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Theorem lighneallem4 41292
Description: Lemma 3 for lighneal 41293. (Contributed by AV, 16-Aug-2021.)
Assertion
Ref Expression
lighneallem4 (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 2 ∥ 𝑁 ∧ ¬ 2 ∥ 𝑀) ∧ ((2↑𝑁) − 1) = (𝑃𝑀)) → 𝑀 = 1)

Proof of Theorem lighneallem4
Dummy variables 𝑘 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 2cnd 11078 . . . . . . . . . 10 (𝑁 ∈ ℕ → 2 ∈ ℂ)
2 nnnn0 11284 . . . . . . . . . 10 (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)
31, 2expcld 12991 . . . . . . . . 9 (𝑁 ∈ ℕ → (2↑𝑁) ∈ ℂ)
433ad2ant3 1082 . . . . . . . 8 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (2↑𝑁) ∈ ℂ)
5 1cnd 10041 . . . . . . . 8 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 1 ∈ ℂ)
6 eldifi 3724 . . . . . . . . . . 11 (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℙ)
7 prmnn 15369 . . . . . . . . . . 11 (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
8 nncn 11013 . . . . . . . . . . 11 (𝑃 ∈ ℕ → 𝑃 ∈ ℂ)
96, 7, 83syl 18 . . . . . . . . . 10 (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℂ)
1093ad2ant1 1080 . . . . . . . . 9 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑃 ∈ ℂ)
11 nnnn0 11284 . . . . . . . . . 10 (𝑀 ∈ ℕ → 𝑀 ∈ ℕ0)
12113ad2ant2 1081 . . . . . . . . 9 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑀 ∈ ℕ0)
1310, 12expcld 12991 . . . . . . . 8 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑃𝑀) ∈ ℂ)
144, 5, 133jca 1240 . . . . . . 7 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((2↑𝑁) ∈ ℂ ∧ 1 ∈ ℂ ∧ (𝑃𝑀) ∈ ℂ))
1514adantr 481 . . . . . 6 (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 2 ∥ 𝑀) → ((2↑𝑁) ∈ ℂ ∧ 1 ∈ ℂ ∧ (𝑃𝑀) ∈ ℂ))
16 subadd2 10270 . . . . . 6 (((2↑𝑁) ∈ ℂ ∧ 1 ∈ ℂ ∧ (𝑃𝑀) ∈ ℂ) → (((2↑𝑁) − 1) = (𝑃𝑀) ↔ ((𝑃𝑀) + 1) = (2↑𝑁)))
1715, 16syl 17 . . . . 5 (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 2 ∥ 𝑀) → (((2↑𝑁) − 1) = (𝑃𝑀) ↔ ((𝑃𝑀) + 1) = (2↑𝑁)))
1810adantr 481 . . . . . . . 8 (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 2 ∥ 𝑀) → 𝑃 ∈ ℂ)
19 simpl2 1063 . . . . . . . 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)
2118, 19, 20oddpwp1fsum 15096 . . . . . . 7 (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 2 ∥ 𝑀) → ((𝑃𝑀) + 1) = ((𝑃 + 1) · Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃𝑘))))
2221eqeq1d 2622 . . . . . 6 (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 2 ∥ 𝑀) → (((𝑃𝑀) + 1) = (2↑𝑁) ↔ ((𝑃 + 1) · Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃𝑘))) = (2↑𝑁)))
23 peano2nn 11017 . . . . . . . . . . . . . 14 (𝑃 ∈ ℕ → (𝑃 + 1) ∈ ℕ)
2423nnzd 11466 . . . . . . . . . . . . 13 (𝑃 ∈ ℕ → (𝑃 + 1) ∈ ℤ)
256, 7, 243syl 18 . . . . . . . . . . . 12 (𝑃 ∈ (ℙ ∖ {2}) → (𝑃 + 1) ∈ ℤ)
26253ad2ant1 1080 . . . . . . . . . . 11 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑃 + 1) ∈ ℤ)
27 fzfid 12755 . . . . . . . . . . . 12 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (0...(𝑀 − 1)) ∈ Fin)
28 neg1z 11398 . . . . . . . . . . . . . . 15 -1 ∈ ℤ
2928a1i 11 . . . . . . . . . . . . . 14 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → -1 ∈ ℤ)
30 elfznn0 12417 . . . . . . . . . . . . . 14 (𝑘 ∈ (0...(𝑀 − 1)) → 𝑘 ∈ ℕ0)
31 zexpcl 12858 . . . . . . . . . . . . . 14 ((-1 ∈ ℤ ∧ 𝑘 ∈ ℕ0) → (-1↑𝑘) ∈ ℤ)
