Theorem pwbdvdslemn 12969

Description: Lemma for pwbdvds 12970. If a natural number has some power of a base which does not divide it, there is a highest power of the base which does divide it. (Contributed by Jim Kingdon, 14-Nov-2021.) (Revised by Jim Kingdon, 17-Aug-2026.)

Hypotheses

Ref	Expression
pwbdvdslemn.n	⊢ (𝜑 → 𝑁 ∈ ℕ)
pwbdvdslemn.a	⊢ (𝜑 → 𝐴 ∈ ℕ)
pwbdvdslemn.b	⊢ (𝜑 → 𝐵 ∈ ℕ)
pwbdvdslemn.dvds	⊢ (𝜑 → ¬ (𝐵↑𝐴) ∥ 𝑁)

Assertion

Ref	Expression
pwbdvdslemn	⊢ (𝜑 → ∃𝑚 ∈ ℕ0 ((𝐵↑𝑚) ∥ 𝑁 ∧ ¬ (𝐵↑(𝑚 + 1)) ∥ 𝑁))

Distinct variable groups:   𝑚,𝑁   𝐵,𝑚   𝜑,𝑚

Allowed substitution hint:   𝐴(𝑚)

Proof of Theorem pwbdvdslemn

Dummy variables 𝑤 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pwbdvdslemn.dvds	. 2 ⊢ (𝜑 → ¬ (𝐵↑𝐴) ∥ 𝑁)
2		pwbdvdslemn.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ℕ)
3	2	adantr 276	. . 3 ⊢ ((𝜑 ∧ ¬ (𝐵↑𝐴) ∥ 𝑁) → 𝐴 ∈ ℕ)
4		oveq2 6096	. . . . . . . 8 ⊢ (𝑤 = 1 → (𝐵↑𝑤) = (𝐵↑1))
5	4	breq1d 4141	. . . . . . 7 ⊢ (𝑤 = 1 → ((𝐵↑𝑤) ∥ 𝑁 ↔ (𝐵↑1) ∥ 𝑁))
6	5	notbid 677	. . . . . 6 ⊢ (𝑤 = 1 → (¬ (𝐵↑𝑤) ∥ 𝑁 ↔ ¬ (𝐵↑1) ∥ 𝑁))
7	6	anbi2d 468	. . . . 5 ⊢ (𝑤 = 1 → ((𝜑 ∧ ¬ (𝐵↑𝑤) ∥ 𝑁) ↔ (𝜑 ∧ ¬ (𝐵↑1) ∥ 𝑁)))
8	7	imbi1d 231	. . . 4 ⊢ (𝑤 = 1 → (((𝜑 ∧ ¬ (𝐵↑𝑤) ∥ 𝑁) → ∃𝑚 ∈ ℕ0 ((𝐵↑𝑚) ∥ 𝑁 ∧ ¬ (𝐵↑(𝑚 + 1)) ∥ 𝑁)) ↔ ((𝜑 ∧ ¬ (𝐵↑1) ∥ 𝑁) → ∃𝑚 ∈ ℕ0 ((𝐵↑𝑚) ∥ 𝑁 ∧ ¬ (𝐵↑(𝑚 + 1)) ∥ 𝑁))))
9		oveq2 6096	. . . . . . . 8 ⊢ (𝑤 = 𝑘 → (𝐵↑𝑤) = (𝐵↑𝑘))
