nn0sumshdiglemA - Mathbox for Alexander van der Vekens

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Theorem nn0sumshdiglemA 44686

Description: Lemma for nn0sumshdig 44690 (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 11907	. . . 4 ⊢ ((𝑎 / 2) ∈ ℕ → (𝑎 / 2) ∈ ℕ₀)
2		blennn0em1 44658	. . . 4 ⊢ ((𝑎 ∈ ℕ ∧ (𝑎 / 2) ∈ ℕ₀) → (#_b‘(𝑎 / 2)) = ((#_b‘𝑎) − 1))
3	1, 2	sylan2 594	. . 3 ⊢ ((𝑎 ∈ ℕ ∧ (𝑎 / 2) ∈ ℕ) → (#_b‘(𝑎 / 2)) = ((#_b‘𝑎) − 1))
4		fveqeq2 6681	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑎 / 2) → ((#_b‘𝑥) = 𝑦 ↔ (#_b‘(𝑎 / 2)) = 𝑦))
5		id 22	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑎 / 2) → 𝑥 = (𝑎 / 2))
6		oveq2 7166	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑎 / 2) → (𝑘(digit‘2)𝑥) = (𝑘(digit‘2)(𝑎 / 2)))
7	6	oveq1d 7173	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑎 / 2) → ((𝑘(digit‘2)𝑥) · (2↑𝑘)) = ((𝑘(digit‘2)(𝑎 / 2)) · (2↑𝑘)))
8	7	adantr 483	. . . . . . . . . . . . 13 ⊢ ((𝑥 = (𝑎 / 2) ∧ 𝑘 ∈ (0..^𝑦)) → ((𝑘(digit‘2)𝑥) · (2↑𝑘)) = ((𝑘(digit‘2)(𝑎 / 2)) · (2↑𝑘)))
9	8	sumeq2dv 15062	. . . . . . . . . . . 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 2839	. . . . . . . . . . 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 3623	. . . . . . . . 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 487	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) → (#_b‘𝑎) = (𝑦 + 1))
14	13	oveq1d 7173	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) → ((#_b‘𝑎) − 1) = ((𝑦 + 1) − 1))
15		nncn 11648	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℂ)
16		pncan1 11066	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ℂ → ((𝑦 + 1) − 1) = 𝑦)
17	15, 16	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℕ → ((𝑦 + 1) − 1) = 𝑦)
18	14, 17	sylan9eq 2878	. . . . . . . . . . . . . . 15 ⊢ (((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) → ((#_b‘𝑎) − 1) = 𝑦)
19	18	eqeq2d 2834	. . . . . . . . . . . . . 14 ⊢ (((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) → ((#_b‘(𝑎 / 2)) = ((#_b‘𝑎) − 1) ↔ (#_b‘(𝑎 / 2)) = 𝑦))
20		nnz 12007	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℤ)
21	20	adantl 484	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) → 𝑦 ∈ ℤ)
22		fzval3 13109	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ ℤ → (0...𝑦) = (0..^(𝑦 + 1)))
23	21, 22	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) → (0...𝑦) = (0..^(𝑦 + 1)))
24	23	eqcomd 2829	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) → (0..^(𝑦 + 1)) = (0...𝑦))
25	24	sumeq1d 15060	. . . . . . . . . . . . . . . . . . . 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 11907	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℕ₀)
27		elnn0uz 12286	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ ℕ₀ ↔ 𝑦 ∈ (ℤ_≥‘0))
28	26, 27	sylib 220	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ (ℤ_≥‘0))
29	28	adantl 484	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) → 𝑦 ∈ (ℤ_≥‘0))
30		2nn 11713	. . . . . . . . . . . . . . . . . . . . . . . . 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 12911	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑘 ∈ (0...𝑦) → 𝑘 ∈ ℤ)
33	32	adantl 484	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑘 ∈ (0...𝑦)) → 𝑘 ∈ ℤ)
34		nnnn0 11907	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑎 ∈ ℕ → 𝑎 ∈ ℕ₀)
35		nn0rp0 12846	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑎 ∈ ℕ₀ → 𝑎 ∈ (0[,)+∞))
36	34, 35	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑎 ∈ ℕ → 𝑎 ∈ (0[,)+∞))
37	36	ad4antlr 731	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑘 ∈ (0...𝑦)) → 𝑎 ∈ (0[,)+∞))
38		digvalnn0 44666	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((2 ∈ ℕ ∧ 𝑘 ∈ ℤ ∧ 𝑎 ∈ (0[,)+∞)) → (𝑘(digit‘2)𝑎) ∈ ℕ₀)
39	31, 33, 37, 38	syl3anc 1367	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑘 ∈ (0...𝑦)) → (𝑘(digit‘2)𝑎) ∈ ℕ₀)
40	39	nn0cnd 11960	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑘 ∈ (0...𝑦)) → (𝑘(digit‘2)𝑎) ∈ ℂ)
