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Theorem lgseisen 26260

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 3878	. . . 4 ⊢ (𝜑 → 𝑄 ∈ ℙ)
3		prmz 16232	. . . 4 ⊢ (𝑄 ∈ ℙ → 𝑄 ∈ ℤ)
4	2, 3	syl 17	. . 3 ⊢ (𝜑 → 𝑄 ∈ ℤ)
5		lgseisen.1	. . 3 ⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))
6		lgsval3 26196	. . 3 ⊢ ((𝑄 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) → (𝑄 /_L 𝑃) = ((((𝑄↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1))
7	4, 5, 6	syl2anc 587	. 2 ⊢ (𝜑 → (𝑄 /_L 𝑃) = ((((𝑄↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1))
8	1	gausslemma2dlem0a 26237	. . . . . . . 8 ⊢ (𝜑 → 𝑄 ∈ ℕ)
9		oddprm 16363	. . . . . . . . . 10 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((𝑃 − 1) / 2) ∈ ℕ)
10	5, 9	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃 − 1) / 2) ∈ ℕ)
11	10	nnnn0d 12150	. . . . . . . 8 ⊢ (𝜑 → ((𝑃 − 1) / 2) ∈ ℕ₀)
12	8, 11	nnexpcld 13812	. . . . . . 7 ⊢ (𝜑 → (𝑄↑((𝑃 − 1) / 2)) ∈ ℕ)
13	12	nnred 11845	. . . . . 6 ⊢ (𝜑 → (𝑄↑((𝑃 − 1) / 2)) ∈ ℝ)
14		neg1rr 11945	. . . . . . . 8 ⊢ -1 ∈ ℝ
15	14	a1i 11	. . . . . . 7 ⊢ (𝜑 → -1 ∈ ℝ)
16		neg1ne0 11946	. . . . . . . 8 ⊢ -1 ≠ 0
17	16	a1i 11	. . . . . . 7 ⊢ (𝜑 → -1 ≠ 0)
18		fzfid 13546	. . . . . . . 8 ⊢ (𝜑 → (1...((𝑃 − 1) / 2)) ∈ Fin)
19	8	nnred 11845	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑄 ∈ ℝ)
20	5	gausslemma2dlem0a 26237	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑃 ∈ ℕ)
21	19, 20	nndivred 11884	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑄 / 𝑃) ∈ ℝ)
22	21	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑄 / 𝑃) ∈ ℝ)
23		2re 11904	. . . . . . . . . . 11 ⊢ 2 ∈ ℝ
24		elfznn 13141	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (1...((𝑃 − 1) / 2)) → 𝑥 ∈ ℕ)
25	24	adantl 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑥 ∈ ℕ)
26	25	nnred 11845	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑥 ∈ ℝ)
27		remulcl 10814	. . . . . . . . . . 11 ⊢ ((2 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (2 · 𝑥) ∈ ℝ)
28	23, 26, 27	sylancr 590	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (2 · 𝑥) ∈ ℝ)
29	22, 28	remulcld 10863	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝑄 / 𝑃) · (2 · 𝑥)) ∈ ℝ)
30	29	flcld 13373	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))) ∈ ℤ)
31	18, 30	fsumzcl 15299	. . . . . . 7 ⊢ (𝜑 → Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))) ∈ ℤ)
32	15, 17, 31	reexpclzd 13816	. . . . . 6 ⊢ (𝜑 → (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ∈ ℝ)
33		1re 10833	. . . . . . 7 ⊢ 1 ∈ ℝ
34	33	a1i 11	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℝ)
35	20	nnrpd 12626	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℝ⁺)
36		lgseisen.3	. . . . . . 7 ⊢ (𝜑 → 𝑃 ≠ 𝑄)
37		eqid 2737	. . . . . . 7 ⊢ ((𝑄 · (2 · 𝑥)) mod 𝑃) = ((𝑄 · (2 · 𝑥)) mod 𝑃)
38		eqid 2737	. . . . . . 7 ⊢ (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))
39		eqid 2737	. . . . . . 7 ⊢ ((𝑄 · (2 · 𝑦)) mod 𝑃) = ((𝑄 · (2 · 𝑦)) mod 𝑃)
40		eqid 2737	. . . . . . 7 ⊢ (ℤ/nℤ‘𝑃) = (ℤ/nℤ‘𝑃)
41		eqid 2737	. . . . . . 7 ⊢ (mulGrp‘(ℤ/nℤ‘𝑃)) = (mulGrp‘(ℤ/nℤ‘𝑃))
42		eqid 2737	. . . . . . 7 ⊢ (ℤRHom‘(ℤ/nℤ‘𝑃)) = (ℤRHom‘(ℤ/nℤ‘𝑃))
43	5, 1, 36, 37, 38, 39, 40, 41, 42	lgseisenlem4 26259	. . . . . 6 ⊢ (𝜑 → ((𝑄↑((𝑃 − 1) / 2)) mod 𝑃) = ((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) mod 𝑃))
44		modadd1 13481	. . . . . 6 ⊢ ((((𝑄↑((𝑃 − 1) / 2)) ∈ ℝ ∧ (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ∈ ℝ) ∧ (1 ∈ ℝ ∧ 𝑃 ∈ ℝ⁺) ∧ ((𝑄↑((𝑃 − 1) / 2)) mod 𝑃) = ((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) mod 𝑃)) → (((𝑄↑((𝑃 − 1) / 2)) + 1) mod 𝑃) = (((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 1) mod 𝑃))
45	13, 32, 34, 35, 43, 44	syl221anc 1383	. . . . 5 ⊢ (𝜑 → (((𝑄↑((𝑃 − 1) / 2)) + 1) mod 𝑃) = (((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 1) mod 𝑃))
46		peano2re 11005	. . . . . . 7 ⊢ ((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ∈ ℝ → ((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 1) ∈ ℝ)
47	32, 46	syl 17	. . . . . 6 ⊢ (𝜑 → ((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 1) ∈ ℝ)
48		df-neg 11065	. . . . . . . 8 ⊢ -1 = (0 − 1)
49		neg1cn 11944	. . . . . . . . . . . . 13 ⊢ -1 ∈ ℂ
50		absexpz 14869	. . . . . . . . . . . . 13 ⊢ ((-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)))))
51	49, 16, 31, 50	mp3an12i 1467	. . . . . . . . . . . 12 ⊢ (𝜑 → (abs‘(-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) = ((abs‘-1)↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))
52		ax-1cn 10787	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℂ
53	52	absnegi 14964	. . . . . . . . . . . . . . 15 ⊢ (abs‘-1) = (abs‘1)
54		abs1 14861	. . . . . . . . . . . . . . 15 ⊢ (abs‘1) = 1
55	53, 54	eqtri 2765	. . . . . . . . . . . . . 14 ⊢ (abs‘-1) = 1
