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Theorem lighneallem4 42229
Description: Lemma 3 for lighneal 42230. (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 11355 . . . . . . . . . 10 (𝑁 ∈ ℕ → 2 ∈ ℂ)
2 nnnn0 11551 . . . . . . . . . 10 (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)
31, 2expcld 13222 . . . . . . . . 9 (𝑁 ∈ ℕ → (2↑𝑁) ∈ ℂ)
433ad2ant3 1165 . . . . . . . 8 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (2↑𝑁) ∈ ℂ)
5 1cnd 10292 . . . . . . . 8 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 1 ∈ ℂ)
6 eldifi 3896 . . . . . . . . . . 11 (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℙ)
7 prmnn 15684 . . . . . . . . . . 11 (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
8 nncn 11288 . . . . . . . . . . 11 (𝑃 ∈ ℕ → 𝑃 ∈ ℂ)
96, 7, 83syl 18 . . . . . . . . . 10 (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℂ)
1093ad2ant1 1163 . . . . . . . . 9 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑃 ∈ ℂ)
11 nnnn0 11551 . . . . . . . . . 10 (𝑀 ∈ ℕ → 𝑀 ∈ ℕ0)
12113ad2ant2 1164 . . . . . . . . 9 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑀 ∈ ℕ0)
1310, 12expcld 13222 . . . . . . . 8 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑃𝑀) ∈ ℂ)
144, 5, 133jca 1158 . . . . . . 7 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((2↑𝑁) ∈ ℂ ∧ 1 ∈ ℂ ∧ (𝑃𝑀) ∈ ℂ))
1514adantr 472 . . . . . 6 (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 2 ∥ 𝑀) → ((2↑𝑁) ∈ ℂ ∧ 1 ∈ ℂ ∧ (𝑃𝑀) ∈ ℂ))
16 subadd2 10544 . . . . . 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 472 . . . . . . . 8 (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 2 ∥ 𝑀) → 𝑃 ∈ ℂ)
19 simpl2 1244 . . . . . . . 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 15413 . . . . . . 7 (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 2 ∥ 𝑀) → ((𝑃𝑀) + 1) = ((𝑃 + 1) · Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃𝑘))))
2221eqeq1d 2767 . . . . . 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 11293 . . . . . . . . . . . . . 14 (𝑃 ∈ ℕ → (𝑃 + 1) ∈ ℕ)
2423nnzd 11734 . . . . . . . . . . . . 13 (𝑃 ∈ ℕ → (𝑃 + 1) ∈ ℤ)
256, 7, 243syl 18 . . . . . . . . . . . 12 (𝑃 ∈ (ℙ ∖ {2}) → (𝑃 + 1) ∈ ℤ)
26253ad2ant1 1163 . . . . . . . . . . 11 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑃 + 1) ∈ ℤ)
27 fzfid 12987 . . . . . . . . . . . 12 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (0...(𝑀 − 1)) ∈ Fin)
28 neg1z 11666 . . . . . . . . . . . . . . 15 -1 ∈ ℤ
2928a1i 11 . . . . . . . . . . . . . 14 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → -1 ∈ ℤ)
30 elfznn0 12647 . . . . . . . . . . . . . 14 (𝑘 ∈ (0...(𝑀 − 1)) → 𝑘 ∈ ℕ0)
31 zexpcl 13089 . . . . . . . . . . . . . 14 ((-1 ∈ ℤ ∧ 𝑘 ∈ ℕ0) → (-1↑𝑘) ∈ ℤ)
3229, 30, 31syl2an 589 . . . . . . . . . . . . 13 (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ (0...(𝑀 − 1))) → (-1↑𝑘) ∈ ℤ)
33 nnz 11652 . . . . . . . . . . . . . . . 16 (𝑃 ∈ ℕ → 𝑃 ∈ ℤ)
346, 7, 333syl 18 . . . . . . . . . . . . . . 15 (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℤ)
35343ad2ant1 1163 . . . . . . . . . . . . . 14 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑃 ∈ ℤ)
36 zexpcl 13089 . . . . . . . . . . . . . 14 ((𝑃 ∈ ℤ ∧ 𝑘 ∈ ℕ0) → (𝑃𝑘) ∈ ℤ)
3735, 30, 36syl2an 589 . . . . . . . . . . . . 13 (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ (0...(𝑀 − 1))) → (𝑃𝑘) ∈ ℤ)
3832, 37zmulcld 11741 . . . . . . . . . . . 12 (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ (0...(𝑀 − 1))) → ((-1↑𝑘) · (𝑃𝑘)) ∈ ℤ)
3927, 38fsumzcl 14767 . . . . . . . . . . 11 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃𝑘)) ∈ ℤ)
