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Theorem lgseisen 27437

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 3974	. . . 4 ⊢ (𝜑 → 𝑄 ∈ ℙ)
3		prmz 16708	. . . 4 ⊢ (𝑄 ∈ ℙ → 𝑄 ∈ ℤ)
4	2, 3	syl 17	. . 3 ⊢ (𝜑 → 𝑄 ∈ ℤ)
5		lgseisen.1	. . 3 ⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))
6		lgsval3 27373	. . 3 ⊢ ((𝑄 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) → (𝑄 /_L 𝑃) = ((((𝑄↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1))
7	4, 5, 6	syl2anc 584	. 2 ⊢ (𝜑 → (𝑄 /_L 𝑃) = ((((𝑄↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1))
8	1	gausslemma2dlem0a 27414	. . . . . . . 8 ⊢ (𝜑 → 𝑄 ∈ ℕ)
9		oddprm 16843	. . . . . . . . . 10 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((𝑃 − 1) / 2) ∈ ℕ)
10	5, 9	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃 − 1) / 2) ∈ ℕ)
11	10	nnnn0d 12584	. . . . . . . 8 ⊢ (𝜑 → ((𝑃 − 1) / 2) ∈ ℕ₀)
12	8, 11	nnexpcld 14280	. . . . . . 7 ⊢ (𝜑 → (𝑄↑((𝑃 − 1) / 2)) ∈ ℕ)
13	12	nnred 12278	. . . . . 6 ⊢ (𝜑 → (𝑄↑((𝑃 − 1) / 2)) ∈ ℝ)
14		neg1rr 12378	. . . . . . . 8 ⊢ -1 ∈ ℝ
15	14	a1i 11	. . . . . . 7 ⊢ (𝜑 → -1 ∈ ℝ)
16		neg1ne0 12379	. . . . . . . 8 ⊢ -1 ≠ 0
17	16	a1i 11	. . . . . . 7 ⊢ (𝜑 → -1 ≠ 0)
18		fzfid 14010	. . . . . . . 8 ⊢ (𝜑 → (1...((𝑃 − 1) / 2)) ∈ Fin)
19	8	nnred 12278	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑄 ∈ ℝ)
20	5	gausslemma2dlem0a 27414	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑃 ∈ ℕ)
21	19, 20	nndivred 12317	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑄 / 𝑃) ∈ ℝ)
22	21	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑄 / 𝑃) ∈ ℝ)
23		2re 12337	. . . . . . . . . . 11 ⊢ 2 ∈ ℝ
24		elfznn 13589	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (1...((𝑃 − 1) / 2)) → 𝑥 ∈ ℕ)
25	24	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑥 ∈ ℕ)
26	25	nnred 12278	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑥 ∈ ℝ)
27		remulcl 11237	. . . . . . . . . . 11 ⊢ ((2 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (2 · 𝑥) ∈ ℝ)
28	23, 26, 27	sylancr 587	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (2 · 𝑥) ∈ ℝ)
29	22, 28	remulcld 11288	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝑄 / 𝑃) · (2 · 𝑥)) ∈ ℝ)
30	29	flcld 13834	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))) ∈ ℤ)
31	18, 30	fsumzcl 15767	. . . . . . 7 ⊢ (𝜑 → Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))) ∈ ℤ)
32	15, 17, 31	reexpclzd 14284	. . . . . 6 ⊢ (𝜑 → (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ∈ ℝ)
33		1re 11258	. . . . . . 7 ⊢ 1 ∈ ℝ
34	33	a1i 11	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℝ)
35	20	nnrpd 13072	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℝ⁺)
36		lgseisen.3	. . . . . . 7 ⊢ (𝜑 → 𝑃 ≠ 𝑄)
37		eqid 2734	. . . . . . 7 ⊢ ((𝑄 · (2 · 𝑥)) mod 𝑃) = ((𝑄 · (2 · 𝑥)) mod 𝑃)
38		eqid 2734	. . . . . . 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 2734	. . . . . . 7 ⊢ ((𝑄 · (2 · 𝑦)) mod 𝑃) = ((𝑄 · (2 · 𝑦)) mod 𝑃)
40		eqid 2734	. . . . . . 7 ⊢ (ℤ/nℤ‘𝑃) = (ℤ/nℤ‘𝑃)
41		eqid 2734	. . . . . . 7 ⊢ (mulGrp‘(ℤ/nℤ‘𝑃)) = (mulGrp‘(ℤ/nℤ‘𝑃))
42		eqid 2734	. . . . . . 7 ⊢ (ℤRHom‘(ℤ/nℤ‘𝑃)) = (ℤRHom‘(ℤ/nℤ‘𝑃))
43	5, 1, 36, 37, 38, 39, 40, 41, 42	lgseisenlem4 27436	. . . . . 6 ⊢ (𝜑 → ((𝑄↑((𝑃 − 1) / 2)) mod 𝑃) = ((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) mod 𝑃))
