Theorem lighneallem4 41527

Description: Lemma 3 for lighneal 41528. (Contributed by AV, 16-Aug-2021.)

Assertion

Ref	Expression
lighneallem4	⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 2 ∥ 𝑁 ∧ ¬ 2 ∥ 𝑀) ∧ ((2↑𝑁) − 1) = (𝑃↑𝑀)) → 𝑀 = 1)

Proof of Theorem lighneallem4

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

Step	Hyp	Ref	Expression
1		2cnd 11093	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℂ)
2		nnnn0 11299	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)
3	1, 2	expcld 13008	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (2↑𝑁) ∈ ℂ)
4	3	3ad2ant3 1084	. . . . . . . 8 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (2↑𝑁) ∈ ℂ)
5		1cnd 10056	. . . . . . . 8 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 1 ∈ ℂ)
6		eldifi 3732	. . . . . . . . . . 11 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℙ)
7		prmnn 15388	. . . . . . . . . . 11 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
8		nncn 11028	. . . . . . . . . . 11 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℂ)
9	6, 7, 8	3syl 18	. . . . . . . . . 10 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℂ)
10	9	3ad2ant1 1082	. . . . . . . . 9 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑃 ∈ ℂ)
11		nnnn0 11299	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℕ0)
12	11	3ad2ant2 1083	. . . . . . . . 9 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑀 ∈ ℕ0)
13	10, 12	expcld 13008	. . . . . . . 8 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑃↑𝑀) ∈ ℂ)
14	4, 5, 13	3jca 1242	. . . . . . 7 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((2↑𝑁) ∈ ℂ ∧ 1 ∈ ℂ ∧ (𝑃↑𝑀) ∈ ℂ))
15	14	adantr 481	. . . . . 6 ⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 2 ∥ 𝑀) → ((2↑𝑁) ∈ ℂ ∧ 1 ∈ ℂ ∧ (𝑃↑𝑀) ∈ ℂ))
16		subadd2 10285	. . . . . 6 ⊢ (((2↑𝑁) ∈ ℂ ∧ 1 ∈ ℂ ∧ (𝑃↑𝑀) ∈ ℂ) → (((2↑𝑁) − 1) = (𝑃↑𝑀) ↔ ((𝑃↑𝑀) + 1) = (2↑𝑁)))
17	15, 16	syl 17	. . . . 5 ⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 2 ∥ 𝑀) → (((2↑𝑁) − 1) = (𝑃↑𝑀) ↔ ((𝑃↑𝑀) + 1) = (2↑𝑁)))
18	10	adantr 481	. . . . . . . 8 ⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 2 ∥ 𝑀) → 𝑃 ∈ ℂ)
19		simpl2 1065	. . . . . . . 8 ⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 2 ∥ 𝑀) → 𝑀 ∈ ℕ)
20		simpr 477	. . . . . . . 8 ⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 2 ∥ 𝑀) → ¬ 2 ∥ 𝑀)
21	18, 19, 20	oddpwp1fsum 15115	. . . . . . 7 ⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 2 ∥ 𝑀) → ((𝑃↑𝑀) + 1) = ((𝑃 + 1) · Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃↑𝑘))))
22	21	eqeq1d 2624	. . . . . 6 ⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 2 ∥ 𝑀) → (((𝑃↑𝑀) + 1) = (2↑𝑁) ↔ ((𝑃 + 1) · Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃↑𝑘))) = (2↑𝑁)))
23		peano2nn 11032	. . . . . . . . . . . . . 14 ⊢ (𝑃 ∈ ℕ → (𝑃 + 1) ∈ ℕ)
24	23	nnzd 11481	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ ℕ → (𝑃 + 1) ∈ ℤ)
25	6, 7, 24	3syl 18	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (𝑃 + 1) ∈ ℤ)
26	25	3ad2ant1 1082	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑃 + 1) ∈ ℤ)
27		fzfid 12772	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (0...(𝑀 − 1)) ∈ Fin)
28		neg1z 11413	. . . . . . . . . . . . . . 15 ⊢ -1 ∈ ℤ
29	28	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → -1 ∈ ℤ)
30		elfznn0 12433	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (0...(𝑀 − 1)) → 𝑘 ∈ ℕ0)
31		zexpcl 12875	. . . . . . . . . . . . . 14 ⊢ ((-1 ∈ ℤ ∧ 𝑘 ∈ ℕ0) → (-1↑𝑘) ∈ ℤ)
32	29, 30, 31	syl2an 494	. . . . . . . . . . . . 13 ⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ (0...(𝑀 − 1))) → (-1↑𝑘) ∈ ℤ)