4026, 39jca 507 . . . . . . . . . 10 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝑃 + 1) ∈ ℤ ∧ Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃𝑘)) ∈ ℤ))
4140ad2antrr 717 . . . . . . . . 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 15305 . . . . . . . . 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 4815 . . . . . . . . . 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 473 . . . . . . . . 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 15701 . . . . . . . . . . . . . . . 16 2 ∈ ℙ
48 prmuz2 15704 . . . . . . . . . . . . . . . . . . . 20 (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ‘2))
496, 48syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ (ℤ‘2))
50493ad2ant1 1163 . . . . . . . . . . . . . . . . . 18 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑃 ∈ (ℤ‘2))
5150adantr 472 . . . . . . . . . . . . . . . . 17 (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) → 𝑃 ∈ (ℤ‘2))
52 df-ne 2938 . . . . . . . . . . . . . . . . . . . . . 22 (𝑀 ≠ 1 ↔ ¬ 𝑀 = 1)
53 eluz2b3 11970 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑀 ∈ (ℤ‘2) ↔ (𝑀 ∈ ℕ ∧ 𝑀 ≠ 1))
5453simplbi2 494 . . . . . . . . . . . . . . . . . . . . . 22 (𝑀 ∈ ℕ → (𝑀 ≠ 1 → 𝑀 ∈ (ℤ‘2)))
5552, 54syl5bir 234 . . . . . . . . . . . . . . . . . . . . 21 (𝑀 ∈ ℕ → (¬ 𝑀 = 1 → 𝑀 ∈ (ℤ‘2)))
56553ad2ant2 1164 . . . . . . . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . . . . . 18 ((¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀) → ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑀 ∈ (ℤ‘2)))
5958impcom 396 . . . . . . . . . . . . . . . . 17 (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) → 𝑀 ∈ (ℤ‘2))
60 simprr 789 . . . . . . . . . . . . . . . . 17 (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) → ¬ 2 ∥ 𝑀)
61 lighneallem4b 42228 . . . . . . . . . . . . . . . . 17 ((𝑃 ∈ (ℤ‘2) ∧ 𝑀 ∈ (ℤ‘2) ∧ ¬ 2 ∥ 𝑀) → Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃𝑘)) ∈ (ℤ‘2))
6251, 59, 60, 61syl3anc 1490 . . . . . . . . . . . . . . . 16 (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) → Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃𝑘)) ∈ (ℤ‘2))
6323ad2ant3 1165 . . . . . . . . . . . . . . . . 17 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ0)
6463adantr 472 . . . . . . . . . . . . . . . 16 (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) → 𝑁 ∈ ℕ0)
65 dvdsprmpweqnn 15884 . . . . . . . . . . . . . . . 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 1590 . . . . . . . . . . . . . . 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 11662 . . . . . . . . . . . . . . . . . . 19 2 ∈ ℤ
6867a1i 11 . . . . . . . . . . . . . . . . . 18 (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) → 2 ∈ ℤ)
69 iddvdsexp 15306 . . . . . . . . . . . . . . . . . 18 ((2 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 2 ∥ (2↑𝑛))
7068, 69sylan 575 . . . . . . . . . . . . . . . . 17 ((((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) ∧ 𝑛 ∈ ℕ) → 2 ∥ (2↑𝑛))
71 breq2 4815 . . . . . . . . . . . . . . . . . . . 20 𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃𝑘)) = (2↑𝑛) → (2 ∥ Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃𝑘)) ↔ 2 ∥ (2↑𝑛)))
7271adantl 473 . . . . . . . . . . . . . . . . . . 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 12987 . . . . . . . . . . . . . . . . . . . . . 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 575 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (-1↑𝑘) ∈ ℤ)
76 nnnn0 11551 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑃 ∈ ℕ → 𝑃 ∈ ℕ0)
7776adantr 472 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → 𝑃 ∈ ℕ0)
