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Theorem pw2dvdslemn 12097

Description: Lemma for pw2dvds 12098. 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 985	. 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ ¬ (2↑𝐴) ∥ 𝑁) → (𝑁 ∈ ℕ ∧ ¬ (2↑𝐴) ∥ 𝑁))
2		oveq2 5850	. . . . . . . 8 ⊢ (𝑤 = 1 → (2↑𝑤) = (2↑1))
3	2	breq1d 3992	. . . . . . 7 ⊢ (𝑤 = 1 → ((2↑𝑤) ∥ 𝑁 ↔ (2↑1) ∥ 𝑁))
4	3	notbid 657	. . . . . 6 ⊢ (𝑤 = 1 → (¬ (2↑𝑤) ∥ 𝑁 ↔ ¬ (2↑1) ∥ 𝑁))
5	4	anbi2d 460	. . . . 5 ⊢ (𝑤 = 1 → ((𝑁 ∈ ℕ ∧ ¬ (2↑𝑤) ∥ 𝑁) ↔ (𝑁 ∈ ℕ ∧ ¬ (2↑1) ∥ 𝑁)))
6	5	imbi1d 230	. . . 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 5850	. . . . . . . 8 ⊢ (𝑤 = 𝑘 → (2↑𝑤) = (2↑𝑘))
8	7	breq1d 3992	. . . . . . 7 ⊢ (𝑤 = 𝑘 → ((2↑𝑤) ∥ 𝑁 ↔ (2↑𝑘) ∥ 𝑁))
9	8	notbid 657	. . . . . 6 ⊢ (𝑤 = 𝑘 → (¬ (2↑𝑤) ∥ 𝑁 ↔ ¬ (2↑𝑘) ∥ 𝑁))
10	9	anbi2d 460	. . . . 5 ⊢ (𝑤 = 𝑘 → ((𝑁 ∈ ℕ ∧ ¬ (2↑𝑤) ∥ 𝑁) ↔ (𝑁 ∈ ℕ ∧ ¬ (2↑𝑘) ∥ 𝑁)))
11	10	imbi1d 230	. . . 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 5850	. . . . . . . 8 ⊢ (𝑤 = (𝑘 + 1) → (2↑𝑤) = (2↑(𝑘 + 1)))
13	12	breq1d 3992	. . . . . . 7 ⊢ (𝑤 = (𝑘 + 1) → ((2↑𝑤) ∥ 𝑁 ↔ (2↑(𝑘 + 1)) ∥ 𝑁))
14	13	notbid 657	. . . . . 6 ⊢ (𝑤 = (𝑘 + 1) → (¬ (2↑𝑤) ∥ 𝑁 ↔ ¬ (2↑(𝑘 + 1)) ∥ 𝑁))
15	14	anbi2d 460	. . . . 5 ⊢ (𝑤 = (𝑘 + 1) → ((𝑁 ∈ ℕ ∧ ¬ (2↑𝑤) ∥ 𝑁) ↔ (𝑁 ∈ ℕ ∧ ¬ (2↑(𝑘 + 1)) ∥ 𝑁)))
16	15	imbi1d 230	. . . 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 5850	. . . . . . . 8 ⊢ (𝑤 = 𝐴 → (2↑𝑤) = (2↑𝐴))
18	17	breq1d 3992	. . . . . . 7 ⊢ (𝑤 = 𝐴 → ((2↑𝑤) ∥ 𝑁 ↔ (2↑𝐴) ∥ 𝑁))
19	18	notbid 657	. . . . . 6 ⊢ (𝑤 = 𝐴 → (¬ (2↑𝑤) ∥ 𝑁 ↔ ¬ (2↑𝐴) ∥ 𝑁))
20	19	anbi2d 460	. . . . 5 ⊢ (𝑤 = 𝐴 → ((𝑁 ∈ ℕ ∧ ¬ (2↑𝑤) ∥ 𝑁) ↔ (𝑁 ∈ ℕ ∧ ¬ (2↑𝐴) ∥ 𝑁)))
21	20	imbi1d 230	. . . 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 9129	. . . . . 6 ⊢ 0 ∈ ℕ₀
23	22	a1i 9	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ ¬ (2↑1) ∥ 𝑁) → 0 ∈ ℕ₀)
24		oveq2 5850	. . . . . . . 8 ⊢ (𝑚 = 0 → (2↑𝑚) = (2↑0))
25	24	breq1d 3992	. . . . . . 7 ⊢ (𝑚 = 0 → ((2↑𝑚) ∥ 𝑁 ↔ (2↑0) ∥ 𝑁))
26		oveq1 5849	. . . . . . . . . 10 ⊢ (𝑚 = 0 → (𝑚 + 1) = (0 + 1))
27	26	oveq2d 5858	. . . . . . . . 9 ⊢ (𝑚 = 0 → (2↑(𝑚 + 1)) = (2↑(0 + 1)))
