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Theorem pw2dvdslemn 12531

Description: Lemma for pw2dvds 12532. If a natural number has some power of two which does not divide it, there is a highest power of two which does divide it. (Contributed by Jim Kingdon, 14-Nov-2021.)

**Assertion**
Ref	Expression
pw2dvdslemn	⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ ¬ (2↑𝐴) ∥ 𝑁) → ∃𝑚 ∈ ℕ₀ ((2↑𝑚) ∥ 𝑁 ∧ ¬ (2↑(𝑚 + 1)) ∥ 𝑁))

Distinct variable group: 𝑚,𝑁 Allowed substitution hint: 𝐴(𝑚)

Proof of Theorem pw2dvdslemn Dummy variables 𝑤 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		3simpb 998	. 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ ¬ (2↑𝐴) ∥ 𝑁) → (𝑁 ∈ ℕ ∧ ¬ (2↑𝐴) ∥ 𝑁))
2		oveq2 5959	. . . . . . . 8 ⊢ (𝑤 = 1 → (2↑𝑤) = (2↑1))
3	2	breq1d 4057	. . . . . . 7 ⊢ (𝑤 = 1 → ((2↑𝑤) ∥ 𝑁 ↔ (2↑1) ∥ 𝑁))
4	3	notbid 669	. . . . . 6 ⊢ (𝑤 = 1 → (¬ (2↑𝑤) ∥ 𝑁 ↔ ¬ (2↑1) ∥ 𝑁))
5	4	anbi2d 464	. . . . 5 ⊢ (𝑤 = 1 → ((𝑁 ∈ ℕ ∧ ¬ (2↑𝑤) ∥ 𝑁) ↔ (𝑁 ∈ ℕ ∧ ¬ (2↑1) ∥ 𝑁)))
6	5	imbi1d 231	. . . 4 ⊢ (𝑤 = 1 → (((𝑁 ∈ ℕ ∧ ¬ (2↑𝑤) ∥ 𝑁) → ∃𝑚 ∈ ℕ₀ ((2↑𝑚) ∥ 𝑁 ∧ ¬ (2↑(𝑚 + 1)) ∥ 𝑁)) ↔ ((𝑁 ∈ ℕ ∧ ¬ (2↑1) ∥ 𝑁) → ∃𝑚 ∈ ℕ₀ ((2↑𝑚) ∥ 𝑁 ∧ ¬ (2↑(𝑚 + 1)) ∥ 𝑁))))
7		oveq2 5959	. . . . . . . 8 ⊢ (𝑤 = 𝑘 → (2↑𝑤) = (2↑𝑘))
8	7	breq1d 4057	. . . . . . 7 ⊢ (𝑤 = 𝑘 → ((2↑𝑤) ∥ 𝑁 ↔ (2↑𝑘) ∥ 𝑁))
9	8	notbid 669	. . . . . 6 ⊢ (𝑤 = 𝑘 → (¬ (2↑𝑤) ∥ 𝑁 ↔ ¬ (2↑𝑘) ∥ 𝑁))
10	9	anbi2d 464	. . . . 5 ⊢ (𝑤 = 𝑘 → ((𝑁 ∈ ℕ ∧ ¬ (2↑𝑤) ∥ 𝑁) ↔ (𝑁 ∈ ℕ ∧ ¬ (2↑𝑘) ∥ 𝑁)))
11	10	imbi1d 231	. . . 4 ⊢ (𝑤 = 𝑘 → (((𝑁 ∈ ℕ ∧ ¬ (2↑𝑤) ∥ 𝑁) → ∃𝑚 ∈ ℕ₀ ((2↑𝑚) ∥ 𝑁 ∧ ¬ (2↑(𝑚 + 1)) ∥ 𝑁)) ↔ ((𝑁 ∈ ℕ ∧ ¬ (2↑𝑘) ∥ 𝑁) → ∃𝑚 ∈ ℕ₀ ((2↑𝑚) ∥ 𝑁 ∧ ¬ (2↑(𝑚 + 1)) ∥ 𝑁))))