78 simpr 477 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
7977, 78nn0expcld 13245 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (𝑃𝑘) ∈ ℕ0)
8079nn0zd 11733 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (𝑃𝑘) ∈ ℤ)
8175, 80zmulcld 11741 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → ((-1↑𝑘) · (𝑃𝑘)) ∈ ℤ)
8281ex 401 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑃 ∈ ℕ → (𝑘 ∈ ℕ0 → ((-1↑𝑘) · (𝑃𝑘)) ∈ ℤ))
836, 7, 823syl 18 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑃 ∈ (ℙ ∖ {2}) → (𝑘 ∈ ℕ0 → ((-1↑𝑘) · (𝑃𝑘)) ∈ ℤ))
84833ad2ant1 1163 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑘 ∈ ℕ0 → ((-1↑𝑘) · (𝑃𝑘)) ∈ ℤ))
8584ad2antrr 717 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) ∧ 𝑛 ∈ ℕ) → (𝑘 ∈ ℕ0 → ((-1↑𝑘) · (𝑃𝑘)) ∈ ℤ))
8685, 30impel 501 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (0...(𝑀 − 1))) → ((-1↑𝑘) · (𝑃𝑘)) ∈ ℤ)
87 nn0z 11653 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑘 ∈ ℕ0𝑘 ∈ ℤ)
88 m1expcl2 13096 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑘 ∈ ℤ → (-1↑𝑘) ∈ {-1, 1})
8987, 88syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑘 ∈ ℕ0 → (-1↑𝑘) ∈ {-1, 1})
90 ovex 6878 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (-1↑𝑘) ∈ V
9190elpr 4359 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((-1↑𝑘) ∈ {-1, 1} ↔ ((-1↑𝑘) = -1 ∨ (-1↑𝑘) = 1))
92 n2dvdsm1 15403 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ¬ 2 ∥ -1
93 breq2 4815 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((-1↑𝑘) = -1 → (2 ∥ (-1↑𝑘) ↔ 2 ∥ -1))
9492, 93mtbiri 318 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((-1↑𝑘) = -1 → ¬ 2 ∥ (-1↑𝑘))
95 n2dvds1 15402 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ¬ 2 ∥ 1
96 breq2 4815 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((-1↑𝑘) = 1 → (2 ∥ (-1↑𝑘) ↔ 2 ∥ 1))
9795, 96mtbiri 318 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((-1↑𝑘) = 1 → ¬ 2 ∥ (-1↑𝑘))
9894, 97jaoi 883 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 208 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((-1↑𝑘) ∈ {-1, 1} → (𝑘 ∈ ℕ0 → ¬ 2 ∥ (-1↑𝑘)))
10189, 100mpcom 38 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑘 ∈ ℕ0 → ¬ 2 ∥ (-1↑𝑘))
102101adantl 473 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ0) → ¬ 2 ∥ (-1↑𝑘))
103 elnn0 11545 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑘 ∈ ℕ0 ↔ (𝑘 ∈ ℕ ∨ 𝑘 = 0))
104 oddn2prm 15812 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑃 ∈ (ℙ ∖ {2}) → ¬ 2 ∥ 𝑃)
105104adantr 472 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ) → ¬ 2 ∥ 𝑃)
106 simpr 477 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
107 prmdvdsexp 15722 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((2 ∈ ℙ ∧ 𝑃 ∈ ℤ ∧ 𝑘 ∈ ℕ) → (2 ∥ (𝑃𝑘) ↔ 2 ∥ 𝑃))
10847, 34, 106, 107mp3an2ani 1592 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ) → (2 ∥ (𝑃𝑘) ↔ 2 ∥ 𝑃))
109105, 108mtbird 316 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ) → ¬ 2 ∥ (𝑃𝑘))
110109expcom 402 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑘 ∈ ℕ → (𝑃 ∈ (ℙ ∖ {2}) → ¬ 2 ∥ (𝑃𝑘)))
111 oveq2 6854 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑘 = 0 → (𝑃𝑘) = (𝑃↑0))
112111adantr 472 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑘 = 0 ∧ 𝑃 ∈ (ℙ ∖ {2})) → (𝑃𝑘) = (𝑃↑0))
1139adantl 473 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑘 = 0 ∧ 𝑃 ∈ (ℙ ∖ {2})) → 𝑃 ∈ ℂ)
114113exp0d 13216 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑘 = 0 ∧ 𝑃 ∈ (ℙ ∖ {2})) → (𝑃↑0) = 1)
115112, 114eqtrd 2799 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑘 = 0 ∧ 𝑃 ∈ (ℙ ∖ {2})) → (𝑃𝑘) = 1)