41		2nn0 11917	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 2 ∈ ℕ₀
42	41	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑘 ∈ (0...𝑦) → 2 ∈ ℕ₀)
43		elfznn0 13003	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑘 ∈ (0...𝑦) → 𝑘 ∈ ℕ₀)
44	42, 43	nn0expcld 13610	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 ∈ (0...𝑦) → (2↑𝑘) ∈ ℕ₀)
45	44	nn0cnd 11960	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 ∈ (0...𝑦) → (2↑𝑘) ∈ ℂ)
46	45	adantl 484	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑘 ∈ (0...𝑦)) → (2↑𝑘) ∈ ℂ)
47	40, 46	mulcld 10663	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑘 ∈ (0...𝑦)) → ((𝑘(digit‘2)𝑎) · (2↑𝑘)) ∈ ℂ)
48		oveq1 7165	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 = 0 → (𝑘(digit‘2)𝑎) = (0(digit‘2)𝑎))
49		oveq2 7166	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 = 0 → (2↑𝑘) = (2↑0))
50	48, 49	oveq12d 7176	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 = 0 → ((𝑘(digit‘2)𝑎) · (2↑𝑘)) = ((0(digit‘2)𝑎) · (2↑0)))
51		2cn 11715	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 2 ∈ ℂ
52		exp0 13436	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (2 ∈ ℂ → (2↑0) = 1)
53	51, 52	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (2↑0) = 1
54	53	oveq2i 7169	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((0(digit‘2)𝑎) · (2↑0)) = ((0(digit‘2)𝑎) · 1)
55	50, 54	syl6eq 2874	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 0 → ((𝑘(digit‘2)𝑎) · (2↑𝑘)) = ((0(digit‘2)𝑎) · 1))
56	29, 47, 55	fsum1p 15110	. . . . . . . . . . . . . . . . . . . 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 44679	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑎 ∈ ℕ₀ ∧ (𝑎 / 2) ∈ ℕ₀) → (0(digit‘2)𝑎) = 0)
58	34, 1, 57	syl2anr 598	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) → (0(digit‘2)𝑎) = 0)
59	58	oveq1d 7173	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) → ((0(digit‘2)𝑎) · 1) = (0 · 1))
60		1re 10643	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 1 ∈ ℝ
61		mul02lem2 10819	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (1 ∈ ℝ → (0 · 1) = 0)
62	60, 61	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (0 · 1) = 0
63	59, 62	syl6eq 2874	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) → ((0(digit‘2)𝑎) · 1) = 0)
64	63	adantr 483	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) → ((0(digit‘2)𝑎) · 1) = 0)
65	64	adantr 483	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) → ((0(digit‘2)𝑎) · 1) = 0)
66		1z 12015	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 1 ∈ ℤ
67	66	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) → 1 ∈ ℤ)
68		0p1e1 11762	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (0 + 1) = 1
69	68, 66	eqeltri 2911	. . . . . . . . . . . . . . . . . . . . . . . 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 12911	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑘 ∈ ((0 + 1)...𝑦) → 𝑘 ∈ ℤ)
73	72	adantl 484	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑘 ∈ ((0 + 1)...𝑦)) → 𝑘 ∈ ℤ)
74	36	ad4antlr 731	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑘 ∈ ((0 + 1)...𝑦)) → 𝑎 ∈ (0[,)+∞))
75	71, 73, 74, 38	syl3anc 1367	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑘 ∈ ((0 + 1)...𝑦)) → (𝑘(digit‘2)𝑎) ∈ ℕ₀)
76	75	nn0cnd 11960	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑘 ∈ ((0 + 1)...𝑦)) → (𝑘(digit‘2)𝑎) ∈ ℂ)
77		2cnd 11718	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑘 ∈ ((0 + 1)...𝑦) → 2 ∈ ℂ)
78		elfznn 12939	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑘 ∈ (1...𝑦) → 𝑘 ∈ ℕ)
79	78	nnnn0d 11958	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑘 ∈ (1...𝑦) → 𝑘 ∈ ℕ₀)
80	68	oveq1i 7168	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((0 + 1)...𝑦) = (1...𝑦)
81	79, 80	eleq2s 2933	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑘 ∈ ((0 + 1)...𝑦) → 𝑘 ∈ ℕ₀)
82	77, 81	expcld 13513	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑘 ∈ ((0 + 1)...𝑦) → (2↑𝑘) ∈ ℂ)