3229, 30, 31syl2an 494 . . . . . . . . . . . . 13 (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ (0...(𝑀 − 1))) → (-1↑𝑘) ∈ ℤ)
33 nnz 11384 . . . . . . . . . . . . . . . 16 (𝑃 ∈ ℕ → 𝑃 ∈ ℤ)
346, 7, 333syl 18 . . . . . . . . . . . . . . 15 (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℤ)
35343ad2ant1 1080 . . . . . . . . . . . . . 14 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑃 ∈ ℤ)
36 zexpcl 12858 . . . . . . . . . . . . . 14 ((𝑃 ∈ ℤ ∧ 𝑘 ∈ ℕ0) → (𝑃𝑘) ∈ ℤ)
3735, 30, 36syl2an 494 . . . . . . . . . . . . 13 (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ (0...(𝑀 − 1))) → (𝑃𝑘) ∈ ℤ)
3832, 37zmulcld 11473 . . . . . . . . . . . 12 (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ (0...(𝑀 − 1))) → ((-1↑𝑘) · (𝑃𝑘)) ∈ ℤ)
3927, 38fsumzcl 14447 . . . . . . . . . . 11 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃𝑘)) ∈ ℤ)
4026, 39jca 554 . . . . . . . . . 10 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝑃 + 1) ∈ ℤ ∧ Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃𝑘)) ∈ ℤ))
4140ad2antrr 761 . . . . . . . . 9 ((((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 2 ∥ 𝑀) ∧ ((𝑃 + 1) · Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃𝑘))) = (2↑𝑁)) → ((𝑃 + 1) ∈ ℤ ∧ Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃𝑘)) ∈ ℤ))
42 dvdsmul2 14985 . . . . . . . . 9 (((𝑃 + 1) ∈ ℤ ∧ Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃𝑘)) ∈ ℤ) → Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃𝑘)) ∥ ((𝑃 + 1) · Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃𝑘))))
4341, 42syl 17 . . . . . . . 8 ((((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 2 ∥ 𝑀) ∧ ((𝑃 + 1) · Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃𝑘))) = (2↑𝑁)) → Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃𝑘)) ∥ ((𝑃 + 1) · Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃𝑘))))
44 breq2 4648 . . . . . . . . . 10 (((𝑃 + 1) · Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃𝑘))) = (2↑𝑁) → (Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃𝑘)) ∥ ((𝑃 + 1) · Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃𝑘))) ↔ Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃𝑘)) ∥ (2↑𝑁)))
4544adantl 482 . . . . . . . . 9 ((((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 2 ∥ 𝑀) ∧ ((𝑃 + 1) · Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃𝑘))) = (2↑𝑁)) → (Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃𝑘)) ∥ ((𝑃 + 1) · Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃𝑘))) ↔ Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃𝑘)) ∥ (2↑𝑁)))
46 2a1 28 . . . . . . . . . . 11 (𝑀 = 1 → (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 2 ∥ 𝑀) → (Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃𝑘)) ∥ (2↑𝑁) → 𝑀 = 1)))
47 2prm 15386 . . . . . . . . . . . . . . . 16 2 ∈ ℙ
48 prmuz2 15389 . . . . . . . . . . . . . . . . . . . 20 (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ‘2))
496, 48syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ (ℤ‘2))
50493ad2ant1 1080 . . . . . . . . . . . . . . . . . 18 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑃 ∈ (ℤ‘2))
5150adantr 481 . . . . . . . . . . . . . . . . 17 (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) → 𝑃 ∈ (ℤ‘2))
52 df-ne 2792 . . . . . . . . . . . . . . . . . . . . . 22 (𝑀 ≠ 1 ↔ ¬ 𝑀 = 1)
