nn0sumshdiglemA - Mathbox for Alexander van der Vekens

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Theorem nn0sumshdiglemA 49458

Description: Lemma for nn0sumshdig 49462 (induction step, even multiplier). (Contributed by AV, 3-Jun-2020.)

**Assertion**
Ref	Expression
nn0sumshdiglemA	⊢ (((𝑎 ∈ ℕ ∧ (𝑎 / 2) ∈ ℕ) ∧ 𝑦 ∈ ℕ) → (∀𝑥 ∈ ℕ₀ ((#_b‘𝑥) = 𝑦 → 𝑥 = Σ𝑘 ∈ (0..^𝑦)((𝑘(digit‘2)𝑥) · (2↑𝑘))) → ((#_b‘𝑎) = (𝑦 + 1) → 𝑎 = Σ𝑘 ∈ (0..^(𝑦 + 1))((𝑘(digit‘2)𝑎) · (2↑𝑘)))))

Distinct variable group: 𝑘,𝑎,𝑥,𝑦

Proof of Theorem nn0sumshdiglemA Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nnnn0 12528	. . . 4 ⊢ ((𝑎 / 2) ∈ ℕ → (𝑎 / 2) ∈ ℕ₀)
2		blennn0em1 49430	. . . 4 ⊢ ((𝑎 ∈ ℕ ∧ (𝑎 / 2) ∈ ℕ₀) → (#_b‘(𝑎 / 2)) = ((#_b‘𝑎) − 1))
3	1, 2	sylan2 605	. . 3 ⊢ ((𝑎 ∈ ℕ ∧ (𝑎 / 2) ∈ ℕ) → (#_b‘(𝑎 / 2)) = ((#_b‘𝑎) − 1))
4		fveqeq2 6894	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑎 / 2) → ((#_b‘𝑥) = 𝑦 ↔ (#_b‘(𝑎 / 2)) = 𝑦))
5		id 23	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑎 / 2) → 𝑥 = (𝑎 / 2))
6		oveq2 7427	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑎 / 2) → (𝑘(digit‘2)𝑥) = (𝑘(digit‘2)(𝑎 / 2)))
7	6	oveq1d 7434	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑎 / 2) → ((𝑘(digit‘2)𝑥) · (2↑𝑘)) = ((𝑘(digit‘2)(𝑎 / 2)) · (2↑𝑘)))
8	7	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝑥 = (𝑎 / 2) ∧ 𝑘 ∈ (0..^𝑦)) → ((𝑘(digit‘2)𝑥) · (2↑𝑘)) = ((𝑘(digit‘2)(𝑎 / 2)) · (2↑𝑘)))
9	8	sumeq2dv 15779	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑎 / 2) → Σ𝑘 ∈ (0..^𝑦)((𝑘(digit‘2)𝑥) · (2↑𝑘)) = Σ𝑘 ∈ (0..^𝑦)((𝑘(digit‘2)(𝑎 / 2)) · (2↑𝑘)))
10	5, 9	eqeq12d 2781	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑎 / 2) → (𝑥 = Σ𝑘 ∈ (0..^𝑦)((𝑘(digit‘2)𝑥) · (2↑𝑘)) ↔ (𝑎 / 2) = Σ𝑘 ∈ (0..^𝑦)((𝑘(digit‘2)(𝑎 / 2)) · (2↑𝑘))))
11	4, 10	imbi12d 347	. . . . . . . . . 10 ⊢ (𝑥 = (𝑎 / 2) → (((#_b‘𝑥) = 𝑦 → 𝑥 = Σ𝑘 ∈ (0..^𝑦)((𝑘(digit‘2)𝑥) · (2↑𝑘))) ↔ ((#_b‘(𝑎 / 2)) = 𝑦 → (𝑎 / 2) = Σ𝑘 ∈ (0..^𝑦)((𝑘(digit‘2)(𝑎 / 2)) · (2↑𝑘)))))
12	11	rspcva 3581	. . . . . . . . 9 ⊢ (((𝑎 / 2) ∈ ℕ₀ ∧ ∀𝑥 ∈ ℕ₀ ((#_b‘𝑥) = 𝑦 → 𝑥 = Σ𝑘 ∈ (0..^𝑦)((𝑘(digit‘2)𝑥) · (2↑𝑘)))) → ((#_b‘(𝑎 / 2)) = 𝑦 → (𝑎 / 2) = Σ𝑘 ∈ (0..^𝑦)((𝑘(digit‘2)(𝑎 / 2)) · (2↑𝑘))))
13		simpr 490	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) → (#_b‘𝑎) = (𝑦 + 1))
14	13	oveq1d 7434	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) → ((#_b‘𝑎) − 1) = ((𝑦 + 1) − 1))
15		nncn 12258	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℂ)
16		pncan1 11655	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ℂ → ((𝑦 + 1) − 1) = 𝑦)
17	15, 16	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℕ → ((𝑦 + 1) − 1) = 𝑦)
18	14, 17	sylan9eq 2820	. . . . . . . . . . . . . . 15 ⊢ (((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) → ((#_b‘𝑎) − 1) = 𝑦)
19	18	eqeq2d 2776	. . . . . . . . . . . . . 14 ⊢ (((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) → ((#_b‘(𝑎 / 2)) = ((#_b‘𝑎) − 1) ↔ (#_b‘(𝑎 / 2)) = 𝑦))
20		nnz 12629	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℤ)
21	20	adantl 487	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) → 𝑦 ∈ ℤ)
22		fzval3 13782	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ ℤ → (0...𝑦) = (0..^(𝑦 + 1)))
23	21, 22	syl 18	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) → (0...𝑦) = (0..^(𝑦 + 1)))
24	23	eqcomd 2771	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) → (0..^(𝑦 + 1)) = (0...𝑦))
25	24	sumeq1d 15777	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) → Σ𝑘 ∈ (0..^(𝑦 + 1))((𝑘(digit‘2)𝑎) · (2↑𝑘)) = Σ𝑘 ∈ (0...𝑦)((𝑘(digit‘2)𝑎) · (2↑𝑘)))
