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Theorem lgseisen 15231

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 3165	. . . 4 ⊢ (𝜑 → 𝑄 ∈ ℙ)
3		prmz 12252	. . . 4 ⊢ (𝑄 ∈ ℙ → 𝑄 ∈ ℤ)
4	2, 3	syl 14	. . 3 ⊢ (𝜑 → 𝑄 ∈ ℤ)
5		lgseisen.1	. . 3 ⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))
6		lgsval3 15175	. . 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 15206	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ ℕ)
9		oddprm 12400	. . . . . . . . 9 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((𝑃 − 1) / 2) ∈ ℕ)
10	5, 9	syl 14	. . . . . . . 8 ⊢ (𝜑 → ((𝑃 − 1) / 2) ∈ ℕ)
11	10	nnnn0d 9296	. . . . . . 7 ⊢ (𝜑 → ((𝑃 − 1) / 2) ∈ ℕ₀)
12	8, 11	nnexpcld 10769	. . . . . 6 ⊢ (𝜑 → (𝑄↑((𝑃 − 1) / 2)) ∈ ℕ)
13		nnq 9701	. . . . . 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 9347	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℤ)
16	15	znegcld 9444	. . . . . . 7 ⊢ (𝜑 → -1 ∈ ℤ)
17		zq 9694	. . . . . . 7 ⊢ (-1 ∈ ℤ → -1 ∈ ℚ)
18	16, 17	syl 14	. . . . . 6 ⊢ (𝜑 → -1 ∈ ℚ)
19		neg1ne0 9091	. . . . . . 7 ⊢ -1 ≠ 0
20	19	a1i 9	. . . . . 6 ⊢ (𝜑 → -1 ≠ 0)
21	10	nnzd 9441	. . . . . . . 8 ⊢ (𝜑 → ((𝑃 − 1) / 2) ∈ ℤ)
22	15, 21	fzfigd 10505	. . . . . . 7 ⊢ (𝜑 → (1...((𝑃 − 1) / 2)) ∈ Fin)
23	5	gausslemma2dlem0a 15206	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ ℕ)
24		znq 9692	. . . . . . . . . 10 ⊢ ((𝑄 ∈ ℤ ∧ 𝑃 ∈ ℕ) → (𝑄 / 𝑃) ∈ ℚ)
25	4, 23, 24	syl2anc 411	. . . . . . . . 9 ⊢ (𝜑 → (𝑄 / 𝑃) ∈ ℚ)
26		2z 9348	. . . . . . . . . . . 12 ⊢ 2 ∈ ℤ
27	26	a1i 9	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 2 ∈ ℤ)
28		elfznn 10123	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (1...((𝑃 − 1) / 2)) → 𝑥 ∈ ℕ)
29	28	adantl 277	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑥 ∈ ℕ)
30	29	nnzd 9441	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑥 ∈ ℤ)
31	27, 30	zmulcld 9448	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (2 · 𝑥) ∈ ℤ)
32		zq 9694	. . . . . . . . . 10 ⊢ ((2 · 𝑥) ∈ ℤ → (2 · 𝑥) ∈ ℚ)
33	31, 32	syl 14	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (2 · 𝑥) ∈ ℚ)
34		qmulcl 9705	. . . . . . . . 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 10349	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))) ∈ ℤ)
37	22, 36	fsumzcl 11548	. . . . . 6 ⊢ (𝜑 → Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))) ∈ ℤ)
38		qexpclz 10634	. . . . . 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 9346	. . . . . 6 ⊢ 1 ∈ ℤ
41		zq 9694	. . . . . 6 ⊢ (1 ∈ ℤ → 1 ∈ ℚ)
42	40, 41	mp1i 10	. . . . 5 ⊢ (𝜑 → 1 ∈ ℚ)
43		nnq 9701	. . . . . 6 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℚ)
44	23, 43	syl 14	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℚ)
45	23	nngt0d 9028	. . . . 5 ⊢ (𝜑 → 0 < 𝑃)
46		lgseisen.3	. . . . . 6 ⊢ (𝜑 → 𝑃 ≠ 𝑄)
47		eqid 2193	. . . . . 6 ⊢ ((𝑄 · (2 · 𝑥)) mod 𝑃) = ((𝑄 · (2 · 𝑥)) mod 𝑃)
48		eqid 2193	. . . . . 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 2193	. . . . . 6 ⊢ ((𝑄 · (2 · 𝑦)) mod 𝑃) = ((𝑄 · (2 · 𝑦)) mod 𝑃)
50		eqid 2193	. . . . . 6 ⊢ (ℤ/nℤ‘𝑃) = (ℤ/nℤ‘𝑃)
51		eqid 2193	. . . . . 6 ⊢ (mulGrp‘(ℤ/nℤ‘𝑃)) = (mulGrp‘(ℤ/nℤ‘𝑃))
52		eqid 2193	. . . . . 6 ⊢ (ℤRHom‘(ℤ/nℤ‘𝑃)) = (ℤRHom‘(ℤ/nℤ‘𝑃))
53	5, 1, 46, 47, 48, 49, 50, 51, 52	lgseisenlem4 15230	. . . . 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 10435	. . . 4 ⊢ (𝜑 → (((𝑄↑((𝑃 − 1) / 2)) + 1) mod 𝑃) = (((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 1) mod 𝑃))
55		qaddcl 9703	. . . . . 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 8195	. . . . . . 7 ⊢ -1 = (0 − 1)
58		neg1cn 9089	. . . . . . . . . . . 12 ⊢ -1 ∈ ℂ
59		neg1ap0 9093	. . . . . . . . . . . 12 ⊢ -1 # 0
60		absexpzap 11227	. . . . . . . . . . . 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 7967	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℂ
63	62	absnegi 11294	. . . . . . . . . . . . . 14 ⊢ (abs‘-1) = (abs‘1)
