Theorem lighneallem4 44950

Description: Lemma 3 for lighneal 44951. (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 11981	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℂ)
2		nnnn0 12170	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)
3	1, 2	expcld 13792	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (2↑𝑁) ∈ ℂ)
4	3	3ad2ant3 1133	. . . . . . . 8 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (2↑𝑁) ∈ ℂ)
5		1cnd 10901	. . . . . . . 8 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 1 ∈ ℂ)
6		eldifi 4057	. . . . . . . . . . 11 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℙ)
7		prmnn 16307	. . . . . . . . . . 11 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
8		nncn 11911	. . . . . . . . . . 11 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℂ)
9	6, 7, 8	3syl 18	. . . . . . . . . 10 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℂ)
10	9	3ad2ant1 1131	. . . . . . . . 9 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑃 ∈ ℂ)
11		nnnn0 12170	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℕ0)
12	11	3ad2ant2 1132	. . . . . . . . 9 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑀 ∈ ℕ0)
13	10, 12	expcld 13792	. . . . . . . 8 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑃↑𝑀) ∈ ℂ)
14	4, 5, 13	3jca 1126	. . . . . . 7 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((2↑𝑁) ∈ ℂ ∧ 1 ∈ ℂ ∧ (𝑃↑𝑀) ∈ ℂ))
15	14	adantr 480	. . . . . 6 ⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 2 ∥ 𝑀) → ((2↑𝑁) ∈ ℂ ∧ 1 ∈ ℂ ∧ (𝑃↑𝑀) ∈ ℂ))
16		subadd2 11155	. . . . . 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 480	. . . . . . . 8 ⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 2 ∥ 𝑀) → 𝑃 ∈ ℂ)
19		simpl2 1190	. . . . . . . 8 ⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 2 ∥ 𝑀) → 𝑀 ∈ ℕ)
20		simpr 484	. . . . . . . 8 ⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 2 ∥ 𝑀) → ¬ 2 ∥ 𝑀)
21	18, 19, 20	oddpwp1fsum 16029	. . . . . . 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 2740	. . . . . 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 11915	. . . . . . . . . . . . . 14 ⊢ (𝑃 ∈ ℕ → (𝑃 + 1) ∈ ℕ)
24	23	nnzd 12354	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ ℕ → (𝑃 + 1) ∈ ℤ)
25	6, 7, 24	3syl 18	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (𝑃 + 1) ∈ ℤ)
26	25	3ad2ant1 1131	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑃 + 1) ∈ ℤ)
27		fzfid 13621	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (0...(𝑀 − 1)) ∈ Fin)
28		neg1z 12286	. . . . . . . . . . . . . . 15 ⊢ -1 ∈ ℤ
29	28	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → -1 ∈ ℤ)
30		elfznn0 13278	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (0...(𝑀 − 1)) → 𝑘 ∈ ℕ0)
31		zexpcl 13725	. . . . . . . . . . . . . 14 ⊢ ((-1 ∈ ℤ ∧ 𝑘 ∈ ℕ0) → (-1↑𝑘) ∈ ℤ)
32	29, 30, 31	syl2an 595	. . . . . . . . . . . . 13 ⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ (0...(𝑀 − 1))) → (-1↑𝑘) ∈ ℤ)
33		nnz 12272	. . . . . . . . . . . . . . . 16 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℤ)
34	6, 7, 33	3syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℤ)
35	34	3ad2ant1 1131	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑃 ∈ ℤ)
36		zexpcl 13725	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ∈ ℤ ∧ 𝑘 ∈ ℕ0) → (𝑃↑𝑘) ∈ ℤ)
37	35, 30, 36	syl2an 595	. . . . . . . . . . . . 13 ⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ (0...(𝑀 − 1))) → (𝑃↑𝑘) ∈ ℤ)
38	32, 37	zmulcld 12361	. . . . . . . . . . . 12 ⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ (0...(𝑀 − 1))) → ((-1↑𝑘) · (𝑃↑𝑘)) ∈ ℤ)
39	27, 38	fsumzcl 15375	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃↑𝑘)) ∈ ℤ)
40	26, 39	jca 511	. . . . . . . . . 10 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝑃 + 1) ∈ ℤ ∧ Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃↑𝑘)) ∈ ℤ))
41	40	ad2antrr 722	. . . . . . . . 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 15916	. . . . . . . . 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 5074	. . . . . . . . . 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 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)) ∥ ((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 16325	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℙ
