lgseisen - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > lgseisen		GIF version

Theorem lgseisen 16364

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 3231	. . . 4 ⊢ (𝜑 → 𝑄 ∈ ℙ)
3		prmz 12908	. . . 4 ⊢ (𝑄 ∈ ℙ → 𝑄 ∈ ℤ)
4	2, 3	syl 14	. . 3 ⊢ (𝜑 → 𝑄 ∈ ℤ)
5		lgseisen.1	. . 3 ⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))
6		lgsval3 16308	. . 3 ⊢ ((𝑄 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) → (𝑄 /_L 𝑃) = ((((𝑄↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1))
7	4, 5, 6	syl2anc 415	. 2 ⊢ (𝜑 → (𝑄 /_L 𝑃) = ((((𝑄↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1))
8	1	gausslemma2dlem0a 16339	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ ℕ)
9		oddprm 13061	. . . . . . . . 9 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((𝑃 − 1) / 2) ∈ ℕ)
10	5, 9	syl 14	. . . . . . . 8 ⊢ (𝜑 → ((𝑃 − 1) / 2) ∈ ℕ)
11	10	nnnn0d 9625	. . . . . . 7 ⊢ (𝜑 → ((𝑃 − 1) / 2) ∈ ℕ₀)
12	8, 11	nnexpcld 11148	. . . . . 6 ⊢ (𝜑 → (𝑄↑((𝑃 − 1) / 2)) ∈ ℕ)
13		nnq 10043	. . . . . 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 9676	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℤ)
16	15	znegcld 9775	. . . . . . 7 ⊢ (𝜑 → -1 ∈ ℤ)
17		zq 10036	. . . . . . 7 ⊢ (-1 ∈ ℤ → -1 ∈ ℚ)
18	16, 17	syl 14	. . . . . 6 ⊢ (𝜑 → -1 ∈ ℚ)
19		neg1ne0 9414	. . . . . . 7 ⊢ -1 ≠ 0
20	19	a1i 9	. . . . . 6 ⊢ (𝜑 → -1 ≠ 0)
21	10	nnzd 9772	. . . . . . . 8 ⊢ (𝜑 → ((𝑃 − 1) / 2) ∈ ℤ)
22	15, 21	fzfigd 10883	. . . . . . 7 ⊢ (𝜑 → (1...((𝑃 − 1) / 2)) ∈ Fin)
23	5	gausslemma2dlem0a 16339	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ ℕ)
24		znq 10034	. . . . . . . . . 10 ⊢ ((𝑄 ∈ ℤ ∧ 𝑃 ∈ ℕ) → (𝑄 / 𝑃) ∈ ℚ)
25	4, 23, 24	syl2anc 415	. . . . . . . . 9 ⊢ (𝜑 → (𝑄 / 𝑃) ∈ ℚ)
26		2z 9677	. . . . . . . . . . . 12 ⊢ 2 ∈ ℤ
27	26	a1i 9	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 2 ∈ ℤ)
28		elfznn 10471	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (1...((𝑃 − 1) / 2)) → 𝑥 ∈ ℕ)
29	28	adantl 277	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑥 ∈ ℕ)
30	29	nnzd 9772	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑥 ∈ ℤ)
31	27, 30	zmulcld 9779	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (2 · 𝑥) ∈ ℤ)
32		zq 10036	. . . . . . . . . 10 ⊢ ((2 · 𝑥) ∈ ℤ → (2 · 𝑥) ∈ ℚ)
33	31, 32	syl 14	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (2 · 𝑥) ∈ ℚ)
34		qmulcl 10047	. . . . . . . . 9 ⊢ (((𝑄 / 𝑃) ∈ ℚ ∧ (2 · 𝑥) ∈ ℚ) → ((𝑄 / 𝑃) · (2 · 𝑥)) ∈ ℚ)
35	25, 33, 34	syl2an2r 603	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝑄 / 𝑃) · (2 · 𝑥)) ∈ ℚ)
36	35	flqcld 10725	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))) ∈ ℤ)
37	22, 36	fsumzcl 12188	. . . . . 6 ⊢ (𝜑 → Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))) ∈ ℤ)