33		nnz 11399	. . . . . . . . . . . . . . . 16 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℤ)
34	6, 7, 33	3syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℤ)
35	34	3ad2ant1 1082	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑃 ∈ ℤ)
36		zexpcl 12875	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ∈ ℤ ∧ 𝑘 ∈ ℕ0) → (𝑃↑𝑘) ∈ ℤ)
37	35, 30, 36	syl2an 494	. . . . . . . . . . . . 13 ⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ (0...(𝑀 − 1))) → (𝑃↑𝑘) ∈ ℤ)
38	32, 37	zmulcld 11488	. . . . . . . . . . . 12 ⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ (0...(𝑀 − 1))) → ((-1↑𝑘) · (𝑃↑𝑘)) ∈ ℤ)
39	27, 38	fsumzcl 14466	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃↑𝑘)) ∈ ℤ)
40	26, 39	jca 554	. . . . . . . . . 10 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝑃 + 1) ∈ ℤ ∧ Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃↑𝑘)) ∈ ℤ))
41	40	ad2antrr 762	. . . . . . . . 9 ⊢ ((((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 2 ∥ 𝑀) ∧ ((𝑃 + 1) · Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃↑𝑘))) = (2↑𝑁)) → ((𝑃 + 1) ∈ ℤ ∧ Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃↑𝑘)) ∈ ℤ))
42		dvdsmul2 15004	. . . . . . . . 9 ⊢ (((𝑃 + 1) ∈ ℤ ∧ Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃↑𝑘)) ∈ ℤ) → Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃↑𝑘)) ∥ ((𝑃 + 1) · Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃↑𝑘))))
43	41, 42	syl 17	. . . . . . . 8 ⊢ ((((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 2 ∥ 𝑀) ∧ ((𝑃 + 1) · Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃↑𝑘))) = (2↑𝑁)) → Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃↑𝑘)) ∥ ((𝑃 + 1) · Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃↑𝑘))))
44		breq2 4657	. . . . . . . . . 10 ⊢ (((𝑃 + 1) · Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃↑𝑘))) = (2↑𝑁) → (Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃↑𝑘)) ∥ ((𝑃 + 1) · Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃↑𝑘))) ↔ Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃↑𝑘)) ∥ (2↑𝑁)))
45	44	adantl 482	. . . . . . . . 9 ⊢ ((((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 2 ∥ 𝑀) ∧ ((𝑃 + 1) · Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃↑𝑘))) = (2↑𝑁)) → (Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃↑𝑘)) ∥ ((𝑃 + 1) · Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃↑𝑘))) ↔ Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃↑𝑘)) ∥ (2↑𝑁)))
46		2a1 28	. . . . . . . . . . 11 ⊢ (𝑀 = 1 → (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 2 ∥ 𝑀) → (Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃↑𝑘)) ∥ (2↑𝑁) → 𝑀 = 1)))
47		2prm 15405	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℙ
48		prmuz2 15408	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ≥‘2))
49	6, 48	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ (ℤ≥‘2))
50	49	3ad2ant1 1082	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑃 ∈ (ℤ≥‘2))
51	50	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) → 𝑃 ∈ (ℤ≥‘2))
52		df-ne 2795	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑀 ≠ 1 ↔ ¬ 𝑀 = 1)
53		eluz2b3 11762	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑀 ∈ (ℤ≥‘2) ↔ (𝑀 ∈ ℕ ∧ 𝑀 ≠ 1))
54	53	simplbi2 655	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑀 ∈ ℕ → (𝑀 ≠ 1 → 𝑀 ∈ (ℤ≥‘2)))
55	52, 54	syl5bir 233	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑀 ∈ ℕ → (¬ 𝑀 = 1 → 𝑀 ∈ (ℤ≥‘2)))
56	55	3ad2ant2 1083	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (¬ 𝑀 = 1 → 𝑀 ∈ (ℤ≥‘2)))
57	56	com12 32	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ 𝑀 = 1 → ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑀 ∈ (ℤ≥‘2)))