53 eluz2b3 11747 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑀 ∈ (ℤ‘2) ↔ (𝑀 ∈ ℕ ∧ 𝑀 ≠ 1))
5453simplbi2 654 . . . . . . . . . . . . . . . . . . . . . 22 (𝑀 ∈ ℕ → (𝑀 ≠ 1 → 𝑀 ∈ (ℤ‘2)))
5552, 54syl5bir 233 . . . . . . . . . . . . . . . . . . . . 21 (𝑀 ∈ ℕ → (¬ 𝑀 = 1 → 𝑀 ∈ (ℤ‘2)))
56553ad2ant2 1081 . . . . . . . . . . . . . . . . . . . 20 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (¬ 𝑀 = 1 → 𝑀 ∈ (ℤ‘2)))
5756com12 32 . . . . . . . . . . . . . . . . . . 19 𝑀 = 1 → ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑀 ∈ (ℤ‘2)))
5857adantr 481 . . . . . . . . . . . . . . . . . 18 ((¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀) → ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑀 ∈ (ℤ‘2)))
5958impcom 446 . . . . . . . . . . . . . . . . 17 (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) → 𝑀 ∈ (ℤ‘2))
60 simprr 795 . . . . . . . . . . . . . . . . 17 (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) → ¬ 2 ∥ 𝑀)
61 lighneallem4b 41291 . . . . . . . . . . . . . . . . 17 ((𝑃 ∈ (ℤ‘2) ∧ 𝑀 ∈ (ℤ‘2) ∧ ¬ 2 ∥ 𝑀) → Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃𝑘)) ∈ (ℤ‘2))
6251, 59, 60, 61syl3anc 1324 . . . . . . . . . . . . . . . 16 (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) → Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃𝑘)) ∈ (ℤ‘2))
6323ad2ant3 1082 . . . . . . . . . . . . . . . . 17 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ0)
6463adantr 481 . . . . . . . . . . . . . . . 16 (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) → 𝑁 ∈ ℕ0)
65 dvdsprmpweqnn 15570 . . . . . . . . . . . . . . . 16 ((2 ∈ ℙ ∧ Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃𝑘)) ∈ (ℤ‘2) ∧ 𝑁 ∈ ℕ0) → (Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃𝑘)) ∥ (2↑𝑁) → ∃𝑛 ∈ ℕ Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃𝑘)) = (2↑𝑛)))
6647, 62, 64, 65mp3an2i 1427 . . . . . . . . . . . . . . 15 (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) → (Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃𝑘)) ∥ (2↑𝑁) → ∃𝑛 ∈ ℕ Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃𝑘)) = (2↑𝑛)))
67 2z 11394 . . . . . . . . . . . . . . . . . . 19 2 ∈ ℤ
6867a1i 11 . . . . . . . . . . . . . . . . . 18 (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) → 2 ∈ ℤ)
69 iddvdsexp 14986 . . . . . . . . . . . . . . . . . 18 ((2 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 2 ∥ (2↑𝑛))
7068, 69sylan 488 . . . . . . . . . . . . . . . . 17 ((((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) ∧ 𝑛 ∈ ℕ) → 2 ∥ (2↑𝑛))
71 breq2 4648 . . . . . . . . . . . . . . . . . . . 20 𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃𝑘)) = (2↑𝑛) → (2 ∥ Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃𝑘)) ↔ 2 ∥ (2↑𝑛)))
7271adantl 482 . . . . . . . . . . . . . . . . . . 19 (((((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) ∧ 𝑛 ∈ ℕ) ∧ Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃𝑘)) = (2↑𝑛)) → (2 ∥ Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃𝑘)) ↔ 2 ∥ (2↑𝑛)))
73 fzfid 12755 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) ∧ 𝑛 ∈ ℕ) → (0...(𝑀 − 1)) ∈ Fin)
7428a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑃 ∈ ℕ → -1 ∈ ℤ)
7574, 31sylan 488 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (-1↑𝑘) ∈ ℤ)
