lgseisen - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > lgseisen		Structured version Visualization version GIF version

Theorem lgseisen 26764

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 3925	. . . 4 ⊢ (𝜑 → 𝑄 ∈ ℙ)
3		prmz 16562	. . . 4 ⊢ (𝑄 ∈ ℙ → 𝑄 ∈ ℤ)
4	2, 3	syl 17	. . 3 ⊢ (𝜑 → 𝑄 ∈ ℤ)
5		lgseisen.1	. . 3 ⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))
6		lgsval3 26700	. . 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 26741	. . . . . . . 8 ⊢ (𝜑 → 𝑄 ∈ ℕ)
9		oddprm 16693	. . . . . . . . . 10 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((𝑃 − 1) / 2) ∈ ℕ)
10	5, 9	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃 − 1) / 2) ∈ ℕ)
11	10	nnnn0d 12482	. . . . . . . 8 ⊢ (𝜑 → ((𝑃 − 1) / 2) ∈ ℕ₀)
12	8, 11	nnexpcld 14158	. . . . . . 7 ⊢ (𝜑 → (𝑄↑((𝑃 − 1) / 2)) ∈ ℕ)
13	12	nnred 12177	. . . . . 6 ⊢ (𝜑 → (𝑄↑((𝑃 − 1) / 2)) ∈ ℝ)
14		neg1rr 12277	. . . . . . . 8 ⊢ -1 ∈ ℝ
15	14	a1i 11	. . . . . . 7 ⊢ (𝜑 → -1 ∈ ℝ)
16		neg1ne0 12278	. . . . . . . 8 ⊢ -1 ≠ 0
17	16	a1i 11	. . . . . . 7 ⊢ (𝜑 → -1 ≠ 0)
18		fzfid 13888	. . . . . . . 8 ⊢ (𝜑 → (1...((𝑃 − 1) / 2)) ∈ Fin)
19	8	nnred 12177	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑄 ∈ ℝ)
20	5	gausslemma2dlem0a 26741	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑃 ∈ ℕ)
21	19, 20	nndivred 12216	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑄 / 𝑃) ∈ ℝ)
22	21	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑄 / 𝑃) ∈ ℝ)
23		2re 12236	. . . . . . . . . . 11 ⊢ 2 ∈ ℝ
24		elfznn 13480	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (1...((𝑃 − 1) / 2)) → 𝑥 ∈ ℕ)
25	24	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑥 ∈ ℕ)
26	25	nnred 12177	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑥 ∈ ℝ)
27		remulcl 11145	. . . . . . . . . . 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 11194	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝑄 / 𝑃) · (2 · 𝑥)) ∈ ℝ)
30	29	flcld 13713	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))) ∈ ℤ)
31	18, 30	fsumzcl 15631	. . . . . . 7 ⊢ (𝜑 → Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))) ∈ ℤ)
32	15, 17, 31	reexpclzd 14162	. . . . . 6 ⊢ (𝜑 → (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ∈ ℝ)
33		1re 11164	. . . . . . 7 ⊢ 1 ∈ ℝ
34	33	a1i 11	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℝ)
35	20	nnrpd 12964	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℝ⁺)
36		lgseisen.3	. . . . . . 7 ⊢ (𝜑 → 𝑃 ≠ 𝑄)
37		eqid 2731	. . . . . . 7 ⊢ ((𝑄 · (2 · 𝑥)) mod 𝑃) = ((𝑄 · (2 · 𝑥)) mod 𝑃)
38		eqid 2731	. . . . . . 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 2731	. . . . . . 7 ⊢ ((𝑄 · (2 · 𝑦)) mod 𝑃) = ((𝑄 · (2 · 𝑦)) mod 𝑃)
40		eqid 2731	. . . . . . 7 ⊢ (ℤ/nℤ‘𝑃) = (ℤ/nℤ‘𝑃)
41		eqid 2731	. . . . . . 7 ⊢ (mulGrp‘(ℤ/nℤ‘𝑃)) = (mulGrp‘(ℤ/nℤ‘𝑃))
42		eqid 2731	. . . . . . 7 ⊢ (ℤRHom‘(ℤ/nℤ‘𝑃)) = (ℤRHom‘(ℤ/nℤ‘𝑃))
43	5, 1, 36, 37, 38, 39, 40, 41, 42	lgseisenlem4 26763	. . . . . 6 ⊢ (𝜑 → ((𝑄↑((𝑃 − 1) / 2)) mod 𝑃) = ((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) mod 𝑃))
44		modadd1 13823	. . . . . 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 1381	. . . . 5 ⊢ (𝜑 → (((𝑄↑((𝑃 − 1) / 2)) + 1) mod 𝑃) = (((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 1) mod 𝑃))
46		peano2re 11337	. . . . . . 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 11397	. . . . . . . 8 ⊢ -1 = (0 − 1)
49		neg1cn 12276	. . . . . . . . . . . . 13 ⊢ -1 ∈ ℂ
50		absexpz 15202	. . . . . . . . . . . . 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 1465	. . . . . . . . . . . 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 11118	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℂ
53	52	absnegi 15297	. . . . . . . . . . . . . . 15 ⊢ (abs‘-1) = (abs‘1)
54		abs1 15194	. . . . . . . . . . . . . . 15 ⊢ (abs‘1) = 1
55	53, 54	eqtri 2759	. . . . . . . . . . . . . 14 ⊢ (abs‘-1) = 1
