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Theorem lgseisen 15399

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 3168	. . . 4 ⊢ (𝜑 → 𝑄 ∈ ℙ)
3		prmz 12304	. . . 4 ⊢ (𝑄 ∈ ℙ → 𝑄 ∈ ℤ)
4	2, 3	syl 14	. . 3 ⊢ (𝜑 → 𝑄 ∈ ℤ)
5		lgseisen.1	. . 3 ⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))
6		lgsval3 15343	. . 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 15374	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ ℕ)
9		oddprm 12453	. . . . . . . . 9 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((𝑃 − 1) / 2) ∈ ℕ)
10	5, 9	syl 14	. . . . . . . 8 ⊢ (𝜑 → ((𝑃 − 1) / 2) ∈ ℕ)
11	10	nnnn0d 9319	. . . . . . 7 ⊢ (𝜑 → ((𝑃 − 1) / 2) ∈ ℕ₀)
12	8, 11	nnexpcld 10804	. . . . . 6 ⊢ (𝜑 → (𝑄↑((𝑃 − 1) / 2)) ∈ ℕ)
13		nnq 9724	. . . . . 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 9370	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℤ)
16	15	znegcld 9467	. . . . . . 7 ⊢ (𝜑 → -1 ∈ ℤ)
17		zq 9717	. . . . . . 7 ⊢ (-1 ∈ ℤ → -1 ∈ ℚ)
18	16, 17	syl 14	. . . . . 6 ⊢ (𝜑 → -1 ∈ ℚ)
19		neg1ne0 9114	. . . . . . 7 ⊢ -1 ≠ 0
20	19	a1i 9	. . . . . 6 ⊢ (𝜑 → -1 ≠ 0)
21	10	nnzd 9464	. . . . . . . 8 ⊢ (𝜑 → ((𝑃 − 1) / 2) ∈ ℤ)
22	15, 21	fzfigd 10540	. . . . . . 7 ⊢ (𝜑 → (1...((𝑃 − 1) / 2)) ∈ Fin)
23	5	gausslemma2dlem0a 15374	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ ℕ)
24		znq 9715	. . . . . . . . . 10 ⊢ ((𝑄 ∈ ℤ ∧ 𝑃 ∈ ℕ) → (𝑄 / 𝑃) ∈ ℚ)
25	4, 23, 24	syl2anc 411	. . . . . . . . 9 ⊢ (𝜑 → (𝑄 / 𝑃) ∈ ℚ)
26		2z 9371	. . . . . . . . . . . 12 ⊢ 2 ∈ ℤ
27	26	a1i 9	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 2 ∈ ℤ)
28		elfznn 10146	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (1...((𝑃 − 1) / 2)) → 𝑥 ∈ ℕ)
29	28	adantl 277	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑥 ∈ ℕ)
30	29	nnzd 9464	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑥 ∈ ℤ)
31	27, 30	zmulcld 9471	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (2 · 𝑥) ∈ ℤ)
32		zq 9717	. . . . . . . . . 10 ⊢ ((2 · 𝑥) ∈ ℤ → (2 · 𝑥) ∈ ℚ)
33	31, 32	syl 14	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (2 · 𝑥) ∈ ℚ)
34		qmulcl 9728	. . . . . . . . 9 ⊢ (((𝑄 / 𝑃) ∈ ℚ ∧ (2 · 𝑥) ∈ ℚ) → ((𝑄 / 𝑃) · (2 · 𝑥)) ∈ ℚ)
35	25, 33, 34	syl2an2r 595	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝑄 / 𝑃) · (2 · 𝑥)) ∈ ℚ)
36	35	flqcld 10384	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))) ∈ ℤ)
37	22, 36	fsumzcl 11584	. . . . . 6 ⊢ (𝜑 → Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))) ∈ ℤ)
38		qexpclz 10669	. . . . . 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 1249	. . . . 5 ⊢ (𝜑 → (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ∈ ℚ)
40		1z 9369	. . . . . 6 ⊢ 1 ∈ ℤ
41		zq 9717	. . . . . 6 ⊢ (1 ∈ ℤ → 1 ∈ ℚ)
42	40, 41	mp1i 10	. . . . 5 ⊢ (𝜑 → 1 ∈ ℚ)
43		nnq 9724	. . . . . 6 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℚ)