48		prmuz2 16329	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ≥‘2))
49	6, 48	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ (ℤ≥‘2))
50	49	3ad2ant1 1131	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑃 ∈ (ℤ≥‘2))
51	50	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) → 𝑃 ∈ (ℤ≥‘2))
52		df-ne 2943	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑀 ≠ 1 ↔ ¬ 𝑀 = 1)
53		eluz2b3 12591	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑀 ∈ (ℤ≥‘2) ↔ (𝑀 ∈ ℕ ∧ 𝑀 ≠ 1))
54	53	simplbi2 500	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑀 ∈ ℕ → (𝑀 ≠ 1 → 𝑀 ∈ (ℤ≥‘2)))
55	52, 54	syl5bir 242	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑀 ∈ ℕ → (¬ 𝑀 = 1 → 𝑀 ∈ (ℤ≥‘2)))
56	55	3ad2ant2 1132	. . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀) → ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑀 ∈ (ℤ≥‘2)))
59	58	impcom 407	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) → 𝑀 ∈ (ℤ≥‘2))
60		simprr 769	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) → ¬ 2 ∥ 𝑀)
61		lighneallem4b 44949	. . . . . . . . . . . . . . . . 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 1369	. . . . . . . . . . . . . . . 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 1133	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ0)
64	63	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) → 𝑁 ∈ ℕ0)
65		dvdsprmpweqnn 16514	. . . . . . . . . . . . . . . 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 1464	. . . . . . . . . . . . . . 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 12282	. . . . . . . . . . . . . . . . . . 19 ⊢ 2 ∈ ℤ
68	67	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) → 2 ∈ ℤ)
69		iddvdsexp 15917	. . . . . . . . . . . . . . . . . 18 ⊢ ((2 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 2 ∥ (2↑𝑛))
70	68, 69	sylan 579	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) ∧ 𝑛 ∈ ℕ) → 2 ∥ (2↑𝑛))
71		breq2 5074	. . . . . . . . . . . . . . . . . . . 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 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)) ↔ 2 ∥ (2↑𝑛)))
73		fzfid 13621	. . . . . . . . . . . . . . . . . . . . . 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 579	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (-1↑𝑘) ∈ ℤ)
76		nnnn0 12170	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℕ0)
77	76	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → 𝑃 ∈ ℕ0)
78		simpr 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
79	77, 78	nn0expcld 13889	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (𝑃↑𝑘) ∈ ℕ0)
80	79	nn0zd 12353	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (𝑃↑𝑘) ∈ ℤ)
81	75, 80	zmulcld 12361	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → ((-1↑𝑘) · (𝑃↑𝑘)) ∈ ℤ)
82	81	ex 412	. . . . . . . . . . . . . . . . . . . . . . . . . 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 1131	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑘 ∈ ℕ0 → ((-1↑𝑘) · (𝑃↑𝑘)) ∈ ℤ))
85	84	ad2antrr 722	. . . . . . . . . . . . . . . . . . . . . . 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 505	. . . . . . . . . . . . . . . . . . . . . 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 12273	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℤ)
88		m1expcl2 13732	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑘 ∈ ℤ → (-1↑𝑘) ∈ {-1, 1})
89	87, 88	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑘 ∈ ℕ0 → (-1↑𝑘) ∈ {-1, 1})
90		ovex 7288	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (-1↑𝑘) ∈ V
91	90	elpr 4581	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((-1↑𝑘) ∈ {-1, 1} ↔ ((-1↑𝑘) = -1 ∨ (-1↑𝑘) = 1))
92		n2dvdsm1 16006	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ¬ 2 ∥ -1
93		breq2 5074	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((-1↑𝑘) = -1 → (2 ∥ (-1↑𝑘) ↔ 2 ∥ -1))
94	92, 93	mtbiri 326	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((-1↑𝑘) = -1 → ¬ 2 ∥ (-1↑𝑘))