56	55	oveq1i 7372	. . . . . . . . . . . . 13 ⊢ ((abs‘-1)↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) = (1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))
57		1exp 14007	. . . . . . . . . . . . . 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 2783	. . . . . . . . . . . 12 ⊢ (𝜑 → ((abs‘-1)↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) = 1)
60	51, 59	eqtrd 2771	. . . . . . . . . . 11 ⊢ (𝜑 → (abs‘(-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) = 1)
61		1le1 11792	. . . . . . . . . . 11 ⊢ 1 ≤ 1
62	60, 61	eqbrtrdi 5149	. . . . . . . . . 10 ⊢ (𝜑 → (abs‘(-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) ≤ 1)
63		absle 15212	. . . . . . . . . . 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 231	. . . . . . . . 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 495	. . . . . . . 8 ⊢ (𝜑 → -1 ≤ (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))
67	48, 66	eqbrtrrid 5146	. . . . . . 7 ⊢ (𝜑 → (0 − 1) ≤ (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))
68		0red 11167	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℝ)
69	68, 34, 32	lesubaddd 11761	. . . . . . 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 231	. . . . . 6 ⊢ (𝜑 → 0 ≤ ((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 1))
71	20	nnred 12177	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ ℝ)
72		peano2rem 11477	. . . . . . . . 9 ⊢ (𝑃 ∈ ℝ → (𝑃 − 1) ∈ ℝ)
73	71, 72	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝑃 − 1) ∈ ℝ)
74	65	simprd 496	. . . . . . . 8 ⊢ (𝜑 → (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ≤ 1)
75		df-2 12225	. . . . . . . . . 10 ⊢ 2 = (1 + 1)
76	23	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 2 ∈ ℝ)
77	5	eldifad 3925	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑃 ∈ ℙ)
78		prmuz2 16583	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ_≥‘2))
79		eluzle 12785	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ (ℤ_≥‘2) → 2 ≤ 𝑃)
80	77, 78, 79	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 2 ≤ 𝑃)
81		eldifsni 4755	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ≠ 2)
82	5, 81	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ≠ 2)
83	76, 71, 80, 82	leneltd 11318	. . . . . . . . . 10 ⊢ (𝜑 → 2 < 𝑃)
84	75, 83	eqbrtrrid 5146	. . . . . . . . 9 ⊢ (𝜑 → (1 + 1) < 𝑃)
85	34, 34, 71	ltaddsubd 11764	. . . . . . . . 9 ⊢ (𝜑 → ((1 + 1) < 𝑃 ↔ 1 < (𝑃 − 1)))
86	84, 85	mpbid 231	. . . . . . . 8 ⊢ (𝜑 → 1 < (𝑃 − 1))
87	32, 34, 73, 74, 86	lelttrd 11322	. . . . . . 7 ⊢ (𝜑 → (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) < (𝑃 − 1))
88	32, 34, 71	ltaddsubd 11764	. . . . . . 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 256	. . . . . 6 ⊢ (𝜑 → ((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 1) < 𝑃)
90		modid 13811	. . . . . 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 837	. . . . 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 2771	. . . 4 ⊢ (𝜑 → (((𝑄↑((𝑃 − 1) / 2)) + 1) mod 𝑃) = ((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 1))
93	92	oveq1d 7377	. . 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 11192	. . . 4 ⊢ (𝜑 → (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ∈ ℂ)
95		pncan 11416	. . . 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 2771	. 2 ⊢ (𝜑 → ((((𝑄↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1) = (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))
98	7, 97	eqtrd 2771	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 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2939 ∖ cdif 3910 {csn 4591 class class class wbr 5110 ↦ cmpt 5193 ‘cfv 6501 (class class class)co 7362 ℂcc 11058 ℝcr 11059 0cc0 11060 1c1 11061 + caddc 11063 · cmul 11065 < clt 11198 ≤ cle 11199 − cmin 11394 -cneg 11395 / cdiv 11821 ℕcn 12162 2c2 12217 ℤcz 12508 ℤ_≥cuz 12772 ℝ⁺crp 12924 ...cfz 13434 ⌊cfl 13705 mod cmo 13784 ↑cexp 13977 abscabs 15131 Σcsu 15582 ℙcprime 16558 mulGrpcmgp 19910 ℤRHomczrh 20937 ℤ/nℤczn 20940 /_L clgs 26679