26		nnnn0 12528	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℕ₀)
27		elnn0uz 12921	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ ℕ₀ ↔ 𝑦 ∈ (ℤ_≥‘0))
28	26, 27	sylib 221	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ (ℤ_≥‘0))
29	28	adantl 487	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) → 𝑦 ∈ (ℤ_≥‘0))
30		2nn 12331	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 2 ∈ ℕ
31	30	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑘 ∈ (0...𝑦)) → 2 ∈ ℕ)
32		elfzelz 13570	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑘 ∈ (0...𝑦) → 𝑘 ∈ ℤ)
33	32	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑘 ∈ (0...𝑦)) → 𝑘 ∈ ℤ)
34		nnnn0 12528	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑎 ∈ ℕ → 𝑎 ∈ ℕ₀)
35		nn0rp0 13500	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑎 ∈ ℕ₀ → 𝑎 ∈ (0[,)+∞))
36	34, 35	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑎 ∈ ℕ → 𝑎 ∈ (0[,)+∞))
37	36	ad4antlr 746	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑘 ∈ (0...𝑦)) → 𝑎 ∈ (0[,)+∞))
38		digvalnn0 49438	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((2 ∈ ℕ ∧ 𝑘 ∈ ℤ ∧ 𝑎 ∈ (0[,)+∞)) → (𝑘(digit‘2)𝑎) ∈ ℕ₀)
39	31, 33, 37, 38	syl3anc 1398	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑘 ∈ (0...𝑦)) → (𝑘(digit‘2)𝑎) ∈ ℕ₀)
40	39	nn0cnd 12584	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑘 ∈ (0...𝑦)) → (𝑘(digit‘2)𝑎) ∈ ℂ)
41		2nn0 12538	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 2 ∈ ℕ₀
42	41	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑘 ∈ (0...𝑦) → 2 ∈ ℕ₀)
43		elfznn0 13667	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑘 ∈ (0...𝑦) → 𝑘 ∈ ℕ₀)
44	42, 43	nn0expcld 14302	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 ∈ (0...𝑦) → (2↑𝑘) ∈ ℕ₀)
45	44	nn0cnd 12584	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 ∈ (0...𝑦) → (2↑𝑘) ∈ ℂ)
46	45	adantl 487	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑘 ∈ (0...𝑦)) → (2↑𝑘) ∈ ℂ)
47	40, 46	mulcld 11246	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑘 ∈ (0...𝑦)) → ((𝑘(digit‘2)𝑎) · (2↑𝑘)) ∈ ℂ)
48		oveq1 7426	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 = 0 → (𝑘(digit‘2)𝑎) = (0(digit‘2)𝑎))
49		oveq2 7427	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 = 0 → (2↑𝑘) = (2↑0))
50	48, 49	oveq12d 7437	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 = 0 → ((𝑘(digit‘2)𝑎) · (2↑𝑘)) = ((0(digit‘2)𝑎) · (2↑0)))
51		2cn 12333	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 2 ∈ ℂ
52		exp0 14121	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (2 ∈ ℂ → (2↑0) = 1)
53	51, 52	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (2↑0) = 1
54	53	oveq2i 7430	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((0(digit‘2)𝑎) · (2↑0)) = ((0(digit‘2)𝑎) · 1)
55	50, 54	eqtrdi 2816	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 0 → ((𝑘(digit‘2)𝑎) · (2↑𝑘)) = ((0(digit‘2)𝑎) · 1))
56	29, 47, 55	fsum1p 15829	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) → Σ𝑘 ∈ (0...𝑦)((𝑘(digit‘2)𝑎) · (2↑𝑘)) = (((0(digit‘2)𝑎) · 1) + Σ𝑘 ∈ ((0 + 1)...𝑦)((𝑘(digit‘2)𝑎) · (2↑𝑘))))
57		0dig2nn0e 49451	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑎 ∈ ℕ₀ ∧ (𝑎 / 2) ∈ ℕ₀) → (0(digit‘2)𝑎) = 0)
58	34, 1, 57	syl2anr 609	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) → (0(digit‘2)𝑎) = 0)
59	58	oveq1d 7434	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) → ((0(digit‘2)𝑎) · 1) = (0 · 1))
60		1re 11225	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 1 ∈ ℝ
61		mul02lem2 11404	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (1 ∈ ℝ → (0 · 1) = 0)