58	57	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀) → ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑀 ∈ (ℤ≥‘2)))
59	58	impcom 446	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) → 𝑀 ∈ (ℤ≥‘2))
60		simprr 796	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) → ¬ 2 ∥ 𝑀)
61		lighneallem4b 41526	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ (ℤ≥‘2) ∧ ¬ 2 ∥ 𝑀) → Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃↑𝑘)) ∈ (ℤ≥‘2))
62	51, 59, 60, 61	syl3anc 1326	. . . . . . . . . . . . . . . 16 ⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) → Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃↑𝑘)) ∈ (ℤ≥‘2))
63	2	3ad2ant3 1084	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ0)
64	63	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) → 𝑁 ∈ ℕ0)
65		dvdsprmpweqnn 15589	. . . . . . . . . . . . . . . 16 ⊢ ((2 ∈ ℙ ∧ Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃↑𝑘)) ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → (Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃↑𝑘)) ∥ (2↑𝑁) → ∃𝑛 ∈ ℕ Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃↑𝑘)) = (2↑𝑛)))
66	47, 62, 64, 65	mp3an2i 1429	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) → (Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃↑𝑘)) ∥ (2↑𝑁) → ∃𝑛 ∈ ℕ Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃↑𝑘)) = (2↑𝑛)))
67		2z 11409	. . . . . . . . . . . . . . . . . . 19 ⊢ 2 ∈ ℤ
68	67	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) → 2 ∈ ℤ)
69		iddvdsexp 15005	. . . . . . . . . . . . . . . . . 18 ⊢ ((2 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 2 ∥ (2↑𝑛))
70	68, 69	sylan 488	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) ∧ 𝑛 ∈ ℕ) → 2 ∥ (2↑𝑛))
71		breq2 4657	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃↑𝑘)) = (2↑𝑛) → (2 ∥ Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃↑𝑘)) ↔ 2 ∥ (2↑𝑛)))
72	71	adantl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) ∧ 𝑛 ∈ ℕ) ∧ Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃↑𝑘)) = (2↑𝑛)) → (2 ∥ Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃↑𝑘)) ↔ 2 ∥ (2↑𝑛)))
73		fzfid 12772	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) ∧ 𝑛 ∈ ℕ) → (0...(𝑀 − 1)) ∈ Fin)
74	28	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑃 ∈ ℕ → -1 ∈ ℤ)
75	74, 31	sylan 488	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (-1↑𝑘) ∈ ℤ)
76		nnnn0 11299	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℕ0)
77	76	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → 𝑃 ∈ ℕ0)
78		simpr 477	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
79	77, 78	nn0expcld 13031	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (𝑃↑𝑘) ∈ ℕ0)
80	79	nn0zd 11480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (𝑃↑𝑘) ∈ ℤ)
81	75, 80	zmulcld 11488	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → ((-1↑𝑘) · (𝑃↑𝑘)) ∈ ℤ)
82	81	ex 450	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑃 ∈ ℕ → (𝑘 ∈ ℕ0 → ((-1↑𝑘) · (𝑃↑𝑘)) ∈ ℤ))
83	6, 7, 82	3syl 18	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (𝑘 ∈ ℕ0 → ((-1↑𝑘) · (𝑃↑𝑘)) ∈ ℤ))
84	83	3ad2ant1 1082	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑘 ∈ ℕ0 → ((-1↑𝑘) · (𝑃↑𝑘)) ∈ ℤ))
85	84	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) ∧ 𝑛 ∈ ℕ) → (𝑘 ∈ ℕ0 → ((-1↑𝑘) · (𝑃↑𝑘)) ∈ ℤ))
86	85, 30	impel 485	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (0...(𝑀 − 1))) → ((-1↑𝑘) · (𝑃↑𝑘)) ∈ ℤ)
87		nn0z 11400	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℤ)
88		m1expcl2 12882	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑘 ∈ ℤ → (-1↑𝑘) ∈ {-1, 1})
