nn0sumshdiglemA - Mathbox for Alexander van der Vekens

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Theorem nn0sumshdiglemA 46695

Description: Lemma for nn0sumshdig 46699 (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 12420	. . . 4 ⊢ ((𝑎 / 2) ∈ ℕ → (𝑎 / 2) ∈ ℕ₀)
2		blennn0em1 46667	. . . 4 ⊢ ((𝑎 ∈ ℕ ∧ (𝑎 / 2) ∈ ℕ₀) → (#_b‘(𝑎 / 2)) = ((#_b‘𝑎) − 1))
3	1, 2	sylan2 593	. . 3 ⊢ ((𝑎 ∈ ℕ ∧ (𝑎 / 2) ∈ ℕ) → (#_b‘(𝑎 / 2)) = ((#_b‘𝑎) − 1))
4		fveqeq2 6851	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑎 / 2) → ((#_b‘𝑥) = 𝑦 ↔ (#_b‘(𝑎 / 2)) = 𝑦))
5		id 22	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑎 / 2) → 𝑥 = (𝑎 / 2))
6		oveq2 7365	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑎 / 2) → (𝑘(digit‘2)𝑥) = (𝑘(digit‘2)(𝑎 / 2)))
7	6	oveq1d 7372	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑎 / 2) → ((𝑘(digit‘2)𝑥) · (2↑𝑘)) = ((𝑘(digit‘2)(𝑎 / 2)) · (2↑𝑘)))
8	7	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝑥 = (𝑎 / 2) ∧ 𝑘 ∈ (0..^𝑦)) → ((𝑘(digit‘2)𝑥) · (2↑𝑘)) = ((𝑘(digit‘2)(𝑎 / 2)) · (2↑𝑘)))
9	8	sumeq2dv 15588	. . . . . . . . . . . 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 2752	. . . . . . . . . . 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 344	. . . . . . . . . 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 3579	. . . . . . . . 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 485	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) → (#_b‘𝑎) = (𝑦 + 1))
14	13	oveq1d 7372	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) → ((#_b‘𝑎) − 1) = ((𝑦 + 1) − 1))
15		nncn 12161	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℂ)
16		pncan1 11579	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ℂ → ((𝑦 + 1) − 1) = 𝑦)
17	15, 16	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℕ → ((𝑦 + 1) − 1) = 𝑦)
18	14, 17	sylan9eq 2796	. . . . . . . . . . . . . . 15 ⊢ (((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) → ((#_b‘𝑎) − 1) = 𝑦)
19	18	eqeq2d 2747	. . . . . . . . . . . . . 14 ⊢ (((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) → ((#_b‘(𝑎 / 2)) = ((#_b‘𝑎) − 1) ↔ (#_b‘(𝑎 / 2)) = 𝑦))
20		nnz 12520	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℤ)
21	20	adantl 482	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) → 𝑦 ∈ ℤ)
22		fzval3 13641	. . . . . . . . . . . . . . . . . . . . . . 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 2742	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) → (0..^(𝑦 + 1)) = (0...𝑦))
25	24	sumeq1d 15586	. . . . . . . . . . . . . . . . . . . 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 12420	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℕ₀)
27		elnn0uz 12808	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ ℕ₀ ↔ 𝑦 ∈ (ℤ_≥‘0))
28	26, 27	sylib 217	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ (ℤ_≥‘0))
29	28	adantl 482	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) → 𝑦 ∈ (ℤ_≥‘0))
30		2nn 12226	. . . . . . . . . . . . . . . . . . . . . . . . 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 13441	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑘 ∈ (0...𝑦) → 𝑘 ∈ ℤ)
33	32	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑘 ∈ (0...𝑦)) → 𝑘 ∈ ℤ)
34		nnnn0 12420	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑎 ∈ ℕ → 𝑎 ∈ ℕ₀)
35		nn0rp0 13372	. . . . . . . . . . . . . . . . . . . . . . . . . 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 46675	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((2 ∈ ℕ ∧ 𝑘 ∈ ℤ ∧ 𝑎 ∈ (0[,)+∞)) → (𝑘(digit‘2)𝑎) ∈ ℕ₀)
39	31, 33, 37, 38	syl3anc 1371	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑘 ∈ (0...𝑦)) → (𝑘(digit‘2)𝑎) ∈ ℕ₀)
40	39	nn0cnd 12475	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑘 ∈ (0...𝑦)) → (𝑘(digit‘2)𝑎) ∈ ℂ)
41		2nn0 12430	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 2 ∈ ℕ₀
42	41	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑘 ∈ (0...𝑦) → 2 ∈ ℕ₀)
43		elfznn0 13534	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑘 ∈ (0...𝑦) → 𝑘 ∈ ℕ₀)
44	42, 43	nn0expcld 14149	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 ∈ (0...𝑦) → (2↑𝑘) ∈ ℕ₀)
45	44	nn0cnd 12475	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 ∈ (0...𝑦) → (2↑𝑘) ∈ ℂ)
46	45	adantl 482	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑘 ∈ (0...𝑦)) → (2↑𝑘) ∈ ℂ)
47	40, 46	mulcld 11175	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑘 ∈ (0...𝑦)) → ((𝑘(digit‘2)𝑎) · (2↑𝑘)) ∈ ℂ)
48		oveq1 7364	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 = 0 → (𝑘(digit‘2)𝑎) = (0(digit‘2)𝑎))