38		qexpclz 11012	. . . . . 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 1278	. . . . 5 ⊢ (𝜑 → (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ∈ ℚ)
40		1z 9675	. . . . . 6 ⊢ 1 ∈ ℤ
41		zq 10036	. . . . . 6 ⊢ (1 ∈ ℤ → 1 ∈ ℚ)
42	40, 41	mp1i 10	. . . . 5 ⊢ (𝜑 → 1 ∈ ℚ)
43		nnq 10043	. . . . . 6 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℚ)
44	23, 43	syl 14	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℚ)
45	23	nngt0d 9351	. . . . 5 ⊢ (𝜑 → 0 < 𝑃)
46		lgseisen.3	. . . . . 6 ⊢ (𝜑 → 𝑃 ≠ 𝑄)
47		eqid 2238	. . . . . 6 ⊢ ((𝑄 · (2 · 𝑥)) mod 𝑃) = ((𝑄 · (2 · 𝑥)) mod 𝑃)
48		eqid 2238	. . . . . 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 2238	. . . . . 6 ⊢ ((𝑄 · (2 · 𝑦)) mod 𝑃) = ((𝑄 · (2 · 𝑦)) mod 𝑃)
50		eqid 2238	. . . . . 6 ⊢ (ℤ/nℤ‘𝑃) = (ℤ/nℤ‘𝑃)
51		eqid 2238	. . . . . 6 ⊢ (mulGrp‘(ℤ/nℤ‘𝑃)) = (mulGrp‘(ℤ/nℤ‘𝑃))
52		eqid 2238	. . . . . 6 ⊢ (ℤRHom‘(ℤ/nℤ‘𝑃)) = (ℤRHom‘(ℤ/nℤ‘𝑃))
53	5, 1, 46, 47, 48, 49, 50, 51, 52	lgseisenlem4 16363	. . . . 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 10813	. . . 4 ⊢ (𝜑 → (((𝑄↑((𝑃 − 1) / 2)) + 1) mod 𝑃) = (((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 1) mod 𝑃))
55		qaddcl 10045	. . . . . 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 415	. . . . 5 ⊢ (𝜑 → ((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 1) ∈ ℚ)
57		df-neg 8502	. . . . . . 7 ⊢ -1 = (0 − 1)
58		neg1cn 9412	. . . . . . . . . . . 12 ⊢ -1 ∈ ℂ
59		neg1ap0 9416	. . . . . . . . . . . 12 ⊢ -1 # 0
60		absexpzap 11863	. . . . . . . . . . . 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 1382	. . . . . . . . . . 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 8273	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℂ
63	62	absnegi 11930	. . . . . . . . . . . . . 14 ⊢ (abs‘-1) = (abs‘1)
64		abs1 11854	. . . . . . . . . . . . . 14 ⊢ (abs‘1) = 1
65	63, 64	eqtri 2259	. . . . . . . . . . . . 13 ⊢ (abs‘-1) = 1
66	65	oveq1i 6095	. . . . . . . . . . . 12 ⊢ ((abs‘-1)↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) = (1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))
67		1exp 11020	. . . . . . . . . . . . 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 2283	. . . . . . . . . . 11 ⊢ (𝜑 → ((abs‘-1)↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) = 1)
70	61, 69	eqtrd 2271	. . . . . . . . . 10 ⊢ (𝜑 → (abs‘(-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) = 1)
71		1le1 8903	. . . . . . . . . 10 ⊢ 1 ≤ 1
72	70, 71	eqbrtrdi 4169	. . . . . . . . 9 ⊢ (𝜑 → (abs‘(-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) ≤ 1)
73		neg1rr 9413	. . . . . . . . . . . 12 ⊢ -1 ∈ ℝ
74	73	a1i 9	. . . . . . . . . . 11 ⊢ (𝜑 → -1 ∈ ℝ)
75	59	a1i 9	. . . . . . . . . . 11 ⊢ (𝜑 → -1 # 0)
76	74, 75, 37	reexpclzapd 11151	. . . . . . . . . 10 ⊢ (𝜑 → (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ∈ ℝ)