28	27	breq1d 3992	. . . . . . . 8 ⊢ (𝑚 = 0 → ((2↑(𝑚 + 1)) ∥ 𝑁 ↔ (2↑(0 + 1)) ∥ 𝑁))
29	28	notbid 657	. . . . . . 7 ⊢ (𝑚 = 0 → (¬ (2↑(𝑚 + 1)) ∥ 𝑁 ↔ ¬ (2↑(0 + 1)) ∥ 𝑁))
30	25, 29	anbi12d 465	. . . . . 6 ⊢ (𝑚 = 0 → (((2↑𝑚) ∥ 𝑁 ∧ ¬ (2↑(𝑚 + 1)) ∥ 𝑁) ↔ ((2↑0) ∥ 𝑁 ∧ ¬ (2↑(0 + 1)) ∥ 𝑁)))
31	30	adantl 275	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ ¬ (2↑1) ∥ 𝑁) ∧ 𝑚 = 0) → (((2↑𝑚) ∥ 𝑁 ∧ ¬ (2↑(𝑚 + 1)) ∥ 𝑁) ↔ ((2↑0) ∥ 𝑁 ∧ ¬ (2↑(0 + 1)) ∥ 𝑁)))
32		2cnd 8930	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ ¬ (2↑1) ∥ 𝑁) → 2 ∈ ℂ)
33	32	exp0d 10582	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ ¬ (2↑1) ∥ 𝑁) → (2↑0) = 1)
34		simpl 108	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ ¬ (2↑1) ∥ 𝑁) → 𝑁 ∈ ℕ)
35	34	nnzd 9312	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ ¬ (2↑1) ∥ 𝑁) → 𝑁 ∈ ℤ)
36		1dvds 11745	. . . . . . . 8 ⊢ (𝑁 ∈ ℤ → 1 ∥ 𝑁)
37	35, 36	syl 14	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ ¬ (2↑1) ∥ 𝑁) → 1 ∥ 𝑁)
38	33, 37	eqbrtrd 4004	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ ¬ (2↑1) ∥ 𝑁) → (2↑0) ∥ 𝑁)
39		simpr 109	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ ¬ (2↑1) ∥ 𝑁) → ¬ (2↑1) ∥ 𝑁)
40		0p1e1 8971	. . . . . . . . 9 ⊢ (0 + 1) = 1
41	40	oveq2i 5853	. . . . . . . 8 ⊢ (2↑(0 + 1)) = (2↑1)
42	41	breq1i 3989	. . . . . . 7 ⊢ ((2↑(0 + 1)) ∥ 𝑁 ↔ (2↑1) ∥ 𝑁)
43	39, 42	sylnibr 667	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ ¬ (2↑1) ∥ 𝑁) → ¬ (2↑(0 + 1)) ∥ 𝑁)
44	38, 43	jca 304	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ ¬ (2↑1) ∥ 𝑁) → ((2↑0) ∥ 𝑁 ∧ ¬ (2↑(0 + 1)) ∥ 𝑁))
45	23, 31, 44	rspcedvd 2836	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ ¬ (2↑1) ∥ 𝑁) → ∃𝑚 ∈ ℕ₀ ((2↑𝑚) ∥ 𝑁 ∧ ¬ (2↑(𝑚 + 1)) ∥ 𝑁))
46		simpll 519	. . . . . . . . 9 ⊢ (((𝑘 ∈ ℕ ∧ (𝑁 ∈ ℕ ∧ ¬ (2↑(𝑘 + 1)) ∥ 𝑁)) ∧ (2↑𝑘) ∥ 𝑁) → 𝑘 ∈ ℕ)
47	46	nnnn0d 9167	. . . . . . . 8 ⊢ (((𝑘 ∈ ℕ ∧ (𝑁 ∈ ℕ ∧ ¬ (2↑(𝑘 + 1)) ∥ 𝑁)) ∧ (2↑𝑘) ∥ 𝑁) → 𝑘 ∈ ℕ₀)
48		oveq2 5850	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑘 → (2↑𝑚) = (2↑𝑘))
49	48	breq1d 3992	. . . . . . . . . 10 ⊢ (𝑚 = 𝑘 → ((2↑𝑚) ∥ 𝑁 ↔ (2↑𝑘) ∥ 𝑁))
50		oveq1 5849	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑘 → (𝑚 + 1) = (𝑘 + 1))
