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Theorem lighneallem4 45792
Description: Lemma 3 for lighneal 45793. (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 12231 . . . . . . . . . 10 (𝑁 ∈ ℕ → 2 ∈ ℂ)
2 nnnn0 12420 . . . . . . . . . 10 (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)
31, 2expcld 14051 . . . . . . . . 9 (𝑁 ∈ ℕ → (2↑𝑁) ∈ ℂ)
433ad2ant3 1135 . . . . . . . 8 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (2↑𝑁) ∈ ℂ)
5 1cnd 11150 . . . . . . . 8 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 1 ∈ ℂ)
6 eldifi 4086 . . . . . . . . . . 11 (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℙ)
7 prmnn 16550 . . . . . . . . . . 11 (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
8 nncn 12161 . . . . . . . . . . 11 (𝑃 ∈ ℕ → 𝑃 ∈ ℂ)
96, 7, 83syl 18 . . . . . . . . . 10 (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℂ)
1093ad2ant1 1133 . . . . . . . . 9 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑃 ∈ ℂ)
11 nnnn0 12420 . . . . . . . . . 10 (𝑀 ∈ ℕ → 𝑀 ∈ ℕ0)
12113ad2ant2 1134 . . . . . . . . 9 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑀 ∈ ℕ0)
1310, 12expcld 14051 . . . . . . . 8 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑃𝑀) ∈ ℂ)
144, 5, 133jca 1128 . . . . . . 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 11405 . . . . . 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 1192 . . . . . . . 8 (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 2 ∥ 𝑀) → 𝑀 ∈ ℕ)
20 simpr 485 . . . . . . . 8 (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 2 ∥ 𝑀) → ¬ 2 ∥ 𝑀)
2118, 19, 20oddpwp1fsum 16274 . . . . . . 7 (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 2 ∥ 𝑀) → ((𝑃𝑀) + 1) = ((𝑃 + 1) · Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃𝑘))))
2221eqeq1d 2738 . . . . . 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 12165 . . . . . . . . . . . . . 14 (𝑃 ∈ ℕ → (𝑃 + 1) ∈ ℕ)
2423nnzd 12526 . . . . . . . . . . . . 13 (𝑃 ∈ ℕ → (𝑃 + 1) ∈ ℤ)
256, 7, 243syl 18 . . . . . . . . . . . 12 (𝑃 ∈ (ℙ ∖ {2}) → (𝑃 + 1) ∈ ℤ)
26253ad2ant1 1133 . . . . . . . . . . 11 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑃 + 1) ∈ ℤ)
27 fzfid 13878 . . . . . . . . . . . 12 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (0...(𝑀 − 1)) ∈ Fin)
28 neg1z 12539 . . . . . . . . . . . . . . 15 -1 ∈ ℤ
2928a1i 11 . . . . . . . . . . . . . 14 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → -1 ∈ ℤ)
30 elfznn0 13534 . . . . . . . . . . . . . 14 (𝑘 ∈ (0...(𝑀 − 1)) → 𝑘 ∈ ℕ0)
31 zexpcl 13982 . . . . . . . . . . . . . 14 ((-1 ∈ ℤ ∧ 𝑘 ∈ ℕ0) → (-1↑𝑘) ∈ ℤ)
3229, 30, 31syl2an 596 . . . . . . . . . . . . 13 (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ (0...(𝑀 − 1))) → (-1↑𝑘) ∈ ℤ)
33 nnz 12520 . . . . . . . . . . . . . . . 16 (𝑃 ∈ ℕ → 𝑃 ∈ ℤ)
346, 7, 333syl 18 . . . . . . . . . . . . . . 15 (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℤ)
35343ad2ant1 1133 . . . . . . . . . . . . . 14 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑃 ∈ ℤ)
36 zexpcl 13982 . . . . . . . . . . . . . 14 ((𝑃 ∈ ℤ ∧ 𝑘 ∈ ℕ0) → (𝑃𝑘) ∈ ℤ)
3735, 30, 36syl2an 596 . . . . . . . . . . . . 13 (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ (0...(𝑀 − 1))) → (𝑃𝑘) ∈ ℤ)
3832, 37zmulcld 12613 . . . . . . . . . . . 12 (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ (0...(𝑀 − 1))) → ((-1↑𝑘) · (𝑃𝑘)) ∈ ℤ)
3927, 38fsumzcl 15620 . . . . . . . . . . 11 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃𝑘)) ∈ ℤ)
4026, 39jca 512 . . . . . . . . . 10 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝑃 + 1) ∈ ℤ ∧ Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃𝑘)) ∈ ℤ))
4140ad2antrr 724 . . . . . . . . 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 16161 . . . . . . . . 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 5109 . . . . . . . . . 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 16568 . . . . . . . . . . . . . . . 16 2 ∈ ℙ
