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Theorem lgseisen 16076

Description: Eisenstein's lemma, an expression for (𝑃 /_L 𝑄) when 𝑃, 𝑄 are distinct odd primes. (Contributed by Mario Carneiro, 18-Jun-2015.)

**Hypotheses**
Ref	Expression
lgseisen.1	⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))
lgseisen.2	⊢ (𝜑 → 𝑄 ∈ (ℙ ∖ {2}))
lgseisen.3	⊢ (𝜑 → 𝑃 ≠ 𝑄)

**Assertion**
Ref	Expression
lgseisen	⊢ (𝜑 → (𝑄 /_L 𝑃) = (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))

Distinct variable groups: 𝑥,𝑃 𝜑,𝑥 𝑥,𝑄

Proof of Theorem lgseisen Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lgseisen.2	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ (ℙ ∖ {2}))
2	1	eldifad 3225	. . . 4 ⊢ (𝜑 → 𝑄 ∈ ℙ)
3		prmz 12836	. . . 4 ⊢ (𝑄 ∈ ℙ → 𝑄 ∈ ℤ)
4	2, 3	syl 14	. . 3 ⊢ (𝜑 → 𝑄 ∈ ℤ)
5		lgseisen.1	. . 3 ⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))
6		lgsval3 16020	. . 3 ⊢ ((𝑄 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) → (𝑄 /_L 𝑃) = ((((𝑄↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1))
7	4, 5, 6	syl2anc 411	. 2 ⊢ (𝜑 → (𝑄 /_L 𝑃) = ((((𝑄↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1))
8	1	gausslemma2dlem0a 16051	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ ℕ)
9		oddprm 12985	. . . . . . . . 9 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((𝑃 − 1) / 2) ∈ ℕ)
10	5, 9	syl 14	. . . . . . . 8 ⊢ (𝜑 → ((𝑃 − 1) / 2) ∈ ℕ)
11	10	nnnn0d 9573	. . . . . . 7 ⊢ (𝜑 → ((𝑃 − 1) / 2) ∈ ℕ₀)
12	8, 11	nnexpcld 11085	. . . . . 6 ⊢ (𝜑 → (𝑄↑((𝑃 − 1) / 2)) ∈ ℕ)
13		nnq 9986	. . . . . 6 ⊢ ((𝑄↑((𝑃 − 1) / 2)) ∈ ℕ → (𝑄↑((𝑃 − 1) / 2)) ∈ ℚ)
14	12, 13	syl 14	. . . . 5 ⊢ (𝜑 → (𝑄↑((𝑃 − 1) / 2)) ∈ ℚ)
15		1zzd 9624	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℤ)
16	15	znegcld 9723	. . . . . . 7 ⊢ (𝜑 → -1 ∈ ℤ)
17		zq 9979	. . . . . . 7 ⊢ (-1 ∈ ℤ → -1 ∈ ℚ)
18	16, 17	syl 14	. . . . . 6 ⊢ (𝜑 → -1 ∈ ℚ)
19		neg1ne0 9364	. . . . . . 7 ⊢ -1 ≠ 0
20	19	a1i 9	. . . . . 6 ⊢ (𝜑 → -1 ≠ 0)
21	10	nnzd 9720	. . . . . . . 8 ⊢ (𝜑 → ((𝑃 − 1) / 2) ∈ ℤ)
22	15, 21	fzfigd 10820	. . . . . . 7 ⊢ (𝜑 → (1...((𝑃 − 1) / 2)) ∈ Fin)
23	5	gausslemma2dlem0a 16051	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ ℕ)
24		znq 9977	. . . . . . . . . 10 ⊢ ((𝑄 ∈ ℤ ∧ 𝑃 ∈ ℕ) → (𝑄 / 𝑃) ∈ ℚ)
25	4, 23, 24	syl2anc 411	. . . . . . . . 9 ⊢ (𝜑 → (𝑄 / 𝑃) ∈ ℚ)
26		2z 9625	. . . . . . . . . . . 12 ⊢ 2 ∈ ℤ
27	26	a1i 9	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 2 ∈ ℤ)
28		elfznn 10412	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (1...((𝑃 − 1) / 2)) → 𝑥 ∈ ℕ)
29	28	adantl 277	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑥 ∈ ℕ)
30	29	nnzd 9720	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑥 ∈ ℤ)
31	27, 30	zmulcld 9727	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (2 · 𝑥) ∈ ℤ)
32		zq 9979	. . . . . . . . . 10 ⊢ ((2 · 𝑥) ∈ ℤ → (2 · 𝑥) ∈ ℚ)
33	31, 32	syl 14	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (2 · 𝑥) ∈ ℚ)
34		qmulcl 9990	. . . . . . . . 9 ⊢ (((𝑄 / 𝑃) ∈ ℚ ∧ (2 · 𝑥) ∈ ℚ) → ((𝑄 / 𝑃) · (2 · 𝑥)) ∈ ℚ)
35	25, 33, 34	syl2an2r 599	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝑄 / 𝑃) · (2 · 𝑥)) ∈ ℚ)