44	23, 43	syl 14	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℚ)
45	23	nngt0d 9051	. . . . 5 ⊢ (𝜑 → 0 < 𝑃)
46		lgseisen.3	. . . . . 6 ⊢ (𝜑 → 𝑃 ≠ 𝑄)
47		eqid 2196	. . . . . 6 ⊢ ((𝑄 · (2 · 𝑥)) mod 𝑃) = ((𝑄 · (2 · 𝑥)) mod 𝑃)
48		eqid 2196	. . . . . 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 2196	. . . . . 6 ⊢ ((𝑄 · (2 · 𝑦)) mod 𝑃) = ((𝑄 · (2 · 𝑦)) mod 𝑃)
50		eqid 2196	. . . . . 6 ⊢ (ℤ/nℤ‘𝑃) = (ℤ/nℤ‘𝑃)
51		eqid 2196	. . . . . 6 ⊢ (mulGrp‘(ℤ/nℤ‘𝑃)) = (mulGrp‘(ℤ/nℤ‘𝑃))
52		eqid 2196	. . . . . 6 ⊢ (ℤRHom‘(ℤ/nℤ‘𝑃)) = (ℤRHom‘(ℤ/nℤ‘𝑃))
53	5, 1, 46, 47, 48, 49, 50, 51, 52	lgseisenlem4 15398	. . . . 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 10470	. . . 4 ⊢ (𝜑 → (((𝑄↑((𝑃 − 1) / 2)) + 1) mod 𝑃) = (((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 1) mod 𝑃))
55		qaddcl 9726	. . . . . 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 8217	. . . . . . 7 ⊢ -1 = (0 − 1)
58		neg1cn 9112	. . . . . . . . . . . 12 ⊢ -1 ∈ ℂ
59		neg1ap0 9116	. . . . . . . . . . . 12 ⊢ -1 # 0
60		absexpzap 11262	. . . . . . . . . . . 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 1352	. . . . . . . . . . 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 7989	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℂ
63	62	absnegi 11329	. . . . . . . . . . . . . 14 ⊢ (abs‘-1) = (abs‘1)
64		abs1 11254	. . . . . . . . . . . . . 14 ⊢ (abs‘1) = 1
65	63, 64	eqtri 2217	. . . . . . . . . . . . 13 ⊢ (abs‘-1) = 1
66	65	oveq1i 5935	. . . . . . . . . . . 12 ⊢ ((abs‘-1)↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) = (1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))
67		1exp 10677	. . . . . . . . . . . . 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 2241	. . . . . . . . . . 11 ⊢ (𝜑 → ((abs‘-1)↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) = 1)
70	61, 69	eqtrd 2229	. . . . . . . . . 10 ⊢ (𝜑 → (abs‘(-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) = 1)
71		1le1 8616	. . . . . . . . . 10 ⊢ 1 ≤ 1
72	70, 71	eqbrtrdi 4073	. . . . . . . . 9 ⊢ (𝜑 → (abs‘(-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) ≤ 1)
73		neg1rr 9113	. . . . . . . . . . . 12 ⊢ -1 ∈ ℝ
74	73	a1i 9	. . . . . . . . . . 11 ⊢ (𝜑 → -1 ∈ ℝ)
75	59	a1i 9	. . . . . . . . . . 11 ⊢ (𝜑 → -1 # 0)
76	74, 75, 37	reexpclzapd 10807	. . . . . . . . . 10 ⊢ (𝜑 → (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ∈ ℝ)
77		1re 8042	. . . . . . . . . 10 ⊢ 1 ∈ ℝ
78		absle 11271	. . . . . . . . . 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 4070	. . . . . 6 ⊢ (𝜑 → (0 − 1) ≤ (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))
83		0red 8044	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℝ)
84		1red 8058	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℝ)
85	83, 84, 76	lesubaddd 8586	. . . . . 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 9020	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ ℝ)