48 prmuz2 16572 . . . . . . . . . . . . . . . . . . . 20 (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ‘2))
496, 48syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ (ℤ‘2))
50493ad2ant1 1133 . . . . . . . . . . . . . . . . . 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 2944 . . . . . . . . . . . . . . . . . . . . . 22 (𝑀 ≠ 1 ↔ ¬ 𝑀 = 1)
53 eluz2b3 12847 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑀 ∈ (ℤ‘2) ↔ (𝑀 ∈ ℕ ∧ 𝑀 ≠ 1))
5453simplbi2 501 . . . . . . . . . . . . . . . . . . . . . 22 (𝑀 ∈ ℕ → (𝑀 ≠ 1 → 𝑀 ∈ (ℤ‘2)))
5552, 54biimtrrid 242 . . . . . . . . . . . . . . . . . . . . 21 (𝑀 ∈ ℕ → (¬ 𝑀 = 1 → 𝑀 ∈ (ℤ‘2)))
56553ad2ant2 1134 . . . . . . . . . . . . . . . . . . . 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 408 . . . . . . . . . . . . . . . . 17 (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) → 𝑀 ∈ (ℤ‘2))
60 simprr 771 . . . . . . . . . . . . . . . . 17 (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) → ¬ 2 ∥ 𝑀)
61 lighneallem4b 45791 . . . . . . . . . . . . . . . . 17 ((𝑃 ∈ (ℤ‘2) ∧ 𝑀 ∈ (ℤ‘2) ∧ ¬ 2 ∥ 𝑀) → Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃𝑘)) ∈ (ℤ‘2))
6251, 59, 60, 61syl3anc 1371 . . . . . . . . . . . . . . . 16 (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) → Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃𝑘)) ∈ (ℤ‘2))
6323ad2ant3 1135 . . . . . . . . . . . . . . . . 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 16757 . . . . . . . . . . . . . . . 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 1466 . . . . . . . . . . . . . . 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 12535 . . . . . . . . . . . . . . . . . . 19 2 ∈ ℤ
6867a1i 11 . . . . . . . . . . . . . . . . . 18 (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) → 2 ∈ ℤ)
69 iddvdsexp 16162 . . . . . . . . . . . . . . . . . 18 ((2 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 2 ∥ (2↑𝑛))
7068, 69sylan 580 . . . . . . . . . . . . . . . . 17 ((((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) ∧ 𝑛 ∈ ℕ) → 2 ∥ (2↑𝑛))
71 breq2 5109 . . . . . . . . . . . . . . . . . . . 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 13878 . . . . . . . . . . . . . . . . . . . . . 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 580 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (-1↑𝑘) ∈ ℤ)
76 nnnn0 12420 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑃 ∈ ℕ → 𝑃 ∈ ℕ0)
7776adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → 𝑃 ∈ ℕ0)
78 simpr 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
7977, 78nn0expcld 14149 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (𝑃𝑘) ∈ ℕ0)
8079nn0zd 12525 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (𝑃𝑘) ∈ ℤ)
8175, 80zmulcld 12613 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → ((-1↑𝑘) · (𝑃𝑘)) ∈ ℤ)
8281ex 413 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑃 ∈ ℕ → (𝑘 ∈ ℕ0 → ((-1↑𝑘) · (𝑃𝑘)) ∈ ℤ))
836, 7, 823syl 18 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑃 ∈ (ℙ ∖ {2}) → (𝑘 ∈ ℕ0 → ((-1↑𝑘) · (𝑃𝑘)) ∈ ℤ))
84833ad2ant1 1133 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑘 ∈ ℕ0 → ((-1↑𝑘) · (𝑃𝑘)) ∈ ℤ))
8584ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) ∧ 𝑛 ∈ ℕ) → (𝑘 ∈ ℕ0 → ((-1↑𝑘) · (𝑃𝑘)) ∈ ℤ))
8685, 30impel 506 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (0...(𝑀 − 1))) → ((-1↑𝑘) · (𝑃𝑘)) ∈ ℤ)
87 nn0z 12524 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑘 ∈ ℕ0𝑘 ∈ ℤ)
88 m1expcl2 13991 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑘 ∈ ℤ → (-1↑𝑘) ∈ {-1, 1})
8987, 88syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑘 ∈ ℕ0 → (-1↑𝑘) ∈ {-1, 1})