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-inf2 9586 ax-cnex 11116 ax-resscn 11117 ax-1cn 11118 ax-icn 11119 ax-addcl 11120 ax-addrcl 11121 ax-mulcl 11122 ax-mulrcl 11123 ax-mulcom 11124 ax-addass 11125 ax-mulass 11126 ax-distr 11127 ax-i2m1 11128 ax-1ne0 11129 ax-1rid 11130 ax-rnegex 11131 ax-rrecex 11132 ax-cnre 11133 ax-pre-lttri 11134 ax-pre-lttrn 11135 ax-pre-ltadd 11136 ax-pre-mulgt0 11137 ax-pre-sup 11138 ax-addf 11139 ax-mulf 11140

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3448 df-sbc 3743 df-csb 3859 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-se 5594 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-isom 6510 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-of 7622 df-om 7808 df-1st 7926 df-2nd 7927 df-supp 8098 df-tpos 8162 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-2o 8418 df-oadd 8421 df-er 8655 df-ec 8657 df-qs 8661 df-map 8774 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-fsupp 9313 df-sup 9387 df-inf 9388 df-oi 9455 df-dju 9846 df-card 9884 df-pnf 11200 df-mnf 11201 df-xr 11202 df-ltxr 11203 df-le 11204 df-sub 11396 df-neg 11397 df-div 11822 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12423 df-xnn0 12495 df-z 12509 df-dec 12628 df-uz 12773 df-q 12883 df-rp 12925 df-fz 13435 df-fzo 13578 df-fl 13707 df-mod 13785 df-seq 13917 df-exp 13978 df-hash 14241 df-cj 14996 df-re 14997 df-im 14998 df-sqrt 15132 df-abs 15133 df-clim 15382 df-sum 15583 df-dvds 16148 df-gcd 16386 df-prm 16559 df-phi 16649 df-pc 16720 df-struct 17030 df-sets 17047 df-slot 17065 df-ndx 17077 df-base 17095 df-ress 17124 df-plusg 17160 df-mulr 17161 df-starv 17162 df-sca 17163 df-vsca 17164 df-ip 17165 df-tset 17166 df-ple 17167 df-ds 17169 df-unif 17170 df-0g 17337 df-gsum 17338 df-imas 17404 df-qus 17405 df-mgm 18511 df-sgrp 18560 df-mnd 18571 df-mhm 18615 df-submnd 18616 df-grp 18765 df-minusg 18766 df-sbg 18767 df-mulg 18887 df-subg 18939 df-nsg 18940 df-eqg 18941 df-ghm 19020 df-cntz 19111 df-cmn 19578 df-abl 19579 df-mgp 19911 df-ur 19928 df-ring 19980 df-cring 19981 df-oppr 20063 df-dvdsr 20084 df-unit 20085 df-invr 20115 df-dvr 20126 df-rnghom 20162 df-nzr 20202 df-drng 20227 df-field 20228 df-subrg 20268 df-lmod 20380 df-lss 20450 df-lsp 20490 df-sra 20692 df-rgmod 20693 df-lidl 20694 df-rsp 20695 df-2idl 20761 df-rlreg 20790 df-domn 20791 df-idom 20792 df-cnfld 20834 df-zring 20907 df-zrh 20941 df-zn 20944 df-lgs 26680

This theorem is referenced by: lgsquadlem2 26766

W3C validator