36	35	flqcld 10664	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))) ∈ ℤ)
37	22, 36	fsumzcl 12116	. . . . . 6 ⊢ (𝜑 → Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))) ∈ ℤ)
38		qexpclz 10949	. . . . . 6 ⊢ ((-1 ∈ ℚ ∧ -1 ≠ 0 ∧ Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))) ∈ ℤ) → (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ∈ ℚ)
39	18, 20, 37, 38	syl3anc 1274	. . . . 5 ⊢ (𝜑 → (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ∈ ℚ)
40		1z 9623	. . . . . 6 ⊢ 1 ∈ ℤ
41		zq 9979	. . . . . 6 ⊢ (1 ∈ ℤ → 1 ∈ ℚ)
42	40, 41	mp1i 10	. . . . 5 ⊢ (𝜑 → 1 ∈ ℚ)
43		nnq 9986	. . . . . 6 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℚ)
44	23, 43	syl 14	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℚ)
45	23	nngt0d 9301	. . . . 5 ⊢ (𝜑 → 0 < 𝑃)
46		lgseisen.3	. . . . . 6 ⊢ (𝜑 → 𝑃 ≠ 𝑄)
47		eqid 2234	. . . . . 6 ⊢ ((𝑄 · (2 · 𝑥)) mod 𝑃) = ((𝑄 · (2 · 𝑥)) mod 𝑃)
48		eqid 2234	. . . . . 6 ⊢ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ ((((-1↑((𝑄 · (2 · 𝑥)) mod 𝑃)) · ((𝑄 · (2 · 𝑥)) mod 𝑃)) mod 𝑃) / 2)) = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ ((((-1↑((𝑄 · (2 · 𝑥)) mod 𝑃)) · ((𝑄 · (2 · 𝑥)) mod 𝑃)) mod 𝑃) / 2))
49		eqid 2234	. . . . . 6 ⊢ ((𝑄 · (2 · 𝑦)) mod 𝑃) = ((𝑄 · (2 · 𝑦)) mod 𝑃)
50		eqid 2234	. . . . . 6 ⊢ (ℤ/nℤ‘𝑃) = (ℤ/nℤ‘𝑃)
51		eqid 2234	. . . . . 6 ⊢ (mulGrp‘(ℤ/nℤ‘𝑃)) = (mulGrp‘(ℤ/nℤ‘𝑃))
52		eqid 2234	. . . . . 6 ⊢ (ℤRHom‘(ℤ/nℤ‘𝑃)) = (ℤRHom‘(ℤ/nℤ‘𝑃))
53	5, 1, 46, 47, 48, 49, 50, 51, 52	lgseisenlem4 16075	. . . . 5 ⊢ (𝜑 → ((𝑄↑((𝑃 − 1) / 2)) mod 𝑃) = ((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) mod 𝑃))
54	14, 39, 42, 44, 45, 53	modqadd1 10750	. . . 4 ⊢ (𝜑 → (((𝑄↑((𝑃 − 1) / 2)) + 1) mod 𝑃) = (((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 1) mod 𝑃))
55		qaddcl 9988	. . . . . 6 ⊢ (((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ∈ ℚ ∧ 1 ∈ ℚ) → ((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 1) ∈ ℚ)
56	39, 42, 55	syl2anc 411	. . . . 5 ⊢ (𝜑 → ((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 1) ∈ ℚ)
57		df-neg 8464	. . . . . . 7 ⊢ -1 = (0 − 1)
58		neg1cn 9362	. . . . . . . . . . . 12 ⊢ -1 ∈ ℂ
59		neg1ap0 9366	. . . . . . . . . . . 12 ⊢ -1 # 0
60		absexpzap 11793	. . . . . . . . . . . 12 ⊢ ((-1 ∈ ℂ ∧ -1 # 0 ∧ Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))) ∈ ℤ) → (abs‘(-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) = ((abs‘-1)↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))
61	58, 59, 37, 60	mp3an12i 1378	. . . . . . . . . . 11 ⊢ (𝜑 → (abs‘(-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) = ((abs‘-1)↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))
62		ax-1cn 8236	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℂ
63	62	absnegi 11860	. . . . . . . . . . . . . 14 ⊢ (abs‘-1) = (abs‘1)
64		abs1 11785	. . . . . . . . . . . . . 14 ⊢ (abs‘1) = 1
65	63, 64	eqtri 2255	. . . . . . . . . . . . 13 ⊢ (abs‘-1) = 1
66	65	oveq1i 6068	. . . . . . . . . . . 12 ⊢ ((abs‘-1)↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) = (1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))
