = 𝑦 → ((1 + 𝑏)↑_𝑐-𝐶) = ((1 + 𝑦)↑_𝑐-𝐶))
28	22, 23, 24, 25, 27	cbvmptf 5257	. . 3 ⊢ (𝑏 ∈ 𝐷 ↦ ((1 + 𝑏)↑_𝑐-𝐶)) = (𝑦 ∈ 𝐷 ↦ ((1 + 𝑦)↑_𝑐-𝐶))
29	28	oveq2i 7428	. 2 ⊢ (ℂ D (𝑏 ∈ 𝐷 ↦ ((1 + 𝑏)↑_𝑐-𝐶))) = (ℂ D (𝑦 ∈ 𝐷 ↦ ((1 + 𝑦)↑_𝑐-𝐶)))
30		cnelprrecn 11231	. . . . 5 ⊢ ℂ ∈ {ℝ, ℂ}
31	30	a1i 11	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) → ℂ ∈ {ℝ, ℂ})
32		1cnd 11239	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑦 ∈ 𝐷) → 1 ∈ ℂ)
33		cnvimass 6085	. . . . . . . . . 10 ⊢ (^◡abs “ (0[,)𝑅)) ⊆ dom abs
34	1, 33	eqsstri 4012	. . . . . . . . 9 ⊢ 𝐷 ⊆ dom abs
35		absf 15316	. . . . . . . . . 10 ⊢ abs:ℂ⟶ℝ
36	35	fdmi 6732	. . . . . . . . 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 3977	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑦 ∈ 𝐷) → 𝑦 ∈ ℂ)
40	32, 39	addcld 11263	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑦 ∈ 𝐷) → (1 + 𝑦) ∈ ℂ)
41		simpr 483	. . . . . . 7 ⊢ ((((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑦 ∈ 𝐷) ∧ (1 + 𝑦) ∈ ℝ) → (1 + 𝑦) ∈ ℝ)
42		1cnd 11239	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑦 ∈ 𝐷) ∧ (1 + 𝑦) ∈ ℝ) → 1 ∈ ℂ)
43	39	adantr 479	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑦 ∈ 𝐷) ∧ (1 + 𝑦) ∈ ℝ) → 𝑦 ∈ ℂ)
44	42, 43	pncan2d 11603	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑦 ∈ 𝐷) ∧ (1 + 𝑦) ∈ ℝ) → ((1 + 𝑦) − 1) = 𝑦)
45		1red 11245	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑦 ∈ 𝐷) ∧ (1 + 𝑦) ∈ ℝ) → 1 ∈ ℝ)
46	41, 45	resubcld 11672	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑦 ∈ 𝐷) ∧ (1 + 𝑦) ∈ ℝ) → ((1 + 𝑦) − 1) ∈ ℝ)
47	44, 46	eqeltrrd 2826	. . . . . . . 8 ⊢ ((((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑦 ∈ 𝐷) ∧ (1 + 𝑦) ∈ ℝ) → 𝑦 ∈ ℝ)
48		1pneg1e0 12361	. . . . . . . . 9 ⊢ (1 + -1) = 0
49		1red 11245	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑦 ∈ 𝐷) ∧ 𝑦 ∈ ℝ) → 1 ∈ ℝ)
50	49	renegcld 11671	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑦 ∈ 𝐷) ∧ 𝑦 ∈ ℝ) → -1 ∈ ℝ)
51		simpr 483	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑦 ∈ 𝐷) ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ)
52		ffn 6721	. . . . . . . . . . . . . . . . . . . 20 ⊢ (abs:ℂ⟶ℝ → abs Fn ℂ)
53		elpreima 7064	. . . . . . . . . . . . . . . . . . . 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 495	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ (^◡abs “ (0[,)𝑅)) → (abs‘𝑦) ∈ (0[,)𝑅))
56	55, 1	eleq2s 2843	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ 𝐷 → (abs‘𝑦) ∈ (0[,)𝑅))
57		0re 11246	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ ℝ
58		ssrab2 4074	. . . . . . . . . . . . . . . . . . . . 21 ⊢ {𝑟 ∈ ℝ ∣ seq0( + , (𝑆‘𝑟)) ∈ dom ⇝ } ⊆ ℝ
59		ressxr 11288	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ℝ ⊆ ℝ^*
60	58, 59	sstri 3987	. . . . . . . . . . . . . . . . . . . 20 ⊢ {𝑟 ∈ ℝ ∣ seq0( + , (𝑆‘𝑟)) ∈ dom ⇝ } ⊆ ℝ^*
