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Theorem pw2dvdslemn 12760

Description: Lemma for pw2dvds 12761. 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 1021	. 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ ¬ (2↑𝐴) ∥ 𝑁) → (𝑁 ∈ ℕ ∧ ¬ (2↑𝐴) ∥ 𝑁))
2		oveq2 6031	. . . . . . . 8 ⊢ (𝑤 = 1 → (2↑𝑤) = (2↑1))
3	2	breq1d 4099	. . . . . . 7 ⊢ (𝑤 = 1 → ((2↑𝑤) ∥ 𝑁 ↔ (2↑1) ∥ 𝑁))
4	3	notbid 673	. . . . . 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 6031	. . . . . . . 8 ⊢ (𝑤 = 𝑘 → (2↑𝑤) = (2↑𝑘))
8	7	breq1d 4099	. . . . . . 7 ⊢ (𝑤 = 𝑘 → ((2↑𝑤) ∥ 𝑁 ↔ (2↑𝑘) ∥ 𝑁))
9	8	notbid 673	. . . . . 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 6031	. . . . . . . 8 ⊢ (𝑤 = (𝑘 + 1) → (2↑𝑤) = (2↑(𝑘 + 1)))
13	12	breq1d 4099	. . . . . . 7 ⊢ (𝑤 = (𝑘 + 1) → ((2↑𝑤) ∥ 𝑁 ↔ (2↑(𝑘 + 1)) ∥ 𝑁))
14	13	notbid 673	. . . . . 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 6031	. . . . . . . 8 ⊢ (𝑤 = 𝐴 → (2↑𝑤) = (2↑𝐴))
18	17	breq1d 4099	. . . . . . 7 ⊢ (𝑤 = 𝐴 → ((2↑𝑤) ∥ 𝑁 ↔ (2↑𝐴) ∥ 𝑁))
19	18	notbid 673	. . . . . 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 9422	. . . . . 6 ⊢ 0 ∈ ℕ₀
23	22	a1i 9	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ ¬ (2↑1) ∥ 𝑁) → 0 ∈ ℕ₀)
24		oveq2 6031	. . . . . . . 8 ⊢ (𝑚 = 0 → (2↑𝑚) = (2↑0))
25	24	breq1d 4099	. . . . . . 7 ⊢ (𝑚 = 0 → ((2↑𝑚) ∥ 𝑁 ↔ (2↑0) ∥ 𝑁))
26		oveq1 6030	. . . . . . . . . 10 ⊢ (𝑚 = 0 → (𝑚 + 1) = (0 + 1))
27	26	oveq2d 6039	. . . . . . . . 9 ⊢ (𝑚 = 0 → (2↑(𝑚 + 1)) = (2↑(0 + 1)))
28	27	breq1d 4099	. . . . . . . 8 ⊢ (𝑚 = 0 → ((2↑(𝑚 + 1)) ∥ 𝑁 ↔ (2↑(0 + 1)) ∥ 𝑁))
29	28	notbid 673	. . . . . . 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 9221	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ ¬ (2↑1) ∥ 𝑁) → 2 ∈ ℂ)
33	32	exp0d 10935	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ ¬ (2↑1) ∥ 𝑁) → (2↑0) = 1)
34		simpl 109	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ ¬ (2↑1) ∥ 𝑁) → 𝑁 ∈ ℕ)
35	34	nnzd 9606	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ ¬ (2↑1) ∥ 𝑁) → 𝑁 ∈ ℤ)
36		1dvds 12389	. . . . . . . 8 ⊢ (𝑁 ∈ ℤ → 1 ∥ 𝑁)
37	35, 36	syl 14	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ ¬ (2↑1) ∥ 𝑁) → 1 ∥ 𝑁)
38	33, 37	eqbrtrd 4111	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ ¬ (2↑1) ∥ 𝑁) → (2↑0) ∥ 𝑁)
39		simpr 110	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ ¬ (2↑1) ∥ 𝑁) → ¬ (2↑1) ∥ 𝑁)
40		0p1e1 9262	. . . . . . . . 9 ⊢ (0 + 1) = 1
41	40	oveq2i 6034	. . . . . . . 8 ⊢ (2↑(0 + 1)) = (2↑1)
42	41	breq1i 4096	. . . . . . 7 ⊢ ((2↑(0 + 1)) ∥ 𝑁 ↔ (2↑1) ∥ 𝑁)
43	39, 42	sylnibr 683	. . . . . 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 2915	. . . 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 9460	. . . . . . . 8 ⊢ (((𝑘 ∈ ℕ ∧ (𝑁 ∈ ℕ ∧ ¬ (2↑(𝑘 + 1)) ∥ 𝑁)) ∧ (2↑𝑘) ∥ 𝑁) → 𝑘 ∈ ℕ₀)
48		oveq2 6031	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑘 → (2↑𝑚) = (2↑𝑘))
49	48	breq1d 4099	. . . . . . . . . 10 ⊢ (𝑚 = 𝑘 → ((2↑𝑚) ∥ 𝑁 ↔ (2↑𝑘) ∥ 𝑁))
50		oveq1 6030	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑘 → (𝑚 + 1) = (𝑘 + 1))
51	50	oveq2d 6039	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑘 → (2↑(𝑚 + 1)) = (2↑(𝑘 + 1)))
52	51	breq1d 4099	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑘 → ((2↑(𝑚 + 1)) ∥ 𝑁 ↔ (2↑(𝑘 + 1)) ∥ 𝑁))
53	52	notbid 673	. . . . . . . . . 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 538	. . . . . . . . 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 2915	. . . . . . 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 531	. . . . . . . 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 536	. . . . . . 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 9310	. . . . . . . . 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 9460	. . . . . . . . 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 10820	. . . . . . . . 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 9606	. . . . . . . 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 12382	. . . . . . . 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 843	. . . . . . 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 805	. . . . 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 9164	. . 3 ⊢ (𝐴 ∈ ℕ → ((𝑁 ∈ ℕ ∧ ¬ (2↑𝐴) ∥ 𝑁) → ∃𝑚 ∈ ℕ₀ ((2↑𝑚) ∥ 𝑁 ∧ ¬ (2↑(𝑚 + 1)) ∥ 𝑁)))
78	77	3ad2ant2 1045	. 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 715 DECID wdc 841 ∧ w3a 1004 = wceq 1397 ∈ wcel 2201 ∃wrex 2510 class class class wbr 4089 (class class class)co 6023 0cc0 8037 1c1 8038 + caddc 8040 ℕcn 9148 2c2 9199 ℕ₀cn0 9407 ℤcz 9484 ↑cexp 10806 ∥ cdvds 12371

