Theorem lighneallem4 48091

Description: Lemma 3 for lighneal 48092. (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 12254	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℂ)
2		nnnn0 12439	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)
3	1, 2	expcld 14103	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (2↑𝑁) ∈ ℂ)
4	3	3ad2ant3 1136	. . . . . . . 8 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (2↑𝑁) ∈ ℂ)
5		1cnd 11134	. . . . . . . 8 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 1 ∈ ℂ)
6		eldifi 4072	. . . . . . . . . . 11 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℙ)
7		prmnn 16638	. . . . . . . . . . 11 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
8		nncn 12177	. . . . . . . . . . 11 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℂ)
9	6, 7, 8	3syl 18	. . . . . . . . . 10 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℂ)
10	9	3ad2ant1 1134	. . . . . . . . 9 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑃 ∈ ℂ)
11		nnnn0 12439	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℕ0)
12	11	3ad2ant2 1135	. . . . . . . . 9 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑀 ∈ ℕ0)
13	10, 12	expcld 14103	. . . . . . . 8 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑃↑𝑀) ∈ ℂ)
14	4, 5, 13	3jca 1129	. . . . . . 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 11392	. . . . . 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 1194	. . . . . . . 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 16356	. . . . . . 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 2739	. . . . . 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 12181	. . . . . . . . . . . . . 14 ⊢ (𝑃 ∈ ℕ → (𝑃 + 1) ∈ ℕ)
24	23	nnzd 12545	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ ℕ → (𝑃 + 1) ∈ ℤ)
25	6, 7, 24	3syl 18	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (𝑃 + 1) ∈ ℤ)
26	25	3ad2ant1 1134	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑃 + 1) ∈ ℤ)
27		fzfid 13930	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (0...(𝑀 − 1)) ∈ Fin)
28		neg1z 12558	. . . . . . . . . . . . . . 15 ⊢ -1 ∈ ℤ
29	28	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → -1 ∈ ℤ)
30		elfznn0 13569	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (0...(𝑀 − 1)) → 𝑘 ∈ ℕ0)
31		zexpcl 14033	. . . . . . . . . . . . . 14 ⊢ ((-1 ∈ ℤ ∧ 𝑘 ∈ ℕ0) → (-1↑𝑘) ∈ ℤ)
32	29, 30, 31	syl2an 597	. . . . . . . . . . . . 13 ⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ (0...(𝑀 − 1))) → (-1↑𝑘) ∈ ℤ)
33		nnz 12540	. . . . . . . . . . . . . . . 16 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℤ)
34	6, 7, 33	3syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℤ)
35	34	3ad2ant1 1134	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑃 ∈ ℤ)
36		zexpcl 14033	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ∈ ℤ ∧ 𝑘 ∈ ℕ0) → (𝑃↑𝑘) ∈ ℤ)
37	35, 30, 36	syl2an 597	. . . . . . . . . . . . 13 ⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ (0...(𝑀 − 1))) → (𝑃↑𝑘) ∈ ℤ)
38	32, 37	zmulcld 12634	. . . . . . . . . . . 12 ⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ (0...(𝑀 − 1))) → ((-1↑𝑘) · (𝑃↑𝑘)) ∈ ℤ)
39	27, 38	fsumzcl 15692	. . . . . . . . . . 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 727	. . . . . . . . 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 16242	. . . . . . . . 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 5090	. . . . . . . . . 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 16656	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℙ
48		prmuz2 16660	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ≥‘2))
49	6, 48	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ (ℤ≥‘2))
50	49	3ad2ant1 1134	. . . . . . . . . . . . . . . . . 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 2934	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑀 ≠ 1 ↔ ¬ 𝑀 = 1)
53		eluz2b3 12867	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑀 ∈ (ℤ≥‘2) ↔ (𝑀 ∈ ℕ ∧ 𝑀 ≠ 1))
54	53	simplbi2 500	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑀 ∈ ℕ → (𝑀 ≠ 1 → 𝑀 ∈ (ℤ≥‘2)))
55	52, 54	biimtrrid 243	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑀 ∈ ℕ → (¬ 𝑀 = 1 → 𝑀 ∈ (ℤ≥‘2)))
56	55	3ad2ant2 1135	. . . . . . . . . . . . . . . . . . . 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 773	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) → ¬ 2 ∥ 𝑀)