62	60, 61	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (0 · 1) = 0
63	59, 62	eqtrdi 2816	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) → ((0(digit‘2)𝑎) · 1) = 0)
64	63	adantr 486	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) → ((0(digit‘2)𝑎) · 1) = 0)
65	64	adantr 486	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) → ((0(digit‘2)𝑎) · 1) = 0)
66		1z 12641	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 1 ∈ ℤ
67	66	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) → 1 ∈ ℤ)
68		0p1e1 12378	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (0 + 1) = 1
69	68, 66	eqeltri 2861	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (0 + 1) ∈ ℤ
70	69	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) → (0 + 1) ∈ ℤ)
71	30	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑘 ∈ ((0 + 1)...𝑦)) → 2 ∈ ℕ)
72		elfzelz 13570	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑘 ∈ ((0 + 1)...𝑦) → 𝑘 ∈ ℤ)
73	72	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑘 ∈ ((0 + 1)...𝑦)) → 𝑘 ∈ ℤ)
74	36	ad4antlr 746	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑘 ∈ ((0 + 1)...𝑦)) → 𝑎 ∈ (0[,)+∞))
75	71, 73, 74, 38	syl3anc 1398	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑘 ∈ ((0 + 1)...𝑦)) → (𝑘(digit‘2)𝑎) ∈ ℕ₀)
76	75	nn0cnd 12584	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑘 ∈ ((0 + 1)...𝑦)) → (𝑘(digit‘2)𝑎) ∈ ℂ)
77		2cnd 12336	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑘 ∈ ((0 + 1)...𝑦) → 2 ∈ ℂ)
78		elfznn 13600	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑘 ∈ (1...𝑦) → 𝑘 ∈ ℕ)
79	78	nnnn0d 12582	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑘 ∈ (1...𝑦) → 𝑘 ∈ ℕ₀)
80	68	oveq1i 7429	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((0 + 1)...𝑦) = (1...𝑦)
81	79, 80	eleq2s 2883	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑘 ∈ ((0 + 1)...𝑦) → 𝑘 ∈ ℕ₀)
82	77, 81	expcld 14202	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑘 ∈ ((0 + 1)...𝑦) → (2↑𝑘) ∈ ℂ)
83	82	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑘 ∈ ((0 + 1)...𝑦)) → (2↑𝑘) ∈ ℂ)
84	76, 83	mulcld 11246	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑘 ∈ ((0 + 1)...𝑦)) → ((𝑘(digit‘2)𝑎) · (2↑𝑘)) ∈ ℂ)
85		oveq1 7426	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 = (𝑖 + 1) → (𝑘(digit‘2)𝑎) = ((𝑖 + 1)(digit‘2)𝑎))
86		oveq2 7427	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 = (𝑖 + 1) → (2↑𝑘) = (2↑(𝑖 + 1)))
87	85, 86	oveq12d 7437	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 = (𝑖 + 1) → ((𝑘(digit‘2)𝑎) · (2↑𝑘)) = (((𝑖 + 1)(digit‘2)𝑎) · (2↑(𝑖 + 1))))
88	67, 70, 21, 84, 87	fsumshftm 15857	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) → Σ𝑘 ∈ ((0 + 1)...𝑦)((𝑘(digit‘2)𝑎) · (2↑𝑘)) = Σ𝑖 ∈ (((0 + 1) − 1)...(𝑦 − 1))(((𝑖 + 1)(digit‘2)𝑎) · (2↑(𝑖 + 1))))
89	65, 88	oveq12d 7437	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) → (((0(digit‘2)𝑎) · 1) + Σ𝑘 ∈ ((0 + 1)...𝑦)((𝑘(digit‘2)𝑎) · (2↑𝑘))) = (0 + Σ𝑖 ∈ (((0 + 1) − 1)...(𝑦 − 1))(((𝑖 + 1)(digit‘2)𝑎) · (2↑(𝑖 + 1)))))
90	1	ad4antr 745	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑦)) → (𝑎 / 2) ∈ ℕ₀)
91	34	ad4antlr 746	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑦)) → 𝑎 ∈ ℕ₀)
92		elfzonn0 13755	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑖 ∈ (0..^𝑦) → 𝑖 ∈ ℕ₀)
93	92	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑦)) → 𝑖 ∈ ℕ₀)
94		dignn0ehalf 49456	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑎 / 2) ∈ ℕ₀ ∧ 𝑎 ∈ ℕ₀ ∧ 𝑖 ∈ ℕ₀) → ((𝑖 + 1)(digit‘2)𝑎) = (𝑖(digit‘2)(𝑎 / 2)))