67		1exp 10957	. . . . . . . . . . . . 13 ⊢ (Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))) ∈ ℤ → (1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) = 1)
68	37, 67	syl 14	. . . . . . . . . . . 12 ⊢ (𝜑 → (1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) = 1)
69	66, 68	eqtrid 2279	. . . . . . . . . . 11 ⊢ (𝜑 → ((abs‘-1)↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) = 1)
70	61, 69	eqtrd 2267	. . . . . . . . . 10 ⊢ (𝜑 → (abs‘(-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) = 1)
71		1le1 8864	. . . . . . . . . 10 ⊢ 1 ≤ 1
72	70, 71	eqbrtrdi 4153	. . . . . . . . 9 ⊢ (𝜑 → (abs‘(-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) ≤ 1)
73		neg1rr 9363	. . . . . . . . . . . 12 ⊢ -1 ∈ ℝ
74	73	a1i 9	. . . . . . . . . . 11 ⊢ (𝜑 → -1 ∈ ℝ)
75	59	a1i 9	. . . . . . . . . . 11 ⊢ (𝜑 → -1 # 0)
76	74, 75, 37	reexpclzapd 11088	. . . . . . . . . 10 ⊢ (𝜑 → (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ∈ ℝ)
77		1re 8289	. . . . . . . . . 10 ⊢ 1 ∈ ℝ
78		absle 11802	. . . . . . . . . 10 ⊢ (((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ∈ ℝ ∧ 1 ∈ ℝ) → ((abs‘(-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) ≤ 1 ↔ (-1 ≤ (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ∧ (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ≤ 1)))
79	76, 77, 78	sylancl 413	. . . . . . . . 9 ⊢ (𝜑 → ((abs‘(-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) ≤ 1 ↔ (-1 ≤ (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ∧ (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ≤ 1)))
80	72, 79	mpbid 147	. . . . . . . 8 ⊢ (𝜑 → (-1 ≤ (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ∧ (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ≤ 1))
81	80	simpld 112	. . . . . . 7 ⊢ (𝜑 → -1 ≤ (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))
82	57, 81	eqbrtrrid 4150	. . . . . 6 ⊢ (𝜑 → (0 − 1) ≤ (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))
83		0red 8291	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℝ)
84		1red 8305	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℝ)
85	83, 84, 76	lesubaddd 8834	. . . . . 6 ⊢ (𝜑 → ((0 − 1) ≤ (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ↔ 0 ≤ ((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 1)))
86	82, 85	mpbid 147	. . . . 5 ⊢ (𝜑 → 0 ≤ ((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 1))
87	23	nnred 9270	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ ℝ)
88		peano2rem 8557	. . . . . . . 8 ⊢ (𝑃 ∈ ℝ → (𝑃 − 1) ∈ ℝ)
89	87, 88	syl 14	. . . . . . 7 ⊢ (𝜑 → (𝑃 − 1) ∈ ℝ)
90	80	simprd 114	. . . . . . 7 ⊢ (𝜑 → (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ≤ 1)
91		df-2 9316	. . . . . . . . 9 ⊢ 2 = (1 + 1)
92		eldifsni 3827	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ≠ 2)
93	5, 92	syl 14	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ≠ 2)
94	23	nnzd 9720	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑃 ∈ ℤ)
95		zapne 9672	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ ℤ ∧ 2 ∈ ℤ) → (𝑃 # 2 ↔ 𝑃 ≠ 2))
96	94, 26, 95	sylancl 413	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑃 # 2 ↔ 𝑃 ≠ 2))