61		lighneallem4b 48090	. . . . . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . . . . 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 1136	. . . . . . . . . . . . . . . . 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 16851	. . . . . . . . . . . . . . . 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 1469	. . . . . . . . . . . . . . 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 12554	. . . . . . . . . . . . . . . . . . 19 ⊢ 2 ∈ ℤ
68	67	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) → 2 ∈ ℤ)
69		iddvdsexp 16243	. . . . . . . . . . . . . . . . . 18 ⊢ ((2 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 2 ∥ (2↑𝑛))
70	68, 69	sylan 581	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) ∧ 𝑛 ∈ ℕ) → 2 ∥ (2↑𝑛))
71		breq2 5090	. . . . . . . . . . . . . . . . . . . 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 13930	. . . . . . . . . . . . . . . . . . . . . 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 581	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (-1↑𝑘) ∈ ℤ)
76		nnnn0 12439	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 14203	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (𝑃↑𝑘) ∈ ℕ0)
80	79	nn0zd 12544	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (𝑃↑𝑘) ∈ ℤ)
81	75, 80	zmulcld 12634	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 1134	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑘 ∈ ℕ0 → ((-1↑𝑘) · (𝑃↑𝑘)) ∈ ℤ))
85	84	ad2antrr 727	. . . . . . . . . . . . . . . . . . . . . . 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 12543	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℤ)
88		m1expcl2 14042	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑘 ∈ ℤ → (-1↑𝑘) ∈ {-1, 1})
89	87, 88	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑘 ∈ ℕ0 → (-1↑𝑘) ∈ {-1, 1})
90		ovex 7395	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (-1↑𝑘) ∈ V
91	90	elpr 4593	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((-1↑𝑘) ∈ {-1, 1} ↔ ((-1↑𝑘) = -1 ∨ (-1↑𝑘) = 1))
92		n2dvdsm1 16333	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ¬ 2 ∥ -1
93		breq2 5090	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((-1↑𝑘) = -1 → (2 ∥ (-1↑𝑘) ↔ 2 ∥ -1))
94	92, 93	mtbiri 327	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((-1↑𝑘) = -1 → ¬ 2 ∥ (-1↑𝑘))
95		n2dvds1 16332	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ¬ 2 ∥ 1
96		breq2 5090	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((-1↑𝑘) = 1 → (2 ∥ (-1↑𝑘) ↔ 2 ∥ 1))
97	95, 96	mtbiri 327	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((-1↑𝑘) = 1 → ¬ 2 ∥ (-1↑𝑘))
98	94, 97	jaoi 858	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 217	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 12434	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑘 ∈ ℕ0 ↔ (𝑘 ∈ ℕ ∨ 𝑘 = 0))
104		oddn2prm 16778	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 16680	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((2 ∈ ℙ ∧ 𝑃 ∈ ℤ ∧ 𝑘 ∈ ℕ) → (2 ∥ (𝑃↑𝑘) ↔ 2 ∥ 𝑃))
108	47, 34, 106, 107	mp3an2ani 1471	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ) → (2 ∥ (𝑃↑𝑘) ↔ 2 ∥ 𝑃))
109	105, 108	mtbird 325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ) → ¬ 2 ∥ (𝑃↑𝑘))
110	109	expcom 413	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑘 ∈ ℕ → (𝑃 ∈ (ℙ ∖ {2}) → ¬ 2 ∥ (𝑃↑𝑘)))
111		oveq2 7370	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 14097	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝑘 = 0 ∧ 𝑃 ∈ (ℙ ∖ {2})) → (𝑃↑0) = 1)
115	112, 114	eqtrd 2772	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑘 = 0 ∧ 𝑃 ∈ (ℙ ∖ {2})) → (𝑃↑𝑘) = 1)
116	115	breq2d 5098	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑘 = 0 ∧ 𝑃 ∈ (ℙ ∖ {2})) → (2 ∥ (𝑃↑𝑘) ↔ 2 ∥ 1))
117	95, 116	mtbiri 327	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 858	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑘 ∈ ℕ ∨ 𝑘 = 0) → (𝑃 ∈ (ℙ ∖ {2}) → ¬ 2 ∥ (𝑃↑𝑘)))
120	103, 119	sylbi 217	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑘 ∈ ℕ0 → (𝑃 ∈ (ℙ ∖ {2}) → ¬ 2 ∥ (𝑃↑𝑘)))
121	120	impcom 407	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ0) → ¬ 2 ∥ (𝑃↑𝑘))
122		ioran 986	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (¬ (2 ∥ (-1↑𝑘) ∨ 2 ∥ (𝑃↑𝑘)) ↔ (¬ 2 ∥ (-1↑𝑘) ∧ ¬ 2 ∥ (𝑃↑𝑘)))
123	102, 121, 122	sylanbrc 584	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ0) → ¬ (2 ∥ (-1↑𝑘) ∨ 2 ∥ (𝑃↑𝑘)))