95	90, 91, 93, 94	syl3anc 1398	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑦)) → ((𝑖 + 1)(digit‘2)𝑎) = (𝑖(digit‘2)(𝑎 / 2)))
96		2cnd 12336	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑖 ∈ (0..^𝑦) → 2 ∈ ℂ)
97	96, 92	expp1d 14203	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑖 ∈ (0..^𝑦) → (2↑(𝑖 + 1)) = ((2↑𝑖) · 2))
98	97	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑦)) → (2↑(𝑖 + 1)) = ((2↑𝑖) · 2))
99	95, 98	oveq12d 7437	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑦)) → (((𝑖 + 1)(digit‘2)𝑎) · (2↑(𝑖 + 1))) = ((𝑖(digit‘2)(𝑎 / 2)) · ((2↑𝑖) · 2)))
100	30	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑦)) → 2 ∈ ℕ)
101		elfzoelz 13706	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑖 ∈ (0..^𝑦) → 𝑖 ∈ ℤ)
102	101	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑦)) → 𝑖 ∈ ℤ)
103		nn0rp0 13500	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑎 / 2) ∈ ℕ₀ → (𝑎 / 2) ∈ (0[,)+∞))
104	1, 103	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑎 / 2) ∈ ℕ → (𝑎 / 2) ∈ (0[,)+∞))
105	104	ad4antr 745	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑦)) → (𝑎 / 2) ∈ (0[,)+∞))
106		digvalnn0 49438	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((2 ∈ ℕ ∧ 𝑖 ∈ ℤ ∧ (𝑎 / 2) ∈ (0[,)+∞)) → (𝑖(digit‘2)(𝑎 / 2)) ∈ ℕ₀)
107	100, 102, 105, 106	syl3anc 1398	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑦)) → (𝑖(digit‘2)(𝑎 / 2)) ∈ ℕ₀)
108	107	nn0cnd 12584	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑦)) → (𝑖(digit‘2)(𝑎 / 2)) ∈ ℂ)
109		2re 12332	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ 2 ∈ ℝ
110	109	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑖 ∈ (0..^𝑦) → 2 ∈ ℝ)
111	110, 92	reexpcld 14219	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑖 ∈ (0..^𝑦) → (2↑𝑖) ∈ ℝ)
112	111	recnd 11254	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑖 ∈ (0..^𝑦) → (2↑𝑖) ∈ ℂ)
113	112	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑦)) → (2↑𝑖) ∈ ℂ)
114		2cnd 12336	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑦)) → 2 ∈ ℂ)
115		mulass 11205	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑖(digit‘2)(𝑎 / 2)) ∈ ℂ ∧ (2↑𝑖) ∈ ℂ ∧ 2 ∈ ℂ) → (((𝑖(digit‘2)(𝑎 / 2)) · (2↑𝑖)) · 2) = ((𝑖(digit‘2)(𝑎 / 2)) · ((2↑𝑖) · 2)))
116	115	eqcomd 2771	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑖(digit‘2)(𝑎 / 2)) ∈ ℂ ∧ (2↑𝑖) ∈ ℂ ∧ 2 ∈ ℂ) → ((𝑖(digit‘2)(𝑎 / 2)) · ((2↑𝑖) · 2)) = (((𝑖(digit‘2)(𝑎 / 2)) · (2↑𝑖)) · 2))
117	108, 113, 114, 116	syl3anc 1398	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑦)) → ((𝑖(digit‘2)(𝑎 / 2)) · ((2↑𝑖) · 2)) = (((𝑖(digit‘2)(𝑎 / 2)) · (2↑𝑖)) · 2))
118	99, 117	eqtrd 2800	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑦)) → (((𝑖 + 1)(digit‘2)𝑎) · (2↑(𝑖 + 1))) = (((𝑖(digit‘2)(𝑎 / 2)) · (2↑𝑖)) · 2))
119	118	sumeq2dv 15779	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) → Σ𝑖 ∈ (0..^𝑦)(((𝑖 + 1)(digit‘2)𝑎) · (2↑(𝑖 + 1))) = Σ𝑖 ∈ (0..^𝑦)(((𝑖(digit‘2)(𝑎 / 2)) · (2↑𝑖)) · 2))
120		0cn 11215	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ 0 ∈ ℂ
121		pncan1 11655	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (0 ∈ ℂ → ((0 + 1) − 1) = 0)
122	120, 121	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((0 + 1) − 1) = 0
123	122	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑦 ∈ ℕ → ((0 + 1) − 1) = 0)
124	123	oveq1d 7434	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑦 ∈ ℕ → (((0 + 1) − 1)...(𝑦 − 1)) = (0...(𝑦 − 1)))
125		fzoval 13707	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑦 ∈ ℤ → (0..^𝑦) = (0...(𝑦 − 1)))