49		oveq2 7365	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 = 0 → (2↑𝑘) = (2↑0))
50	48, 49	oveq12d 7375	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 = 0 → ((𝑘(digit‘2)𝑎) · (2↑𝑘)) = ((0(digit‘2)𝑎) · (2↑0)))
51		2cn 12228	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 2 ∈ ℂ
52		exp0 13971	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (2 ∈ ℂ → (2↑0) = 1)
53	51, 52	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (2↑0) = 1
54	53	oveq2i 7368	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((0(digit‘2)𝑎) · (2↑0)) = ((0(digit‘2)𝑎) · 1)
55	50, 54	eqtrdi 2792	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 0 → ((𝑘(digit‘2)𝑎) · (2↑𝑘)) = ((0(digit‘2)𝑎) · 1))
56	29, 47, 55	fsum1p 15638	. . . . . . . . . . . . . . . . . . . 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 46688	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑎 ∈ ℕ₀ ∧ (𝑎 / 2) ∈ ℕ₀) → (0(digit‘2)𝑎) = 0)
58	34, 1, 57	syl2anr 597	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) → (0(digit‘2)𝑎) = 0)
59	58	oveq1d 7372	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) → ((0(digit‘2)𝑎) · 1) = (0 · 1))
60		1re 11155	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 1 ∈ ℝ
61		mul02lem2 11332	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (1 ∈ ℝ → (0 · 1) = 0)
62	60, 61	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (0 · 1) = 0
63	59, 62	eqtrdi 2792	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) → ((0(digit‘2)𝑎) · 1) = 0)
64	63	adantr 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) → ((0(digit‘2)𝑎) · 1) = 0)
65	64	adantr 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) → ((0(digit‘2)𝑎) · 1) = 0)
66		1z 12533	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 1 ∈ ℤ
67	66	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) → 1 ∈ ℤ)
68		0p1e1 12275	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (0 + 1) = 1
69	68, 66	eqeltri 2834	. . . . . . . . . . . . . . . . . . . . . . . 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 13441	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑘 ∈ ((0 + 1)...𝑦) → 𝑘 ∈ ℤ)
73	72	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . 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 1371	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑘 ∈ ((0 + 1)...𝑦)) → (𝑘(digit‘2)𝑎) ∈ ℕ₀)
76	75	nn0cnd 12475	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑘 ∈ ((0 + 1)...𝑦)) → (𝑘(digit‘2)𝑎) ∈ ℂ)
77		2cnd 12231	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑘 ∈ ((0 + 1)...𝑦) → 2 ∈ ℂ)
78		elfznn 13470	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑘 ∈ (1...𝑦) → 𝑘 ∈ ℕ)
79	78	nnnn0d 12473	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑘 ∈ (1...𝑦) → 𝑘 ∈ ℕ₀)
80	68	oveq1i 7367	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((0 + 1)...𝑦) = (1...𝑦)
81	79, 80	eleq2s 2856	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑘 ∈ ((0 + 1)...𝑦) → 𝑘 ∈ ℕ₀)
82	77, 81	expcld 14051	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑘 ∈ ((0 + 1)...𝑦) → (2↑𝑘) ∈ ℂ)
83	82	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑘 ∈ ((0 + 1)...𝑦)) → (2↑𝑘) ∈ ℂ)
84	76, 83	mulcld 11175	. . . . . . . . . . . . . . . . . . . . . . 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 7364	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 = (𝑖 + 1) → (𝑘(digit‘2)𝑎) = ((𝑖 + 1)(digit‘2)𝑎))
86		oveq2 7365	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 = (𝑖 + 1) → (2↑𝑘) = (2↑(𝑖 + 1)))
87	85, 86	oveq12d 7375	. . . . . . . . . . . . . . . . . . . . . . 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 15666	. . . . . . . . . . . . . . . . . . . . . 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 7375	. . . . . . . . . . . . . . . . . . . . 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 13617	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑖 ∈ (0..^𝑦) → 𝑖 ∈ ℕ₀)
93	92	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑦)) → 𝑖 ∈ ℕ₀)
94		dignn0ehalf 46693	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑎 / 2) ∈ ℕ₀ ∧ 𝑎 ∈ ℕ₀ ∧ 𝑖 ∈ ℕ₀) → ((𝑖 + 1)(digit‘2)𝑎) = (𝑖(digit‘2)(𝑎 / 2)))
95	90, 91, 93, 94	syl3anc 1371	. . . . . . . . . . . . . . . . . . . . . . . . 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 12231	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑖 ∈ (0..^𝑦) → 2 ∈ ℂ)
97	96, 92	expp1d 14052	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑖 ∈ (0..^𝑦) → (2↑(𝑖 + 1)) = ((2↑𝑖) · 2))