12		oveq2 5959	. . . . . . . 8 ⊢ (𝑤 = (𝑘 + 1) → (2↑𝑤) = (2↑(𝑘 + 1)))
13	12	breq1d 4057	. . . . . . 7 ⊢ (𝑤 = (𝑘 + 1) → ((2↑𝑤) ∥ 𝑁 ↔ (2↑(𝑘 + 1)) ∥ 𝑁))
14	13	notbid 669	. . . . . 6 ⊢ (𝑤 = (𝑘 + 1) → (¬ (2↑𝑤) ∥ 𝑁 ↔ ¬ (2↑(𝑘 + 1)) ∥ 𝑁))
15	14	anbi2d 464	. . . . 5 ⊢ (𝑤 = (𝑘 + 1) → ((𝑁 ∈ ℕ ∧ ¬ (2↑𝑤) ∥ 𝑁) ↔ (𝑁 ∈ ℕ ∧ ¬ (2↑(𝑘 + 1)) ∥ 𝑁)))
16	15	imbi1d 231	. . . 4 ⊢ (𝑤 = (𝑘 + 1) → (((𝑁 ∈ ℕ ∧ ¬ (2↑𝑤) ∥ 𝑁) → ∃𝑚 ∈ ℕ₀ ((2↑𝑚) ∥ 𝑁 ∧ ¬ (2↑(𝑚 + 1)) ∥ 𝑁)) ↔ ((𝑁 ∈ ℕ ∧ ¬ (2↑(𝑘 + 1)) ∥ 𝑁) → ∃𝑚 ∈ ℕ₀ ((2↑𝑚) ∥ 𝑁 ∧ ¬ (2↑(𝑚 + 1)) ∥ 𝑁))))
17		oveq2 5959	. . . . . . . 8 ⊢ (𝑤 = 𝐴 → (2↑𝑤) = (2↑𝐴))
18	17	breq1d 4057	. . . . . . 7 ⊢ (𝑤 = 𝐴 → ((2↑𝑤) ∥ 𝑁 ↔ (2↑𝐴) ∥ 𝑁))
19	18	notbid 669	. . . . . 6 ⊢ (𝑤 = 𝐴 → (¬ (2↑𝑤) ∥ 𝑁 ↔ ¬ (2↑𝐴) ∥ 𝑁))
20	19	anbi2d 464	. . . . 5 ⊢ (𝑤 = 𝐴 → ((𝑁 ∈ ℕ ∧ ¬ (2↑𝑤) ∥ 𝑁) ↔ (𝑁 ∈ ℕ ∧ ¬ (2↑𝐴) ∥ 𝑁)))
21	20	imbi1d 231	. . . 4 ⊢ (𝑤 = 𝐴 → (((𝑁 ∈ ℕ ∧ ¬ (2↑𝑤) ∥ 𝑁) → ∃𝑚 ∈ ℕ₀ ((2↑𝑚) ∥ 𝑁 ∧ ¬ (2↑(𝑚 + 1)) ∥ 𝑁)) ↔ ((𝑁 ∈ ℕ ∧ ¬ (2↑𝐴) ∥ 𝑁) → ∃𝑚 ∈ ℕ₀ ((2↑𝑚) ∥ 𝑁 ∧ ¬ (2↑(𝑚 + 1)) ∥ 𝑁))))
22		0nn0 9317	. . . . . 6 ⊢ 0 ∈ ℕ₀
23	22	a1i 9	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ ¬ (2↑1) ∥ 𝑁) → 0 ∈ ℕ₀)
24		oveq2 5959	. . . . . . . 8 ⊢ (𝑚 = 0 → (2↑𝑚) = (2↑0))
25	24	breq1d 4057	. . . . . . 7 ⊢ (𝑚 = 0 → ((2↑𝑚) ∥ 𝑁 ↔ (2↑0) ∥ 𝑁))
26		oveq1 5958	. . . . . . . . . 10 ⊢ (𝑚 = 0 → (𝑚 + 1) = (0 + 1))
27	26	oveq2d 5967	. . . . . . . . 9 ⊢ (𝑚 = 0 → (2↑(𝑚 + 1)) = (2↑(0 + 1)))
28	27	breq1d 4057	. . . . . . . 8 ⊢ (𝑚 = 0 → ((2↑(𝑚 + 1)) ∥ 𝑁 ↔ (2↑(0 + 1)) ∥ 𝑁))
29	28	notbid 669	. . . . . . 7 ⊢ (𝑚 = 0 → (¬ (2↑(𝑚 + 1)) ∥ 𝑁 ↔ ¬ (2↑(0 + 1)) ∥ 𝑁))
30	25, 29	anbi12d 473	. . . . . 6 ⊢ (𝑚 = 0 → (((2↑𝑚) ∥ 𝑁 ∧ ¬ (2↑(𝑚 + 1)) ∥ 𝑁) ↔ ((2↑0) ∥ 𝑁 ∧ ¬ (2↑(0 + 1)) ∥ 𝑁)))