88		peano2rem 8310	. . . . . . . 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 9066	. . . . . . . . 9 ⊢ 2 = (1 + 1)
92		eldifsni 3752	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ≠ 2)
93	5, 92	syl 14	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ≠ 2)
94	23	nnzd 9464	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑃 ∈ ℤ)
95		zapne 9417	. . . . . . . . . . . 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 9077	. . . . . . . . . . . 12 ⊢ 2 ∈ ℝ
99	98	a1i 9	. . . . . . . . . . 11 ⊢ (𝜑 → 2 ∈ ℝ)
100	5	eldifad 3168	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑃 ∈ ℙ)
101		prmuz2 12324	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ_≥‘2))
102		eluzle 9630	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ (ℤ_≥‘2) → 2 ≤ 𝑃)
103	100, 101, 102	3syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 2 ≤ 𝑃)
104	99, 87, 103	leltapd 8683	. . . . . . . . . 10 ⊢ (𝜑 → (2 < 𝑃 ↔ 𝑃 # 2))
105	97, 104	mpbird 167	. . . . . . . . 9 ⊢ (𝜑 → 2 < 𝑃)
106	91, 105	eqbrtrrid 4070	. . . . . . . 8 ⊢ (𝜑 → (1 + 1) < 𝑃)
107	84, 84, 87	ltaddsubd 8589	. . . . . . . 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 8168	. . . . . 6 ⊢ (𝜑 → (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) < (𝑃 − 1))
110	76, 84, 87	ltaddsubd 8589	. . . . . 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 10458	. . . . 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 1250	. . . 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 2229	. . 3 ⊢ (𝜑 → (((𝑄↑((𝑃 − 1) / 2)) + 1) mod 𝑃) = ((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 1))
115	114	oveq1d 5940	. 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 8072	. . 3 ⊢ (𝜑 → (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ∈ ℂ)
117		pncan 8249	. . 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 2233	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 1364 ∈ wcel 2167 ≠ wne 2367 ∖ cdif 3154 {csn 3623 class class class wbr 4034 ↦ cmpt 4095 ‘cfv 5259 (class class class)co 5925 ℂcc 7894 ℝcr 7895 0cc0 7896 1c1 7897 + caddc 7899 · cmul 7901 < clt 8078 ≤ cle 8079 − cmin 8214 -cneg 8215 # cap 8625 / cdiv 8716 ℕcn 9007 2c2 9058 ℤcz 9343 ℤ_≥cuz 9618 ℚcq 9710 ...cfz 10100 ⌊cfl 10375 mod cmo 10431 ↑cexp 10647 abscabs 11179 Σcsu 11535 ℙcprime 12300 mulGrpcmgp 13552 ℤRHomczrh 14243 ℤ/nℤczn 14245 /_L clgs 15322

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 615 ax-in2 616 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4149 ax-sep 4152 ax-nul 4160 ax-pow 4208 ax-pr 4243 ax-un 4469 ax-setind 4574 ax-iinf 4625 ax-cnex 7987 ax-resscn 7988 ax-1cn 7989 ax-1re 7990 ax-icn 7991 ax-addcl 7992 ax-addrcl 7993 ax-mulcl 7994 ax-mulrcl 7995 ax-addcom 7996 ax-mulcom 7997 ax-addass 7998 ax-mulass 7999 ax-distr 8000 ax-i2m1 8001 ax-0lt1 8002 ax-1rid 8003 ax-0id 8004 ax-rnegex 8005 ax-precex 8006 ax-cnre 8007 ax-pre-ltirr 8008 ax-pre-ltwlin 8009 ax-pre-lttrn 8010 ax-pre-apti 8011 ax-pre-ltadd 8012 ax-pre-mulgt0 8013 ax-pre-mulext 8014 ax-arch 8015 ax-caucvg 8016 ax-addf 8018 ax-mulf 8019