64		abs1 11219	. . . . . . . . . . . . . 14 ⊢ (abs‘1) = 1
65	63, 64	eqtri 2214	. . . . . . . . . . . . 13 ⊢ (abs‘-1) = 1
66	65	oveq1i 5929	. . . . . . . . . . . 12 ⊢ ((abs‘-1)↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) = (1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))
67		1exp 10642	. . . . . . . . . . . . 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 2238	. . . . . . . . . . 11 ⊢ (𝜑 → ((abs‘-1)↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) = 1)
70	61, 69	eqtrd 2226	. . . . . . . . . 10 ⊢ (𝜑 → (abs‘(-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) = 1)
71		1le1 8593	. . . . . . . . . 10 ⊢ 1 ≤ 1
72	70, 71	eqbrtrdi 4069	. . . . . . . . 9 ⊢ (𝜑 → (abs‘(-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) ≤ 1)
73		neg1rr 9090	. . . . . . . . . . . 12 ⊢ -1 ∈ ℝ
74	73	a1i 9	. . . . . . . . . . 11 ⊢ (𝜑 → -1 ∈ ℝ)
75	59	a1i 9	. . . . . . . . . . 11 ⊢ (𝜑 → -1 # 0)
76	74, 75, 37	reexpclzapd 10772	. . . . . . . . . 10 ⊢ (𝜑 → (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ∈ ℝ)
77		1re 8020	. . . . . . . . . 10 ⊢ 1 ∈ ℝ
78		absle 11236	. . . . . . . . . 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 4066	. . . . . 6 ⊢ (𝜑 → (0 − 1) ≤ (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))
83		0red 8022	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℝ)
84		1red 8036	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℝ)
85	83, 84, 76	lesubaddd 8563	. . . . . 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 8997	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ ℝ)
88		peano2rem 8288	. . . . . . . 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 9043	. . . . . . . . 9 ⊢ 2 = (1 + 1)
92		eldifsni 3748	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ≠ 2)
93	5, 92	syl 14	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ≠ 2)
94	23	nnzd 9441	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑃 ∈ ℤ)
95		zapne 9394	. . . . . . . . . . . 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 9054	. . . . . . . . . . . 12 ⊢ 2 ∈ ℝ
99	98	a1i 9	. . . . . . . . . . 11 ⊢ (𝜑 → 2 ∈ ℝ)
100	5	eldifad 3165	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑃 ∈ ℙ)
101		prmuz2 12272	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ_≥‘2))
102		eluzle 9607	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ (ℤ_≥‘2) → 2 ≤ 𝑃)
103	100, 101, 102	3syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 2 ≤ 𝑃)
104	99, 87, 103	leltapd 8660	. . . . . . . . . 10 ⊢ (𝜑 → (2 < 𝑃 ↔ 𝑃 # 2))
105	97, 104	mpbird 167	. . . . . . . . 9 ⊢ (𝜑 → 2 < 𝑃)
106	91, 105	eqbrtrrid 4066	. . . . . . . 8 ⊢ (𝜑 → (1 + 1) < 𝑃)
107	84, 84, 87	ltaddsubd 8566	. . . . . . . 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 8146	. . . . . 6 ⊢ (𝜑 → (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) < (𝑃 − 1))
110	76, 84, 87	ltaddsubd 8566	. . . . . 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 10423	. . . . 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 2226	. . 3 ⊢ (𝜑 → (((𝑄↑((𝑃 − 1) / 2)) + 1) mod 𝑃) = ((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 1))
115	114	oveq1d 5934	. 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 8050	. . 3 ⊢ (𝜑 → (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ∈ ℂ)
117		pncan 8227	. . 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 2230	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 2164 ≠ wne 2364 ∖ cdif 3151 {csn 3619 class class class wbr 4030 ↦ cmpt 4091 ‘cfv 5255 (class class class)co 5919 ℂcc 7872 ℝcr 7873 0cc0 7874 1c1 7875 + caddc 7877 · cmul 7879 < clt 8056 ≤ cle 8057 − cmin 8192 -cneg 8193 # cap 8602 / cdiv 8693 ℕcn 8984 2c2 9035 ℤcz 9320 ℤ_≥cuz 9595 ℚcq 9687 ...cfz 10077 ⌊cfl 10340 mod cmo 10396 ↑cexp 10612 abscabs 11144 Σcsu 11499 ℙcprime 12248 mulGrpcmgp 13419 ℤRHomczrh 14110 ℤ/nℤczn 14112 /_L clgs 15154