61		supxrcl 13326	. . . . . . . . . . . . . . . . . . . 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 2821	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑅 ∈ ℝ^*
64		elico2 13420	. . . . . . . . . . . . . . . . . 18 ⊢ ((0 ∈ ℝ ∧ 𝑅 ∈ ℝ^*) → ((abs‘𝑦) ∈ (0[,)𝑅) ↔ ((abs‘𝑦) ∈ ℝ ∧ 0 ≤ (abs‘𝑦) ∧ (abs‘𝑦) < 𝑅)))
65	57, 63, 64	mp2an 690	. . . . . . . . . . . . . . . . 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 1141	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ 𝐷 → (abs‘𝑦) < 𝑅)
68	67	adantl 480	. . . . . . . . . . . . . 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 43854	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) → 𝑅 = 1)
75	74	adantr 479	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑦 ∈ 𝐷) → 𝑅 = 1)
76	68, 75	breqtrd 5174	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑦 ∈ 𝐷) → (abs‘𝑦) < 1)
77	76	adantr 479	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑦 ∈ 𝐷) ∧ 𝑦 ∈ ℝ) → (abs‘𝑦) < 1)
78	51, 49	absltd 15408	. . . . . . . . . . . 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 493	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑦 ∈ 𝐷) ∧ 𝑦 ∈ ℝ) → -1 < 𝑦)
81	50, 51, 49, 80	ltadd2dd 11403	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑦 ∈ 𝐷) ∧ 𝑦 ∈ ℝ) → (1 + -1) < (1 + 𝑦))
82	48, 81	eqbrtrrid 5184	. . . . . . . 8 ⊢ ((((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑦 ∈ 𝐷) ∧ 𝑦 ∈ ℝ) → 0 < (1 + 𝑦))
83	47, 82	syldan 589	. . . . . . 7 ⊢ ((((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑦 ∈ 𝐷) ∧ (1 + 𝑦) ∈ ℝ) → 0 < (1 + 𝑦))
84	41, 83	elrpd 13045	. . . . . 6 ⊢ ((((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑦 ∈ 𝐷) ∧ (1 + 𝑦) ∈ ℝ) → (1 + 𝑦) ∈ ℝ⁺)
85	84	ex 411	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑦 ∈ 𝐷) → ((1 + 𝑦) ∈ ℝ → (1 + 𝑦) ∈ ℝ⁺))
86		eqid 2725	. . . . . 6 ⊢ (ℂ ∖ (-∞(,]0)) = (ℂ ∖ (-∞(,]0))
87	86	ellogdm 26603	. . . . 5 ⊢ ((1 + 𝑦) ∈ (ℂ ∖ (-∞(,]0)) ↔ ((1 + 𝑦) ∈ ℂ ∧ ((1 + 𝑦) ∈ ℝ → (1 + 𝑦) ∈ ℝ⁺)))
88	40, 85, 87	sylanbrc 581	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑦 ∈ 𝐷) → (1 + 𝑦) ∈ (ℂ ∖ (-∞(,]0)))
89		eldifi 4124	. . . . . 6 ⊢ (𝑥 ∈ (ℂ ∖ (-∞(,]0)) → 𝑥 ∈ ℂ)
90	89	adantl 480	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑥 ∈ (ℂ ∖ (-∞(,]0))) → 𝑥 ∈ ℂ)
91	72	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) → 𝐶 ∈ ℂ)
92	91	negcld 11588	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) → -𝐶 ∈ ℂ)
93	92	adantr 479	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑥 ∈ (ℂ ∖ (-∞(,]0))) → -𝐶 ∈ ℂ)
94	90, 93	cxpcld 26672	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑥 ∈ (ℂ ∖ (-∞(,]0))) → (𝑥↑_𝑐-𝐶) ∈ ℂ)
95		ovexd 7452	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑥 ∈ (ℂ ∖ (-∞(,]0))) → (-𝐶 · (𝑥↑_𝑐(-𝐶 − 1))) ∈ V)
96		1cnd 11239	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑥 ∈ ℂ) → 1 ∈ ℂ)