51	50	oveq2d 5858	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑘 → (2↑(𝑚 + 1)) = (2↑(𝑘 + 1)))
52	51	breq1d 3992	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑘 → ((2↑(𝑚 + 1)) ∥ 𝑁 ↔ (2↑(𝑘 + 1)) ∥ 𝑁))
53	52	notbid 657	. . . . . . . . . 10 ⊢ (𝑚 = 𝑘 → (¬ (2↑(𝑚 + 1)) ∥ 𝑁 ↔ ¬ (2↑(𝑘 + 1)) ∥ 𝑁))
54	49, 53	anbi12d 465	. . . . . . . . 9 ⊢ (𝑚 = 𝑘 → (((2↑𝑚) ∥ 𝑁 ∧ ¬ (2↑(𝑚 + 1)) ∥ 𝑁) ↔ ((2↑𝑘) ∥ 𝑁 ∧ ¬ (2↑(𝑘 + 1)) ∥ 𝑁)))
55	54	adantl 275	. . . . . . . 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 109	. . . . . . . . 9 ⊢ (((𝑘 ∈ ℕ ∧ (𝑁 ∈ ℕ ∧ ¬ (2↑(𝑘 + 1)) ∥ 𝑁)) ∧ (2↑𝑘) ∥ 𝑁) → (2↑𝑘) ∥ 𝑁)
57		simplrr 526	. . . . . . . . 9 ⊢ (((𝑘 ∈ ℕ ∧ (𝑁 ∈ ℕ ∧ ¬ (2↑(𝑘 + 1)) ∥ 𝑁)) ∧ (2↑𝑘) ∥ 𝑁) → ¬ (2↑(𝑘 + 1)) ∥ 𝑁)
58	56, 57	jca 304	. . . . . . . 8 ⊢ (((𝑘 ∈ ℕ ∧ (𝑁 ∈ ℕ ∧ ¬ (2↑(𝑘 + 1)) ∥ 𝑁)) ∧ (2↑𝑘) ∥ 𝑁) → ((2↑𝑘) ∥ 𝑁 ∧ ¬ (2↑(𝑘 + 1)) ∥ 𝑁))
59	47, 55, 58	rspcedvd 2836	. . . . . . 7 ⊢ (((𝑘 ∈ ℕ ∧ (𝑁 ∈ ℕ ∧ ¬ (2↑(𝑘 + 1)) ∥ 𝑁)) ∧ (2↑𝑘) ∥ 𝑁) → ∃𝑚 ∈ ℕ₀ ((2↑𝑚) ∥ 𝑁 ∧ ¬ (2↑(𝑚 + 1)) ∥ 𝑁))
60	59	adantllr 473	. . . . . 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 521	. . . . . . . 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 338	. . . . . . 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 524	. . . . . . 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 9018	. . . . . . . . 9 ⊢ 2 ∈ ℕ
66		simpll 519	. . . . . . . . . 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 9167	. . . . . . . . 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 10468	. . . . . . . . 9 ⊢ ((2 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) → (2↑𝑘) ∈ ℕ)
69	65, 67, 68	sylancr 411	. . . . . . . 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 9312	. . . . . . . 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 11738	. . . . . . . 8 ⊢ (((2↑𝑘) ∈ ℕ ∧ 𝑁 ∈ ℤ) → DECID (2↑𝑘) ∥ 𝑁)
72	69, 70, 71	syl2anc 409	. . . . . . 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 826	. . . . . . 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 788	. . . . 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 362	. . . 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 8873	. . 3 ⊢ (𝐴 ∈ ℕ → ((𝑁 ∈ ℕ ∧ ¬ (2↑𝐴) ∥ 𝑁) → ∃𝑚 ∈ ℕ₀ ((2↑𝑚) ∥ 𝑁 ∧ ¬ (2↑(𝑚 + 1)) ∥ 𝑁)))