116115breq2d 4823 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑘 = 0 ∧ 𝑃 ∈ (ℙ ∖ {2})) → (2 ∥ (𝑃𝑘) ↔ 2 ∥ 1))
11795, 116mtbiri 318 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑘 = 0 ∧ 𝑃 ∈ (ℙ ∖ {2})) → ¬ 2 ∥ (𝑃𝑘))
118117ex 401 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑘 = 0 → (𝑃 ∈ (ℙ ∖ {2}) → ¬ 2 ∥ (𝑃𝑘)))
119110, 118jaoi 883 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑘 ∈ ℕ ∨ 𝑘 = 0) → (𝑃 ∈ (ℙ ∖ {2}) → ¬ 2 ∥ (𝑃𝑘)))
120103, 119sylbi 208 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑘 ∈ ℕ0 → (𝑃 ∈ (ℙ ∖ {2}) → ¬ 2 ∥ (𝑃𝑘)))
121120impcom 396 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ0) → ¬ 2 ∥ (𝑃𝑘))
122 ioran 1006 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (¬ (2 ∥ (-1↑𝑘) ∨ 2 ∥ (𝑃𝑘)) ↔ (¬ 2 ∥ (-1↑𝑘) ∧ ¬ 2 ∥ (𝑃𝑘)))
123102, 121, 122sylanbrc 578 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ0) → ¬ (2 ∥ (-1↑𝑘) ∨ 2 ∥ (𝑃𝑘)))
12428, 31mpan 681 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑘 ∈ ℕ0 → (-1↑𝑘) ∈ ℤ)
125124adantl 473 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ0) → (-1↑𝑘) ∈ ℤ)
1266, 7, 763syl 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℕ0)
127126adantr 472 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ0) → 𝑃 ∈ ℕ0)
128 simpr 477 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
129127, 128nn0expcld 13245 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ0) → (𝑃𝑘) ∈ ℕ0)
130129nn0zd 11733 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ0) → (𝑃𝑘) ∈ ℤ)
131 euclemma 15720 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((2 ∈ ℙ ∧ (-1↑𝑘) ∈ ℤ ∧ (𝑃𝑘) ∈ ℤ) → (2 ∥ ((-1↑𝑘) · (𝑃𝑘)) ↔ (2 ∥ (-1↑𝑘) ∨ 2 ∥ (𝑃𝑘))))
13247, 125, 130, 131mp3an2i 1590 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ0) → (2 ∥ ((-1↑𝑘) · (𝑃𝑘)) ↔ (2 ∥ (-1↑𝑘) ∨ 2 ∥ (𝑃𝑘))))
133123, 132mtbird 316 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑘 ∈ ℕ0) → ¬ 2 ∥ ((-1↑𝑘) · (𝑃𝑘)))
134133ex 401 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑃 ∈ (ℙ ∖ {2}) → (𝑘 ∈ ℕ0 → ¬ 2 ∥ ((-1↑𝑘) · (𝑃𝑘))))
1351343ad2ant1 1163 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑘 ∈ ℕ0 → ¬ 2 ∥ ((-1↑𝑘) · (𝑃𝑘))))
136135ad2antrr 717 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) ∧ 𝑛 ∈ ℕ) → (𝑘 ∈ ℕ0 → ¬ 2 ∥ ((-1↑𝑘) · (𝑃𝑘))))
137136, 30impel 501 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (0...(𝑀 − 1))) → ¬ 2 ∥ ((-1↑𝑘) · (𝑃𝑘)))
138 nnm1nn0 11586 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑀 ∈ ℕ → (𝑀 − 1) ∈ ℕ0)
139 hashfz0 13427 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑀 − 1) ∈ ℕ0 → (♯‘(0...(𝑀 − 1))) = ((𝑀 − 1) + 1))
140138, 139syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑀 ∈ ℕ → (♯‘(0...(𝑀 − 1))) = ((𝑀 − 1) + 1))
141 nncn 11288 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑀 ∈ ℕ → 𝑀 ∈ ℂ)
142 npcan1 10714 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑀 ∈ ℂ → ((𝑀 − 1) + 1) = 𝑀)
143141, 142syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑀 ∈ ℕ → ((𝑀 − 1) + 1) = 𝑀)
144140, 143eqtr2d 2800 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑀 ∈ ℕ → 𝑀 = (♯‘(0...(𝑀 − 1))))
1451443ad2ant2 1164 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑀 = (♯‘(0...(𝑀 − 1))))
146145adantr 472 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 𝑀 = 1) → 𝑀 = (♯‘(0...(𝑀 − 1))))
147146breq2d 4823 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 𝑀 = 1) → (2 ∥ 𝑀 ↔ 2 ∥ (♯‘(0...(𝑀 − 1)))))
148147notbid 309 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 𝑀 = 1) → (¬ 2 ∥ 𝑀 ↔ ¬ 2 ∥ (♯‘(0...(𝑀 − 1)))))
149148biimpd 220 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 𝑀 = 1) → (¬ 2 ∥ 𝑀 → ¬ 2 ∥ (♯‘(0...(𝑀 − 1)))))
150149impr 446 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) → ¬ 2 ∥ (♯‘(0...(𝑀 − 1))))