76 nnnn0 11284 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑃 ∈ ℕ → 𝑃 ∈ ℕ0)
7776adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → 𝑃 ∈ ℕ0)
78 simpr 477 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
7977, 78nn0expcld 13014 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (𝑃𝑘) ∈ ℕ0)
8079nn0zd 11465 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (𝑃𝑘) ∈ ℤ)
8175, 80zmulcld 11473 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → ((-1↑𝑘) · (𝑃𝑘)) ∈ ℤ)
8281ex 450 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑃 ∈ ℕ → (𝑘 ∈ ℕ0 → ((-1↑𝑘) · (𝑃𝑘)) ∈ ℤ))
836, 7, 823syl 18 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑃 ∈ (ℙ ∖ {2}) → (𝑘 ∈ ℕ0 → ((-1↑𝑘) · (𝑃𝑘)) ∈ ℤ))
84833ad2ant1 1080 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑘 ∈ ℕ0 → ((-1↑𝑘) · (𝑃𝑘)) ∈ ℤ))
8584ad2antrr 761 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) ∧ 𝑛 ∈ ℕ) → (𝑘 ∈ ℕ0 → ((-1↑𝑘) · (𝑃𝑘)) ∈ ℤ))
8685, 30impel 485 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (0...(𝑀 − 1))) → ((-1↑𝑘) · (𝑃𝑘)) ∈ ℤ)
87 nn0z 11385 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑘 ∈ ℕ0𝑘 ∈ ℤ)
88 m1expcl2 12865 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑘 ∈ ℤ → (-1↑𝑘) ∈ {-1, 1})
8987, 88syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑘 ∈ ℕ0 → (-1↑𝑘) ∈ {-1, 1})
90 ovex 6663 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (-1↑𝑘) ∈ V
9190elpr 4189 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((-1↑𝑘) ∈ {-1, 1} ↔ ((-1↑𝑘) = -1 ∨ (-1↑𝑘) = 1))
92 n2dvdsm1 15086 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ¬ 2 ∥ -1
93 breq2 4648 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((-1↑𝑘) = -1 → (2 ∥ (-1↑𝑘) ↔ 2 ∥ -1))
9492, 93mtbiri 317 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((-1↑𝑘) = -1 → ¬ 2 ∥ (-1↑𝑘))
95 n2dvds1 15085 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ¬ 2 ∥ 1
96 breq2 4648 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((-1↑𝑘) = 1 → (2 ∥ (-1↑𝑘) ↔ 2 ∥ 1))
9795, 96mtbiri 317 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((-1↑𝑘) = 1 → ¬ 2 ∥ (-1↑𝑘))
9894, 97jaoi 394 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((-1↑𝑘) = -1 ∨ (-1↑𝑘) = 1) → ¬ 2 ∥ (-1↑𝑘))
9998a1d 25 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((-1↑𝑘) = -1 ∨ (-1↑𝑘) = 1) → (𝑘 ∈ ℕ0 → ¬ 2 ∥ (-1↑𝑘)))
10091, 99sylbi 207 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((-1↑𝑘) ∈ {-1, 1} → (𝑘 ∈ ℕ0 → ¬ 2 ∥ (-1↑𝑘)))
10189, 100mpcom 38 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑘 ∈ ℕ0 → ¬ 2 ∥ (-1↑𝑘))
102101adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ0) → ¬ 2 ∥ (-1↑𝑘))
103 elnn0 11279 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑘 ∈ ℕ0 ↔ (𝑘 ∈ ℕ ∨ 𝑘 = 0))
104 oddn2prm 15498 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑃 ∈ (ℙ ∖ {2}) → ¬ 2 ∥ 𝑃)
105104adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ) → ¬ 2 ∥ 𝑃)
106 simpr 477 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
107 prmdvdsexp 15408 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((2 ∈ ℙ ∧ 𝑃 ∈ ℤ ∧ 𝑘 ∈ ℕ) → (2 ∥ (𝑃𝑘) ↔ 2 ∥ 𝑃))
10847, 34, 106, 107mp3an2ani 1429 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ) → (2 ∥ (𝑃𝑘) ↔ 2 ∥ 𝑃))
109105, 108mtbird 315 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ) → ¬ 2 ∥ (𝑃𝑘))