31	30	adantl 277	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ ¬ (2↑1) ∥ 𝑁) ∧ 𝑚 = 0) → (((2↑𝑚) ∥ 𝑁 ∧ ¬ (2↑(𝑚 + 1)) ∥ 𝑁) ↔ ((2↑0) ∥ 𝑁 ∧ ¬ (2↑(0 + 1)) ∥ 𝑁)))
32		2cnd 9116	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ ¬ (2↑1) ∥ 𝑁) → 2 ∈ ℂ)
33	32	exp0d 10819	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ ¬ (2↑1) ∥ 𝑁) → (2↑0) = 1)
34		simpl 109	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ ¬ (2↑1) ∥ 𝑁) → 𝑁 ∈ ℕ)
35	34	nnzd 9501	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ ¬ (2↑1) ∥ 𝑁) → 𝑁 ∈ ℤ)
36		1dvds 12160	. . . . . . . 8 ⊢ (𝑁 ∈ ℤ → 1 ∥ 𝑁)
37	35, 36	syl 14	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ ¬ (2↑1) ∥ 𝑁) → 1 ∥ 𝑁)
38	33, 37	eqbrtrd 4069	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ ¬ (2↑1) ∥ 𝑁) → (2↑0) ∥ 𝑁)
39		simpr 110	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ ¬ (2↑1) ∥ 𝑁) → ¬ (2↑1) ∥ 𝑁)
40		0p1e1 9157	. . . . . . . . 9 ⊢ (0 + 1) = 1
41	40	oveq2i 5962	. . . . . . . 8 ⊢ (2↑(0 + 1)) = (2↑1)
42	41	breq1i 4054	. . . . . . 7 ⊢ ((2↑(0 + 1)) ∥ 𝑁 ↔ (2↑1) ∥ 𝑁)
43	39, 42	sylnibr 679	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ ¬ (2↑1) ∥ 𝑁) → ¬ (2↑(0 + 1)) ∥ 𝑁)
44	38, 43	jca 306	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ ¬ (2↑1) ∥ 𝑁) → ((2↑0) ∥ 𝑁 ∧ ¬ (2↑(0 + 1)) ∥ 𝑁))
45	23, 31, 44	rspcedvd 2884	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ ¬ (2↑1) ∥ 𝑁) → ∃𝑚 ∈ ℕ₀ ((2↑𝑚) ∥ 𝑁 ∧ ¬ (2↑(𝑚 + 1)) ∥ 𝑁))
46		simpll 527	. . . . . . . . 9 ⊢ (((𝑘 ∈ ℕ ∧ (𝑁 ∈ ℕ ∧ ¬ (2↑(𝑘 + 1)) ∥ 𝑁)) ∧ (2↑𝑘) ∥ 𝑁) → 𝑘 ∈ ℕ)
47	46	nnnn0d 9355	. . . . . . . 8 ⊢ (((𝑘 ∈ ℕ ∧ (𝑁 ∈ ℕ ∧ ¬ (2↑(𝑘 + 1)) ∥ 𝑁)) ∧ (2↑𝑘) ∥ 𝑁) → 𝑘 ∈ ℕ₀)