89	87, 88	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑘 ∈ ℕ0 → (-1↑𝑘) ∈ {-1, 1})
90		ovex 6678	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (-1↑𝑘) ∈ V
91	90	elpr 4198	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((-1↑𝑘) ∈ {-1, 1} ↔ ((-1↑𝑘) = -1 ∨ (-1↑𝑘) = 1))
92		n2dvdsm1 15105	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ¬ 2 ∥ -1
93		breq2 4657	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((-1↑𝑘) = -1 → (2 ∥ (-1↑𝑘) ↔ 2 ∥ -1))
94	92, 93	mtbiri 317	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((-1↑𝑘) = -1 → ¬ 2 ∥ (-1↑𝑘))
95		n2dvds1 15104	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ¬ 2 ∥ 1
96		breq2 4657	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((-1↑𝑘) = 1 → (2 ∥ (-1↑𝑘) ↔ 2 ∥ 1))
97	95, 96	mtbiri 317	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((-1↑𝑘) = 1 → ¬ 2 ∥ (-1↑𝑘))
98	94, 97	jaoi 394	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((-1↑𝑘) = -1 ∨ (-1↑𝑘) = 1) → ¬ 2 ∥ (-1↑𝑘))
99	98	a1d 25	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((-1↑𝑘) = -1 ∨ (-1↑𝑘) = 1) → (𝑘 ∈ ℕ0 → ¬ 2 ∥ (-1↑𝑘)))
100	91, 99	sylbi 207	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((-1↑𝑘) ∈ {-1, 1} → (𝑘 ∈ ℕ0 → ¬ 2 ∥ (-1↑𝑘)))
101	89, 100	mpcom 38	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑘 ∈ ℕ0 → ¬ 2 ∥ (-1↑𝑘))
102	101	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ0) → ¬ 2 ∥ (-1↑𝑘))
103		elnn0 11294	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑘 ∈ ℕ0 ↔ (𝑘 ∈ ℕ ∨ 𝑘 = 0))
104		oddn2prm 15517	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ¬ 2 ∥ 𝑃)
105	104	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ) → ¬ 2 ∥ 𝑃)
106		simpr 477	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
107		prmdvdsexp 15427	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((2 ∈ ℙ ∧ 𝑃 ∈ ℤ ∧ 𝑘 ∈ ℕ) → (2 ∥ (𝑃↑𝑘) ↔ 2 ∥ 𝑃))
108	47, 34, 106, 107	mp3an2ani 1431	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ) → (2 ∥ (𝑃↑𝑘) ↔ 2 ∥ 𝑃))
109	105, 108	mtbird 315	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ) → ¬ 2 ∥ (𝑃↑𝑘))
110	109	expcom 451	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑘 ∈ ℕ → (𝑃 ∈ (ℙ ∖ {2}) → ¬ 2 ∥ (𝑃↑𝑘)))
111		oveq2 6658	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑘 = 0 → (𝑃↑𝑘) = (𝑃↑0))
112	111	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝑘 = 0 ∧ 𝑃 ∈ (ℙ ∖ {2})) → (𝑃↑𝑘) = (𝑃↑0))
113	9	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝑘 = 0 ∧ 𝑃 ∈ (ℙ ∖ {2})) → 𝑃 ∈ ℂ)
114	113	exp0d 13002	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝑘 = 0 ∧ 𝑃 ∈ (ℙ ∖ {2})) → (𝑃↑0) = 1)
115	112, 114	eqtrd 2656	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑘 = 0 ∧ 𝑃 ∈ (ℙ ∖ {2})) → (𝑃↑𝑘) = 1)
116	115	breq2d 4665	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑘 = 0 ∧ 𝑃 ∈ (ℙ ∖ {2})) → (2 ∥ (𝑃↑𝑘) ↔ 2 ∥ 1))
117	95, 116	mtbiri 317	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑘 = 0 ∧ 𝑃 ∈ (ℙ ∖ {2})) → ¬ 2 ∥ (𝑃↑𝑘))
118	117	ex 450	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑘 = 0 → (𝑃 ∈ (ℙ ∖ {2}) → ¬ 2 ∥ (𝑃↑𝑘)))
119	110, 118	jaoi 394	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑘 ∈ ℕ ∨ 𝑘 = 0) → (𝑃 ∈ (ℙ ∖ {2}) → ¬ 2 ∥ (𝑃↑𝑘)))
120	103, 119	sylbi 207	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑘 ∈ ℕ0 → (𝑃 ∈ (ℙ ∖ {2}) → ¬ 2 ∥ (𝑃↑𝑘)))
121	120	impcom 446	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ0) → ¬ 2 ∥ (𝑃↑𝑘))
122		ioran 511	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (¬ (2 ∥ (-1↑𝑘) ∨ 2 ∥ (𝑃↑𝑘)) ↔ (¬ 2 ∥ (-1↑𝑘) ∧ ¬ 2 ∥ (𝑃↑𝑘)))
123	102, 121, 122	sylanbrc 698	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ0) → ¬ (2 ∥ (-1↑𝑘) ∨ 2 ∥ (𝑃↑𝑘)))