126	125	eqcomd 2771	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑦 ∈ ℤ → (0...(𝑦 − 1)) = (0..^𝑦))
127	20, 126	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑦 ∈ ℕ → (0...(𝑦 − 1)) = (0..^𝑦))
128	124, 127	eqtrd 2800	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 ∈ ℕ → (((0 + 1) − 1)...(𝑦 − 1)) = (0..^𝑦))
129	128	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) → (((0 + 1) − 1)...(𝑦 − 1)) = (0..^𝑦))
130	129	sumeq1d 15777	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) → Σ𝑖 ∈ (((0 + 1) − 1)...(𝑦 − 1))(((𝑖 + 1)(digit‘2)𝑎) · (2↑(𝑖 + 1))) = Σ𝑖 ∈ (0..^𝑦)(((𝑖 + 1)(digit‘2)𝑎) · (2↑(𝑖 + 1))))
131	130	oveq2d 7435	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) → (0 + Σ𝑖 ∈ (((0 + 1) − 1)...(𝑦 − 1))(((𝑖 + 1)(digit‘2)𝑎) · (2↑(𝑖 + 1)))) = (0 + Σ𝑖 ∈ (0..^𝑦)(((𝑖 + 1)(digit‘2)𝑎) · (2↑(𝑖 + 1)))))
132		fzofi 14030	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (0..^𝑦) ∈ Fin
133	132	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) → (0..^𝑦) ∈ Fin)
134	101	peano2zd 12721	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑖 ∈ (0..^𝑦) → (𝑖 + 1) ∈ ℤ)
135	134	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑦)) → (𝑖 + 1) ∈ ℤ)
136	36	ad4antlr 746	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑦)) → 𝑎 ∈ (0[,)+∞))
137		digvalnn0 49438	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((2 ∈ ℕ ∧ (𝑖 + 1) ∈ ℤ ∧ 𝑎 ∈ (0[,)+∞)) → ((𝑖 + 1)(digit‘2)𝑎) ∈ ℕ₀)
138	100, 135, 136, 137	syl3anc 1398	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑦)) → ((𝑖 + 1)(digit‘2)𝑎) ∈ ℕ₀)
139	138	nn0cnd 12584	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑦)) → ((𝑖 + 1)(digit‘2)𝑎) ∈ ℂ)
140	41	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑖 ∈ (0..^𝑦) → 2 ∈ ℕ₀)
141		peano2nn0 12561	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑖 ∈ ℕ₀ → (𝑖 + 1) ∈ ℕ₀)
142	92, 141	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑖 ∈ (0..^𝑦) → (𝑖 + 1) ∈ ℕ₀)
143	140, 142	nn0expcld 14302	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑖 ∈ (0..^𝑦) → (2↑(𝑖 + 1)) ∈ ℕ₀)
144	143	nn0cnd 12584	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑖 ∈ (0..^𝑦) → (2↑(𝑖 + 1)) ∈ ℂ)
145	144	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑦)) → (2↑(𝑖 + 1)) ∈ ℂ)
146	139, 145	mulcld 11246	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑦)) → (((𝑖 + 1)(digit‘2)𝑎) · (2↑(𝑖 + 1))) ∈ ℂ)
147	133, 146	fsumcl 15809	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) → Σ𝑖 ∈ (0..^𝑦)(((𝑖 + 1)(digit‘2)𝑎) · (2↑(𝑖 + 1))) ∈ ℂ)
148	147	addlidd 11428	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) → (0 + Σ𝑖 ∈ (0..^𝑦)(((𝑖 + 1)(digit‘2)𝑎) · (2↑(𝑖 + 1)))) = Σ𝑖 ∈ (0..^𝑦)(((𝑖 + 1)(digit‘2)𝑎) · (2↑(𝑖 + 1))))
149	131, 148	eqtrd 2800	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) → (0 + Σ𝑖 ∈ (((0 + 1) − 1)...(𝑦 − 1))(((𝑖 + 1)(digit‘2)𝑎) · (2↑(𝑖 + 1)))) = Σ𝑖 ∈ (0..^𝑦)(((𝑖 + 1)(digit‘2)𝑎) · (2↑(𝑖 + 1))))
150		2cnd 12336	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) → 2 ∈ ℂ)
151	140, 92	nn0expcld 14302	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑖 ∈ (0..^𝑦) → (2↑𝑖) ∈ ℕ₀)
152	151	nn0cnd 12584	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑖 ∈ (0..^𝑦) → (2↑𝑖) ∈ ℂ)
153	152	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑦)) → (2↑𝑖) ∈ ℂ)
154	108, 153	mulcld 11246	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑦)) → ((𝑖(digit‘2)(𝑎 / 2)) · (2↑𝑖)) ∈ ℂ)
155	133, 150, 154	fsummulc1 15861	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) → (Σ𝑖 ∈ (0..^𝑦)((𝑖(digit‘2)(𝑎 / 2)) · (2↑𝑖)) · 2) = Σ𝑖 ∈ (0..^𝑦)(((𝑖(digit‘2)(𝑎 / 2)) · (2↑𝑖)) · 2))