97	93, 96	mpbird 167	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 # 2)
98		2re 9327	. . . . . . . . . . . 12 ⊢ 2 ∈ ℝ
99	98	a1i 9	. . . . . . . . . . 11 ⊢ (𝜑 → 2 ∈ ℝ)
100	5	eldifad 3225	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑃 ∈ ℙ)
101		prmuz2 12856	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ_≥‘2))
102		eluzle 9887	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ (ℤ_≥‘2) → 2 ≤ 𝑃)
103	100, 101, 102	3syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 2 ≤ 𝑃)
104	99, 87, 103	leltapd 8931	. . . . . . . . . 10 ⊢ (𝜑 → (2 < 𝑃 ↔ 𝑃 # 2))
105	97, 104	mpbird 167	. . . . . . . . 9 ⊢ (𝜑 → 2 < 𝑃)
106	91, 105	eqbrtrrid 4150	. . . . . . . 8 ⊢ (𝜑 → (1 + 1) < 𝑃)
107	84, 84, 87	ltaddsubd 8837	. . . . . . . 8 ⊢ (𝜑 → ((1 + 1) < 𝑃 ↔ 1 < (𝑃 − 1)))
108	106, 107	mpbid 147	. . . . . . 7 ⊢ (𝜑 → 1 < (𝑃 − 1))
109	76, 84, 89, 90, 108	lelttrd 8415	. . . . . 6 ⊢ (𝜑 → (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) < (𝑃 − 1))
110	76, 84, 87	ltaddsubd 8837	. . . . . 6 ⊢ (𝜑 → (((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 1) < 𝑃 ↔ (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) < (𝑃 − 1)))
111	109, 110	mpbird 167	. . . . 5 ⊢ (𝜑 → ((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 1) < 𝑃)
112		modqid 10738	. . . . 5 ⊢ (((((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 1) ∈ ℚ ∧ 𝑃 ∈ ℚ) ∧ (0 ≤ ((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 1) ∧ ((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 1) < 𝑃)) → (((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 1) mod 𝑃) = ((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 1))
113	56, 44, 86, 111, 112	syl22anc 1275	. . . 4 ⊢ (𝜑 → (((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 1) mod 𝑃) = ((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 1))
114	54, 113	eqtrd 2267	. . 3 ⊢ (𝜑 → (((𝑄↑((𝑃 − 1) / 2)) + 1) mod 𝑃) = ((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 1))
115	114	oveq1d 6073	. 2 ⊢ (𝜑 → ((((𝑄↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1) = (((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 1) − 1))
116	76	recnd 8318	. . 3 ⊢ (𝜑 → (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ∈ ℂ)
117		pncan 8496	. . 3 ⊢ (((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ∈ ℂ ∧ 1 ∈ ℂ) → (((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 1) − 1) = (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))
118	116, 62, 117	sylancl 413	. 2 ⊢ (𝜑 → (((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 1) − 1) = (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))
119	7, 115, 118	3eqtrd 2271	1 ⊢ (𝜑 → (𝑄 /_L 𝑃) = (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1398 ∈ wcel 2205 ≠ wne 2414 ∖ cdif 3211 {csn 3694 class class class wbr 4114 ↦ cmpt 4176 ‘cfv 5357 (class class class)co 6058 ℂcc 8141 ℝcr 8142 0cc0 8143 1c1 8144 + caddc 8146 · cmul 8148 < clt 8324 ≤ cle 8325 − cmin 8461 -cneg 8462 # cap 8873 / cdiv 8966 ℕcn 9257 2c2 9308 ℤcz 9597 ℤ_≥cuz 9874 ℚcq 9972 ...cfz 10364 ⌊cfl 10655 mod cmo 10711 ↑cexp 10927 abscabs 11710 Σcsu 12066 ℙcprime 12832 mulGrpcmgp 14162 ℤRHomczrh 14888 ℤ/nℤczn 14890 /_L clgs 15999