124	28, 31	mpan 691	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 14203	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ0) → (𝑃↑𝑘) ∈ ℕ0)
130	129	nn0zd 12544	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ0) → (𝑃↑𝑘) ∈ ℤ)
131		euclemma 16678	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1469	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ0) → (2 ∥ ((-1↑𝑘) · (𝑃↑𝑘)) ↔ (2 ∥ (-1↑𝑘) ∨ 2 ∥ (𝑃↑𝑘))))
133	123, 132	mtbird 325	. . . . . . . . . . . . . . . . . . . . . . . . . 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 1134	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑘 ∈ ℕ0 → ¬ 2 ∥ ((-1↑𝑘) · (𝑃↑𝑘))))
136	135	ad2antrr 727	. . . . . . . . . . . . . . . . . . . . . . 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 12473	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑀 ∈ ℕ → (𝑀 − 1) ∈ ℕ0)
139		hashfz0 14389	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 12177	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℂ)
142		npcan1 11570	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑀 ∈ ℂ → ((𝑀 − 1) + 1) = 𝑀)
143	141, 142	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑀 ∈ ℕ → ((𝑀 − 1) + 1) = 𝑀)
144	140, 143	eqtr2d 2773	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑀 ∈ ℕ → 𝑀 = (♯‘(0...(𝑀 − 1))))
145	144	3ad2ant2 1135	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5098	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 𝑀 = 1) → (2 ∥ 𝑀 ↔ 2 ∥ (♯‘(0...(𝑀 − 1)))))
148	147	notbid 318	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 𝑀 = 1) → (¬ 2 ∥ 𝑀 ↔ ¬ 2 ∥ (♯‘(0...(𝑀 − 1)))))
149	148	biimpd 229	. . . . . . . . . . . . . . . . . . . . . . . 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 16354	. . . . . . . . . . . . . . . . . . . . 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 260	. . . . . . . . . . . . . . . . . 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 3139	. . . . . . . . . . . . . . 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 240	. . . . . . . 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 240	. . . . 5 ⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 2 ∥ 𝑀) → (((𝑃↑𝑀) + 1) = (2↑𝑁) → 𝑀 = 1))
169	17, 168	sylbid 240	. . . 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 1111	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 206   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∃wrex 3062   ∖ cdif 3887  {csn 4568  {cpr 4570   class class class wbr 5086  ‘cfv 6494  (class class class)co 7362  ℂcc 11031  0cc0 11033  1c1 11034   + caddc 11036   · cmul 11038   − cmin 11372  -cneg 11373  ℕcn 12169  2c2 12231  ℕ₀cn0 12432  ℤcz 12519  ℤ_≥cuz 12783  ...cfz 13456  ↑cexp 14018  ♯chash 14287  Σcsu 15643   ∥ cdvds 16216  ℙcprime 16635

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5304  ax-pr 5372  ax-un 7684  ax-inf2 9557  ax-cnex 11089  ax-resscn 11090  ax-1cn 11091  ax-icn 11092  ax-addcl 11093  ax-addrcl 11094  ax-mulcl 11095  ax-mulrcl 11096  ax-mulcom 11097  ax-addass 11098  ax-mulass 11099  ax-distr 11100  ax-i2m1 11101  ax-1ne0 11102  ax-1rid 11103  ax-rnegex 11104  ax-rrecex 11105  ax-cnre 11106  ax-pre-lttri 11107  ax-pre-lttrn 11108  ax-pre-ltadd 11109  ax-pre-mulgt0 11110  ax-pre-sup 11111

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5521  df-eprel 5526  df-po 5534  df-so 5535  df-fr 5579  df-se 5580  df-we 5581  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-pred 6261  df-ord 6322  df-on 6323  df-lim 6324  df-suc 6325  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-isom 6503  df-riota 7319  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7813  df-1st 7937  df-2nd 7938  df-frecs 8226  df-wrecs 8257  df-recs 8306  df-rdg 8344  df-1o 8400  df-2o 8401  df-oadd 8404  df-er 8638  df-en 8889  df-dom 8890  df-sdom 8891  df-fin 8892  df-sup 9350  df-inf 9351  df-oi 9420  df-dju 9820  df-card 9858  df-pnf 11176  df-mnf 11177  df-xr 11178  df-ltxr 11179  df-le 11180  df-sub 11374  df-neg 11375  df-div 11803  df-nn 12170  df-2 12239  df-3 12240  df-4 12241  df-n0 12433  df-z 12520  df-uz 12784  df-q 12894  df-rp 12938  df-fz 13457  df-fzo 13604  df-fl 13746  df-mod 13824  df-seq 13959  df-exp 14019  df-hash 14288  df-cj 15056  df-re 15057  df-im 15058  df-sqrt 15192  df-abs 15193  df-clim 15445  df-sum 15644  df-dvds 16217  df-gcd 16459  df-prm 16636  df-pc 16803

This theorem is referenced by:  lighneal  48092