44		modadd1 13944	. . . . . 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 1380	. . . . 5 ⊢ (𝜑 → (((𝑄↑((𝑃 − 1) / 2)) + 1) mod 𝑃) = (((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 1) mod 𝑃))
46		peano2re 11431	. . . . . . 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 11492	. . . . . . . 8 ⊢ -1 = (0 − 1)
49		neg1cn 12377	. . . . . . . . . . . . 13 ⊢ -1 ∈ ℂ
50		absexpz 15340	. . . . . . . . . . . . 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 1464	. . . . . . . . . . . 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 11210	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℂ
53	52	absnegi 15435	. . . . . . . . . . . . . . 15 ⊢ (abs‘-1) = (abs‘1)
54		abs1 15332	. . . . . . . . . . . . . . 15 ⊢ (abs‘1) = 1
55	53, 54	eqtri 2762	. . . . . . . . . . . . . 14 ⊢ (abs‘-1) = 1
56	55	oveq1i 7440	. . . . . . . . . . . . 13 ⊢ ((abs‘-1)↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) = (1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))
57		1exp 14128	. . . . . . . . . . . . . 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	eqtrid 2786	. . . . . . . . . . . 12 ⊢ (𝜑 → ((abs‘-1)↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) = 1)
60	51, 59	eqtrd 2774	. . . . . . . . . . 11 ⊢ (𝜑 → (abs‘(-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) = 1)
61		1le1 11888	. . . . . . . . . . 11 ⊢ 1 ≤ 1
62	60, 61	eqbrtrdi 5186	. . . . . . . . . 10 ⊢ (𝜑 → (abs‘(-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) ≤ 1)
63		absle 15350	. . . . . . . . . . 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 586	. . . . . . . . . 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 232	. . . . . . . . 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 494	. . . . . . . 8 ⊢ (𝜑 → -1 ≤ (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))
67	48, 66	eqbrtrrid 5183	. . . . . . 7 ⊢ (𝜑 → (0 − 1) ≤ (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))
68		0red 11261	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℝ)
69	68, 34, 32	lesubaddd 11857	. . . . . . 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 232	. . . . . 6 ⊢ (𝜑 → 0 ≤ ((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 1))
71	20	nnred 12278	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ ℝ)
72		peano2rem 11573	. . . . . . . . 9 ⊢ (𝑃 ∈ ℝ → (𝑃 − 1) ∈ ℝ)
73	71, 72	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝑃 − 1) ∈ ℝ)
74	65	simprd 495	. . . . . . . 8 ⊢ (𝜑 → (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ≤ 1)
75		df-2 12326	. . . . . . . . . 10 ⊢ 2 = (1 + 1)
76	23	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 2 ∈ ℝ)
77	5	eldifad 3974	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑃 ∈ ℙ)
78		prmuz2 16729	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ_≥‘2))
79		eluzle 12888	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ (ℤ_≥‘2) → 2 ≤ 𝑃)
80	77, 78, 79	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 2 ≤ 𝑃)
81		eldifsni 4794	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ≠ 2)
82	5, 81	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ≠ 2)
83	76, 71, 80, 82	leneltd 11412	. . . . . . . . . 10 ⊢ (𝜑 → 2 < 𝑃)
84	75, 83	eqbrtrrid 5183	. . . . . . . . 9 ⊢ (𝜑 → (1 + 1) < 𝑃)
85	34, 34, 71	ltaddsubd 11860	. . . . . . . . 9 ⊢ (𝜑 → ((1 + 1) < 𝑃 ↔ 1 < (𝑃 − 1)))