10	9	breq1d 4141	. . . . . . 7 ⊢ (𝑤 = 𝑘 → ((𝐵↑𝑤) ∥ 𝑁 ↔ (𝐵↑𝑘) ∥ 𝑁))
11	10	notbid 677	. . . . . 6 ⊢ (𝑤 = 𝑘 → (¬ (𝐵↑𝑤) ∥ 𝑁 ↔ ¬ (𝐵↑𝑘) ∥ 𝑁))
12	11	anbi2d 468	. . . . 5 ⊢ (𝑤 = 𝑘 → ((𝜑 ∧ ¬ (𝐵↑𝑤) ∥ 𝑁) ↔ (𝜑 ∧ ¬ (𝐵↑𝑘) ∥ 𝑁)))
13	12	imbi1d 231	. . . 4 ⊢ (𝑤 = 𝑘 → (((𝜑 ∧ ¬ (𝐵↑𝑤) ∥ 𝑁) → ∃𝑚 ∈ ℕ0 ((𝐵↑𝑚) ∥ 𝑁 ∧ ¬ (𝐵↑(𝑚 + 1)) ∥ 𝑁)) ↔ ((𝜑 ∧ ¬ (𝐵↑𝑘) ∥ 𝑁) → ∃𝑚 ∈ ℕ0 ((𝐵↑𝑚) ∥ 𝑁 ∧ ¬ (𝐵↑(𝑚 + 1)) ∥ 𝑁))))
14		oveq2 6096	. . . . . . . 8 ⊢ (𝑤 = (𝑘 + 1) → (𝐵↑𝑤) = (𝐵↑(𝑘 + 1)))
15	14	breq1d 4141	. . . . . . 7 ⊢ (𝑤 = (𝑘 + 1) → ((𝐵↑𝑤) ∥ 𝑁 ↔ (𝐵↑(𝑘 + 1)) ∥ 𝑁))
16	15	notbid 677	. . . . . 6 ⊢ (𝑤 = (𝑘 + 1) → (¬ (𝐵↑𝑤) ∥ 𝑁 ↔ ¬ (𝐵↑(𝑘 + 1)) ∥ 𝑁))
17	16	anbi2d 468	. . . . 5 ⊢ (𝑤 = (𝑘 + 1) → ((𝜑 ∧ ¬ (𝐵↑𝑤) ∥ 𝑁) ↔ (𝜑 ∧ ¬ (𝐵↑(𝑘 + 1)) ∥ 𝑁)))
18	17	imbi1d 231	. . . 4 ⊢ (𝑤 = (𝑘 + 1) → (((𝜑 ∧ ¬ (𝐵↑𝑤) ∥ 𝑁) → ∃𝑚 ∈ ℕ0 ((𝐵↑𝑚) ∥ 𝑁 ∧ ¬ (𝐵↑(𝑚 + 1)) ∥ 𝑁)) ↔ ((𝜑 ∧ ¬ (𝐵↑(𝑘 + 1)) ∥ 𝑁) → ∃𝑚 ∈ ℕ0 ((𝐵↑𝑚) ∥ 𝑁 ∧ ¬ (𝐵↑(𝑚 + 1)) ∥ 𝑁))))
19		oveq2 6096	. . . . . . . 8 ⊢ (𝑤 = 𝐴 → (𝐵↑𝑤) = (𝐵↑𝐴))
20	19	breq1d 4141	. . . . . . 7 ⊢ (𝑤 = 𝐴 → ((𝐵↑𝑤) ∥ 𝑁 ↔ (𝐵↑𝐴) ∥ 𝑁))
21	20	notbid 677	. . . . . 6 ⊢ (𝑤 = 𝐴 → (¬ (𝐵↑𝑤) ∥ 𝑁 ↔ ¬ (𝐵↑𝐴) ∥ 𝑁))
22	21	anbi2d 468	. . . . 5 ⊢ (𝑤 = 𝐴 → ((𝜑 ∧ ¬ (𝐵↑𝑤) ∥ 𝑁) ↔ (𝜑 ∧ ¬ (𝐵↑𝐴) ∥ 𝑁)))
23	22	imbi1d 231	. . . 4 ⊢ (𝑤 = 𝐴 → (((𝜑 ∧ ¬ (𝐵↑𝑤) ∥ 𝑁) → ∃𝑚 ∈ ℕ0 ((𝐵↑𝑚) ∥ 𝑁 ∧ ¬ (𝐵↑(𝑚 + 1)) ∥ 𝑁)) ↔ ((𝜑 ∧ ¬ (𝐵↑𝐴) ∥ 𝑁) → ∃𝑚 ∈ ℕ0 ((𝐵↑𝑚) ∥ 𝑁 ∧ ¬ (𝐵↑(𝑚 + 1)) ∥ 𝑁))))
24		oveq2 6096	. . . . . . 7 ⊢ (𝑚 = 0 → (𝐵↑𝑚) = (𝐵↑0))