110109expcom 451 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑘 ∈ ℕ → (𝑃 ∈ (ℙ ∖ {2}) → ¬ 2 ∥ (𝑃𝑘)))
111 oveq2 6643 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑘 = 0 → (𝑃𝑘) = (𝑃↑0))
112111adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑘 = 0 ∧ 𝑃 ∈ (ℙ ∖ {2})) → (𝑃𝑘) = (𝑃↑0))
1139adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑘 = 0 ∧ 𝑃 ∈ (ℙ ∖ {2})) → 𝑃 ∈ ℂ)
114113exp0d 12985 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑘 = 0 ∧ 𝑃 ∈ (ℙ ∖ {2})) → (𝑃↑0) = 1)
115112, 114eqtrd 2654 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑘 = 0 ∧ 𝑃 ∈ (ℙ ∖ {2})) → (𝑃𝑘) = 1)
116115breq2d 4656 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑘 = 0 ∧ 𝑃 ∈ (ℙ ∖ {2})) → (2 ∥ (𝑃𝑘) ↔ 2 ∥ 1))
11795, 116mtbiri 317 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑘 = 0 ∧ 𝑃 ∈ (ℙ ∖ {2})) → ¬ 2 ∥ (𝑃𝑘))
118117ex 450 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑘 = 0 → (𝑃 ∈ (ℙ ∖ {2}) → ¬ 2 ∥ (𝑃𝑘)))
119110, 118jaoi 394 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑘 ∈ ℕ ∨ 𝑘 = 0) → (𝑃 ∈ (ℙ ∖ {2}) → ¬ 2 ∥ (𝑃𝑘)))
120103, 119sylbi 207 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑘 ∈ ℕ0 → (𝑃 ∈ (ℙ ∖ {2}) → ¬ 2 ∥ (𝑃𝑘)))
121120impcom 446 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ0) → ¬ 2 ∥ (𝑃𝑘))
122 ioran 511 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (¬ (2 ∥ (-1↑𝑘) ∨ 2 ∥ (𝑃𝑘)) ↔ (¬ 2 ∥ (-1↑𝑘) ∧ ¬ 2 ∥ (𝑃𝑘)))
123102, 121, 122sylanbrc 697 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ0) → ¬ (2 ∥ (-1↑𝑘) ∨ 2 ∥ (𝑃𝑘)))
12428, 31mpan 705 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑘 ∈ ℕ0 → (-1↑𝑘) ∈ ℤ)
125124adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ0) → (-1↑𝑘) ∈ ℤ)
1266, 7, 763syl 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℕ0)
127126adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ0) → 𝑃 ∈ ℕ0)
128 simpr 477 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
129127, 128nn0expcld 13014 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ0) → (𝑃𝑘) ∈ ℕ0)
130129nn0zd 11465 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ0) → (𝑃𝑘) ∈ ℤ)
131 euclemma 15406 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((2 ∈ ℙ ∧ (-1↑𝑘) ∈ ℤ ∧ (𝑃𝑘) ∈ ℤ) → (2 ∥ ((-1↑𝑘) · (𝑃𝑘)) ↔ (2 ∥ (-1↑𝑘) ∨ 2 ∥ (𝑃𝑘))))
13247, 125, 130, 131mp3an2i 1427 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ0) → (2 ∥ ((-1↑𝑘) · (𝑃𝑘)) ↔ (2 ∥ (-1↑𝑘) ∨ 2 ∥ (𝑃𝑘))))
133123, 132mtbird 315 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ0) → ¬ 2 ∥ ((-1↑𝑘) · (𝑃𝑘)))
134133ex 450 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑃 ∈ (ℙ ∖ {2}) → (𝑘 ∈ ℕ0 → ¬ 2 ∥ ((-1↑𝑘) · (𝑃𝑘))))
1351343ad2ant1 1080 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑘 ∈ ℕ0 → ¬ 2 ∥ ((-1↑𝑘) · (𝑃𝑘))))
136135ad2antrr 761 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) ∧ 𝑛 ∈ ℕ) → (𝑘 ∈ ℕ0 → ¬ 2 ∥ ((-1↑𝑘) · (𝑃𝑘))))
137136, 30impel 485 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (0...(𝑀 − 1))) → ¬ 2 ∥ ((-1↑𝑘) · (𝑃𝑘)))
138 nnm1nn0 11319 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑀 ∈ ℕ → (𝑀 − 1) ∈ ℕ0)
139 hashfz0 13202 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑀 − 1) ∈ ℕ0 → (#‘(0...(𝑀 − 1))) = ((𝑀 − 1) + 1))
140138, 139syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑀 ∈ ℕ → (#‘(0...(𝑀 − 1))) = ((𝑀 − 1) + 1))