156	119, 149, 155	3eqtr4d 2810	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) → (0 + Σ𝑖 ∈ (((0 + 1) − 1)...(𝑦 − 1))(((𝑖 + 1)(digit‘2)𝑎) · (2↑(𝑖 + 1)))) = (Σ𝑖 ∈ (0..^𝑦)((𝑖(digit‘2)(𝑎 / 2)) · (2↑𝑖)) · 2))
157	89, 156	eqtrd 2800	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) → (((0(digit‘2)𝑎) · 1) + Σ𝑘 ∈ ((0 + 1)...𝑦)((𝑘(digit‘2)𝑎) · (2↑𝑘))) = (Σ𝑖 ∈ (0..^𝑦)((𝑖(digit‘2)(𝑎 / 2)) · (2↑𝑖)) · 2))
158	25, 56, 157	3eqtrd 2804	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) → Σ𝑘 ∈ (0..^(𝑦 + 1))((𝑘(digit‘2)𝑎) · (2↑𝑘)) = (Σ𝑖 ∈ (0..^𝑦)((𝑖(digit‘2)(𝑎 / 2)) · (2↑𝑖)) · 2))
159	158	adantl 487	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑎 / 2) = Σ𝑘 ∈ (0..^𝑦)((𝑘(digit‘2)(𝑎 / 2)) · (2↑𝑘)) ∧ ((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ)) → Σ𝑘 ∈ (0..^(𝑦 + 1))((𝑘(digit‘2)𝑎) · (2↑𝑘)) = (Σ𝑖 ∈ (0..^𝑦)((𝑖(digit‘2)(𝑎 / 2)) · (2↑𝑖)) · 2))
160		oveq1 7426	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑘 = 𝑖 → (𝑘(digit‘2)(𝑎 / 2)) = (𝑖(digit‘2)(𝑎 / 2)))
161		oveq2 7427	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑘 = 𝑖 → (2↑𝑘) = (2↑𝑖))
162	160, 161	oveq12d 7437	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 = 𝑖 → ((𝑘(digit‘2)(𝑎 / 2)) · (2↑𝑘)) = ((𝑖(digit‘2)(𝑎 / 2)) · (2↑𝑖)))
163	162	cbvsumv 15773	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Σ𝑘 ∈ (0..^𝑦)((𝑘(digit‘2)(𝑎 / 2)) · (2↑𝑘)) = Σ𝑖 ∈ (0..^𝑦)((𝑖(digit‘2)(𝑎 / 2)) · (2↑𝑖))
164	163	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) → Σ𝑘 ∈ (0..^𝑦)((𝑘(digit‘2)(𝑎 / 2)) · (2↑𝑘)) = Σ𝑖 ∈ (0..^𝑦)((𝑖(digit‘2)(𝑎 / 2)) · (2↑𝑖)))
165	164	eqeq2d 2776	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) → ((𝑎 / 2) = Σ𝑘 ∈ (0..^𝑦)((𝑘(digit‘2)(𝑎 / 2)) · (2↑𝑘)) ↔ (𝑎 / 2) = Σ𝑖 ∈ (0..^𝑦)((𝑖(digit‘2)(𝑎 / 2)) · (2↑𝑖))))
166	165	biimpac 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑎 / 2) = Σ𝑘 ∈ (0..^𝑦)((𝑘(digit‘2)(𝑎 / 2)) · (2↑𝑘)) ∧ ((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ)) → (𝑎 / 2) = Σ𝑖 ∈ (0..^𝑦)((𝑖(digit‘2)(𝑎 / 2)) · (2↑𝑖)))
167	166	eqcomd 2771	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑎 / 2) = Σ𝑘 ∈ (0..^𝑦)((𝑘(digit‘2)(𝑎 / 2)) · (2↑𝑘)) ∧ ((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ)) → Σ𝑖 ∈ (0..^𝑦)((𝑖(digit‘2)(𝑎 / 2)) · (2↑𝑖)) = (𝑎 / 2))
168	167	oveq1d 7434	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑎 / 2) = Σ𝑘 ∈ (0..^𝑦)((𝑘(digit‘2)(𝑎 / 2)) · (2↑𝑘)) ∧ ((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ)) → (Σ𝑖 ∈ (0..^𝑦)((𝑖(digit‘2)(𝑎 / 2)) · (2↑𝑖)) · 2) = ((𝑎 / 2) · 2))
169		nncn 12258	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 ∈ ℕ → 𝑎 ∈ ℂ)
170		2cnd 12336	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 ∈ ℕ → 2 ∈ ℂ)
171		2ne0 12364	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 2 ≠ 0
172	171	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 ∈ ℕ → 2 ≠ 0)
173	169, 170, 172	divcan1d 12009	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 ∈ ℕ → ((𝑎 / 2) · 2) = 𝑎)
174	173	ad3antlr 744	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) → ((𝑎 / 2) · 2) = 𝑎)
175	174	adantl 487	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑎 / 2) = Σ𝑘 ∈ (0..^𝑦)((𝑘(digit‘2)(𝑎 / 2)) · (2↑𝑘)) ∧ ((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ)) → ((𝑎 / 2) · 2) = 𝑎)
176	159, 168, 175	3eqtrrd 2805	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑎 / 2) = Σ𝑘 ∈ (0..^𝑦)((𝑘(digit‘2)(𝑎 / 2)) · (2↑𝑘)) ∧ ((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ)) → 𝑎 = Σ𝑘 ∈ (0..^(𝑦 + 1))((𝑘(digit‘2)𝑎) · (2↑𝑘)))