25	24	breq1d 4141	. . . . . 6 ⊢ (𝑚 = 0 → ((𝐵↑𝑚) ∥ 𝑁 ↔ (𝐵↑0) ∥ 𝑁))
26		oveq1 6095	. . . . . . . . 9 ⊢ (𝑚 = 0 → (𝑚 + 1) = (0 + 1))
27	26	oveq2d 6104	. . . . . . . 8 ⊢ (𝑚 = 0 → (𝐵↑(𝑚 + 1)) = (𝐵↑(0 + 1)))
28	27	breq1d 4141	. . . . . . 7 ⊢ (𝑚 = 0 → ((𝐵↑(𝑚 + 1)) ∥ 𝑁 ↔ (𝐵↑(0 + 1)) ∥ 𝑁))
29	28	notbid 677	. . . . . 6 ⊢ (𝑚 = 0 → (¬ (𝐵↑(𝑚 + 1)) ∥ 𝑁 ↔ ¬ (𝐵↑(0 + 1)) ∥ 𝑁))
30	25, 29	anbi12d 477	. . . . 5 ⊢ (𝑚 = 0 → (((𝐵↑𝑚) ∥ 𝑁 ∧ ¬ (𝐵↑(𝑚 + 1)) ∥ 𝑁) ↔ ((𝐵↑0) ∥ 𝑁 ∧ ¬ (𝐵↑(0 + 1)) ∥ 𝑁)))
31		0nn0 9589	. . . . . 6 ⊢ 0 ∈ ℕ0
32	31	a1i 9	. . . . 5 ⊢ ((𝜑 ∧ ¬ (𝐵↑1) ∥ 𝑁) → 0 ∈ ℕ0)
33		pwbdvdslemn.b	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ ℕ)
34	33	nncnd 9327	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℂ)
35	34	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ (𝐵↑1) ∥ 𝑁) → 𝐵 ∈ ℂ)
36	35	exp0d 11126	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ (𝐵↑1) ∥ 𝑁) → (𝐵↑0) = 1)
37		pwbdvdslemn.n	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℕ)
38	37	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ (𝐵↑1) ∥ 𝑁) → 𝑁 ∈ ℕ)
39	38	nnzd 9778	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ (𝐵↑1) ∥ 𝑁) → 𝑁 ∈ ℤ)
40		1dvds 12597	. . . . . . . 8 ⊢ (𝑁 ∈ ℤ → 1 ∥ 𝑁)
41	39, 40	syl 14	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ (𝐵↑1) ∥ 𝑁) → 1 ∥ 𝑁)
42	36, 41	eqbrtrd 4153	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝐵↑1) ∥ 𝑁) → (𝐵↑0) ∥ 𝑁)