78	77	3ad2ant2 1009	. 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 103 ↔ wb 104 ∨ wo 698 DECID wdc 824 ∧ w3a 968 = wceq 1343 ∈ wcel 2136 ∃wrex 2445 class class class wbr 3982 (class class class)co 5842 0cc0 7753 1c1 7754 + caddc 7756 ℕcn 8857 2c2 8908 ℕ₀cn0 9114 ℤcz 9191 ↑cexp 10454 ∥ cdvds 11727

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-coll 4097 ax-sep 4100 ax-nul 4108 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-setind 4514 ax-iinf 4565 ax-cnex 7844 ax-resscn 7845 ax-1cn 7846 ax-1re 7847 ax-icn 7848 ax-addcl 7849 ax-addrcl 7850 ax-mulcl 7851 ax-mulrcl 7852 ax-addcom 7853 ax-mulcom 7854 ax-addass 7855 ax-mulass 7856 ax-distr 7857 ax-i2m1 7858 ax-0lt1 7859 ax-1rid 7860 ax-0id 7861 ax-rnegex 7862 ax-precex 7863 ax-cnre 7864 ax-pre-ltirr 7865 ax-pre-ltwlin 7866 ax-pre-lttrn 7867 ax-pre-apti 7868 ax-pre-ltadd 7869 ax-pre-mulgt0 7870 ax-pre-mulext 7871 ax-arch 7872

This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 969 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-nel 2432 df-ral 2449 df-rex 2450 df-reu 2451 df-rmo 2452 df-rab 2453 df-v 2728 df-sbc 2952 df-csb 3046 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-nul 3410 df-if 3521 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-int 3825 df-iun 3868 df-br 3983 df-opab 4044 df-mpt 4045 df-tr 4081 df-id 4271 df-po 4274 df-iso 4275 df-iord 4344 df-on 4346 df-ilim 4347 df-suc 4349 df-iom 4568 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-rn 4615 df-res 4616 df-ima 4617 df-iota 5153 df-fun 5190 df-fn 5191 df-f 5192 df-f1 5193 df-fo 5194 df-f1o 5195 df-fv 5196 df-riota 5798 df-ov 5845 df-oprab 5846 df-mpo 5847 df-1st 6108 df-2nd 6109 df-recs 6273 df-frec 6359 df-pnf 7935 df-mnf 7936 df-xr 7937 df-ltxr 7938 df-le 7939 df-sub 8071 df-neg 8072 df-reap 8473 df-ap 8480 df-div 8569 df-inn 8858 df-2 8916 df-n0 9115 df-z 9192 df-uz 9467 df-q 9558 df-rp 9590 df-fl 10205 df-mod 10258 df-seqfrec 10381 df-exp 10455 df-dvds 11728

This theorem is referenced by: pw2dvds 12098

W3C validator