48		oveq2 5959	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑘 → (2↑𝑚) = (2↑𝑘))
49	48	breq1d 4057	. . . . . . . . . 10 ⊢ (𝑚 = 𝑘 → ((2↑𝑚) ∥ 𝑁 ↔ (2↑𝑘) ∥ 𝑁))
50		oveq1 5958	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑘 → (𝑚 + 1) = (𝑘 + 1))
51	50	oveq2d 5967	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑘 → (2↑(𝑚 + 1)) = (2↑(𝑘 + 1)))
52	51	breq1d 4057	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑘 → ((2↑(𝑚 + 1)) ∥ 𝑁 ↔ (2↑(𝑘 + 1)) ∥ 𝑁))
53	52	notbid 669	. . . . . . . . . 10 ⊢ (𝑚 = 𝑘 → (¬ (2↑(𝑚 + 1)) ∥ 𝑁 ↔ ¬ (2↑(𝑘 + 1)) ∥ 𝑁))
54	49, 53	anbi12d 473	. . . . . . . . 9 ⊢ (𝑚 = 𝑘 → (((2↑𝑚) ∥ 𝑁 ∧ ¬ (2↑(𝑚 + 1)) ∥ 𝑁) ↔ ((2↑𝑘) ∥ 𝑁 ∧ ¬ (2↑(𝑘 + 1)) ∥ 𝑁)))
55	54	adantl 277	. . . . . . . 8 ⊢ ((((𝑘 ∈ ℕ ∧ (𝑁 ∈ ℕ ∧ ¬ (2↑(𝑘 + 1)) ∥ 𝑁)) ∧ (2↑𝑘) ∥ 𝑁) ∧ 𝑚 = 𝑘) → (((2↑𝑚) ∥ 𝑁 ∧ ¬ (2↑(𝑚 + 1)) ∥ 𝑁) ↔ ((2↑𝑘) ∥ 𝑁 ∧ ¬ (2↑(𝑘 + 1)) ∥ 𝑁)))
56		simpr 110	. . . . . . . . 9 ⊢ (((𝑘 ∈ ℕ ∧ (𝑁 ∈ ℕ ∧ ¬ (2↑(𝑘 + 1)) ∥ 𝑁)) ∧ (2↑𝑘) ∥ 𝑁) → (2↑𝑘) ∥ 𝑁)
57		simplrr 536	. . . . . . . . 9 ⊢ (((𝑘 ∈ ℕ ∧ (𝑁 ∈ ℕ ∧ ¬ (2↑(𝑘 + 1)) ∥ 𝑁)) ∧ (2↑𝑘) ∥ 𝑁) → ¬ (2↑(𝑘 + 1)) ∥ 𝑁)
58	56, 57	jca 306	. . . . . . . 8 ⊢ (((𝑘 ∈ ℕ ∧ (𝑁 ∈ ℕ ∧ ¬ (2↑(𝑘 + 1)) ∥ 𝑁)) ∧ (2↑𝑘) ∥ 𝑁) → ((2↑𝑘) ∥ 𝑁 ∧ ¬ (2↑(𝑘 + 1)) ∥ 𝑁))
59	47, 55, 58	rspcedvd 2884	. . . . . . 7 ⊢ (((𝑘 ∈ ℕ ∧ (𝑁 ∈ ℕ ∧ ¬ (2↑(𝑘 + 1)) ∥ 𝑁)) ∧ (2↑𝑘) ∥ 𝑁) → ∃𝑚 ∈ ℕ₀ ((2↑𝑚) ∥ 𝑁 ∧ ¬ (2↑(𝑚 + 1)) ∥ 𝑁))
60	59	adantllr 481	. . . . . 6 ⊢ ((((𝑘 ∈ ℕ ∧ ((𝑁 ∈ ℕ ∧ ¬ (2↑𝑘) ∥ 𝑁) → ∃𝑚 ∈ ℕ₀ ((2↑𝑚) ∥ 𝑁 ∧ ¬ (2↑(𝑚 + 1)) ∥ 𝑁))) ∧ (𝑁 ∈ ℕ ∧ ¬ (2↑(𝑘 + 1)) ∥ 𝑁)) ∧ (2↑𝑘) ∥ 𝑁) → ∃𝑚 ∈ ℕ₀ ((2↑𝑚) ∥ 𝑁 ∧ ¬ (2↑(𝑚 + 1)) ∥ 𝑁))