43		simpr 110	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ (𝐵↑1) ∥ 𝑁) → ¬ (𝐵↑1) ∥ 𝑁)
44		0p1e1 9427	. . . . . . . . 9 ⊢ (0 + 1) = 1
45	44	oveq2i 6099	. . . . . . . 8 ⊢ (𝐵↑(0 + 1)) = (𝐵↑1)
46	45	breq1i 4138	. . . . . . 7 ⊢ ((𝐵↑(0 + 1)) ∥ 𝑁 ↔ (𝐵↑1) ∥ 𝑁)
47	43, 46	sylnibr 688	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝐵↑1) ∥ 𝑁) → ¬ (𝐵↑(0 + 1)) ∥ 𝑁)
48	42, 47	jca 306	. . . . 5 ⊢ ((𝜑 ∧ ¬ (𝐵↑1) ∥ 𝑁) → ((𝐵↑0) ∥ 𝑁 ∧ ¬ (𝐵↑(0 + 1)) ∥ 𝑁))
49	30, 32, 48	rspcedvdw 2936	. . . 4 ⊢ ((𝜑 ∧ ¬ (𝐵↑1) ∥ 𝑁) → ∃𝑚 ∈ ℕ0 ((𝐵↑𝑚) ∥ 𝑁 ∧ ¬ (𝐵↑(𝑚 + 1)) ∥ 𝑁))
50		oveq2 6096	. . . . . . . . . 10 ⊢ (𝑚 = 𝑘 → (𝐵↑𝑚) = (𝐵↑𝑘))
51	50	breq1d 4141	. . . . . . . . 9 ⊢ (𝑚 = 𝑘 → ((𝐵↑𝑚) ∥ 𝑁 ↔ (𝐵↑𝑘) ∥ 𝑁))
52		oveq1 6095	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑘 → (𝑚 + 1) = (𝑘 + 1))
53	52	oveq2d 6104	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑘 → (𝐵↑(𝑚 + 1)) = (𝐵↑(𝑘 + 1)))
54	53	breq1d 4141	. . . . . . . . . 10 ⊢ (𝑚 = 𝑘 → ((𝐵↑(𝑚 + 1)) ∥ 𝑁 ↔ (𝐵↑(𝑘 + 1)) ∥ 𝑁))
55	54	notbid 677	. . . . . . . . 9 ⊢ (𝑚 = 𝑘 → (¬ (𝐵↑(𝑚 + 1)) ∥ 𝑁 ↔ ¬ (𝐵↑(𝑘 + 1)) ∥ 𝑁))
56	51, 55	anbi12d 477	. . . . . . . 8 ⊢ (𝑚 = 𝑘 → (((𝐵↑𝑚) ∥ 𝑁 ∧ ¬ (𝐵↑(𝑚 + 1)) ∥ 𝑁) ↔ ((𝐵↑𝑘) ∥ 𝑁 ∧ ¬ (𝐵↑(𝑘 + 1)) ∥ 𝑁)))
57		simpll 531	. . . . . . . . 9 ⊢ (((𝑘 ∈ ℕ ∧ (𝜑 ∧ ¬ (𝐵↑(𝑘 + 1)) ∥ 𝑁)) ∧ (𝐵↑𝑘) ∥ 𝑁) → 𝑘 ∈ ℕ)