83	82	adantl 484	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑘 ∈ ((0 + 1)...𝑦)) → (2↑𝑘) ∈ ℂ)
84	76, 83	mulcld 10663	. . . . . . . . . . . . . . . . . . . . . . 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 7165	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 = (𝑖 + 1) → (𝑘(digit‘2)𝑎) = ((𝑖 + 1)(digit‘2)𝑎))
86		oveq2 7166	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 = (𝑖 + 1) → (2↑𝑘) = (2↑(𝑖 + 1)))
87	85, 86	oveq12d 7176	. . . . . . . . . . . . . . . . . . . . . . 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 15138	. . . . . . . . . . . . . . . . . . . . . 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 7176	. . . . . . . . . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑦)) → (𝑎 / 2) ∈ ℕ₀)
91	34	ad4antlr 731	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑦)) → 𝑎 ∈ ℕ₀)
92		elfzonn0 13085	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑖 ∈ (0..^𝑦) → 𝑖 ∈ ℕ₀)
93	92	adantl 484	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑦)) → 𝑖 ∈ ℕ₀)
94		dignn0ehalf 44684	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑎 / 2) ∈ ℕ₀ ∧ 𝑎 ∈ ℕ₀ ∧ 𝑖 ∈ ℕ₀) → ((𝑖 + 1)(digit‘2)𝑎) = (𝑖(digit‘2)(𝑎 / 2)))
95	90, 91, 93, 94	syl3anc 1367	. . . . . . . . . . . . . . . . . . . . . . . . 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 11718	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑖 ∈ (0..^𝑦) → 2 ∈ ℂ)
97	96, 92	expp1d 13514	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑖 ∈ (0..^𝑦) → (2↑(𝑖 + 1)) = ((2↑𝑖) · 2))
98	97	adantl 484	. . . . . . . . . . . . . . . . . . . . . . . . 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 7176	. . . . . . . . . . . . . . . . . . . . . . . 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 13041	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑖 ∈ (0..^𝑦) → 𝑖 ∈ ℤ)
102	101	adantl 484	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑦)) → 𝑖 ∈ ℤ)
103		nn0rp0 12846	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑎 / 2) ∈ ℕ₀ → (𝑎 / 2) ∈ (0[,)+∞))
104	1, 103	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑎 / 2) ∈ ℕ → (𝑎 / 2) ∈ (0[,)+∞))
105	104	ad4antr 730	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑦)) → (𝑎 / 2) ∈ (0[,)+∞))
106		digvalnn0 44666	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((2 ∈ ℕ ∧ 𝑖 ∈ ℤ ∧ (𝑎 / 2) ∈ (0[,)+∞)) → (𝑖(digit‘2)(𝑎 / 2)) ∈ ℕ₀)
107	100, 102, 105, 106	syl3anc 1367	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑦)) → (𝑖(digit‘2)(𝑎 / 2)) ∈ ℕ₀)
108	107	nn0cnd 11960	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑦)) → (𝑖(digit‘2)(𝑎 / 2)) ∈ ℂ)
109		2re 11714	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ 2 ∈ ℝ
110	109	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑖 ∈ (0..^𝑦) → 2 ∈ ℝ)
111	110, 92	reexpcld 13530	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑖 ∈ (0..^𝑦) → (2↑𝑖) ∈ ℝ)
112	111	recnd 10671	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑖 ∈ (0..^𝑦) → (2↑𝑖) ∈ ℂ)
113	112	adantl 484	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑦)) → (2↑𝑖) ∈ ℂ)
114		2cnd 11718	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑦)) → 2 ∈ ℂ)
115		mulass 10627	. . . . . . . . . . . . . . . . . . . . . . . . . 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 2829	. . . . . . . . . . . . . . . . . . . . . . . . 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 1367	. . . . . . . . . . . . . . . . . . . . . . . 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 2858	. . . . . . . . . . . . . . . . . . . . . . 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 15062	. . . . . . . . . . . . . . . . . . . . . 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 10635	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ 0 ∈ ℂ
121		pncan1 11066	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7173	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑦 ∈ ℕ → (((0 + 1) − 1)...(𝑦 − 1)) = (0...(𝑦 − 1)))
125		fzoval 13042	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑦 ∈ ℤ → (0..^𝑦) = (0...(𝑦 − 1)))