77		1re 8326	. . . . . . . . . 10 ⊢ 1 ∈ ℝ
78		absle 11872	. . . . . . . . . 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 417	. . . . . . . . 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 4166	. . . . . 6 ⊢ (𝜑 → (0 − 1) ≤ (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))
83		0red 8328	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℝ)
84		1red 8342	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℝ)
85	83, 84, 76	lesubaddd 8872	. . . . . 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 9320	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ ℝ)
88		peano2rem 8595	. . . . . . . 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 9366	. . . . . . . . 9 ⊢ 2 = (1 + 1)
92		eldifsni 3843	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ≠ 2)
93	5, 92	syl 14	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ≠ 2)
94	23	nnzd 9772	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑃 ∈ ℤ)
95		zapne 9724	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ ℤ ∧ 2 ∈ ℤ) → (𝑃 # 2 ↔ 𝑃 ≠ 2))
96	94, 26, 95	sylancl 417	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑃 # 2 ↔ 𝑃 ≠ 2))
97	93, 96	mpbird 167	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 # 2)
98		2re 9377	. . . . . . . . . . . 12 ⊢ 2 ∈ ℝ
99	98	a1i 9	. . . . . . . . . . 11 ⊢ (𝜑 → 2 ∈ ℝ)
100	5	eldifad 3231	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑃 ∈ ℙ)
101		prmuz2 12929	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ_≥‘2))
102		eluzle 9944	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ (ℤ_≥‘2) → 2 ≤ 𝑃)
103	100, 101, 102	3syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 2 ≤ 𝑃)
104	99, 87, 103	leltapd 8970	. . . . . . . . . 10 ⊢ (𝜑 → (2 < 𝑃 ↔ 𝑃 # 2))
105	97, 104	mpbird 167	. . . . . . . . 9 ⊢ (𝜑 → 2 < 𝑃)
106	91, 105	eqbrtrrid 4166	. . . . . . . 8 ⊢ (𝜑 → (1 + 1) < 𝑃)
107	84, 84, 87	ltaddsubd 8875	. . . . . . . 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 8453	. . . . . 6 ⊢ (𝜑 → (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) < (𝑃 − 1))
110	76, 84, 87	ltaddsubd 8875	. . . . . 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 10801	. . . . 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 1279	. . . 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 2271	. . 3 ⊢ (𝜑 → (((𝑄↑((𝑃 − 1) / 2)) + 1) mod 𝑃) = ((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 1))
115	114	oveq1d 6100	. 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 8355	. . 3 ⊢ (𝜑 → (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ∈ ℂ)
117		pncan 8534	. . 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 417	. 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 2275	1 ⊢ (𝜑 → (𝑄 /_L 𝑃) = (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))