97		simpr 483	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑥 ∈ ℂ) → 𝑥 ∈ ℂ)
98	96, 97	addcld 11263	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑥 ∈ ℂ) → (1 + 𝑥) ∈ ℂ)
99		c0ex 11238	. . . . . . . . 9 ⊢ 0 ∈ V
100	99	a1i 11	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑥 ∈ ℂ) → 0 ∈ V)
101		1cnd 11239	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) → 1 ∈ ℂ)
102	31, 101	dvmptc 25920	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) → (ℂ D (𝑥 ∈ ℂ ↦ 1)) = (𝑥 ∈ ℂ ↦ 0))
103	31	dvmptid 25919	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) → (ℂ D (𝑥 ∈ ℂ ↦ 𝑥)) = (𝑥 ∈ ℂ ↦ 1))
104	31, 96, 100, 102, 97, 96, 103	dvmptadd 25922	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) → (ℂ D (𝑥 ∈ ℂ ↦ (1 + 𝑥))) = (𝑥 ∈ ℂ ↦ (0 + 1)))
105		0p1e1 12364	. . . . . . . 8 ⊢ (0 + 1) = 1
106	105	mpteq2i 5253	. . . . . . 7 ⊢ (𝑥 ∈ ℂ ↦ (0 + 1)) = (𝑥 ∈ ℂ ↦ 1)
107	104, 106	eqtrdi 2781	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) → (ℂ D (𝑥 ∈ ℂ ↦ (1 + 𝑥))) = (𝑥 ∈ ℂ ↦ 1))
108		fvex 6907	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) ∈ V
109		cnfldtps 24724	. . . . . . . . . 10 ⊢ ℂ_fld ∈ TopSp
110		cnfldbas 21287	. . . . . . . . . . 11 ⊢ ℂ = (Base‘ℂ_fld)
111		eqid 2725	. . . . . . . . . . 11 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
112	110, 111	tpsuni 22868	. . . . . . . . . 10 ⊢ (ℂ_fld ∈ TopSp → ℂ = ∪ (TopOpen‘ℂ_fld))
113	109, 112	ax-mp 5	. . . . . . . . 9 ⊢ ℂ = ∪ (TopOpen‘ℂ_fld)
114	113	restid 17414	. . . . . . . 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 2734	. . . . . 6 ⊢ (TopOpen‘ℂ_fld) = ((TopOpen‘ℂ_fld) ↾_t ℂ)
117	111	cnfldtop 24730	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) ∈ Top
118		eqid 2725	. . . . . . . . . . . 12 ⊢ (abs ∘ − ) = (abs ∘ − )
119	118	cnbl0 24720	. . . . . . . . . . 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 2753	. . . . . . . . 9 ⊢ 𝐷 = (0(ball‘(abs ∘ − ))𝑅)
122		cnxmet 24719	. . . . . . . . . 10 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
123		0cn 11236	. . . . . . . . . 10 ⊢ 0 ∈ ℂ
124	111	cnfldtopn 24728	. . . . . . . . . . 11 ⊢ (TopOpen‘ℂ_fld) = (MetOpen‘(abs ∘ − ))
125	124	blopn 24439	. . . . . . . . . 10 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ 𝑅 ∈ ℝ^*) → (0(ball‘(abs ∘ − ))𝑅) ∈ (TopOpen‘ℂ_fld))
126	122, 123, 63, 125	mp3an 1457	. . . . . . . . 9 ⊢ (0(ball‘(abs ∘ − ))𝑅) ∈ (TopOpen‘ℂ_fld)
127	121, 126	eqeltri 2821	. . . . . . . 8 ⊢ 𝐷 ∈ (TopOpen‘ℂ_fld)
128		isopn3i 23016	. . . . . . . 8 ⊢ (((TopOpen‘ℂ_fld) ∈ Top ∧ 𝐷 ∈ (TopOpen‘ℂ_fld)) → ((int‘(TopOpen‘ℂ_fld))‘𝐷) = 𝐷)
129	117, 127, 128	mp2an 690	. . . . . . 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 25924	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) → (ℂ D (𝑥 ∈ 𝐷 ↦ (1 + 𝑥))) = (𝑥 ∈ 𝐷 ↦ 1))