98	97	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . 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 7375	. . . . . . . . . . . . . . . . . . . . . . . 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 13572	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑖 ∈ (0..^𝑦) → 𝑖 ∈ ℤ)
102	101	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑦)) → 𝑖 ∈ ℤ)
103		nn0rp0 13372	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 46675	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((2 ∈ ℕ ∧ 𝑖 ∈ ℤ ∧ (𝑎 / 2) ∈ (0[,)+∞)) → (𝑖(digit‘2)(𝑎 / 2)) ∈ ℕ₀)
107	100, 102, 105, 106	syl3anc 1371	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑦)) → (𝑖(digit‘2)(𝑎 / 2)) ∈ ℕ₀)
108	107	nn0cnd 12475	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑦)) → (𝑖(digit‘2)(𝑎 / 2)) ∈ ℂ)
109		2re 12227	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ 2 ∈ ℝ
110	109	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑖 ∈ (0..^𝑦) → 2 ∈ ℝ)
111	110, 92	reexpcld 14068	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑖 ∈ (0..^𝑦) → (2↑𝑖) ∈ ℝ)
112	111	recnd 11183	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑖 ∈ (0..^𝑦) → (2↑𝑖) ∈ ℂ)
113	112	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑦)) → (2↑𝑖) ∈ ℂ)
114		2cnd 12231	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑦)) → 2 ∈ ℂ)
115		mulass 11139	. . . . . . . . . . . . . . . . . . . . . . . . . 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 2742	. . . . . . . . . . . . . . . . . . . . . . . . 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 1371	. . . . . . . . . . . . . . . . . . . . . . . 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 2776	. . . . . . . . . . . . . . . . . . . . . . 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 15588	. . . . . . . . . . . . . . . . . . . . . 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 11147	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ 0 ∈ ℂ
121		pncan1 11579	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7372	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑦 ∈ ℕ → (((0 + 1) − 1)...(𝑦 − 1)) = (0...(𝑦 − 1)))
125		fzoval 13573	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑦 ∈ ℤ → (0..^𝑦) = (0...(𝑦 − 1)))
126	125	eqcomd 2742	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2776	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 ∈ ℕ → (((0 + 1) − 1)...(𝑦 − 1)) = (0..^𝑦))
129	128	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) → (((0 + 1) − 1)...(𝑦 − 1)) = (0..^𝑦))
130	129	sumeq1d 15586	. . . . . . . . . . . . . . . . . . . . . . . 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 7373	. . . . . . . . . . . . . . . . . . . . . . 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 13879	. . . . . . . . . . . . . . . . . . . . . . . . . 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 12610	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑖 ∈ (0..^𝑦) → (𝑖 + 1) ∈ ℤ)
135	134	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 46675	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((2 ∈ ℕ ∧ (𝑖 + 1) ∈ ℤ ∧ 𝑎 ∈ (0[,)+∞)) → ((𝑖 + 1)(digit‘2)𝑎) ∈ ℕ₀)
138	100, 135, 136, 137	syl3anc 1371	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑦)) → ((𝑖 + 1)(digit‘2)𝑎) ∈ ℕ₀)
139	138	nn0cnd 12475	. . . . . . . . . . . . . . . . . . . . . . . . . 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 12453	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑖 ∈ ℕ₀ → (𝑖 + 1) ∈ ℕ₀)
142	92, 141	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑖 ∈ (0..^𝑦) → (𝑖 + 1) ∈ ℕ₀)
143	140, 142	nn0expcld 14149	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑖 ∈ (0..^𝑦) → (2↑(𝑖 + 1)) ∈ ℕ₀)
144	143	nn0cnd 12475	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑖 ∈ (0..^𝑦) → (2↑(𝑖 + 1)) ∈ ℂ)
145	144	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑦)) → (2↑(𝑖 + 1)) ∈ ℂ)
146	139, 145	mulcld 11175	. . . . . . . . . . . . . . . . . . . . . . . . 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 15618	. . . . . . . . . . . . . . . . . . . . . . . 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 11356	. . . . . . . . . . . . . . . . . . . . . . 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 2776	. . . . . . . . . . . . . . . . . . . . . 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 12231	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) → 2 ∈ ℂ)
151	140, 92	nn0expcld 14149	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑖 ∈ (0..^𝑦) → (2↑𝑖) ∈ ℕ₀)
152	151	nn0cnd 12475	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑖 ∈ (0..^𝑦) → (2↑𝑖) ∈ ℂ)
153	152	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((𝑎 / 2) ∈ ℕ ∧ 𝑎 ∈ ℕ) ∧ (#_b‘𝑎) = (𝑦 + 1)) ∧ 𝑦 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑦)) → (2↑𝑖) ∈ ℂ)