Colors of variables: wff set class

This proof depends on syntax axioms: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1402 ∈ wcel 2209 ≠ wne 2420 ∖ cdif 3217 {csn 3709 class class class wbr 4130 ↦ cmpt 4192 ‘cfv 5377 (class class class)co 6085 ℂcc 8178 ℝcr 8179 0cc0 8180 1c1 8181 + caddc 8183 · cmul 8185 < clt 8361 ≤ cle 8362 − cmin 8499 -cneg 8500 # cap 8912 / cdiv 9005 ℕcn 9307 2c2 9358 ℤcz 9649 ℤ_≥cuz 9931 ℚcq 10029 ...cfz 10422 ⌊cfl 10714 mod cmo 10774 ↑cexp 10990 abscabs 11779 Σcsu 12138 ℙcprime 12904 mulGrpcmgp 14301 ℤRHomczrh 15030 ℤ/nℤczn 15032 /_L clgs 16287

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4246 ax-sep 4249 ax-nul 4259 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-iinf 4735 ax-cnex 8271 ax-resscn 8272 ax-1cn 8273 ax-1re 8274 ax-icn 8275 ax-addcl 8276 ax-addrcl 8277 ax-mulcl 8278 ax-mulrcl 8279 ax-addcom 8280 ax-mulcom 8281 ax-addass 8282 ax-mulass 8283 ax-distr 8284 ax-i2m1 8285 ax-0lt1 8286 ax-1rid 8287 ax-0id 8288 ax-rnegex 8289 ax-precex 8290 ax-cnre 8291 ax-pre-ltirr 8292 ax-pre-ltwlin 8293 ax-pre-lttrn 8294 ax-pre-apti 8295 ax-pre-ltadd 8296 ax-pre-mulgt0 8297 ax-pre-mulext 8298 ax-arch 8299 ax-caucvg 8300 ax-addf 8302 ax-mulf 8303

This proof depends on definitions: df-bi 117 df-stab 843 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-xor 1425 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-tp 3717 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4131 df-opab 4193 df-mpt 4194 df-tr 4230 df-id 4438 df-po 4441 df-iso 4442 df-iord 4511 df-on 4513 df-ilim 4514 df-suc 4516 df-iom 4738 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-rn 4785 df-res 4786 df-ima 4787 df-iota 5337 df-fun 5379 df-fn 5380 df-f 5381 df-f1 5382 df-fo 5383 df-f1o 5384 df-fv 5385 df-isom 5386 df-riota 6038 df-ov 6088 df-oprab 6089 df-mpo 6090 df-of 6302 df-1st 6374 df-2nd 6375 df-tpos 6516 df-recs 6576 df-irdg 6641 df-frec 6662 df-1o 6687 df-2o 6688 df-oadd 6691 df-er 6807 df-ec 6809 df-qs 6813 df-map 6924 df-en 7023 df-dom 7024 df-fin 7025 df-sup 7325 df-inf 7326 df-pnf 8363 df-mnf 8364 df-xr 8365 df-ltxr 8366 df-le 8367 df-sub 8501 df-neg 8502 df-reap 8906 df-ap 8913 df-div 9006 df-inn 9308 df-2 9366 df-3 9367 df-4 9368 df-5 9369 df-6 9370 df-7 9371 df-8 9372 df-9 9373 df-n0 9569 df-z 9650 df-dec 9783 df-uz 9932 df-q 10030 df-rp 10066 df-fz 10423 df-fzo 10561 df-fl 10716 df-mod 10775 df-seqfrec 10900 df-exp 10991 df-ihash 11231 df-cj 11623 df-re 11624 df-im 11625 df-rsqrt 11780 df-abs 11781 df-clim 12064 df-sumdc 12139 df-proddc 12337 df-dvds 12574 df-gcd 12750 df-prm 12905 df-phi 13012 df-pc 13087 df-struct 13406 df-ndx 13407 df-slot 13408 df-base 13410 df-sets 13411 df-iress 13412 df-plusg 13497 df-mulr 13498 df-starv 13499 df-sca 13500 df-vsca 13501 df-ip 13502 df-tset 13503 df-ple 13504 df-ds 13506 df-unif 13507 df-0g 13665 df-gzsum 13666 df-topgen 13667 df-iimas 13677 df-qus 13678 df-mgm 13729 df-sgrp 13770 df-mnd 13783 df-mhm 13819 df-submnd 13820 df-grp 13861 df-minusg 13862 df-sbg 13863 df-mulg 13976 df-subg 14026 df-nsg 14027 df-eqg 14028 df-ghm 14097 df-cmn 14173 df-abl 14174 df-gsumfi 14235 df-mgp 14302 df-rng 14316 df-ur 14347 df-srg 14352 df-ring 14386 df-cring 14387 df-oppr 14457 df-dvdsr 14479 df-unit 14480 df-invr 14512 df-dvr 14523 df-rhm 14543 df-nzr 14571 df-subrg 14611 df-domn 14651 df-idom 14652 df-lmod 14709 df-lssm 14774 df-lsp 14808 df-sra 14856 df-rgmod 14857 df-lidl 14890 df-rsp 14891 df-2idl 14921 df-bl 14967 df-mopn 14968 df-fg 14970 df-metu 14971 df-cnfld 14978 df-zring 15010 df-zrh 15033 df-zn 15035 df-lgs 16288

This theorem is used by: lgsquadlem2 16368

W3C validator