141 nncn 11013 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑀 ∈ ℕ → 𝑀 ∈ ℂ)
142 npcan1 10440 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑀 ∈ ℂ → ((𝑀 − 1) + 1) = 𝑀)
143141, 142syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑀 ∈ ℕ → ((𝑀 − 1) + 1) = 𝑀)
144140, 143eqtr2d 2655 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑀 ∈ ℕ → 𝑀 = (#‘(0...(𝑀 − 1))))
1451443ad2ant2 1081 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑀 = (#‘(0...(𝑀 − 1))))
146145adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 𝑀 = 1) → 𝑀 = (#‘(0...(𝑀 − 1))))
147146breq2d 4656 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 𝑀 = 1) → (2 ∥ 𝑀 ↔ 2 ∥ (#‘(0...(𝑀 − 1)))))
148147notbid 308 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 𝑀 = 1) → (¬ 2 ∥ 𝑀 ↔ ¬ 2 ∥ (#‘(0...(𝑀 − 1)))))
149148biimpd 219 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 𝑀 = 1) → (¬ 2 ∥ 𝑀 → ¬ 2 ∥ (#‘(0...(𝑀 − 1)))))
150149impr 648 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) → ¬ 2 ∥ (#‘(0...(𝑀 − 1))))
151150adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) ∧ 𝑛 ∈ ℕ) → ¬ 2 ∥ (#‘(0...(𝑀 − 1))))
15273, 86, 137, 151oddsumodd 15094 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) ∧ 𝑛 ∈ ℕ) → ¬ 2 ∥ Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃𝑘)))
153152pm2.21d 118 . . . . . . . . . . . . . . . . . . . 20 ((((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) ∧ 𝑛 ∈ ℕ) → (2 ∥ Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃𝑘)) → 𝑀 = 1))
154153adantr 481 . . . . . . . . . . . . . . . . . . 19 (((((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) ∧ 𝑛 ∈ ℕ) ∧ Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃𝑘)) = (2↑𝑛)) → (2 ∥ Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃𝑘)) → 𝑀 = 1))
15572, 154sylbird 250 . . . . . . . . . . . . . . . . . 18 (((((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) ∧ 𝑛 ∈ ℕ) ∧ Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃𝑘)) = (2↑𝑛)) → (2 ∥ (2↑𝑛) → 𝑀 = 1))
156155ex 450 . . . . . . . . . . . . . . . . 17 ((((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) ∧ 𝑛 ∈ ℕ) → (Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃𝑘)) = (2↑𝑛) → (2 ∥ (2↑𝑛) → 𝑀 = 1)))
15770, 156mpid 44 . . . . . . . . . . . . . . . 16 ((((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) ∧ 𝑛 ∈ ℕ) → (Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃𝑘)) = (2↑𝑛) → 𝑀 = 1))
158157rexlimdva 3027 . . . . . . . . . . . . . . 15 (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) → (∃𝑛 ∈ ℕ Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃𝑘)) = (2↑𝑛) → 𝑀 = 1))
15966, 158syld 47 . . . . . . . . . . . . . 14 (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) → (Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃𝑘)) ∥ (2↑𝑁) → 𝑀 = 1))
160159exp32 630 . . . . . . . . . . . . 13 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (¬ 𝑀 = 1 → (¬ 2 ∥ 𝑀 → (Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃𝑘)) ∥ (2↑𝑁) → 𝑀 = 1))))
161160com12 32 . . . . . . . . . . . 12 𝑀 = 1 → ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (¬ 2 ∥ 𝑀 → (Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃𝑘)) ∥ (2↑𝑁) → 𝑀 = 1))))
162161impd 447 . . . . . . . . . . 11 𝑀 = 1 → (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 2 ∥ 𝑀) → (Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃𝑘)) ∥ (2↑𝑁) → 𝑀 = 1)))