86	84, 85	mpbid 232	. . . . . . . 8 ⊢ (𝜑 → 1 < (𝑃 − 1))
87	32, 34, 73, 74, 86	lelttrd 11416	. . . . . . 7 ⊢ (𝜑 → (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) < (𝑃 − 1))
88	32, 34, 71	ltaddsubd 11860	. . . . . . 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 257	. . . . . 6 ⊢ (𝜑 → ((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 1) < 𝑃)
90		modid 13932	. . . . . 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 2774	. . . 4 ⊢ (𝜑 → (((𝑄↑((𝑃 − 1) / 2)) + 1) mod 𝑃) = ((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 1))
93	92	oveq1d 7445	. . 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 11286	. . . 4 ⊢ (𝜑 → (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ∈ ℂ)
95		pncan 11511	. . . 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 586	. . 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 2774	. 2 ⊢ (𝜑 → ((((𝑄↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1) = (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))
98	7, 97	eqtrd 2774	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 206 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 ∖ cdif 3959 {csn 4630 class class class wbr 5147 ↦ cmpt 5230 ‘cfv 6562 (class class class)co 7430 ℂcc 11150 ℝcr 11151 0cc0 11152 1c1 11153 + caddc 11155 · cmul 11157 < clt 11292 ≤ cle 11293 − cmin 11489 -cneg 11490 / cdiv 11917 ℕcn 12263 2c2 12318 ℤcz 12610 ℤ_≥cuz 12875 ℝ⁺crp 13031 ...cfz 13543 ⌊cfl 13826 mod cmo 13905 ↑cexp 14098 abscabs 15269 Σcsu 15718 ℙcprime 16704 mulGrpcmgp 20151 ℤRHomczrh 21527 ℤ/nℤczn 21530 /_L clgs 27352

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-inf2 9678 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 ax-addf 11231 ax-mulf 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7696 df-om 7887 df-1st 8012 df-2nd 8013 df-supp 8184 df-tpos 8249 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-oadd 8508 df-er 8743 df-ec 8745 df-qs 8749 df-map 8866 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-fsupp 9399 df-sup 9479 df-inf 9480 df-oi 9547 df-dju 9938 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-xnn0 12597 df-z 12611 df-dec 12731 df-uz 12876 df-q 12988 df-rp 13032 df-fz 13544 df-fzo 13691 df-fl 13828 df-mod 13906 df-seq 14039 df-exp 14099 df-hash 14366 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-clim 15520 df-sum 15719 df-dvds 16287 df-gcd 16528 df-prm 16705 df-phi 16799 df-pc 16870 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-0g 17487 df-gsum 17488 df-imas 17554 df-qus 17555 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-mhm 18808 df-submnd 18809 df-grp 18966 df-minusg 18967 df-sbg 18968 df-mulg 19098 df-subg 19153 df-nsg 19154 df-eqg 19155 df-ghm 19243 df-cntz 19347 df-cmn 19814 df-abl 19815 df-mgp 20152 df-rng 20170 df-ur 20199 df-ring 20252 df-cring 20253 df-oppr 20350 df-dvdsr 20373 df-unit 20374 df-invr 20404 df-dvr 20417 df-rhm 20488 df-nzr 20529 df-subrng 20562 df-subrg 20586 df-rlreg 20710 df-domn 20711 df-idom 20712 df-drng 20747 df-field 20748 df-lmod 20876 df-lss 20947 df-lsp 20987 df-sra 21189 df-rgmod 21190 df-lidl 21235 df-rsp 21236 df-2idl 21277 df-cnfld 21382 df-zring 21475 df-zrh 21531 df-zn 21534 df-lgs 27353

This theorem is referenced by: lgsquadlem2 27439

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