132		oveq2 7425	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (1 + 𝑥) = (1 + 𝑦))
133	132	cbvmptv 5261	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 ↦ (1 + 𝑥)) = (𝑦 ∈ 𝐷 ↦ (1 + 𝑦))
134	133	oveq2i 7428	. . . . 5 ⊢ (ℂ D (𝑥 ∈ 𝐷 ↦ (1 + 𝑥))) = (ℂ D (𝑦 ∈ 𝐷 ↦ (1 + 𝑦)))
135		eqidd 2726	. . . . . 6 ⊢ (𝑥 = 𝑦 → 1 = 1)
136	135	cbvmptv 5261	. . . . 5 ⊢ (𝑥 ∈ 𝐷 ↦ 1) = (𝑦 ∈ 𝐷 ↦ 1)
137	131, 134, 136	3eqtr3g 2788	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) → (ℂ D (𝑦 ∈ 𝐷 ↦ (1 + 𝑦))) = (𝑦 ∈ 𝐷 ↦ 1))
138	86	dvcncxp1 26707	. . . . 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 7424	. . . 4 ⊢ (𝑥 = (1 + 𝑦) → (𝑥↑_𝑐-𝐶) = ((1 + 𝑦)↑_𝑐-𝐶))
141		oveq1 7424	. . . . 5 ⊢ (𝑥 = (1 + 𝑦) → (𝑥↑_𝑐(-𝐶 − 1)) = ((1 + 𝑦)↑_𝑐(-𝐶 − 1)))
142	141	oveq2d 7433	. . . 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 25934	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) → (ℂ D (𝑦 ∈ 𝐷 ↦ ((1 + 𝑦)↑_𝑐-𝐶))) = (𝑦 ∈ 𝐷 ↦ ((-𝐶 · ((1 + 𝑦)↑_𝑐(-𝐶 − 1))) · 1)))
144	91	adantr 479	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑦 ∈ 𝐷) → 𝐶 ∈ ℂ)
145	144	negcld 11588	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑦 ∈ 𝐷) → -𝐶 ∈ ℂ)
146	145, 32	subcld 11601	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑦 ∈ 𝐷) → (-𝐶 − 1) ∈ ℂ)
147	40, 146	cxpcld 26672	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑦 ∈ 𝐷) → ((1 + 𝑦)↑_𝑐(-𝐶 − 1)) ∈ ℂ)
148	145, 147	mulcld 11264	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑦 ∈ 𝐷) → (-𝐶 · ((1 + 𝑦)↑_𝑐(-𝐶 − 1))) ∈ ℂ)
149	148	mulridd 11261	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) ∧ 𝑦 ∈ 𝐷) → ((-𝐶 · ((1 + 𝑦)↑_𝑐(-𝐶 − 1))) · 1) = (-𝐶 · ((1 + 𝑦)↑_𝑐(-𝐶 − 1))))
150	149	mpteq2dva 5248	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) → (𝑦 ∈ 𝐷 ↦ ((-𝐶 · ((1 + 𝑦)↑_𝑐(-𝐶 − 1))) · 1)) = (𝑦 ∈ 𝐷 ↦ (-𝐶 · ((1 + 𝑦)↑_𝑐(-𝐶 − 1)))))
151		nfcv 2892	. . . . 5 ⊢ Ⅎ𝑏(-𝐶 · ((1 + 𝑦)↑_𝑐(-𝐶 − 1)))
152		nfcv 2892	. . . . 5 ⊢ Ⅎ𝑦(-𝐶 · ((1 + 𝑏)↑_𝑐(-𝐶 − 1)))
153		oveq2 7425	. . . . . . 7 ⊢ (𝑦 = 𝑏 → (1 + 𝑦) = (1 + 𝑏))
154	153	oveq1d 7432	. . . . . 6 ⊢ (𝑦 = 𝑏 → ((1 + 𝑦)↑_𝑐(-𝐶 − 1)) = ((1 + 𝑏)↑_𝑐(-𝐶 − 1)))
155	154	oveq2d 7433	. . . . 5 ⊢ (𝑦 = 𝑏 → (-𝐶 · ((1 + 𝑦)↑_𝑐(-𝐶 − 1))) = (-𝐶 · ((1 + 𝑏)↑_𝑐(-𝐶 − 1))))
156	23, 22, 151, 152, 155	cbvmptf 5257	. . . 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 2769	. 2 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ ℕ₀) → (ℂ D (𝑦 ∈ 𝐷 ↦ ((1 + 𝑦)↑_𝑐-𝐶))) = (𝑏 ∈ 𝐷 ↦ (-𝐶 · ((1 + 𝑏)↑_𝑐(-𝐶 − 1)))))