124	28, 31	mpan 706	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑘 ∈ ℕ0 → (-1↑𝑘) ∈ ℤ)
125	124	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ0) → (-1↑𝑘) ∈ ℤ)
126	6, 7, 76	3syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℕ0)
127	126	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ0) → 𝑃 ∈ ℕ0)
128		simpr 477	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
129	127, 128	nn0expcld 13031	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ0) → (𝑃↑𝑘) ∈ ℕ0)
130	129	nn0zd 11480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ0) → (𝑃↑𝑘) ∈ ℤ)
131		euclemma 15425	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((2 ∈ ℙ ∧ (-1↑𝑘) ∈ ℤ ∧ (𝑃↑𝑘) ∈ ℤ) → (2 ∥ ((-1↑𝑘) · (𝑃↑𝑘)) ↔ (2 ∥ (-1↑𝑘) ∨ 2 ∥ (𝑃↑𝑘))))
132	47, 125, 130, 131	mp3an2i 1429	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ0) → (2 ∥ ((-1↑𝑘) · (𝑃↑𝑘)) ↔ (2 ∥ (-1↑𝑘) ∨ 2 ∥ (𝑃↑𝑘))))
133	123, 132	mtbird 315	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ0) → ¬ 2 ∥ ((-1↑𝑘) · (𝑃↑𝑘)))
134	133	ex 450	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (𝑘 ∈ ℕ0 → ¬ 2 ∥ ((-1↑𝑘) · (𝑃↑𝑘))))
135	134	3ad2ant1 1082	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑘 ∈ ℕ0 → ¬ 2 ∥ ((-1↑𝑘) · (𝑃↑𝑘))))
136	135	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) ∧ 𝑛 ∈ ℕ) → (𝑘 ∈ ℕ0 → ¬ 2 ∥ ((-1↑𝑘) · (𝑃↑𝑘))))
137	136, 30	impel 485	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (0...(𝑀 − 1))) → ¬ 2 ∥ ((-1↑𝑘) · (𝑃↑𝑘)))
138		nnm1nn0 11334	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑀 ∈ ℕ → (𝑀 − 1) ∈ ℕ0)
139		hashfz0 13219	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑀 − 1) ∈ ℕ0 → (#‘(0...(𝑀 − 1))) = ((𝑀 − 1) + 1))
140	138, 139	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑀 ∈ ℕ → (#‘(0...(𝑀 − 1))) = ((𝑀 − 1) + 1))
141		nncn 11028	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℂ)
142		npcan1 10455	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑀 ∈ ℂ → ((𝑀 − 1) + 1) = 𝑀)
143	141, 142	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑀 ∈ ℕ → ((𝑀 − 1) + 1) = 𝑀)
144	140, 143	eqtr2d 2657	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑀 ∈ ℕ → 𝑀 = (#‘(0...(𝑀 − 1))))
145	144	3ad2ant2 1083	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑀 = (#‘(0...(𝑀 − 1))))
146	145	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 𝑀 = 1) → 𝑀 = (#‘(0...(𝑀 − 1))))
147	146	breq2d 4665	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 𝑀 = 1) → (2 ∥ 𝑀 ↔ 2 ∥ (#‘(0...(𝑀 − 1)))))
148	147	notbid 308	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 𝑀 = 1) → (¬ 2 ∥ 𝑀 ↔ ¬ 2 ∥ (#‘(0...(𝑀 − 1)))))
149	148	biimpd 219	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 𝑀 = 1) → (¬ 2 ∥ 𝑀 → ¬ 2 ∥ (#‘(0...(𝑀 − 1)))))
150	149	impr 649	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) → ¬ 2 ∥ (#‘(0...(𝑀 − 1))))
151	150	adantr 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) ∧ 𝑛 ∈ ℕ) → ¬ 2 ∥ (#‘(0...(𝑀 − 1))))
152	73, 86, 137, 151	oddsumodd 15113	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) ∧ 𝑛 ∈ ℕ) → ¬ 2 ∥ Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃↑𝑘)))
153	152	pm2.21d 118	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) ∧ 𝑛 ∈ ℕ) → (2 ∥ Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃↑𝑘)) → 𝑀 = 1))
154	153	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) ∧ 𝑛 ∈ ℕ) ∧ Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃↑𝑘)) = (2↑𝑛)) → (2 ∥ Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃↑𝑘)) → 𝑀 = 1))
155	72, 154	sylbird 250	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) ∧ 𝑛 ∈ ℕ) ∧ Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃↑𝑘)) = (2↑𝑛)) → (2 ∥ (2↑𝑛) → 𝑀 = 1))