61		simprl 529	. . . . . . . 8 ⊢ (((𝑘 ∈ ℕ ∧ ((𝑁 ∈ ℕ ∧ ¬ (2↑𝑘) ∥ 𝑁) → ∃𝑚 ∈ ℕ₀ ((2↑𝑚) ∥ 𝑁 ∧ ¬ (2↑(𝑚 + 1)) ∥ 𝑁))) ∧ (𝑁 ∈ ℕ ∧ ¬ (2↑(𝑘 + 1)) ∥ 𝑁)) → 𝑁 ∈ ℕ)
62	61	anim1i 340	. . . . . . 7 ⊢ ((((𝑘 ∈ ℕ ∧ ((𝑁 ∈ ℕ ∧ ¬ (2↑𝑘) ∥ 𝑁) → ∃𝑚 ∈ ℕ₀ ((2↑𝑚) ∥ 𝑁 ∧ ¬ (2↑(𝑚 + 1)) ∥ 𝑁))) ∧ (𝑁 ∈ ℕ ∧ ¬ (2↑(𝑘 + 1)) ∥ 𝑁)) ∧ ¬ (2↑𝑘) ∥ 𝑁) → (𝑁 ∈ ℕ ∧ ¬ (2↑𝑘) ∥ 𝑁))
63		simpllr 534	. . . . . . 7 ⊢ ((((𝑘 ∈ ℕ ∧ ((𝑁 ∈ ℕ ∧ ¬ (2↑𝑘) ∥ 𝑁) → ∃𝑚 ∈ ℕ₀ ((2↑𝑚) ∥ 𝑁 ∧ ¬ (2↑(𝑚 + 1)) ∥ 𝑁))) ∧ (𝑁 ∈ ℕ ∧ ¬ (2↑(𝑘 + 1)) ∥ 𝑁)) ∧ ¬ (2↑𝑘) ∥ 𝑁) → ((𝑁 ∈ ℕ ∧ ¬ (2↑𝑘) ∥ 𝑁) → ∃𝑚 ∈ ℕ₀ ((2↑𝑚) ∥ 𝑁 ∧ ¬ (2↑(𝑚 + 1)) ∥ 𝑁)))
64	62, 63	mpd 13	. . . . . 6 ⊢ ((((𝑘 ∈ ℕ ∧ ((𝑁 ∈ ℕ ∧ ¬ (2↑𝑘) ∥ 𝑁) → ∃𝑚 ∈ ℕ₀ ((2↑𝑚) ∥ 𝑁 ∧ ¬ (2↑(𝑚 + 1)) ∥ 𝑁))) ∧ (𝑁 ∈ ℕ ∧ ¬ (2↑(𝑘 + 1)) ∥ 𝑁)) ∧ ¬ (2↑𝑘) ∥ 𝑁) → ∃𝑚 ∈ ℕ₀ ((2↑𝑚) ∥ 𝑁 ∧ ¬ (2↑(𝑚 + 1)) ∥ 𝑁))
65		2nn 9205	. . . . . . . . 9 ⊢ 2 ∈ ℕ
66		simpll 527	. . . . . . . . . 10 ⊢ (((𝑘 ∈ ℕ ∧ ((𝑁 ∈ ℕ ∧ ¬ (2↑𝑘) ∥ 𝑁) → ∃𝑚 ∈ ℕ₀ ((2↑𝑚) ∥ 𝑁 ∧ ¬ (2↑(𝑚 + 1)) ∥ 𝑁))) ∧ (𝑁 ∈ ℕ ∧ ¬ (2↑(𝑘 + 1)) ∥ 𝑁)) → 𝑘 ∈ ℕ)
67	66	nnnn0d 9355	. . . . . . . . 9 ⊢ (((𝑘 ∈ ℕ ∧ ((𝑁 ∈ ℕ ∧ ¬ (2↑𝑘) ∥ 𝑁) → ∃𝑚 ∈ ℕ₀ ((2↑𝑚) ∥ 𝑁 ∧ ¬ (2↑(𝑚 + 1)) ∥ 𝑁))) ∧ (𝑁 ∈ ℕ ∧ ¬ (2↑(𝑘 + 1)) ∥ 𝑁)) → 𝑘 ∈ ℕ₀)
68		nnexpcl 10704	. . . . . . . . 9 ⊢ ((2 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) → (2↑𝑘) ∈ ℕ)
69	65, 67, 68	sylancr 414	. . . . . . . 8 ⊢ (((𝑘 ∈ ℕ ∧ ((𝑁 ∈ ℕ ∧ ¬ (2↑𝑘) ∥ 𝑁) → ∃𝑚 ∈ ℕ₀ ((2↑𝑚) ∥ 𝑁 ∧ ¬ (2↑(𝑚 + 1)) ∥ 𝑁))) ∧ (𝑁 ∈ ℕ ∧ ¬ (2↑(𝑘 + 1)) ∥ 𝑁)) → (2↑𝑘) ∈ ℕ)
70	61	nnzd 9501	. . . . . . . 8 ⊢ (((𝑘 ∈ ℕ ∧ ((𝑁 ∈ ℕ ∧ ¬ (2↑𝑘) ∥ 𝑁) → ∃𝑚 ∈ ℕ₀ ((2↑𝑚) ∥ 𝑁 ∧ ¬ (2↑(𝑚 + 1)) ∥ 𝑁))) ∧ (𝑁 ∈ ℕ ∧ ¬ (2↑(𝑘 + 1)) ∥ 𝑁)) → 𝑁 ∈ ℤ)
71		dvdsdc 12153	. . . . . . . 8 ⊢ (((2↑𝑘) ∈ ℕ ∧ 𝑁 ∈ ℤ) → DECID (2↑𝑘) ∥ 𝑁)