154	108, 153	mulcld 11175	. . . . . . . . . . . . . . . . . . . . . . 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 15670	. . . . . . . . . . . . . . . . . . . . . 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 2786	. . . . . . . . . . . . . . . . . . . . 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 2776	. . . . . . . . . . . . . . . . . . . 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 2780	. . . . . . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . . . . 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 7364	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑘 = 𝑖 → (𝑘(digit‘2)(𝑎 / 2)) = (𝑖(digit‘2)(𝑎 / 2)))
161		oveq2 7365	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑘 = 𝑖 → (2↑𝑘) = (2↑𝑖))
162	160, 161	oveq12d 7375	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 = 𝑖 → ((𝑘(digit‘2)(𝑎 / 2)) · (2↑𝑘)) = ((𝑖(digit‘2)(𝑎 / 2)) · (2↑𝑖)))
163	162	cbvsumv 15581	. . . . . . . . . . . . . . . . . . . . . . 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 2747	. . . . . . . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . . . . . . 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 2742	. . . . . . . . . . . . . . . . . . 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 7372	. . . . . . . . . . . . . . . . . 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 12161	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 ∈ ℕ → 𝑎 ∈ ℂ)
170		2cnd 12231	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 ∈ ℕ → 2 ∈ ℂ)
171		2ne0 12257	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 2 ≠ 0
172	171	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 ∈ ℕ → 2 ≠ 0)
173	169, 170, 172	divcan1d 11932	. . . . . . . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . . . . 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 2781	. . . . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . . . . 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 239	. . . . . . . . . . . . 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 420	. . . . . . . . . . 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 413	. . . . . . 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 415	. . . . 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 408	. . 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 407	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 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ‘cfv 6496 (class class class)co 7357 Fincfn 8883 ℂcc 11049 ℝcr 11050 0cc0 11051 1c1 11052 + caddc 11054 · cmul 11056 +∞cpnf 11186 − cmin 11385 / cdiv 11812 ℕcn 12153 2c2 12208 ℕ₀cn0 12413 ℤcz 12499 ℤ_≥cuz 12763 [,)cico 13266 ...cfz 13424 ..^cfzo 13567 ↑cexp 13967 Σcsu 15570 #_bcblen 46645 digitcdig 46671

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 ax-addf 11130 ax-mulf 11131

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-er 8648 df-map 8767 df-pm 8768 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-fi 9347 df-sup 9378 df-inf 9379 df-oi 9446 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-q 12874 df-rp 12916 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-ioo 13268 df-ioc 13269 df-ico 13270 df-icc 13271 df-fz 13425 df-fzo 13568 df-fl 13697 df-mod 13775 df-seq 13907 df-exp 13968 df-fac 14174 df-bc 14203 df-hash 14231 df-shft 14952 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-limsup 15353 df-clim 15370 df-rlim 15371 df-sum 15571 df-ef 15950 df-sin 15952 df-cos 15953 df-pi 15955 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-starv 17148 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-hom 17157 df-cco 17158 df-rest 17304 df-topn 17305 df-0g 17323 df-gsum 17324 df-topgen 17325 df-pt 17326 df-prds 17329 df-xrs 17384 df-qtop 17389 df-imas 17390 df-xps 17392 df-mre 17466 df-mrc 17467 df-acs 17469 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-submnd 18602 df-mulg 18873 df-cntz 19097 df-cmn 19564 df-psmet 20788 df-xmet 20789 df-met 20790 df-bl 20791 df-mopn 20792 df-fbas 20793 df-fg 20794 df-cnfld 20797 df-top 22243 df-topon 22260 df-topsp 22282 df-bases 22296 df-cld 22370 df-ntr 22371 df-cls 22372 df-nei 22449 df-lp 22487 df-perf 22488 df-cn 22578 df-cnp 22579 df-haus 22666 df-tx 22913 df-hmeo 23106 df-fil 23197 df-fm 23289 df-flim 23290 df-flf 23291 df-xms 23673 df-ms 23674 df-tms 23675 df-cncf 24241 df-limc 25230 df-dv 25231 df-log 25912 df-logb 26115 df-blen 46646 df-dig 46672

This theorem is referenced by: nn0sumshdiglem1 46697

W3C validator