126	125	eqcomd 2829	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑦 ∈ ℤ → (0...(𝑦 − 1)) = (0..^𝑦))
127	20, 126	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑦 ∈ ℕ → (0...(𝑦 − 1)) = (0..^𝑦))
128	124, 127	eqtrd 2858	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 ∈ ℕ → (((0 + 1) − 1)...(𝑦 − 1)) = (0..^𝑦))
129	128	adantl 484	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) → (((0 + 1) − 1)...(𝑦 − 1)) = (0..^𝑦))
130	129	sumeq1d 15060	. . . . . . . . . . . . . . . . . . . . . . . 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 7174	. . . . . . . . . . . . . . . . . . . . . . 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 13345	. . . . . . . . . . . . . . . . . . . . . . . . . 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 12093	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑖 ∈ (0..^𝑦) → (𝑖 + 1) ∈ ℤ)
135	134	adantl 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑦)) → (𝑖 + 1) ∈ ℤ)
136	36	ad4antlr 731	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑦)) → 𝑎 ∈ (0[,)+∞))
137		digvalnn0 44666	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((2 ∈ ℕ ∧ (𝑖 + 1) ∈ ℤ ∧ 𝑎 ∈ (0[,)+∞)) → ((𝑖 + 1)(digit‘2)𝑎) ∈ ℕ₀)
138	100, 135, 136, 137	syl3anc 1367	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑦)) → ((𝑖 + 1)(digit‘2)𝑎) ∈ ℕ₀)
139	138	nn0cnd 11960	. . . . . . . . . . . . . . . . . . . . . . . . . 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 11940	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑖 ∈ ℕ₀ → (𝑖 + 1) ∈ ℕ₀)
142	92, 141	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑖 ∈ (0..^𝑦) → (𝑖 + 1) ∈ ℕ₀)
143	140, 142	nn0expcld 13610	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑖 ∈ (0..^𝑦) → (2↑(𝑖 + 1)) ∈ ℕ₀)
144	143	nn0cnd 11960	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑖 ∈ (0..^𝑦) → (2↑(𝑖 + 1)) ∈ ℂ)
145	144	adantl 484	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑦)) → (2↑(𝑖 + 1)) ∈ ℂ)
146	139, 145	mulcld 10663	. . . . . . . . . . . . . . . . . . . . . . . . 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 15092	. . . . . . . . . . . . . . . . . . . . . . . 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	addid2d 10843	. . . . . . . . . . . . . . . . . . . . . . 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 2858	. . . . . . . . . . . . . . . . . . . . . 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 11718	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) → 2 ∈ ℂ)
151	140, 92	nn0expcld 13610	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑖 ∈ (0..^𝑦) → (2↑𝑖) ∈ ℕ₀)
152	151	nn0cnd 11960	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑖 ∈ (0..^𝑦) → (2↑𝑖) ∈ ℂ)
153	152	adantl 484	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑦)) → (2↑𝑖) ∈ ℂ)
154	108, 153	mulcld 10663	. . . . . . . . . . . . . . . . . . . . . . 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 15142	. . . . . . . . . . . . . . . . . . . . . 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 2868	. . . . . . . . . . . . . . . . . . . . 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 2858	. . . . . . . . . . . . . . . . . . . 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 2862	. . . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . 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 7165	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑘 = 𝑖 → (𝑘(digit‘2)(𝑎 / 2)) = (𝑖(digit‘2)(𝑎 / 2)))
161		oveq2 7166	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑘 = 𝑖 → (2↑𝑘) = (2↑𝑖))
162	160, 161	oveq12d 7176	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 = 𝑖 → ((𝑘(digit‘2)(𝑎 / 2)) · (2↑𝑘)) = ((𝑖(digit‘2)(𝑎 / 2)) · (2↑𝑖)))
163	162	cbvsumv 15055	. . . . . . . . . . . . . . . . . . . . . . 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 2834	. . . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . . 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 2829	. . . . . . . . . . . . . . . . . . 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 7173	. . . . . . . . . . . . . . . . . 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 11648	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 ∈ ℕ → 𝑎 ∈ ℂ)
