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Theorem pw2dvdslemn 12695

Description: Lemma for pw2dvds 12696. 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 1019	. 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ ¬ (2↑𝐴) ∥ 𝑁) → (𝑁 ∈ ℕ ∧ ¬ (2↑𝐴) ∥ 𝑁))
2		oveq2 6015	. . . . . . . 8 ⊢ (𝑤 = 1 → (2↑𝑤) = (2↑1))
3	2	breq1d 4093	. . . . . . 7 ⊢ (𝑤 = 1 → ((2↑𝑤) ∥ 𝑁 ↔ (2↑1) ∥ 𝑁))
4	3	notbid 671	. . . . . 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 6015	. . . . . . . 8 ⊢ (𝑤 = 𝑘 → (2↑𝑤) = (2↑𝑘))
8	7	breq1d 4093	. . . . . . 7 ⊢ (𝑤 = 𝑘 → ((2↑𝑤) ∥ 𝑁 ↔ (2↑𝑘) ∥ 𝑁))
9	8	notbid 671	. . . . . 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 6015	. . . . . . . 8 ⊢ (𝑤 = (𝑘 + 1) → (2↑𝑤) = (2↑(𝑘 + 1)))
13	12	breq1d 4093	. . . . . . 7 ⊢ (𝑤 = (𝑘 + 1) → ((2↑𝑤) ∥ 𝑁 ↔ (2↑(𝑘 + 1)) ∥ 𝑁))
14	13	notbid 671	. . . . . 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 6015	. . . . . . . 8 ⊢ (𝑤 = 𝐴 → (2↑𝑤) = (2↑𝐴))
18	17	breq1d 4093	. . . . . . 7 ⊢ (𝑤 = 𝐴 → ((2↑𝑤) ∥ 𝑁 ↔ (2↑𝐴) ∥ 𝑁))
19	18	notbid 671	. . . . . 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 9392	. . . . . 6 ⊢ 0 ∈ ℕ₀
23	22	a1i 9	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ ¬ (2↑1) ∥ 𝑁) → 0 ∈ ℕ₀)
24		oveq2 6015	. . . . . . . 8 ⊢ (𝑚 = 0 → (2↑𝑚) = (2↑0))
25	24	breq1d 4093	. . . . . . 7 ⊢ (𝑚 = 0 → ((2↑𝑚) ∥ 𝑁 ↔ (2↑0) ∥ 𝑁))
26		oveq1 6014	. . . . . . . . . 10 ⊢ (𝑚 = 0 → (𝑚 + 1) = (0 + 1))
27	26	oveq2d 6023	. . . . . . . . 9 ⊢ (𝑚 = 0 → (2↑(𝑚 + 1)) = (2↑(0 + 1)))
28	27	breq1d 4093	. . . . . . . 8 ⊢ (𝑚 = 0 → ((2↑(𝑚 + 1)) ∥ 𝑁 ↔ (2↑(0 + 1)) ∥ 𝑁))
29	28	notbid 671	. . . . . . 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 9191	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ ¬ (2↑1) ∥ 𝑁) → 2 ∈ ℂ)
33	32	exp0d 10897	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ ¬ (2↑1) ∥ 𝑁) → (2↑0) = 1)
34		simpl 109	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ ¬ (2↑1) ∥ 𝑁) → 𝑁 ∈ ℕ)
35	34	nnzd 9576	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ ¬ (2↑1) ∥ 𝑁) → 𝑁 ∈ ℤ)
36		1dvds 12324	. . . . . . . 8 ⊢ (𝑁 ∈ ℤ → 1 ∥ 𝑁)
37	35, 36	syl 14	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ ¬ (2↑1) ∥ 𝑁) → 1 ∥ 𝑁)
38	33, 37	eqbrtrd 4105	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ ¬ (2↑1) ∥ 𝑁) → (2↑0) ∥ 𝑁)
39		simpr 110	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ ¬ (2↑1) ∥ 𝑁) → ¬ (2↑1) ∥ 𝑁)
40		0p1e1 9232	. . . . . . . . 9 ⊢ (0 + 1) = 1
41	40	oveq2i 6018	. . . . . . . 8 ⊢ (2↑(0 + 1)) = (2↑1)
42	41	breq1i 4090	. . . . . . 7 ⊢ ((2↑(0 + 1)) ∥ 𝑁 ↔ (2↑1) ∥ 𝑁)
43	39, 42	sylnibr 681	. . . . . 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 2913	. . . 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 9430	. . . . . . . 8 ⊢ (((𝑘 ∈ ℕ ∧ (𝑁 ∈ ℕ ∧ ¬ (2↑(𝑘 + 1)) ∥ 𝑁)) ∧ (2↑𝑘) ∥ 𝑁) → 𝑘 ∈ ℕ₀)
48		oveq2 6015	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑘 → (2↑𝑚) = (2↑𝑘))
49	48	breq1d 4093	. . . . . . . . . 10 ⊢ (𝑚 = 𝑘 → ((2↑𝑚) ∥ 𝑁 ↔ (2↑𝑘) ∥ 𝑁))
50		oveq1 6014	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑘 → (𝑚 + 1) = (𝑘 + 1))
51	50	oveq2d 6023	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑘 → (2↑(𝑚 + 1)) = (2↑(𝑘 + 1)))
52	51	breq1d 4093	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑘 → ((2↑(𝑚 + 1)) ∥ 𝑁 ↔ (2↑(𝑘 + 1)) ∥ 𝑁))
53	52	notbid 671	. . . . . . . . . 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 2913	. . . . . . 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 9280	. . . . . . . . 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 9430	. . . . . . . . 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 10782	. . . . . . . . 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 9576	. . . . . . . 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 12317	. . . . . . . 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 841	. . . . . . 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 803	. . . . 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 9134	. . 3 ⊢ (𝐴 ∈ ℕ → ((𝑁 ∈ ℕ ∧ ¬ (2↑𝐴) ∥ 𝑁) → ∃𝑚 ∈ ℕ₀ ((2↑𝑚) ∥ 𝑁 ∧ ¬ (2↑(𝑚 + 1)) ∥ 𝑁)))
78	77	3ad2ant2 1043	. 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 713 DECID wdc 839 ∧ w3a 1002 = wceq 1395 ∈ wcel 2200 ∃wrex 2509 class class class wbr 4083 (class class class)co 6007 0cc0 8007 1c1 8008 + caddc 8010 ℕcn 9118 2c2 9169 ℕ₀cn0 9377 ℤcz 9454 ↑cexp 10768 ∥ cdvds 12306