151150adantr 472 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) ∧ 𝑛 ∈ ℕ) → ¬ 2 ∥ (♯‘(0...(𝑀 − 1))))
15273, 86, 137, 151oddsumodd 15411 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 𝑀 = 1 ∧ ¬ 2 ∥ 𝑀)) ∧ 𝑛 ∈ ℕ) → ¬ 2 ∥ Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝑃𝑘)))
153152pm2.21d 119 . . . . . . . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . . . . . . 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 251 . . . . . . . . . . . . . . . . . 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 401 . . . . . . . . . . . . . . . . 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 3178 . . . . . . . . . . . . . . 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 411 . . . . . . . . . . . . 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 398 . . . . . . . . . . 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 472 . . . . . . . . 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 231 . . . . . . . 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 401 . . . . . 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 231 . . . . 5 (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 2 ∥ 𝑀) → (((𝑃𝑀) + 1) = (2↑𝑁) → 𝑀 = 1))
16917, 168sylbid 231 . . . 4 (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ¬ 2 ∥ 𝑀) → (((2↑𝑁) − 1) = (𝑃𝑀) → 𝑀 = 1))
170169ex 401 . . 3 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (¬ 2 ∥ 𝑀 → (((2↑𝑁) − 1) = (𝑃𝑀) → 𝑀 = 1)))
171170adantld 484 . 2 ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((¬ 2 ∥ 𝑁 ∧ ¬ 2 ∥ 𝑀) → (((2↑𝑁) − 1) = (𝑃𝑀) → 𝑀 = 1)))
1721713imp 1137 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 197  wa 384  wo 873  w3a 1107   = wceq 1652  wcel 2155  wne 2937  wrex 3056  cdif 3731  {csn 4336  {cpr 4338   class class class wbr 4811  cfv 6070  (class class class)co 6846  cc 10191  0cc0 10193  1c1 10194   + caddc 10196   · cmul 10198  cmin 10525  -cneg 10526  cn 11279  2c2 11332  0cn0 11543  cz 11629  cuz 11893  ...cfz 12540  cexp 13074  chash 13328  Σcsu 14717  cdvds 15281  cprime 15681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151  ax-inf2 8757  ax-cnex 10249  ax-resscn 10250  ax-1cn 10251  ax-icn 10252  ax-addcl 10253  ax-addrcl 10254  ax-mulcl 10255  ax-mulrcl 10256  ax-mulcom 10257  ax-addass 10258  ax-mulass 10259  ax-distr 10260  ax-i2m1 10261  ax-1ne0 10262  ax-1rid 10263  ax-rnegex 10264  ax-rrecex 10265  ax-cnre 10266  ax-pre-lttri 10267  ax-pre-lttrn 10268  ax-pre-ltadd 10269  ax-pre-mulgt0 10270  ax-pre-sup 10271
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-int 4636  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-se 5239  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-isom 6079  df-riota 6807  df-ov 6849  df-oprab 6850  df-mpt2 6851  df-om 7268  df-1st 7370  df-2nd 7371  df-wrecs 7614  df-recs 7676  df-rdg 7714  df-1o 7768  df-2o 7769  df-oadd 7772  df-er 7951  df-en 8165  df-dom 8166  df-sdom 8167  df-fin 8168  df-sup 8559  df-inf 8560  df-oi 8626  df-card 9020  df-cda 9247  df-pnf 10334  df-mnf 10335  df-xr 10336  df-ltxr 10337  df-le 10338  df-sub 10527  df-neg 10528  df-div 10944  df-nn 11280  df-2 11340  df-3 11341  df-4 11342  df-n0 11544  df-z 11630  df-uz 11894  df-q 11997  df-rp 12036  df-fz 12541  df-fzo 12681  df-fl 12808  df-mod 12884  df-seq 13016  df-exp 13075  df-hash 13329  df-cj 14140  df-re 14141  df-im 14142  df-sqrt 14276  df-abs 14277  df-clim 14520  df-sum 14718  df-dvds 15282  df-gcd 15514  df-prm 15682  df-pc 15837
This theorem is referenced by:  lighneal  42230
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