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Theorem lighneallem4 40842
Description: Lemma 3 for lighneal 40843. (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 11040 . . . . . . . . . 10 (𝑁 ∈ ℕ → 2 ∈ ℂ)
2 nnnn0 11246 . . . . . . . . . 10 (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)
31, 2expcld 12951 . . . . . . . . 9 (𝑁 ∈ ℕ → (2↑𝑁) ∈ ℂ)
433ad2ant3 1082 . . . . . . . 8 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (2↑𝑁) ∈ ℂ)
5 1cnd 10003 . . . . . . . 8 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 1 ∈ ℂ)
6 eldifi 3712 . . . . . . . . . . 11 (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℙ)
7 prmnn 15315 . . . . . . . . . . 11 (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
8 nncn 10975 . . . . . . . . . . 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 11246 . . . . . . . . . 10 (𝑀 ∈ ℕ → 𝑀 ∈ ℕ0)
12113ad2ant2 1081 . . . . . . . . 9 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑀 ∈ ℕ0)
1310, 12expcld 12951 . . . . . . . 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 10232 . . . . . 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 15042 . . . . . . 7 (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 2 ∥ 𝑀) → ((𝑃𝑀) + 1) = ((𝑃 + 1) · Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃𝑘))))
2221eqeq1d 2623 . . . . . 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 10979 . . . . . . . . . . . . . 14 (𝑃 ∈ ℕ → (𝑃 + 1) ∈ ℕ)
2423nnzd 11428 . . . . . . . . . . . . 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 12715 . . . . . . . . . . . 12 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (0...(𝑀 − 1)) ∈ Fin)
28 neg1z 11360 . . . . . . . . . . . . . . 15 -1 ∈ ℤ
2928a1i 11 . . . . . . . . . . . . . 14 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → -1 ∈ ℤ)
30 elfznn0 12377 . . . . . . . . . . . . . 14 (𝑘 ∈ (0...(𝑀 − 1)) → 𝑘 ∈ ℕ0)
31 zexpcl 12818 . . . . . . . . . . . . . 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 11346 . . . . . . . . . . . . . . . 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 12818 . . . . . . . . . . . . . 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 11435 . . . . . . . . . . . 12 (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ (0...(𝑀 − 1))) → ((-1↑𝑘) · (𝑃𝑘)) ∈ ℤ)
3927, 38fsumzcl 14402 . . . . . . . . . . 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 14931 . . . . . . . . 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 4619 . . . . . . . . . 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 15332 . . . . . . . . . . . . . . . 16 2 ∈ ℙ
48 prmuz2 15335 . . . . . . . . . . . . . . . . . . . 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 2791 . . . . . . . . . . . . . . . . . . . . . 22 (𝑀 ≠ 1 ↔ ¬ 𝑀 = 1)
53 eluz2b3 11709 . . . . . . . . . . . . . . . . . . . . . . 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 40841 . . . . . . . . . . . . . . . . 17 ((𝑃 ∈ (ℤ‘2) ∧ 𝑀 ∈ (ℤ‘2) ∧ ¬ 2 ∥ 𝑀) → Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃𝑘)) ∈ (ℤ‘2))
6251, 59, 60, 61syl3anc 1323 . . . . . . . . . . . . . . . 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 15516 . . . . . . . . . . . . . . . 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 1426 . . . . . . . . . . . . . . 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 11356 . . . . . . . . . . . . . . . . . . 19 2 ∈ ℤ
6867a1i 11 . . . . . . . . . . . . . . . . . 18 (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) → 2 ∈ ℤ)
69 iddvdsexp 14932 . . . . . . . . . . . . . . . . . 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 4619 . . . . . . . . . . . . . . . . . . . 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 12715 . . . . . . . . . . . . . . . . . . . . . 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 11246 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 12974 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (𝑃𝑘) ∈ ℕ0)
8079nn0zd 11427 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (𝑃𝑘) ∈ ℤ)
8175, 80zmulcld 11435 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 11347 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑘 ∈ ℕ0𝑘 ∈ ℤ)
88 m1expcl2 12825 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑘 ∈ ℤ → (-1↑𝑘) ∈ {-1, 1})
8987, 88syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑘 ∈ ℕ0 → (-1↑𝑘) ∈ {-1, 1})
90 ovex 6635 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (-1↑𝑘) ∈ V
9190elpr 4171 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((-1↑𝑘) ∈ {-1, 1} ↔ ((-1↑𝑘) = -1 ∨ (-1↑𝑘) = 1))
92 n2dvdsm1 15032 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ¬ 2 ∥ -1