90 ovex 7390 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (-1↑𝑘) ∈ V
9190elpr 4609 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((-1↑𝑘) ∈ {-1, 1} ↔ ((-1↑𝑘) = -1 ∨ (-1↑𝑘) = 1))
92 n2dvdsm1 16251 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ¬ 2 ∥ -1
93 breq2 5109 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((-1↑𝑘) = -1 → (2 ∥ (-1↑𝑘) ↔ 2 ∥ -1))
9492, 93mtbiri 326 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((-1↑𝑘) = -1 → ¬ 2 ∥ (-1↑𝑘))
95 n2dvds1 16250 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ¬ 2 ∥ 1
96 breq2 5109 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((-1↑𝑘) = 1 → (2 ∥ (-1↑𝑘) ↔ 2 ∥ 1))
9795, 96mtbiri 326 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((-1↑𝑘) = 1 → ¬ 2 ∥ (-1↑𝑘))
9894, 97jaoi 855 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 216 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 12415 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑘 ∈ ℕ0 ↔ (𝑘 ∈ ℕ ∨ 𝑘 = 0))
104 oddn2prm 16684 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑃 ∈ (ℙ ∖ {2}) → ¬ 2 ∥ 𝑃)
105104adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ) → ¬ 2 ∥ 𝑃)
106 simpr 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
107 prmdvdsexp 16591 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((2 ∈ ℙ ∧ 𝑃 ∈ ℤ ∧ 𝑘 ∈ ℕ) → (2 ∥ (𝑃𝑘) ↔ 2 ∥ 𝑃))
10847, 34, 106, 107mp3an2ani 1468 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ) → (2 ∥ (𝑃𝑘) ↔ 2 ∥ 𝑃))
109105, 108mtbird 324 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ) → ¬ 2 ∥ (𝑃𝑘))
110109expcom 414 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑘 ∈ ℕ → (𝑃 ∈ (ℙ ∖ {2}) → ¬ 2 ∥ (𝑃𝑘)))
111 oveq2 7365 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 14045 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑘 = 0 ∧ 𝑃 ∈ (ℙ ∖ {2})) → (𝑃↑0) = 1)
115112, 114eqtrd 2776 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑘 = 0 ∧ 𝑃 ∈ (ℙ ∖ {2})) → (𝑃𝑘) = 1)
116115breq2d 5117 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑘 = 0 ∧ 𝑃 ∈ (ℙ ∖ {2})) → (2 ∥ (𝑃𝑘) ↔ 2 ∥ 1))
11795, 116mtbiri 326 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑘 = 0 ∧ 𝑃 ∈ (ℙ ∖ {2})) → ¬ 2 ∥ (𝑃𝑘))
118117ex 413 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑘 = 0 → (𝑃 ∈ (ℙ ∖ {2}) → ¬ 2 ∥ (𝑃𝑘)))
119110, 118jaoi 855 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑘 ∈ ℕ ∨ 𝑘 = 0) → (𝑃 ∈ (ℙ ∖ {2}) → ¬ 2 ∥ (𝑃𝑘)))
120103, 119sylbi 216 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑘 ∈ ℕ0 → (𝑃 ∈ (ℙ ∖ {2}) → ¬ 2 ∥ (𝑃𝑘)))
121120impcom 408 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ0) → ¬ 2 ∥ (𝑃𝑘))
122 ioran 982 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (¬ (2 ∥ (-1↑𝑘) ∨ 2 ∥ (𝑃𝑘)) ↔ (¬ 2 ∥ (-1↑𝑘) ∧ ¬ 2 ∥ (𝑃𝑘)))
123102, 121, 122sylanbrc 583 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ0) → ¬ (2 ∥ (-1↑𝑘) ∨ 2 ∥ (𝑃𝑘)))
12428, 31mpan 688 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
129127, 128nn0expcld 14149 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ0) → (𝑃𝑘) ∈ ℕ0)
130129nn0zd 12525 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ0) → (𝑃𝑘) ∈ ℤ)
131 euclemma 16589 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((2 ∈ ℙ ∧ (-1↑𝑘) ∈ ℤ ∧ (𝑃𝑘) ∈ ℤ) → (2 ∥ ((-1↑𝑘) · (𝑃𝑘)) ↔ (2 ∥ (-1↑𝑘) ∨ 2 ∥ (𝑃𝑘))))
13247, 125, 130, 131mp3an2i 1466 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ0) → (2 ∥ ((-1↑𝑘) · (𝑃𝑘)) ↔ (2 ∥ (-1↑𝑘) ∨ 2 ∥ (𝑃𝑘))))
133123, 132mtbird 324 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ0) → ¬ 2 ∥ ((-1↑𝑘) · (𝑃𝑘)))
134133ex 413 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑃 ∈ (ℙ ∖ {2}) → (𝑘 ∈ ℕ0 → ¬ 2 ∥ ((-1↑𝑘) · (𝑃𝑘))))
1351343ad2ant1 1133 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑘 ∈ ℕ0 → ¬ 2 ∥ ((-1↑𝑘) · (𝑃𝑘))))
136135ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) ∧ 𝑛 ∈ ℕ) → (𝑘 ∈ ℕ0 → ¬ 2 ∥ ((-1↑𝑘) · (𝑃𝑘))))