95		n2dvds1 16005	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ¬ 2 ∥ 1
96		breq2 5074	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((-1↑𝑘) = 1 → (2 ∥ (-1↑𝑘) ↔ 2 ∥ 1))
97	95, 96	mtbiri 326	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((-1↑𝑘) = 1 → ¬ 2 ∥ (-1↑𝑘))
98	94, 97	jaoi 853	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 216	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((-1↑𝑘) ∈ {-1, 1} → (𝑘 ∈ ℕ0 → ¬ 2 ∥ (-1↑𝑘)))
101	89, 100	mpcom 38	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑘 ∈ ℕ0 → ¬ 2 ∥ (-1↑𝑘))
102	101	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ0) → ¬ 2 ∥ (-1↑𝑘))
103		elnn0 12165	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑘 ∈ ℕ0 ↔ (𝑘 ∈ ℕ ∨ 𝑘 = 0))
104		oddn2prm 16441	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ¬ 2 ∥ 𝑃)
105	104	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ) → ¬ 2 ∥ 𝑃)
106		simpr 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
107		prmdvdsexp 16348	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((2 ∈ ℙ ∧ 𝑃 ∈ ℤ ∧ 𝑘 ∈ ℕ) → (2 ∥ (𝑃↑𝑘) ↔ 2 ∥ 𝑃))
108	47, 34, 106, 107	mp3an2ani 1466	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ) → (2 ∥ (𝑃↑𝑘) ↔ 2 ∥ 𝑃))
109	105, 108	mtbird 324	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ) → ¬ 2 ∥ (𝑃↑𝑘))
110	109	expcom 413	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑘 ∈ ℕ → (𝑃 ∈ (ℙ ∖ {2}) → ¬ 2 ∥ (𝑃↑𝑘)))
111		oveq2 7263	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑘 = 0 → (𝑃↑𝑘) = (𝑃↑0))
112	111	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝑘 = 0 ∧ 𝑃 ∈ (ℙ ∖ {2})) → (𝑃↑𝑘) = (𝑃↑0))
113	9	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝑘 = 0 ∧ 𝑃 ∈ (ℙ ∖ {2})) → 𝑃 ∈ ℂ)
114	113	exp0d 13786	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝑘 = 0 ∧ 𝑃 ∈ (ℙ ∖ {2})) → (𝑃↑0) = 1)
115	112, 114	eqtrd 2778	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑘 = 0 ∧ 𝑃 ∈ (ℙ ∖ {2})) → (𝑃↑𝑘) = 1)
116	115	breq2d 5082	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑘 = 0 ∧ 𝑃 ∈ (ℙ ∖ {2})) → (2 ∥ (𝑃↑𝑘) ↔ 2 ∥ 1))
117	95, 116	mtbiri 326	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑘 = 0 ∧ 𝑃 ∈ (ℙ ∖ {2})) → ¬ 2 ∥ (𝑃↑𝑘))
118	117	ex 412	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑘 = 0 → (𝑃 ∈ (ℙ ∖ {2}) → ¬ 2 ∥ (𝑃↑𝑘)))
119	110, 118	jaoi 853	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑘 ∈ ℕ ∨ 𝑘 = 0) → (𝑃 ∈ (ℙ ∖ {2}) → ¬ 2 ∥ (𝑃↑𝑘)))
120	103, 119	sylbi 216	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑘 ∈ ℕ0 → (𝑃 ∈ (ℙ ∖ {2}) → ¬ 2 ∥ (𝑃↑𝑘)))
121	120	impcom 407	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ0) → ¬ 2 ∥ (𝑃↑𝑘))
122		ioran 980	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (¬ (2 ∥ (-1↑𝑘) ∨ 2 ∥ (𝑃↑𝑘)) ↔ (¬ 2 ∥ (-1↑𝑘) ∧ ¬ 2 ∥ (𝑃↑𝑘)))
123	102, 121, 122	sylanbrc 582	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ0) → ¬ (2 ∥ (-1↑𝑘) ∨ 2 ∥ (𝑃↑𝑘)))
124	28, 31	mpan 686	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑘 ∈ ℕ0 → (-1↑𝑘) ∈ ℤ)
125	124	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ0) → (-1↑𝑘) ∈ ℤ)
126	6, 7, 76	3syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℕ0)
127	126	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ0) → 𝑃 ∈ ℕ0)
128		simpr 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
129	127, 128	nn0expcld 13889	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ0) → (𝑃↑𝑘) ∈ ℕ0)
130	129	nn0zd 12353	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ0) → (𝑃↑𝑘) ∈ ℤ)
131		euclemma 16346	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1464	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ0) → (2 ∥ ((-1↑𝑘) · (𝑃↑𝑘)) ↔ (2 ∥ (-1↑𝑘) ∨ 2 ∥ (𝑃↑𝑘))))
133	123, 132	mtbird 324	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ0) → ¬ 2 ∥ ((-1↑𝑘) · (𝑃↑𝑘)))
134	133	ex 412	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (𝑘 ∈ ℕ0 → ¬ 2 ∥ ((-1↑𝑘) · (𝑃↑𝑘))))
135	134	3ad2ant1 1131	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑘 ∈ ℕ0 → ¬ 2 ∥ ((-1↑𝑘) · (𝑃↑𝑘))))