177	176	ex 418	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 / 2) = Σ𝑘 ∈ (0..^𝑦)((𝑘(digit‘2)(𝑎 / 2)) · (2↑𝑘)) → (((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) → 𝑎 = Σ𝑘 ∈ (0..^(𝑦 + 1))((𝑘(digit‘2)𝑎) · (2↑𝑘))))
178	177	imim2i 17	. . . . . . . . . . . . . . 15 ⊢ (((#_b‘(𝑎 / 2)) = 𝑦 → (𝑎 / 2) = Σ𝑘 ∈ (0..^𝑦)((𝑘(digit‘2)(𝑎 / 2)) · (2↑𝑘))) → ((#_b‘(𝑎 / 2)) = 𝑦 → (((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) → 𝑎 = Σ𝑘 ∈ (0..^(𝑦 + 1))((𝑘(digit‘2)𝑎) · (2↑𝑘)))))
179	178	com13 89	. . . . . . . . . . . . . 14 ⊢ (((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) → ((#_b‘(𝑎 / 2)) = 𝑦 → (((#_b‘(𝑎 / 2)) = 𝑦 → (𝑎 / 2) = Σ𝑘 ∈ (0..^𝑦)((𝑘(digit‘2)(𝑎 / 2)) · (2↑𝑘))) → 𝑎 = Σ𝑘 ∈ (0..^(𝑦 + 1))((𝑘(digit‘2)𝑎) · (2↑𝑘)))))
180	19, 179	sylbid 243	. . . . . . . . . . . . 13 ⊢ (((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) → ((#_b‘(𝑎 / 2)) = ((#_b‘𝑎) − 1) → (((#_b‘(𝑎 / 2)) = 𝑦 → (𝑎 / 2) = Σ𝑘 ∈ (0..^𝑦)((𝑘(digit‘2)(𝑎 / 2)) · (2↑𝑘))) → 𝑎 = Σ𝑘 ∈ (0..^(𝑦 + 1))((𝑘(digit‘2)𝑎) · (2↑𝑘)))))
181	180	com23 87	. . . . . . . . . . . 12 ⊢ (((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) → (((#_b‘(𝑎 / 2)) = 𝑦 → (𝑎 / 2) = Σ𝑘 ∈ (0..^𝑦)((𝑘(digit‘2)(𝑎 / 2)) · (2↑𝑘))) → ((#_b‘(𝑎 / 2)) = ((#_b‘𝑎) − 1) → 𝑎 = Σ𝑘 ∈ (0..^(𝑦 + 1))((𝑘(digit‘2)𝑎) · (2↑𝑘)))))
182	181	exp31 425	. . . . . . . . . . 11 ⊢ (((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) → ((#_b‘𝑎) = (𝑦 + 1) → (𝑦 ∈ ℕ → (((#_b‘(𝑎 / 2)) = 𝑦 → (𝑎 / 2) = Σ𝑘 ∈ (0..^𝑦)((𝑘(digit‘2)(𝑎 / 2)) · (2↑𝑘))) → ((#_b‘(𝑎 / 2)) = ((#_b‘𝑎) − 1) → 𝑎 = Σ𝑘 ∈ (0..^(𝑦 + 1))((𝑘(digit‘2)𝑎) · (2↑𝑘)))))))
183	182	com25 100	. . . . . . . . . 10 ⊢ (((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) → ((#_b‘(𝑎 / 2)) = ((#_b‘𝑎) − 1) → (𝑦 ∈ ℕ → (((#_b‘(𝑎 / 2)) = 𝑦 → (𝑎 / 2) = Σ𝑘 ∈ (0..^𝑦)((𝑘(digit‘2)(𝑎 / 2)) · (2↑𝑘))) → ((#_b‘𝑎) = (𝑦 + 1) → 𝑎 = Σ𝑘 ∈ (0..^(𝑦 + 1))((𝑘(digit‘2)𝑎) · (2↑𝑘)))))))
184	183	com14 97	. . . . . . . . 9 ⊢ (((#_b‘(𝑎 / 2)) = 𝑦 → (𝑎 / 2) = Σ𝑘 ∈ (0..^𝑦)((𝑘(digit‘2)(𝑎 / 2)) · (2↑𝑘))) → ((#_b‘(𝑎 / 2)) = ((#_b‘𝑎) − 1) → (𝑦 ∈ ℕ → (((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) → ((#_b‘𝑎) = (𝑦 + 1) → 𝑎 = Σ𝑘 ∈ (0..^(𝑦 + 1))((𝑘(digit‘2)𝑎) · (2↑𝑘)))))))
185	12, 184	syl 18	. . . . . . . 8 ⊢ (((𝑎 / 2) ∈ ℕ₀ ∧ ∀𝑥 ∈ ℕ₀ ((#_b‘𝑥) = 𝑦 → 𝑥 = Σ𝑘 ∈ (0..^𝑦)((𝑘(digit‘2)𝑥) · (2↑𝑘)))) → ((#_b‘(𝑎 / 2)) = ((#_b‘𝑎) − 1) → (𝑦 ∈ ℕ → (((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) → ((#_b‘𝑎) = (𝑦 + 1) → 𝑎 = Σ𝑘 ∈ (0..^(𝑦 + 1))((𝑘(digit‘2)𝑎) · (2↑𝑘)))))))
186	185	ex 418	. . . . . . 7 ⊢ ((𝑎 / 2) ∈ ℕ₀ → (∀𝑥 ∈ ℕ₀ ((#_b‘𝑥) = 𝑦 → 𝑥 = Σ𝑘 ∈ (0..^𝑦)((𝑘(digit‘2)𝑥) · (2↑𝑘))) → ((#_b‘(𝑎 / 2)) = ((#_b‘𝑎) − 1) → (𝑦 ∈ ℕ → (((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) → ((#_b‘𝑎) = (𝑦 + 1) → 𝑎 = Σ𝑘 ∈ (0..^(𝑦 + 1))((𝑘(digit‘2)𝑎) · (2↑𝑘))))))))
187	186	com25 100	. . . . . 6 ⊢ ((𝑎 / 2) ∈ ℕ₀ → (((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) → ((#_b‘(𝑎 / 2)) = ((#_b‘𝑎) − 1) → (𝑦 ∈ ℕ → (∀𝑥 ∈ ℕ₀ ((#_b‘𝑥) = 𝑦 → 𝑥 = Σ𝑘 ∈ (0..^𝑦)((𝑘(digit‘2)𝑥) · (2↑𝑘))) → ((#_b‘𝑎) = (𝑦 + 1) → 𝑎 = Σ𝑘 ∈ (0..^(𝑦 + 1))((𝑘(digit‘2)𝑎) · (2↑𝑘))))))))
188	187	expdcom 420	. . . . 5 ⊢ ((𝑎 / 2) ∈ ℕ → (𝑎 ∈ ℕ → ((𝑎 / 2) ∈ ℕ₀ → ((#_b‘(𝑎 / 2)) = ((#_b‘𝑎) − 1) → (𝑦 ∈ ℕ → (∀𝑥 ∈ ℕ₀ ((#_b‘𝑥) = 𝑦 → 𝑥 = Σ𝑘 ∈ (0..^𝑦)((𝑘(digit‘2)𝑥) · (2↑𝑘))) → ((#_b‘𝑎) = (𝑦 + 1) → 𝑎 = Σ𝑘 ∈ (0..^(𝑦 + 1))((𝑘(digit‘2)𝑎) · (2↑𝑘)))))))))
189	1, 188	mpid 45	. . . 4 ⊢ ((𝑎 / 2) ∈ ℕ → (𝑎 ∈ ℕ → ((#_b‘(𝑎 / 2)) = ((#_b‘𝑎) − 1) → (𝑦 ∈ ℕ → (∀𝑥 ∈ ℕ₀ ((#_b‘𝑥) = 𝑦 → 𝑥 = Σ𝑘 ∈ (0..^𝑦)((𝑘(digit‘2)𝑥) · (2↑𝑘))) → ((#_b‘𝑎) = (𝑦 + 1) → 𝑎 = Σ𝑘 ∈ (0..^(𝑦 + 1))((𝑘(digit‘2)𝑎) · (2↑𝑘))))))))