93 breq2 4619 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((-1↑𝑘) = -1 → (2 ∥ (-1↑𝑘) ↔ 2 ∥ -1))
9492, 93mtbiri 317 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((-1↑𝑘) = -1 → ¬ 2 ∥ (-1↑𝑘))
95 n2dvds1 15031 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ¬ 2 ∥ 1
96 breq2 4619 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 11241 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑘 ∈ ℕ0 ↔ (𝑘 ∈ ℕ ∨ 𝑘 = 0))
104 oddn2prm 15444 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 15354 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((2 ∈ ℙ ∧ 𝑃 ∈ ℤ ∧ 𝑘 ∈ ℕ) → (2 ∥ (𝑃𝑘) ↔ 2 ∥ 𝑃))
10847, 34, 106, 107mp3an2ani 1428 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6615 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 12945 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑘 = 0 ∧ 𝑃 ∈ (ℙ ∖ {2})) → (𝑃↑0) = 1)
115112, 114eqtrd 2655 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑘 = 0 ∧ 𝑃 ∈ (ℙ ∖ {2})) → (𝑃𝑘) = 1)
116115breq2d 4627 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 12974 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ0) → (𝑃𝑘) ∈ ℕ0)
130129nn0zd 11427 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ0) → (𝑃𝑘) ∈ ℤ)
131 euclemma 15352 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((2 ∈ ℙ ∧ (-1↑𝑘) ∈ ℤ ∧ (𝑃𝑘) ∈ ℤ) → (2 ∥ ((-1↑𝑘) · (𝑃𝑘)) ↔ (2 ∥ (-1↑𝑘) ∨ 2 ∥ (𝑃𝑘))))
13247, 125, 130, 131mp3an2i 1426 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 11281 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑀 ∈ ℕ → (𝑀 − 1) ∈ ℕ0)
139 hashfz0 13162 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑀 − 1) ∈ ℕ0 → (#‘(0...(𝑀 − 1))) = ((𝑀 − 1) + 1))
140138, 139syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑀 ∈ ℕ → (#‘(0...(𝑀 − 1))) = ((𝑀 − 1) + 1))
141 nncn 10975 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑀 ∈ ℕ → 𝑀 ∈ ℂ)
142 npcan1 10402 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑀 ∈ ℂ → ((𝑀 − 1) + 1) = 𝑀)
143141, 142syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑀 ∈ ℕ → ((𝑀 − 1) + 1) = 𝑀)
144140, 143eqtr2d 2656 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4627 . . . . . . . . . . . . . . . . . . . . . . . . . 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 15040 . . . . . . . . . . . . . . . . . . . . 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 3024 . . . . . . . . . . . . . . 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 1480  wcel 1987  wne 2790  wrex 2908  cdif 3553  {csn 4150  {cpr 4152   class class class wbr 4615  cfv 5849  (class class class)co 6607  cc 9881  0cc0 9883  1c1 9884   + caddc 9886   · cmul 9888  cmin 10213  -cneg 10214  cn 10967  2c2 11017  0cn0 11239  cz 11324  cuz 11634  ...cfz 12271  cexp 12803  #chash 13060  Σcsu 14353  cdvds 14910  cprime 15312
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905  ax-inf2 8485  ax-cnex 9939  ax-resscn 9940  ax-1cn 9941  ax-icn 9942  ax-addcl 9943  ax-addrcl 9944  ax-mulcl 9945  ax-mulrcl 9946  ax-mulcom 9947  ax-addass 9948  ax-mulass 9949  ax-distr 9950  ax-i2m1 9951  ax-1ne0 9952  ax-1rid 9953  ax-rnegex 9954  ax-rrecex 9955  ax-cnre 9956  ax-pre-lttri 9957  ax-pre-lttrn 9958  ax-pre-ltadd 9959  ax-pre-mulgt0 9960  ax-pre-sup 9961
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-pss 3572  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-tp 4155  df-op 4157  df-uni 4405  df-int 4443  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-tr 4715  df-eprel 4987  df-id 4991  df-po 4997  df-so 4998  df-fr 5035  df-se 5036  df-we 5037  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-pred 5641  df-ord 5687  df-on 5688  df-lim 5689  df-suc 5690  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-isom 5858  df-riota 6568  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-om 7016  df-1st 7116  df-2nd 7117  df-wrecs 7355  df-recs 7416  df-rdg 7454  df-1o 7508  df-2o 7509  df-oadd 7512  df-er 7690  df-en 7903  df-dom 7904  df-sdom 7905  df-fin 7906  df-sup 8295  df-inf 8296  df-oi 8362  df-card 8712  df-cda 8937  df-pnf 10023  df-mnf 10024  df-xr 10025  df-ltxr 10026  df-le 10027  df-sub 10215  df-neg 10216  df-div 10632  df-nn 10968  df-2 11026  df-3 11027  df-4 11028  df-n0 11240  df-z 11325  df-uz 11635  df-q 11736  df-rp 11780  df-fz 12272  df-fzo 12410  df-fl 12536  df-mod 12612  df-seq 12745  df-exp 12804  df-hash 13061  df-cj 13776  df-re 13777  df-im 13778  df-sqrt 13912  df-abs 13913  df-clim 14156  df-sum 14354  df-dvds 14911  df-gcd 15144  df-prm 15313  df-pc 15469
This theorem is referenced by:  lighneal  40843
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