This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-xor 1387 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rmo 2483 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3452 df-if 3563 df-pw 3608 df-sn 3629 df-pr 3630 df-tp 3631 df-op 3632 df-uni 3841 df-int 3876 df-iun 3919 df-br 4035 df-opab 4096 df-mpt 4097 df-tr 4133 df-id 4329 df-po 4332 df-iso 4333 df-iord 4402 df-on 4404 df-ilim 4405 df-suc 4407 df-iom 4628 df-xp 4670 df-rel 4671 df-cnv 4672 df-co 4673 df-dm 4674 df-rn 4675 df-res 4676 df-ima 4677 df-iota 5220 df-fun 5261 df-fn 5262 df-f 5263 df-f1 5264 df-fo 5265 df-f1o 5266 df-fv 5267 df-isom 5268 df-riota 5880 df-ov 5928 df-oprab 5929 df-mpo 5930 df-of 6139 df-1st 6207 df-2nd 6208 df-tpos 6312 df-recs 6372 df-irdg 6437 df-frec 6458 df-1o 6483 df-2o 6484 df-oadd 6487 df-er 6601 df-ec 6603 df-qs 6607 df-map 6718 df-en 6809 df-dom 6810 df-fin 6811 df-sup 7059 df-inf 7060 df-pnf 8080 df-mnf 8081 df-xr 8082 df-ltxr 8083 df-le 8084 df-sub 8216 df-neg 8217 df-reap 8619 df-ap 8626 df-div 8717 df-inn 9008 df-2 9066 df-3 9067 df-4 9068 df-5 9069 df-6 9070 df-7 9071 df-8 9072 df-9 9073 df-n0 9267 df-z 9344 df-dec 9475 df-uz 9619 df-q 9711 df-rp 9746 df-fz 10101 df-fzo 10235 df-fl 10377 df-mod 10432 df-seqfrec 10557 df-exp 10648 df-ihash 10885 df-cj 11024 df-re 11025 df-im 11026 df-rsqrt 11180 df-abs 11181 df-clim 11461 df-sumdc 11536 df-proddc 11733 df-dvds 11970 df-gcd 12146 df-prm 12301 df-phi 12404 df-pc 12479 df-struct 12705 df-ndx 12706 df-slot 12707 df-base 12709 df-sets 12710 df-iress 12711 df-plusg 12793 df-mulr 12794 df-starv 12795 df-sca 12796 df-vsca 12797 df-ip 12798 df-tset 12799 df-ple 12800 df-ds 12802 df-unif 12803 df-0g 12960 df-igsum 12961 df-topgen 12962 df-iimas 13004 df-qus 13005 df-mgm 13058 df-sgrp 13104 df-mnd 13119 df-mhm 13161 df-submnd 13162 df-grp 13205 df-minusg 13206 df-sbg 13207 df-mulg 13326 df-subg 13376 df-nsg 13377 df-eqg 13378 df-ghm 13447 df-cmn 13492 df-abl 13493 df-mgp 13553 df-rng 13565 df-ur 13592 df-srg 13596 df-ring 13630 df-cring 13631 df-oppr 13700 df-dvdsr 13721 df-unit 13722 df-invr 13753 df-dvr 13764 df-rhm 13784 df-nzr 13812 df-subrg 13851 df-domn 13891 df-idom 13892 df-lmod 13921 df-lssm 13985 df-lsp 14019 df-sra 14067 df-rgmod 14068 df-lidl 14101 df-rsp 14102 df-2idl 14132 df-bl 14178 df-mopn 14179 df-fg 14181 df-metu 14182 df-cnfld 14189 df-zring 14223 df-zrh 14246 df-zn 14248 df-lgs 15323

This theorem is referenced by: lgsquadlem2 15403

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