159	29, 158	eqtrid 2777	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 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 {crab 3419 Vcvv 3463 ∖ cdif 3942 ⊆ wss 3945 {cpr 4631 ∪ cuni 4908 class class class wbr 5148 ↦ cmpt 5231 ^◡ccnv 5676 dom cdm 5677 “ cima 5680 ∘ ccom 5681 Fn wfn 6542 ⟶wf 6543 ‘cfv 6547 (class class class)co 7417 supcsup 9463 ℂcc 11136 ℝcr 11137 0cc0 11138 1c1 11139 + caddc 11141 · cmul 11143 -∞cmnf 11276 ℝ^*cxr 11277 < clt 11278 ≤ cle 11279 − cmin 11474 -cneg 11475 ℕcn 12242 ℕ₀cn0 12502 ℝ⁺crp 13006 (,]cioc 13357 [,)cico 13358 seqcseq 13998 ↑cexp 14058 abscabs 15213 ⇝ cli 15460 ↾_t crest 17401 TopOpenctopn 17402 ∞Metcxmet 21268 ballcbl 21270 ℂ_fldccnfld 21283 Topctop 22825 TopSpctps 22864 intcnt 22951 D cdv 25822 ↑_𝑐ccxp 26519 C_𝑐cbcc 43838

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5364 ax-pr 5428 ax-un 7739 ax-inf2 9664 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 ax-addf 11217

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3965 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 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 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6499 df-fun 6549 df-fn 6550 df-f 6551 df-f1 6552 df-fo 6553 df-f1o 6554 df-fv 6555 df-isom 6556 df-riota 7373 df-ov 7420 df-oprab 7421 df-mpo 7422 df-of 7683 df-om 7870 df-1st 7992 df-2nd 7993 df-supp 8164 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-er 8723 df-map 8845 df-pm 8846 df-ixp 8915 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-fsupp 9386 df-fi 9434 df-sup 9465 df-inf 9466 df-oi 9533 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-q 12963 df-rp 13007 df-xneg 13124 df-xadd 13125 df-xmul 13126 df-ioo 13360 df-ioc 13361 df-ico 13362 df-icc 13363 df-fz 13517 df-fzo 13660 df-fl 13789 df-mod 13867 df-seq 13999 df-exp 14059 df-fac 14265 df-bc 14294 df-hash 14322 df-shft 15046 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-abs 15215 df-limsup 15447 df-clim 15464 df-rlim 15465 df-sum 15665 df-prod 15882 df-fallfac 15983 df-ef 16043 df-sin 16045 df-cos 16046 df-tan 16047 df-pi 16048 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-starv 17247 df-sca 17248 df-vsca 17249 df-ip 17250 df-tset 17251 df-ple 17252 df-ds 17254 df-unif 17255 df-hom 17256 df-cco 17257 df-rest 17403 df-topn 17404 df-0g 17422 df-gsum 17423 df-topgen 17424 df-pt 17425 df-prds 17428 df-xrs 17483 df-qtop 17488 df-imas 17489 df-xps 17491 df-mre 17565 df-mrc 17566 df-acs 17568 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18740 df-mulg 19028 df-cntz 19272 df-cmn 19741 df-psmet 21275 df-xmet 21276 df-met 21277 df-bl 21278 df-mopn 21279 df-fbas 21280 df-fg 21281 df-cnfld 21284 df-top 22826 df-topon 22843 df-topsp 22865 df-bases 22879 df-cld 22953 df-ntr 22954 df-cls 22955 df-nei 23032 df-lp 23070 df-perf 23071 df-cn 23161 df-cnp 23162 df-haus 23249 df-cmp 23321 df-tx 23496 df-hmeo 23689 df-fil 23780 df-fm 23872 df-flim 23873 df-flf 23874 df-xms 24256 df-ms 24257 df-tms 24258 df-cncf 24828 df-limc 25825 df-dv 25826 df-log 26520 df-cxp 26521 df-bcc 43839

This theorem is referenced by: binomcxplemnotnn0 43858

W3C validator