72	69, 70, 71	syl2anc 411	. . . . . . 7 ⊢ (((𝑘 ∈ ℕ ∧ ((𝑁 ∈ ℕ ∧ ¬ (2↑𝑘) ∥ 𝑁) → ∃𝑚 ∈ ℕ₀ ((2↑𝑚) ∥ 𝑁 ∧ ¬ (2↑(𝑚 + 1)) ∥ 𝑁))) ∧ (𝑁 ∈ ℕ ∧ ¬ (2↑(𝑘 + 1)) ∥ 𝑁)) → DECID (2↑𝑘) ∥ 𝑁)
73		exmiddc 838	. . . . . . 7 ⊢ (DECID (2↑𝑘) ∥ 𝑁 → ((2↑𝑘) ∥ 𝑁 ∨ ¬ (2↑𝑘) ∥ 𝑁))
74	72, 73	syl 14	. . . . . 6 ⊢ (((𝑘 ∈ ℕ ∧ ((𝑁 ∈ ℕ ∧ ¬ (2↑𝑘) ∥ 𝑁) → ∃𝑚 ∈ ℕ₀ ((2↑𝑚) ∥ 𝑁 ∧ ¬ (2↑(𝑚 + 1)) ∥ 𝑁))) ∧ (𝑁 ∈ ℕ ∧ ¬ (2↑(𝑘 + 1)) ∥ 𝑁)) → ((2↑𝑘) ∥ 𝑁 ∨ ¬ (2↑𝑘) ∥ 𝑁))
75	60, 64, 74	mpjaodan 800	. . . . 5 ⊢ (((𝑘 ∈ ℕ ∧ ((𝑁 ∈ ℕ ∧ ¬ (2↑𝑘) ∥ 𝑁) → ∃𝑚 ∈ ℕ₀ ((2↑𝑚) ∥ 𝑁 ∧ ¬ (2↑(𝑚 + 1)) ∥ 𝑁))) ∧ (𝑁 ∈ ℕ ∧ ¬ (2↑(𝑘 + 1)) ∥ 𝑁)) → ∃𝑚 ∈ ℕ₀ ((2↑𝑚) ∥ 𝑁 ∧ ¬ (2↑(𝑚 + 1)) ∥ 𝑁))
76	75	exp31 364	. . . 4 ⊢ (𝑘 ∈ ℕ → (((𝑁 ∈ ℕ ∧ ¬ (2↑𝑘) ∥ 𝑁) → ∃𝑚 ∈ ℕ₀ ((2↑𝑚) ∥ 𝑁 ∧ ¬ (2↑(𝑚 + 1)) ∥ 𝑁)) → ((𝑁 ∈ ℕ ∧ ¬ (2↑(𝑘 + 1)) ∥ 𝑁) → ∃𝑚 ∈ ℕ₀ ((2↑𝑚) ∥ 𝑁 ∧ ¬ (2↑(𝑚 + 1)) ∥ 𝑁))))
77	6, 11, 16, 21, 45, 76	nnind 9059	. . 3 ⊢ (𝐴 ∈ ℕ → ((𝑁 ∈ ℕ ∧ ¬ (2↑𝐴) ∥ 𝑁) → ∃𝑚 ∈ ℕ₀ ((2↑𝑚) ∥ 𝑁 ∧ ¬ (2↑(𝑚 + 1)) ∥ 𝑁)))
78	77	3ad2ant2 1022	. 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ ¬ (2↑𝐴) ∥ 𝑁) → ((𝑁 ∈ ℕ ∧ ¬ (2↑𝐴) ∥ 𝑁) → ∃𝑚 ∈ ℕ₀ ((2↑𝑚) ∥ 𝑁 ∧ ¬ (2↑(𝑚 + 1)) ∥ 𝑁)))
79	1, 78	mpd 13	1 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ ¬ (2↑𝐴) ∥ 𝑁) → ∃𝑚 ∈ ℕ₀ ((2↑𝑚) ∥ 𝑁 ∧ ¬ (2↑(𝑚 + 1)) ∥ 𝑁))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 710 DECID wdc 836 ∧ w3a 981 = wceq 1373 ∈ wcel 2177 ∃wrex 2486 class class class wbr 4047 (class class class)co 5951 0cc0 7932 1c1 7933 + caddc 7935 ℕcn 9043 2c2 9094 ℕ₀cn0 9302 ℤcz 9379 ↑cexp 10690 ∥ cdvds 12142