56	55	oveq1i 7223	. . . . . . . . . . . . 13 ⊢ ((abs‘-1)↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) = (1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))
57		1exp 13664	. . . . . . . . . . . . . 14 ⊢ (Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))) ∈ ℤ → (1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) = 1)
58	31, 57	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) = 1)
59	56, 58	syl5eq 2790	. . . . . . . . . . . 12 ⊢ (𝜑 → ((abs‘-1)↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) = 1)
60	51, 59	eqtrd 2777	. . . . . . . . . . 11 ⊢ (𝜑 → (abs‘(-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) = 1)
61		1le1 11460	. . . . . . . . . . 11 ⊢ 1 ≤ 1
62	60, 61	eqbrtrdi 5092	. . . . . . . . . 10 ⊢ (𝜑 → (abs‘(-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) ≤ 1)
63		absle 14879	. . . . . . . . . . 11 ⊢ (((-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)))
64	32, 33, 63	sylancl 589	. . . . . . . . . 10 ⊢ (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)))
65	62, 64	mpbid 235	. . . . . . . . 9 ⊢ (𝜑 → (-1 ≤ (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ∧ (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ≤ 1))
66	65	simpld 498	. . . . . . . 8 ⊢ (𝜑 → -1 ≤ (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))
67	48, 66	eqbrtrrid 5089	. . . . . . 7 ⊢ (𝜑 → (0 − 1) ≤ (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))
68		0red 10836	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℝ)
69	68, 34, 32	lesubaddd 11429	. . . . . . 7 ⊢ (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)))
70	67, 69	mpbid 235	. . . . . 6 ⊢ (𝜑 → 0 ≤ ((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 1))
71	20	nnred 11845	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ ℝ)
72		peano2rem 11145	. . . . . . . . 9 ⊢ (𝑃 ∈ ℝ → (𝑃 − 1) ∈ ℝ)
73	71, 72	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝑃 − 1) ∈ ℝ)
74	65	simprd 499	. . . . . . . 8 ⊢ (𝜑 → (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ≤ 1)
75		df-2 11893	. . . . . . . . . 10 ⊢ 2 = (1 + 1)
76	23	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 2 ∈ ℝ)
77	5	eldifad 3878	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑃 ∈ ℙ)
78		prmuz2 16253	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ_≥‘2))
79		eluzle 12451	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ (ℤ_≥‘2) → 2 ≤ 𝑃)
80	77, 78, 79	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 2 ≤ 𝑃)
81		eldifsni 4703	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ≠ 2)
82	5, 81	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ≠ 2)
83	76, 71, 80, 82	leneltd 10986	. . . . . . . . . 10 ⊢ (𝜑 → 2 < 𝑃)
84	75, 83	eqbrtrrid 5089	. . . . . . . . 9 ⊢ (𝜑 → (1 + 1) < 𝑃)
85	34, 34, 71	ltaddsubd 11432	. . . . . . . . 9 ⊢ (𝜑 → ((1 + 1) < 𝑃 ↔ 1 < (𝑃 − 1)))
86	84, 85	mpbid 235	. . . . . . . 8 ⊢ (𝜑 → 1 < (𝑃 − 1))
87	32, 34, 73, 74, 86	lelttrd 10990	. . . . . . 7 ⊢ (𝜑 → (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) < (𝑃 − 1))
88	32, 34, 71	ltaddsubd 11432	. . . . . . 7 ⊢ (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)))
89	87, 88	mpbird 260	. . . . . 6 ⊢ (𝜑 → ((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 1) < 𝑃)
90		modid 13469	. . . . . 6 ⊢ (((((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 1) ∈ ℝ ∧ 𝑃 ∈ ℝ⁺) ∧ (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))
91	47, 35, 70, 89, 90	syl22anc 839	. . . . 5 ⊢ (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))
92	45, 91	eqtrd 2777	. . . 4 ⊢ (𝜑 → (((𝑄↑((𝑃 − 1) / 2)) + 1) mod 𝑃) = ((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 1))
93	92	oveq1d 7228	. . 3 ⊢ (𝜑 → ((((𝑄↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1) = (((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 1) − 1))
94	32	recnd 10861	. . . 4 ⊢ (𝜑 → (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ∈ ℂ)
95		pncan 11084	. . . 4 ⊢ (((-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)))))
96	94, 52, 95	sylancl 589	. . 3 ⊢ (𝜑 → (((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 1) − 1) = (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))
97	93, 96	eqtrd 2777	. 2 ⊢ (𝜑 → ((((𝑄↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1) = (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))
98	7, 97	eqtrd 2777	1 ⊢ (𝜑 → (𝑄 /_L 𝑃) = (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ≠ wne 2940 ∖ cdif 3863 {csn 4541 class class class wbr 5053 ↦ cmpt 5135 ‘cfv 6380 (class class class)co 7213 ℂcc 10727 ℝcr 10728 0cc0 10729 1c1 10730 + caddc 10732 · cmul 10734 < clt 10867 ≤ cle 10868 − cmin 11062 -cneg 11063 / cdiv 11489 ℕcn 11830 2c2 11885 ℤcz 12176 ℤ_≥cuz 12438 ℝ⁺crp 12586 ...cfz 13095 ⌊cfl 13365 mod cmo 13442 ↑cexp 13635 abscabs 14797 Σcsu 15249 ℙcprime 16228 mulGrpcmgp 19504 ℤRHomczrh 20466 ℤ/nℤczn 20469 /_L clgs 26175

