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Theorem pw2dvdslemn 12739

Description: Lemma for pw2dvds 12740. 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 6026	. . . . . . . 8 ⊢ (𝑤 = 1 → (2↑𝑤) = (2↑1))
3	2	breq1d 4098	. . . . . . 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 6026	. . . . . . . 8 ⊢ (𝑤 = 𝑘 → (2↑𝑤) = (2↑𝑘))
8	7	breq1d 4098	. . . . . . 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 6026	. . . . . . . 8 ⊢ (𝑤 = (𝑘 + 1) → (2↑𝑤) = (2↑(𝑘 + 1)))
13	12	breq1d 4098	. . . . . . 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 6026	. . . . . . . 8 ⊢ (𝑤 = 𝐴 → (2↑𝑤) = (2↑𝐴))
18	17	breq1d 4098	. . . . . . 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 9417	. . . . . 6 ⊢ 0 ∈ ℕ₀
23	22	a1i 9	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ ¬ (2↑1) ∥ 𝑁) → 0 ∈ ℕ₀)
24		oveq2 6026	. . . . . . . 8 ⊢ (𝑚 = 0 → (2↑𝑚) = (2↑0))
25	24	breq1d 4098	. . . . . . 7 ⊢ (𝑚 = 0 → ((2↑𝑚) ∥ 𝑁 ↔ (2↑0) ∥ 𝑁))
26		oveq1 6025	. . . . . . . . . 10 ⊢ (𝑚 = 0 → (𝑚 + 1) = (0 + 1))
27	26	oveq2d 6034	. . . . . . . . 9 ⊢ (𝑚 = 0 → (2↑(𝑚 + 1)) = (2↑(0 + 1)))
28	27	breq1d 4098	. . . . . . . 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 9216	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ ¬ (2↑1) ∥ 𝑁) → 2 ∈ ℂ)
33	32	exp0d 10930	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ ¬ (2↑1) ∥ 𝑁) → (2↑0) = 1)
34		simpl 109	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ ¬ (2↑1) ∥ 𝑁) → 𝑁 ∈ ℕ)
35	34	nnzd 9601	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ ¬ (2↑1) ∥ 𝑁) → 𝑁 ∈ ℤ)
36		1dvds 12368	. . . . . . . 8 ⊢ (𝑁 ∈ ℤ → 1 ∥ 𝑁)
37	35, 36	syl 14	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ ¬ (2↑1) ∥ 𝑁) → 1 ∥ 𝑁)
38	33, 37	eqbrtrd 4110	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ ¬ (2↑1) ∥ 𝑁) → (2↑0) ∥ 𝑁)
39		simpr 110	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ ¬ (2↑1) ∥ 𝑁) → ¬ (2↑1) ∥ 𝑁)
40		0p1e1 9257	. . . . . . . . 9 ⊢ (0 + 1) = 1
41	40	oveq2i 6029	. . . . . . . 8 ⊢ (2↑(0 + 1)) = (2↑1)
42	41	breq1i 4095	. . . . . . 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 2916	. . . 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 9455	. . . . . . . 8 ⊢ (((𝑘 ∈ ℕ ∧ (𝑁 ∈ ℕ ∧ ¬ (2↑(𝑘 + 1)) ∥ 𝑁)) ∧ (2↑𝑘) ∥ 𝑁) → 𝑘 ∈ ℕ₀)
48		oveq2 6026	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑘 → (2↑𝑚) = (2↑𝑘))
49	48	breq1d 4098	. . . . . . . . . 10 ⊢ (𝑚 = 𝑘 → ((2↑𝑚) ∥ 𝑁 ↔ (2↑𝑘) ∥ 𝑁))
50		oveq1 6025	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑘 → (𝑚 + 1) = (𝑘 + 1))
51	50	oveq2d 6034	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑘 → (2↑(𝑚 + 1)) = (2↑(𝑘 + 1)))
52	51	breq1d 4098	. . . . . . . . . . 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 2916	. . . . . . 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 9305	. . . . . . . . 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 9455	. . . . . . . . 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 10815	. . . . . . . . 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 9601	. . . . . . . 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 12361	. . . . . . . 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 9159	. . 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 2202 ∃wrex 2511 class class class wbr 4088 (class class class)co 6018 0cc0 8032 1c1 8033 + caddc 8035 ℕcn 9143 2c2 9194 ℕ₀cn0 9402 ℤcz 9479 ↑cexp 10801 ∥ cdvds 12350

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 2204 ax-14 2205 ax-ext 2213 ax-coll 4204 ax-sep 4207 ax-nul 4215 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-iinf 4686 ax-cnex 8123 ax-resscn 8124 ax-1cn 8125 ax-1re 8126 ax-icn 8127 ax-addcl 8128 ax-addrcl 8129 ax-mulcl 8130 ax-mulrcl 8131 ax-addcom 8132 ax-mulcom 8133 ax-addass 8134 ax-mulass 8135 ax-distr 8136 ax-i2m1 8137 ax-0lt1 8138 ax-1rid 8139 ax-0id 8140 ax-rnegex 8141 ax-precex 8142 ax-cnre 8143 ax-pre-ltirr 8144 ax-pre-ltwlin 8145 ax-pre-lttrn 8146 ax-pre-apti 8147 ax-pre-ltadd 8148 ax-pre-mulgt0 8149 ax-pre-mulext 8150 ax-arch 8151

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 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-nel 2498 df-ral 2515 df-rex 2516 df-reu 2517 df-rmo 2518 df-rab 2519 df-v 2804 df-sbc 3032 df-csb 3128 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-nul 3495 df-if 3606 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-int 3929 df-iun 3972 df-br 4089 df-opab 4151 df-mpt 4152 df-tr 4188 df-id 4390 df-po 4393 df-iso 4394 df-iord 4463 df-on 4465 df-ilim 4466 df-suc 4468 df-iom 4689 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-rn 4736 df-res 4737 df-ima 4738 df-iota 5286 df-fun 5328 df-fn 5329 df-f 5330 df-f1 5331 df-fo 5332 df-f1o 5333 df-fv 5334 df-riota 5971 df-ov 6021 df-oprab 6022 df-mpo 6023 df-1st 6303 df-2nd 6304 df-recs 6471 df-frec 6557 df-pnf 8216 df-mnf 8217 df-xr 8218 df-ltxr 8219 df-le 8220 df-sub 8352 df-neg 8353 df-reap 8755 df-ap 8762 df-div 8853 df-inn 9144 df-2 9202 df-n0 9403 df-z 9480 df-uz 9756 df-q 9854 df-rp 9889 df-fl 10531 df-mod 10586 df-seqfrec 10711 df-exp 10802 df-dvds 12351

This theorem is referenced by: pw2dvds 12740

W3C validator