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2207 ax-14 2208 ax-ext 2216 ax-coll 4230 ax-sep 4233 ax-nul 4241 ax-pow 4292 ax-pr 4327 ax-un 4559 ax-setind 4664 ax-iinf 4715 ax-cnex 8234 ax-resscn 8235 ax-1cn 8236 ax-1re 8237 ax-icn 8238 ax-addcl 8239 ax-addrcl 8240 ax-mulcl 8241 ax-mulrcl 8242 ax-addcom 8243 ax-mulcom 8244 ax-addass 8245 ax-mulass 8246 ax-distr 8247 ax-i2m1 8248 ax-0lt1 8249 ax-1rid 8250 ax-0id 8251 ax-rnegex 8252 ax-precex 8253 ax-cnre 8254 ax-pre-ltirr 8255 ax-pre-ltwlin 8256 ax-pre-lttrn 8257 ax-pre-apti 8258 ax-pre-ltadd 8259 ax-pre-mulgt0 8260 ax-pre-mulext 8261 ax-arch 8262 ax-caucvg 8263 ax-addf 8265 ax-mulf 8266

This theorem depends on definitions: df-bi 117 df-stab 839 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-xor 1421 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-reu 2529 df-rmo 2530 df-rab 2531 df-v 2817 df-sbc 3046 df-csb 3142 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-nul 3513 df-if 3625 df-pw 3676 df-sn 3700 df-pr 3701 df-tp 3702 df-op 3703 df-uni 3920 df-int 3955 df-iun 3998 df-br 4115 df-opab 4177 df-mpt 4178 df-tr 4214 df-id 4419 df-po 4422 df-iso 4423 df-iord 4492 df-on 4494 df-ilim 4495 df-suc 4497 df-iom 4718 df-xp 4760 df-rel 4761 df-cnv 4762 df-co 4763 df-dm 4764 df-rn 4765 df-res 4766 df-ima 4767 df-iota 5317 df-fun 5359 df-fn 5360 df-f 5361 df-f1 5362 df-fo 5363 df-f1o 5364 df-fv 5365 df-isom 5366 df-riota 6011 df-ov 6061 df-oprab 6062 df-mpo 6063 df-of 6275 df-1st 6347 df-2nd 6348 df-tpos 6489 df-recs 6549 df-irdg 6614 df-frec 6635 df-1o 6660 df-2o 6661 df-oadd 6664 df-er 6780 df-ec 6782 df-qs 6786 df-map 6897 df-en 6989 df-dom 6990 df-fin 6991 df-sup 7288 df-inf 7289 df-pnf 8326 df-mnf 8327 df-xr 8328 df-ltxr 8329 df-le 8330 df-sub 8463 df-neg 8464 df-reap 8867 df-ap 8874 df-div 8967 df-inn 9258 df-2 9316 df-3 9317 df-4 9318 df-5 9319 df-6 9320 df-7 9321 df-8 9322 df-9 9323 df-n0 9517 df-z 9598 df-dec 9731 df-uz 9875 df-q 9973 df-rp 10008 df-fz 10365 df-fzo 10502 df-fl 10657 df-mod 10712 df-seqfrec 10837 df-exp 10928 df-ihash 11167 df-cj 11555 df-re 11556 df-im 11557 df-rsqrt 11711 df-abs 11712 df-clim 11992 df-sumdc 12067 df-proddc 12265 df-dvds 12502 df-gcd 12678 df-prm 12833 df-phi 12936 df-pc 13011 df-struct 13301 df-ndx 13302 df-slot 13303 df-base 13305 df-sets 13306 df-iress 13307 df-plusg 13390 df-mulr 13391 df-starv 13392 df-sca 13393 df-vsca 13394 df-ip 13395 df-tset 13396 df-ple 13397 df-ds 13399 df-unif 13400 df-0g 13558 df-igsum 13559 df-topgen 13560 df-iimas 13570 df-qus 13571 df-mgm 13622 df-sgrp 13668 df-mnd 13681 df-mhm 13717 df-submnd 13718 df-grp 13761 df-minusg 13762 df-sbg 13763 df-mulg 13876 df-subg 13926 df-nsg 13927 df-eqg 13928 df-ghm 13997 df-cmn 14042 df-abl 14043 df-mgp 14163 df-rng 14175 df-ur 14206 df-srg 14210 df-ring 14244 df-cring 14245 df-oppr 14314 df-dvdsr 14336 df-unit 14337 df-invr 14369 df-dvr 14380 df-rhm 14400 df-nzr 14428 df-subrg 14468 df-domn 14508 df-idom 14509 df-lmod 14566 df-lssm 14630 df-lsp 14664 df-sra 14712 df-rgmod 14713 df-lidl 14746 df-rsp 14747 df-2idl 14777 df-bl 14823 df-mopn 14824 df-fg 14826 df-metu 14827 df-cnfld 14834 df-zring 14868 df-zrh 14891 df-zn 14893 df-lgs 16000

This theorem is referenced by: lgsquadlem2 16080

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