190	189	impcom 413	. . 3 ⊢ ((𝑎 ∈ ℕ ∧ (𝑎 / 2) ∈ ℕ) → ((#_b‘(𝑎 / 2)) = ((#_b‘𝑎) − 1) → (𝑦 ∈ ℕ → (∀𝑥 ∈ ℕ₀ ((#_b‘𝑥) = 𝑦 → 𝑥 = Σ𝑘 ∈ (0..^𝑦)((𝑘(digit‘2)𝑥) · (2↑𝑘))) → ((#_b‘𝑎) = (𝑦 + 1) → 𝑎 = Σ𝑘 ∈ (0..^(𝑦 + 1))((𝑘(digit‘2)𝑎) · (2↑𝑘)))))))
191	3, 190	mpd 16	. 2 ⊢ ((𝑎 ∈ ℕ ∧ (𝑎 / 2) ∈ ℕ) → (𝑦 ∈ ℕ → (∀𝑥 ∈ ℕ₀ ((#_b‘𝑥) = 𝑦 → 𝑥 = Σ𝑘 ∈ (0..^𝑦)((𝑘(digit‘2)𝑥) · (2↑𝑘))) → ((#_b‘𝑎) = (𝑦 + 1) → 𝑎 = Σ𝑘 ∈ (0..^(𝑦 + 1))((𝑘(digit‘2)𝑎) · (2↑𝑘))))))
192	191	imp 412	1 ⊢ (((𝑎 ∈ ℕ ∧ (𝑎 / 2) ∈ ℕ) ∧ 𝑦 ∈ ℕ) → (∀𝑥 ∈ ℕ₀ ((#_b‘𝑥) = 𝑦 → 𝑥 = Σ𝑘 ∈ (0..^𝑦)((𝑘(digit‘2)𝑥) · (2↑𝑘))) → ((#_b‘𝑎) = (𝑦 + 1) → 𝑎 = Σ𝑘 ∈ (0..^(𝑦 + 1))((𝑘(digit‘2)𝑎) · (2↑𝑘)))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2146 ≠ wne 2960 ∀wral 3081 ‘cfv 6540 (class class class)co 7419 Fincfn 8949 ℂcc 11115 ℝcr 11116 0cc0 11117 1c1 11118 + caddc 11120 · cmul 11122 +∞cpnf 11257 − cmin 11458 / cdiv 11888 ℕcn 12250 2c2 12312 ℕ₀cn0 12521 ℤcz 12608 ℤ_≥cuz 12880 [,)cico 13392 ...cfz 13553 ..^cfzo 13701 ↑cexp 14117 Σcsu 15763 #_bcblen 49408 digitcdig 49434

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-inf2 9617 ax-cnex 11173 ax-resscn 11174 ax-1cn 11175 ax-icn 11176 ax-addcl 11177 ax-addrcl 11178 ax-mulcl 11179 ax-mulrcl 11180 ax-mulcom 11181 ax-addass 11182 ax-mulass 11183 ax-distr 11184 ax-i2m1 11185 ax-1ne0 11186 ax-1rid 11187 ax-rnegex 11188 ax-rrecex 11189 ax-cnre 11190 ax-pre-lttri 11191 ax-pre-lttrn 11192 ax-pre-ltadd 11193 ax-pre-mulgt0 11194 ax-pre-sup 11195 ax-addf 11196

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-iin 4961 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7684 df-om 7869 df-1st 7992 df-2nd 7993 df-supp 8163 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-2o 8460 df-er 8700 df-map 8832 df-pm 8833 df-ixp 8902 df-en 8950 df-dom 8951 df-sdom 8952 df-fin 8953 df-fsupp 9329 df-fi 9378 df-sup 9409 df-inf 9410 df-oi 9479 df-card 9941 df-pnf 11262 df-mnf 11263 df-xr 11264 df-ltxr 11265 df-le 11266 df-sub 11460 df-neg 11461 df-div 11889 df-nn 12251 df-2 12320 df-3 12321 df-4 12322 df-5 12323 df-6 12324 df-7 12325 df-8 12326 df-9 12327 df-n0 12522 df-z 12609 df-dec 12730 df-uz 12881 df-q 12991 df-rp 13035 df-xneg 13155 df-xadd 13156 df-xmul 13157 df-ioo 13394 df-ioc 13395 df-ico 13396 df-icc 13397 df-fz 13554 df-fzo 13702 df-fl 13845 df-mod 13923 df-seq 14058 df-exp 14118 df-fac 14330 df-bc 14359 df-hash 14387 df-shft 15130 df-cj 15176 df-re 15177 df-im 15178 df-sqrt 15312 df-abs 15313 df-limsup 15548 df-clim 15565 df-rlim 15566 df-sum 15764 df-ef 16145 df-sin 16147 df-cos 16148 df-pi 16150 df-struct 17231 df-sets 17248 df-slot 17266 df-ndx 17278 df-base 17294 df-ress 17315 df-plusg 17347 df-mulr 17348 df-starv 17349 df-sca 17350 df-vsca 17351 df-ip 17352 df-tset 17353 df-ple 17354 df-ds 17356 df-unif 17357 df-hom 17358 df-cco 17359 df-rest 17499 df-topn 17500 df-0g 17518 df-gsum 17519 df-topgen 17520 df-pt 17521 df-prds 17524 df-xrs 17580 df-qtop 17585 df-imas 17586 df-xps 17588 df-mre 17662 df-mrc 17663 df-acs 17665 df-mgm 18722 df-sgrp 18811 df-mnd 18827 df-submnd 18881 df-mulg 19180 df-cntz 19433 df-cmn 19898 df-psmet 21566 df-xmet 21567 df-met 21568 df-bl 21569 df-mopn 21570 df-fbas 21571 df-fg 21572 df-cnfld 21575 df-top 23103 df-topon 23120 df-topsp 23142 df-bases 23155 df-cld 23228 df-ntr 23229 df-cls 23230 df-nei 23307 df-lp 23345 df-perf 23346 df-cn 23436 df-cnp 23437 df-haus 23524 df-tx 23772 df-hmeo 23965 df-fil 24056 df-fm 24148 df-flim 24149 df-flf 24150 df-xms 24530 df-ms 24531 df-tms 24532 df-cncf 25090 df-limc 26078 df-dv 26079 df-log 26774 df-logb 26983 df-blen 49409 df-dig 49435

This theorem is used by: nn0sumshdiglem1 49460

W3C validator