136	135	ad2antrr 722	. . . . . . . . . . . . . . . . . . . . . . 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 505	. . . . . . . . . . . . . . . . . . . . . 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 12204	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑀 ∈ ℕ → (𝑀 − 1) ∈ ℕ0)
139		hashfz0 14075	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 11911	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℂ)
142		npcan1 11330	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑀 ∈ ℂ → ((𝑀 − 1) + 1) = 𝑀)
143	141, 142	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑀 ∈ ℕ → ((𝑀 − 1) + 1) = 𝑀)
144	140, 143	eqtr2d 2779	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑀 ∈ ℕ → 𝑀 = (♯‘(0...(𝑀 − 1))))
145	144	3ad2ant2 1132	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑀 = (♯‘(0...(𝑀 − 1))))
146	145	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 𝑀 = 1) → 𝑀 = (♯‘(0...(𝑀 − 1))))
147	146	breq2d 5082	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 𝑀 = 1) → (2 ∥ 𝑀 ↔ 2 ∥ (♯‘(0...(𝑀 − 1)))))
148	147	notbid 317	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 𝑀 = 1) → (¬ 2 ∥ 𝑀 ↔ ¬ 2 ∥ (♯‘(0...(𝑀 − 1)))))
149	148	biimpd 228	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 𝑀 = 1) → (¬ 2 ∥ 𝑀 → ¬ 2 ∥ (♯‘(0...(𝑀 − 1)))))
150	149	impr 454	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) → ¬ 2 ∥ (♯‘(0...(𝑀 − 1))))
151	150	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) ∧ 𝑛 ∈ ℕ) → ¬ 2 ∥ (♯‘(0...(𝑀 − 1))))
152	73, 86, 137, 151	oddsumodd 16027	. . . . . . . . . . . . . . . . . . . . 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 121	. . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . 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 259	. . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . . 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 3212	. . . . . . . . . . . . . . 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 420	. . . . . . . . . . . . 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 410	. . . . . . . . . . 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 182	. . . . . . . . . 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 480	. . . . . . . . 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 239	. . . . . . . 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 412	. . . . . 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 239	. . . . 5 ⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 2 ∥ 𝑀) → (((𝑃↑𝑀) + 1) = (2↑𝑁) → 𝑀 = 1))
169	17, 168	sylbid 239	. . . 4 ⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 2 ∥ 𝑀) → (((2↑𝑁) − 1) = (𝑃↑𝑀) → 𝑀 = 1))
170	169	ex 412	. . 3 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (¬ 2 ∥ 𝑀 → (((2↑𝑁) − 1) = (𝑃↑𝑀) → 𝑀 = 1)))
171	170	adantld 490	. 2 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((¬ 2 ∥ 𝑁 ∧ ¬ 2 ∥ 𝑀) → (((2↑𝑁) − 1) = (𝑃↑𝑀) → 𝑀 = 1)))
172	171	3imp 1109	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 205   ∧ wa 395   ∨ wo 843   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∃wrex 3064   ∖ cdif 3880  {csn 4558  {cpr 4560   class class class wbr 5070  ‘cfv 6418  (class class class)co 7255  ℂcc 10800  0cc0 10802  1c1 10803   + caddc 10805   · cmul 10807   − cmin 11135  -cneg 11136  ℕcn 11903  2c2 11958  ℕ₀cn0 12163  ℤcz 12249  ℤ_≥cuz 12511  ...cfz 13168  ↑cexp 13710  ♯chash 13972  Σcsu 15325   ∥ cdvds 15891  ℙcprime 16304

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-inf2 9329  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879  ax-pre-sup 10880

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-2o 8268  df-oadd 8271  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-sup 9131  df-inf 9132  df-oi 9199  df-dju 9590  df-card 9628  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-div 11563  df-nn 11904  df-2 11966  df-3 11967  df-4 11968  df-n0 12164  df-z 12250  df-uz 12512  df-q 12618  df-rp 12660  df-fz 13169  df-fzo 13312  df-fl 13440  df-mod 13518  df-seq 13650  df-exp 13711  df-hash 13973  df-cj 14738  df-re 14739  df-im 14740  df-sqrt 14874  df-abs 14875  df-clim 15125  df-sum 15326  df-dvds 15892  df-gcd 16130  df-prm 16305  df-pc 16466

This theorem is referenced by:  lighneal  44951