137136, 30impel 506 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (0...(𝑀 − 1))) → ¬ 2 ∥ ((-1↑𝑘) · (𝑃𝑘)))
138 nnm1nn0 12454 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑀 ∈ ℕ → (𝑀 − 1) ∈ ℕ0)
139 hashfz0 14332 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑀 − 1) ∈ ℕ0 → (♯‘(0...(𝑀 − 1))) = ((𝑀 − 1) + 1))
140138, 139syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑀 ∈ ℕ → (♯‘(0...(𝑀 − 1))) = ((𝑀 − 1) + 1))
141 nncn 12161 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑀 ∈ ℕ → 𝑀 ∈ ℂ)
142 npcan1 11580 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑀 ∈ ℂ → ((𝑀 − 1) + 1) = 𝑀)
143141, 142syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑀 ∈ ℕ → ((𝑀 − 1) + 1) = 𝑀)
144140, 143eqtr2d 2777 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑀 ∈ ℕ → 𝑀 = (♯‘(0...(𝑀 − 1))))
1451443ad2ant2 1134 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5117 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 𝑀 = 1) → (2 ∥ 𝑀 ↔ 2 ∥ (♯‘(0...(𝑀 − 1)))))
148147notbid 317 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 𝑀 = 1) → (¬ 2 ∥ 𝑀 ↔ ¬ 2 ∥ (♯‘(0...(𝑀 − 1)))))
149148biimpd 228 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 𝑀 = 1) → (¬ 2 ∥ 𝑀 → ¬ 2 ∥ (♯‘(0...(𝑀 − 1)))))
150149impr 455 . . . . . . . . . . . . . . . . . . . . . . 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 16272 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) ∧ 𝑛 ∈ ℕ) → ¬ 2 ∥ Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃𝑘)))
153152pm2.21d 121 . . . . . . . . . . . . . . . . . . . 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 259 . . . . . . . . . . . . . . . . . 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 413 . . . . . . . . . . . . . . . . 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 3152 . . . . . . . . . . . . . . 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 421 . . . . . . . . . . . . 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 411 . . . . . . . . . . 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 182 . . . . . . . . . 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 239 . . . . . . . 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 413 . . . . . 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 239 . . . . 5 (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 2 ∥ 𝑀) → (((𝑃𝑀) + 1) = (2↑𝑁) → 𝑀 = 1))
16917, 168sylbid 239 . . . 4 (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 2 ∥ 𝑀) → (((2↑𝑁) − 1) = (𝑃𝑀) → 𝑀 = 1))
170169ex 413 . . 3 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (¬ 2 ∥ 𝑀 → (((2↑𝑁) − 1) = (𝑃𝑀) → 𝑀 = 1)))
171170adantld 491 . 2 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((¬ 2 ∥ 𝑁 ∧ ¬ 2 ∥ 𝑀) → (((2↑𝑁) − 1) = (𝑃𝑀) → 𝑀 = 1)))
1721713imp 1111 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 205  wa 396  wo 845  w3a 1087   = wceq 1541  wcel 2106  wne 2943  wrex 3073  cdif 3907  {csn 4586  {cpr 4588   class class class wbr 5105  cfv 6496  (class class class)co 7357  cc 11049  0cc0 11051  1c1 11052   + caddc 11054   · cmul 11056  cmin 11385  -cneg 11386  cn 12153  2c2 12208  0cn0 12413  cz 12499  cuz 12763  ...cfz 13424  cexp 13967  chash 14230  Σcsu 15570  cdvds 16136  cprime 16547
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-oadd 8416  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-inf 9379  df-oi 9446  df-dju 9837  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-n0 12414  df-z 12500  df-uz 12764  df-q 12874  df-rp 12916  df-fz 13425  df-fzo 13568  df-fl 13697  df-mod 13775  df-seq 13907  df-exp 13968  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-clim 15370  df-sum 15571  df-dvds 16137  df-gcd 16375  df-prm 16548  df-pc 16709
This theorem is referenced by:  lighneal  45793
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