58	57	nnnn0d 9631	. . . . . . . 8 ⊢ (((𝑘 ∈ ℕ ∧ (𝜑 ∧ ¬ (𝐵↑(𝑘 + 1)) ∥ 𝑁)) ∧ (𝐵↑𝑘) ∥ 𝑁) → 𝑘 ∈ ℕ0)
59		simpr 110	. . . . . . . . 9 ⊢ (((𝑘 ∈ ℕ ∧ (𝜑 ∧ ¬ (𝐵↑(𝑘 + 1)) ∥ 𝑁)) ∧ (𝐵↑𝑘) ∥ 𝑁) → (𝐵↑𝑘) ∥ 𝑁)
60		simplrr 542	. . . . . . . . 9 ⊢ (((𝑘 ∈ ℕ ∧ (𝜑 ∧ ¬ (𝐵↑(𝑘 + 1)) ∥ 𝑁)) ∧ (𝐵↑𝑘) ∥ 𝑁) → ¬ (𝐵↑(𝑘 + 1)) ∥ 𝑁)
61	59, 60	jca 306	. . . . . . . 8 ⊢ (((𝑘 ∈ ℕ ∧ (𝜑 ∧ ¬ (𝐵↑(𝑘 + 1)) ∥ 𝑁)) ∧ (𝐵↑𝑘) ∥ 𝑁) → ((𝐵↑𝑘) ∥ 𝑁 ∧ ¬ (𝐵↑(𝑘 + 1)) ∥ 𝑁))
62	56, 58, 61	rspcedvdw 2936	. . . . . . 7 ⊢ (((𝑘 ∈ ℕ ∧ (𝜑 ∧ ¬ (𝐵↑(𝑘 + 1)) ∥ 𝑁)) ∧ (𝐵↑𝑘) ∥ 𝑁) → ∃𝑚 ∈ ℕ0 ((𝐵↑𝑚) ∥ 𝑁 ∧ ¬ (𝐵↑(𝑚 + 1)) ∥ 𝑁))
63	62	adantllr 485	. . . . . 6 ⊢ ((((𝑘 ∈ ℕ ∧ ((𝜑 ∧ ¬ (𝐵↑𝑘) ∥ 𝑁) → ∃𝑚 ∈ ℕ0 ((𝐵↑𝑚) ∥ 𝑁 ∧ ¬ (𝐵↑(𝑚 + 1)) ∥ 𝑁))) ∧ (𝜑 ∧ ¬ (𝐵↑(𝑘 + 1)) ∥ 𝑁)) ∧ (𝐵↑𝑘) ∥ 𝑁) → ∃𝑚 ∈ ℕ0 ((𝐵↑𝑚) ∥ 𝑁 ∧ ¬ (𝐵↑(𝑚 + 1)) ∥ 𝑁))
64		simprl 535	. . . . . . . 8 ⊢ (((𝑘 ∈ ℕ ∧ ((𝜑 ∧ ¬ (𝐵↑𝑘) ∥ 𝑁) → ∃𝑚 ∈ ℕ0 ((𝐵↑𝑚) ∥ 𝑁 ∧ ¬ (𝐵↑(𝑚 + 1)) ∥ 𝑁))) ∧ (𝜑 ∧ ¬ (𝐵↑(𝑘 + 1)) ∥ 𝑁)) → 𝜑)
65	64	anim1i 340	. . . . . . 7 ⊢ ((((𝑘 ∈ ℕ ∧ ((𝜑 ∧ ¬ (𝐵↑𝑘) ∥ 𝑁) → ∃𝑚 ∈ ℕ0 ((𝐵↑𝑚) ∥ 𝑁 ∧ ¬ (𝐵↑(𝑚 + 1)) ∥ 𝑁))) ∧ (𝜑 ∧ ¬ (𝐵↑(𝑘 + 1)) ∥ 𝑁)) ∧ ¬ (𝐵↑𝑘) ∥ 𝑁) → (𝜑 ∧ ¬ (𝐵↑𝑘) ∥ 𝑁))
66		simpllr 540	. . . . . . 7 ⊢ ((((𝑘 ∈ ℕ ∧ ((𝜑 ∧ ¬ (𝐵↑𝑘) ∥ 𝑁) → ∃𝑚 ∈ ℕ0 ((𝐵↑𝑚) ∥ 𝑁 ∧ ¬ (𝐵↑(𝑚 + 1)) ∥ 𝑁))) ∧ (𝜑 ∧ ¬ (𝐵↑(𝑘 + 1)) ∥ 𝑁)) ∧ ¬ (𝐵↑𝑘) ∥ 𝑁) → ((𝜑 ∧ ¬ (𝐵↑𝑘) ∥ 𝑁) → ∃𝑚 ∈ ℕ0 ((𝐵↑𝑚) ∥ 𝑁 ∧ ¬ (𝐵↑(𝑚 + 1)) ∥ 𝑁)))
67	65, 66	mpd 13	. . . . . 6 ⊢ ((((𝑘 ∈ ℕ ∧ ((𝜑 ∧ ¬ (𝐵↑𝑘) ∥ 𝑁) → ∃𝑚 ∈ ℕ0 ((𝐵↑𝑚) ∥ 𝑁 ∧ ¬ (𝐵↑(𝑚 + 1)) ∥ 𝑁))) ∧ (𝜑 ∧ ¬ (𝐵↑(𝑘 + 1)) ∥ 𝑁)) ∧ ¬ (𝐵↑𝑘) ∥ 𝑁) → ∃𝑚 ∈ ℕ0 ((𝐵↑𝑚) ∥ 𝑁 ∧ ¬ (𝐵↑(𝑚 + 1)) ∥ 𝑁))