156	155	ex 450	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) ∧ 𝑛 ∈ ℕ) → (Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃↑𝑘)) = (2↑𝑛) → (2 ∥ (2↑𝑛) → 𝑀 = 1)))
157	70, 156	mpid 44	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) ∧ 𝑛 ∈ ℕ) → (Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃↑𝑘)) = (2↑𝑛) → 𝑀 = 1))
158	157	rexlimdva 3031	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) → (∃𝑛 ∈ ℕ Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃↑𝑘)) = (2↑𝑛) → 𝑀 = 1))
159	66, 158	syld 47	. . . . . . . . . . . . . 14 ⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) → (Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃↑𝑘)) ∥ (2↑𝑁) → 𝑀 = 1))
160	159	exp32 631	. . . . . . . . . . . . 13 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (¬ 𝑀 = 1 → (¬ 2 ∥ 𝑀 → (Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃↑𝑘)) ∥ (2↑𝑁) → 𝑀 = 1))))
161	160	com12 32	. . . . . . . . . . . 12 ⊢ (¬ 𝑀 = 1 → ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (¬ 2 ∥ 𝑀 → (Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃↑𝑘)) ∥ (2↑𝑁) → 𝑀 = 1))))
162	161	impd 447	. . . . . . . . . . 11 ⊢ (¬ 𝑀 = 1 → (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 2 ∥ 𝑀) → (Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃↑𝑘)) ∥ (2↑𝑁) → 𝑀 = 1)))
163	46, 162	pm2.61i 176	. . . . . . . . . 10 ⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 2 ∥ 𝑀) → (Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃↑𝑘)) ∥ (2↑𝑁) → 𝑀 = 1))
164	163	adantr 481	. . . . . . . . 9 ⊢ ((((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 2 ∥ 𝑀) ∧ ((𝑃 + 1) · Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃↑𝑘))) = (2↑𝑁)) → (Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃↑𝑘)) ∥ (2↑𝑁) → 𝑀 = 1))
165	45, 164	sylbid 230	. . . . . . . 8 ⊢ ((((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 2 ∥ 𝑀) ∧ ((𝑃 + 1) · Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃↑𝑘))) = (2↑𝑁)) → (Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃↑𝑘)) ∥ ((𝑃 + 1) · Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃↑𝑘))) → 𝑀 = 1))
166	43, 165	mpd 15	. . . . . . 7 ⊢ ((((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 2 ∥ 𝑀) ∧ ((𝑃 + 1) · Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃↑𝑘))) = (2↑𝑁)) → 𝑀 = 1)
167	166	ex 450	. . . . . 6 ⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 2 ∥ 𝑀) → (((𝑃 + 1) · Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃↑𝑘))) = (2↑𝑁) → 𝑀 = 1))
168	22, 167	sylbid 230	. . . . 5 ⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 2 ∥ 𝑀) → (((𝑃↑𝑀) + 1) = (2↑𝑁) → 𝑀 = 1))
169	17, 168	sylbid 230	. . . 4 ⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 2 ∥ 𝑀) → (((2↑𝑁) − 1) = (𝑃↑𝑀) → 𝑀 = 1))
170	169	ex 450	. . 3 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (¬ 2 ∥ 𝑀 → (((2↑𝑁) − 1) = (𝑃↑𝑀) → 𝑀 = 1)))
171	170	adantld 483	. 2 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((¬ 2 ∥ 𝑁 ∧ ¬ 2 ∥ 𝑀) → (((2↑𝑁) − 1) = (𝑃↑𝑀) → 𝑀 = 1)))
172	171	3imp 1256	1 ⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 2 ∥ 𝑁 ∧ ¬ 2 ∥ 𝑀) ∧ ((2↑𝑁) − 1) = (𝑃↑𝑀)) → 𝑀 = 1)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∃wrex 2913   ∖ cdif 3571  {csn 4177  {cpr 4179   class class class wbr 4653  ‘cfv 5888  (class class class)co 6650  ℂcc 9934  0cc0 9936  1c1 9937   + caddc 9939   · cmul 9941   − cmin 10266  -cneg 10267  ℕcn 11020  2c2 11070  ℕ₀cn0 11292  ℤcz 11377  ℤ_≥cuz 11687  ...cfz 12326  ↑cexp 12860  #chash 13117  Σcsu 14416   ∥ cdvds 14983  ℙcprime 15385

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-dvds 14984  df-gcd 15217  df-prm 15386  df-pc 15542

This theorem is referenced by:  lighneal  41528