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 619 ax-in2 620 ax-io 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2203 ax-14 2204 ax-ext 2212 ax-coll 4205 ax-sep 4208 ax-nul 4216 ax-pow 4266 ax-pr 4301 ax-un 4532 ax-setind 4637 ax-iinf 4688 ax-cnex 8128 ax-resscn 8129 ax-1cn 8130 ax-1re 8131 ax-icn 8132 ax-addcl 8133 ax-addrcl 8134 ax-mulcl 8135 ax-mulrcl 8136 ax-addcom 8137 ax-mulcom 8138 ax-addass 8139 ax-mulass 8140 ax-distr 8141 ax-i2m1 8142 ax-0lt1 8143 ax-1rid 8144 ax-0id 8145 ax-rnegex 8146 ax-precex 8147 ax-cnre 8148 ax-pre-ltirr 8149 ax-pre-ltwlin 8150 ax-pre-lttrn 8151 ax-pre-apti 8152 ax-pre-ltadd 8153 ax-pre-mulgt0 8154 ax-pre-mulext 8155 ax-arch 8156

This theorem depends on definitions: df-bi 117 df-dc 842 df-3or 1005 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1810 df-eu 2081 df-mo 2082 df-clab 2217 df-cleq 2223 df-clel 2226 df-nfc 2362 df-ne 2402 df-nel 2497 df-ral 2514 df-rex 2515 df-reu 2516 df-rmo 2517 df-rab 2518 df-v 2803 df-sbc 3031 df-csb 3127 df-dif 3201 df-un 3203 df-in 3205 df-ss 3212 df-nul 3494 df-if 3605 df-pw 3655 df-sn 3676 df-pr 3677 df-op 3679 df-uni 3895 df-int 3930 df-iun 3973 df-br 4090 df-opab 4152 df-mpt 4153 df-tr 4189 df-id 4392 df-po 4395 df-iso 4396 df-iord 4465 df-on 4467 df-ilim 4468 df-suc 4470 df-iom 4691 df-xp 4733 df-rel 4734 df-cnv 4735 df-co 4736 df-dm 4737 df-rn 4738 df-res 4739 df-ima 4740 df-iota 5288 df-fun 5330 df-fn 5331 df-f 5332 df-f1 5333 df-fo 5334 df-f1o 5335 df-fv 5336 df-riota 5976 df-ov 6026 df-oprab 6027 df-mpo 6028 df-1st 6308 df-2nd 6309 df-recs 6476 df-frec 6562 df-pnf 8221 df-mnf 8222 df-xr 8223 df-ltxr 8224 df-le 8225 df-sub 8357 df-neg 8358 df-reap 8760 df-ap 8767 df-div 8858 df-inn 9149 df-2 9207 df-n0 9408 df-z 9485 df-uz 9761 df-q 9859 df-rp 9894 df-fl 10536 df-mod 10591 df-seqfrec 10716 df-exp 10807 df-dvds 12372

This theorem is referenced by: pw2dvds 12761

W3C validator