68	33	ad2antrl 494	. . . . . . . . 9 ⊢ (((𝑘 ∈ ℕ ∧ ((𝜑 ∧ ¬ (𝐵↑𝑘) ∥ 𝑁) → ∃𝑚 ∈ ℕ0 ((𝐵↑𝑚) ∥ 𝑁 ∧ ¬ (𝐵↑(𝑚 + 1)) ∥ 𝑁))) ∧ (𝜑 ∧ ¬ (𝐵↑(𝑘 + 1)) ∥ 𝑁)) → 𝐵 ∈ ℕ)
69		nnnn0 9581	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ0)
70	69	ad2antrr 492	. . . . . . . . 9 ⊢ (((𝑘 ∈ ℕ ∧ ((𝜑 ∧ ¬ (𝐵↑𝑘) ∥ 𝑁) → ∃𝑚 ∈ ℕ0 ((𝐵↑𝑚) ∥ 𝑁 ∧ ¬ (𝐵↑(𝑚 + 1)) ∥ 𝑁))) ∧ (𝜑 ∧ ¬ (𝐵↑(𝑘 + 1)) ∥ 𝑁)) → 𝑘 ∈ ℕ0)
71	68, 70	nnexpcld 11154	. . . . . . . 8 ⊢ (((𝑘 ∈ ℕ ∧ ((𝜑 ∧ ¬ (𝐵↑𝑘) ∥ 𝑁) → ∃𝑚 ∈ ℕ0 ((𝐵↑𝑚) ∥ 𝑁 ∧ ¬ (𝐵↑(𝑚 + 1)) ∥ 𝑁))) ∧ (𝜑 ∧ ¬ (𝐵↑(𝑘 + 1)) ∥ 𝑁)) → (𝐵↑𝑘) ∈ ℕ)
72	37	ad2antrl 494	. . . . . . . . 9 ⊢ (((𝑘 ∈ ℕ ∧ ((𝜑 ∧ ¬ (𝐵↑𝑘) ∥ 𝑁) → ∃𝑚 ∈ ℕ0 ((𝐵↑𝑚) ∥ 𝑁 ∧ ¬ (𝐵↑(𝑚 + 1)) ∥ 𝑁))) ∧ (𝜑 ∧ ¬ (𝐵↑(𝑘 + 1)) ∥ 𝑁)) → 𝑁 ∈ ℕ)
73	72	nnzd 9778	. . . . . . . 8 ⊢ (((𝑘 ∈ ℕ ∧ ((𝜑 ∧ ¬ (𝐵↑𝑘) ∥ 𝑁) → ∃𝑚 ∈ ℕ0 ((𝐵↑𝑚) ∥ 𝑁 ∧ ¬ (𝐵↑(𝑚 + 1)) ∥ 𝑁))) ∧ (𝜑 ∧ ¬ (𝐵↑(𝑘 + 1)) ∥ 𝑁)) → 𝑁 ∈ ℤ)
74		dvdsdc 12590	. . . . . . . 8 ⊢ (((𝐵↑𝑘) ∈ ℕ ∧ 𝑁 ∈ ℤ) → DECID (𝐵↑𝑘) ∥ 𝑁)
75	71, 73, 74	syl2anc 415	. . . . . . 7 ⊢ (((𝑘 ∈ ℕ ∧ ((𝜑 ∧ ¬ (𝐵↑𝑘) ∥ 𝑁) → ∃𝑚 ∈ ℕ0 ((𝐵↑𝑚) ∥ 𝑁 ∧ ¬ (𝐵↑(𝑚 + 1)) ∥ 𝑁))) ∧ (𝜑 ∧ ¬ (𝐵↑(𝑘 + 1)) ∥ 𝑁)) → DECID (𝐵↑𝑘) ∥ 𝑁)
76		exmiddc 848	. . . . . . 7 ⊢ (DECID (𝐵↑𝑘) ∥ 𝑁 → ((𝐵↑𝑘) ∥ 𝑁 ∨ ¬ (𝐵↑𝑘) ∥ 𝑁))
77	75, 76	syl 14	. . . . . 6 ⊢ (((𝑘 ∈ ℕ ∧ ((𝜑 ∧ ¬ (𝐵↑𝑘) ∥ 𝑁) → ∃𝑚 ∈ ℕ0 ((𝐵↑𝑚) ∥ 𝑁 ∧ ¬ (𝐵↑(𝑚 + 1)) ∥ 𝑁))) ∧ (𝜑 ∧ ¬ (𝐵↑(𝑘 + 1)) ∥ 𝑁)) → ((𝐵↑𝑘) ∥ 𝑁 ∨ ¬ (𝐵↑𝑘) ∥ 𝑁))