170		2cnd 11718	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 ∈ ℕ → 2 ∈ ℂ)
171		2ne0 11744	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 2 ≠ 0
172	171	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 ∈ ℕ → 2 ≠ 0)
173	169, 170, 172	divcan1d 11419	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 ∈ ℕ → ((𝑎 / 2) · 2) = 𝑎)
174	173	ad3antlr 729	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) → ((𝑎 / 2) · 2) = 𝑎)
175	174	adantl 484	. . . . . . . . . . . . . . . . . 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 2863	. . . . . . . . . . . . . . . . 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 415	. . . . . . . . . . . . . . . 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 16	. . . . . . . . . . . . . . 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 88	. . . . . . . . . . . . . 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 242	. . . . . . . . . . . . 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 86	. . . . . . . . . . . 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 422	. . . . . . . . . . 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 99	. . . . . . . . . 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 96	. . . . . . . . 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 17	. . . . . . . 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 415	. . . . . . 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 99	. . . . . 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 417	. . . . 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 44	. . . 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 410	. . 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 15	. 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 409	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

Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∀wral 3140 ‘cfv 6357 (class class class)co 7158 Fincfn 8511 ℂcc 10537 ℝcr 10538 0cc0 10539 1c1 10540 + caddc 10542 · cmul 10544 +∞cpnf 10674 − cmin 10872 / cdiv 11299 ℕcn 11640 2c2 11695 ℕ₀cn0 11900 ℤcz 11984 ℤ_≥cuz 12246 [,)cico 12743 ...cfz 12895 ..^cfzo 13036 ↑cexp 13432 Σcsu 15044 #_bcblen 44636 digitcdig 44662

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 ax-addf 10618 ax-mulf 10619

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-map 8410 df-pm 8411 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-fi 8877 df-sup 8908 df-inf 8909 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ioo 12745 df-ioc 12746 df-ico 12747 df-icc 12748 df-fz 12896 df-fzo 13037 df-fl 13165 df-mod 13241 df-seq 13373 df-exp 13433 df-fac 13637 df-bc 13666 df-hash 13694 df-shft 14428 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-limsup 14830 df-clim 14847 df-rlim 14848 df-sum 15045 df-ef 15423 df-sin 15425 df-cos 15426 df-pi 15428 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-hom 16591 df-cco 16592 df-rest 16698 df-topn 16699 df-0g 16717 df-gsum 16718 df-topgen 16719 df-pt 16720 df-prds 16723 df-xrs 16777 df-qtop 16782 df-imas 16783 df-xps 16785 df-mre 16859 df-mrc 16860 df-acs 16862 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-submnd 17959 df-mulg 18227 df-cntz 18449 df-cmn 18910 df-psmet 20539 df-xmet 20540 df-met 20541 df-bl 20542 df-mopn 20543 df-fbas 20544 df-fg 20545 df-cnfld 20548 df-top 21504 df-topon 21521 df-topsp 21543 df-bases 21556 df-cld 21629 df-ntr 21630 df-cls 21631 df-nei 21708 df-lp 21746 df-perf 21747 df-cn 21837 df-cnp 21838 df-haus 21925 df-tx 22172 df-hmeo 22365 df-fil 22456 df-fm 22548 df-flim 22549 df-flf 22550 df-xms 22932 df-ms 22933 df-tms 22934 df-cncf 23488 df-limc 24466 df-dv 24467 df-log 25142 df-logb 25345 df-blen 44637 df-dig 44663

This theorem is referenced by: nn0sumshdiglem1 44688

W3C validator