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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4145 ax-sep 4148 ax-nul 4156 ax-pow 4204 ax-pr 4239 ax-un 4465 ax-setind 4570 ax-iinf 4621 ax-cnex 7965 ax-resscn 7966 ax-1cn 7967 ax-1re 7968 ax-icn 7969 ax-addcl 7970 ax-addrcl 7971 ax-mulcl 7972 ax-mulrcl 7973 ax-addcom 7974 ax-mulcom 7975 ax-addass 7976 ax-mulass 7977 ax-distr 7978 ax-i2m1 7979 ax-0lt1 7980 ax-1rid 7981 ax-0id 7982 ax-rnegex 7983 ax-precex 7984 ax-cnre 7985 ax-pre-ltirr 7986 ax-pre-ltwlin 7987 ax-pre-lttrn 7988 ax-pre-apti 7989 ax-pre-ltadd 7990 ax-pre-mulgt0 7991 ax-pre-mulext 7992 ax-arch 7993 ax-caucvg 7994 ax-addf 7996 ax-mulf 7997

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 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2987 df-csb 3082 df-dif 3156 df-un 3158 df-in 3160 df-ss 3167 df-nul 3448 df-if 3559 df-pw 3604 df-sn 3625 df-pr 3626 df-tp 3627 df-op 3628 df-uni 3837 df-int 3872 df-iun 3915 df-br 4031 df-opab 4092 df-mpt 4093 df-tr 4129 df-id 4325 df-po 4328 df-iso 4329 df-iord 4398 df-on 4400 df-ilim 4401 df-suc 4403 df-iom 4624 df-xp 4666 df-rel 4667 df-cnv 4668 df-co 4669 df-dm 4670 df-rn 4671 df-res 4672 df-ima 4673 df-iota 5216 df-fun 5257 df-fn 5258 df-f 5259 df-f1 5260 df-fo 5261 df-f1o 5262 df-fv 5263 df-isom 5264 df-riota 5874 df-ov 5922 df-oprab 5923 df-mpo 5924 df-of 6132 df-1st 6195 df-2nd 6196 df-tpos 6300 df-recs 6360 df-irdg 6425 df-frec 6446 df-1o 6471 df-2o 6472 df-oadd 6475 df-er 6589 df-ec 6591 df-qs 6595 df-map 6706 df-en 6797 df-dom 6798 df-fin 6799 df-sup 7045 df-inf 7046 df-pnf 8058 df-mnf 8059 df-xr 8060 df-ltxr 8061 df-le 8062 df-sub 8194 df-neg 8195 df-reap 8596 df-ap 8603 df-div 8694 df-inn 8985 df-2 9043 df-3 9044 df-4 9045 df-5 9046 df-6 9047 df-7 9048 df-8 9049 df-9 9050 df-n0 9244 df-z 9321 df-dec 9452 df-uz 9596 df-q 9688 df-rp 9723 df-fz 10078 df-fzo 10212 df-fl 10342 df-mod 10397 df-seqfrec 10522 df-exp 10613 df-ihash 10850 df-cj 10989 df-re 10990 df-im 10991 df-rsqrt 11145 df-abs 11146 df-clim 11425 df-sumdc 11500 df-proddc 11697 df-dvds 11934 df-gcd 12083 df-prm 12249 df-phi 12352 df-pc 12426 df-struct 12623 df-ndx 12624 df-slot 12625 df-base 12627 df-sets 12628 df-iress 12629 df-plusg 12711 df-mulr 12712 df-starv 12713 df-sca 12714 df-vsca 12715 df-ip 12716 df-tset 12717 df-ple 12718 df-ds 12720 df-unif 12721 df-0g 12872 df-igsum 12873 df-topgen 12874 df-iimas 12888 df-qus 12889 df-mgm 12942 df-sgrp 12988 df-mnd 13001 df-mhm 13034 df-submnd 13035 df-grp 13078 df-minusg 13079 df-sbg 13080 df-mulg 13193 df-subg 13243 df-nsg 13244 df-eqg 13245 df-ghm 13314 df-cmn 13359 df-abl 13360 df-mgp 13420 df-rng 13432 df-ur 13459 df-srg 13463 df-ring 13497 df-cring 13498 df-oppr 13567 df-dvdsr 13588 df-unit 13589 df-invr 13620 df-dvr 13631 df-rhm 13651 df-nzr 13679 df-subrg 13718 df-domn 13758 df-idom 13759 df-lmod 13788 df-lssm 13852 df-lsp 13886 df-sra 13934 df-rgmod 13935 df-lidl 13968 df-rsp 13969 df-2idl 13999 df-bl 14045 df-mopn 14046 df-fg 14048 df-metu 14049 df-cnfld 14056 df-zring 14090 df-zrh 14113 df-zn 14115 df-lgs 15155

This theorem is referenced by: lgsquadlem2 15235

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