16346, 162pm2.61i 176 . . . . . . . . . 10 (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 2 ∥ 𝑀) → (Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃𝑘)) ∥ (2↑𝑁) → 𝑀 = 1))
164163adantr 481 . . . . . . . . 9 ((((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 2 ∥ 𝑀) ∧ ((𝑃 + 1) · Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃𝑘))) = (2↑𝑁)) → (Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃𝑘)) ∥ (2↑𝑁) → 𝑀 = 1))
16545, 164sylbid 230 . . . . . . . 8 ((((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 2 ∥ 𝑀) ∧ ((𝑃 + 1) · Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃𝑘))) = (2↑𝑁)) → (Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃𝑘)) ∥ ((𝑃 + 1) · Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃𝑘))) → 𝑀 = 1))
16643, 165mpd 15 . . . . . . 7 ((((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 2 ∥ 𝑀) ∧ ((𝑃 + 1) · Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃𝑘))) = (2↑𝑁)) → 𝑀 = 1)
167166ex 450 . . . . . 6 (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 2 ∥ 𝑀) → (((𝑃 + 1) · Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃𝑘))) = (2↑𝑁) → 𝑀 = 1))
16822, 167sylbid 230 . . . . 5 (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 2 ∥ 𝑀) → (((𝑃𝑀) + 1) = (2↑𝑁) → 𝑀 = 1))
16917, 168sylbid 230 . . . 4 (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 2 ∥ 𝑀) → (((2↑𝑁) − 1) = (𝑃𝑀) → 𝑀 = 1))
170169ex 450 . . 3 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (¬ 2 ∥ 𝑀 → (((2↑𝑁) − 1) = (𝑃𝑀) → 𝑀 = 1)))
171170adantld 483 . 2 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((¬ 2 ∥ 𝑁 ∧ ¬ 2 ∥ 𝑀) → (((2↑𝑁) − 1) = (𝑃𝑀) → 𝑀 = 1)))
1721713imp 1254 1 (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 2 ∥ 𝑁 ∧ ¬ 2 ∥ 𝑀) ∧ ((2↑𝑁) − 1) = (𝑃𝑀)) → 𝑀 = 1)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384  w3a 1036   = wceq 1481  wcel 1988  wne 2791  wrex 2910  cdif 3564  {csn 4168  {cpr 4170   class class class wbr 4644  cfv 5876  (class class class)co 6635  cc 9919  0cc0 9921  1c1 9922   + caddc 9924   · cmul 9926  cmin 10251  -cneg 10252  cn 11005  2c2 11055  0cn0 11277  cz 11362  cuz 11672  ...cfz 12311  cexp 12843  #chash 13100  Σcsu 14397  cdvds 14964  cprime 15366
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-inf2 8523  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998  ax-pre-sup 9999
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-fal 1487  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-se 5064  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-isom 5885  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-2o 7546  df-oadd 7549  df-er 7727  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-sup 8333  df-inf 8334  df-oi 8400  df-card 8750  df-cda 8975  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-div 10670  df-nn 11006  df-2 11064  df-3 11065  df-4 11066  df-n0 11278  df-z 11363  df-uz 11673  df-q 11774  df-rp 11818  df-fz 12312  df-fzo 12450  df-fl 12576  df-mod 12652  df-seq 12785  df-exp 12844  df-hash 13101  df-cj 13820  df-re 13821  df-im 13822  df-sqrt 13956  df-abs 13957  df-clim 14200  df-sum 14398  df-dvds 14965  df-gcd 15198  df-prm 15367  df-pc 15523
This theorem is referenced by:  lighneal  41293
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