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4199 ax-sep 4202 ax-nul 4210 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-setind 4629 ax-iinf 4680 ax-cnex 8098 ax-resscn 8099 ax-1cn 8100 ax-1re 8101 ax-icn 8102 ax-addcl 8103 ax-addrcl 8104 ax-mulcl 8105 ax-mulrcl 8106 ax-addcom 8107 ax-mulcom 8108 ax-addass 8109 ax-mulass 8110 ax-distr 8111 ax-i2m1 8112 ax-0lt1 8113 ax-1rid 8114 ax-0id 8115 ax-rnegex 8116 ax-precex 8117 ax-cnre 8118 ax-pre-ltirr 8119 ax-pre-ltwlin 8120 ax-pre-lttrn 8121 ax-pre-apti 8122 ax-pre-ltadd 8123 ax-pre-mulgt0 8124 ax-pre-mulext 8125 ax-arch 8126

This theorem depends on definitions: df-bi 117 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-if 3603 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-int 3924 df-iun 3967 df-br 4084 df-opab 4146 df-mpt 4147 df-tr 4183 df-id 4384 df-po 4387 df-iso 4388 df-iord 4457 df-on 4459 df-ilim 4460 df-suc 4462 df-iom 4683 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-rn 4730 df-res 4731 df-ima 4732 df-iota 5278 df-fun 5320 df-fn 5321 df-f 5322 df-f1 5323 df-fo 5324 df-f1o 5325 df-fv 5326 df-riota 5960 df-ov 6010 df-oprab 6011 df-mpo 6012 df-1st 6292 df-2nd 6293 df-recs 6457 df-frec 6543 df-pnf 8191 df-mnf 8192 df-xr 8193 df-ltxr 8194 df-le 8195 df-sub 8327 df-neg 8328 df-reap 8730 df-ap 8737 df-div 8828 df-inn 9119 df-2 9177 df-n0 9378 df-z 9455 df-uz 9731 df-q 9823 df-rp 9858 df-fl 10498 df-mod 10553 df-seqfrec 10678 df-exp 10769 df-dvds 12307

This theorem is referenced by: pw2dvds 12696

W3C validator