78	63, 67, 77	mpjaodan 810	. . . . 5 ⊢ (((𝑘 ∈ ℕ ∧ ((𝜑 ∧ ¬ (𝐵↑𝑘) ∥ 𝑁) → ∃𝑚 ∈ ℕ0 ((𝐵↑𝑚) ∥ 𝑁 ∧ ¬ (𝐵↑(𝑚 + 1)) ∥ 𝑁))) ∧ (𝜑 ∧ ¬ (𝐵↑(𝑘 + 1)) ∥ 𝑁)) → ∃𝑚 ∈ ℕ0 ((𝐵↑𝑚) ∥ 𝑁 ∧ ¬ (𝐵↑(𝑚 + 1)) ∥ 𝑁))
79	78	exp31 364	. . . 4 ⊢ (𝑘 ∈ ℕ → (((𝜑 ∧ ¬ (𝐵↑𝑘) ∥ 𝑁) → ∃𝑚 ∈ ℕ0 ((𝐵↑𝑚) ∥ 𝑁 ∧ ¬ (𝐵↑(𝑚 + 1)) ∥ 𝑁)) → ((𝜑 ∧ ¬ (𝐵↑(𝑘 + 1)) ∥ 𝑁) → ∃𝑚 ∈ ℕ0 ((𝐵↑𝑚) ∥ 𝑁 ∧ ¬ (𝐵↑(𝑚 + 1)) ∥ 𝑁))))
80	8, 13, 18, 23, 49, 79	nnind 9329	. . 3 ⊢ (𝐴 ∈ ℕ → ((𝜑 ∧ ¬ (𝐵↑𝐴) ∥ 𝑁) → ∃𝑚 ∈ ℕ0 ((𝐵↑𝑚) ∥ 𝑁 ∧ ¬ (𝐵↑(𝑚 + 1)) ∥ 𝑁)))
81	3, 80	mpcom 36	. 2 ⊢ ((𝜑 ∧ ¬ (𝐵↑𝐴) ∥ 𝑁) → ∃𝑚 ∈ ℕ0 ((𝐵↑𝑚) ∥ 𝑁 ∧ ¬ (𝐵↑(𝑚 + 1)) ∥ 𝑁))
82	1, 81	mpdan 425	1 ⊢ (𝜑 → ∃𝑚 ∈ ℕ0 ((𝐵↑𝑚) ∥ 𝑁 ∧ ¬ (𝐵↑(𝑚 + 1)) ∥ 𝑁))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 720  DECID wdc 846   = wceq 1402   ∈ wcel 2209  ∃wrex 2529   class class class wbr 4131  (class class class)co 6088  ℂcc 8183  0cc0 8185  1c1 8186   + caddc 8188  ℕcn 9313  ℕ₀cn0 9574  ℤcz 9655  ↑cexp 10996   ∥ cdvds 12579

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302  ax-pre-mulext 8303  ax-arch 8304

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-frec 6666  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-reap 8911  df-ap 8918  df-div 9011  df-inn 9314  df-n0 9575  df-z 9656  df-uz 9938  df-q 10036  df-rp 10072  df-fl 10722  df-mod 10781  df-seqfrec 10906  df-exp 10997  df-dvds 12580

This theorem is used by:  pwbdvds  12970