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2179 ax-14 2180 ax-ext 2188 ax-coll 4163 ax-sep 4166 ax-nul 4174 ax-pow 4222 ax-pr 4257 ax-un 4484 ax-setind 4589 ax-iinf 4640 ax-cnex 8023 ax-resscn 8024 ax-1cn 8025 ax-1re 8026 ax-icn 8027 ax-addcl 8028 ax-addrcl 8029 ax-mulcl 8030 ax-mulrcl 8031 ax-addcom 8032 ax-mulcom 8033 ax-addass 8034 ax-mulass 8035 ax-distr 8036 ax-i2m1 8037 ax-0lt1 8038 ax-1rid 8039 ax-0id 8040 ax-rnegex 8041 ax-precex 8042 ax-cnre 8043 ax-pre-ltirr 8044 ax-pre-ltwlin 8045 ax-pre-lttrn 8046 ax-pre-apti 8047 ax-pre-ltadd 8048 ax-pre-mulgt0 8049 ax-pre-mulext 8050 ax-arch 8051

This theorem depends on definitions: df-bi 117 df-dc 837 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2193 df-cleq 2199 df-clel 2202 df-nfc 2338 df-ne 2378 df-nel 2473 df-ral 2490 df-rex 2491 df-reu 2492 df-rmo 2493 df-rab 2494 df-v 2775 df-sbc 3000 df-csb 3095 df-dif 3169 df-un 3171 df-in 3173 df-ss 3180 df-nul 3462 df-if 3573 df-pw 3619 df-sn 3640 df-pr 3641 df-op 3643 df-uni 3853 df-int 3888 df-iun 3931 df-br 4048 df-opab 4110 df-mpt 4111 df-tr 4147 df-id 4344 df-po 4347 df-iso 4348 df-iord 4417 df-on 4419 df-ilim 4420 df-suc 4422 df-iom 4643 df-xp 4685 df-rel 4686 df-cnv 4687 df-co 4688 df-dm 4689 df-rn 4690 df-res 4691 df-ima 4692 df-iota 5237 df-fun 5278 df-fn 5279 df-f 5280 df-f1 5281 df-fo 5282 df-f1o 5283 df-fv 5284 df-riota 5906 df-ov 5954 df-oprab 5955 df-mpo 5956 df-1st 6233 df-2nd 6234 df-recs 6398 df-frec 6484 df-pnf 8116 df-mnf 8117 df-xr 8118 df-ltxr 8119 df-le 8120 df-sub 8252 df-neg 8253 df-reap 8655 df-ap 8662 df-div 8753 df-inn 9044 df-2 9102 df-n0 9303 df-z 9380 df-uz 9656 df-q 9748 df-rp 9783 df-fl 10420 df-mod 10475 df-seqfrec 10600 df-exp 10691 df-dvds 12143

This theorem is referenced by: pw2dvds 12532

W3C validator