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-inf2 9256 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807 ax-addf 10808 ax-mulf 10809

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-se 5510 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-isom 6389 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-of 7469 df-om 7645 df-1st 7761 df-2nd 7762 df-supp 7904 df-tpos 7968 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-2o 8203 df-oadd 8206 df-er 8391 df-ec 8393 df-qs 8397 df-map 8510 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-fsupp 8986 df-sup 9058 df-inf 9059 df-oi 9126 df-dju 9517 df-card 9555 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-7 11898 df-8 11899 df-9 11900 df-n0 12091 df-xnn0 12163 df-z 12177 df-dec 12294 df-uz 12439 df-q 12545 df-rp 12587 df-fz 13096 df-fzo 13239 df-fl 13367 df-mod 13443 df-seq 13575 df-exp 13636 df-hash 13897 df-cj 14662 df-re 14663 df-im 14664 df-sqrt 14798 df-abs 14799 df-clim 15049 df-sum 15250 df-dvds 15816 df-gcd 16054 df-prm 16229 df-phi 16319 df-pc 16390 df-struct 16700 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ress 16785 df-plusg 16815 df-mulr 16816 df-starv 16817 df-sca 16818 df-vsca 16819 df-ip 16820 df-tset 16821 df-ple 16822 df-ds 16824 df-unif 16825 df-0g 16946 df-gsum 16947 df-imas 17013 df-qus 17014 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-mhm 18218 df-submnd 18219 df-grp 18368 df-minusg 18369 df-sbg 18370 df-mulg 18489 df-subg 18540 df-nsg 18541 df-eqg 18542 df-ghm 18620 df-cntz 18711 df-cmn 19172 df-abl 19173 df-mgp 19505 df-ur 19517 df-ring 19564 df-cring 19565 df-oppr 19641 df-dvdsr 19659 df-unit 19660 df-invr 19690 df-dvr 19701 df-rnghom 19735 df-drng 19769 df-field 19770 df-subrg 19798 df-lmod 19901 df-lss 19969 df-lsp 20009 df-sra 20209 df-rgmod 20210 df-lidl 20211 df-rsp 20212 df-2idl 20270 df-nzr 20296 df-rlreg 20321 df-domn 20322 df-idom 20323 df-cnfld 20364 df-zring 20436 df-zrh 20470 df-zn 20473 df-lgs 26176

This theorem is referenced by: lgsquadlem2 26262

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