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Theorem lgseisen 16312

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 12905	. . . 4 ⊢ (𝑄 ∈ ℙ → 𝑄 ∈ ℤ)
4	2, 3	syl 14	. . 3 ⊢ (𝜑 → 𝑄 ∈ ℤ)
5		lgseisen.1	. . 3 ⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))
6		lgsval3 16256	. . 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 16287	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ ℕ)
9		oddprm 13058	. . . . . . . . 9 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((𝑃 − 1) / 2) ∈ ℕ)
10	5, 9	syl 14	. . . . . . . 8 ⊢ (𝜑 → ((𝑃 − 1) / 2) ∈ ℕ)
11	10	nnnn0d 9624	. . . . . . 7 ⊢ (𝜑 → ((𝑃 − 1) / 2) ∈ ℕ₀)
12	8, 11	nnexpcld 11146	. . . . . 6 ⊢ (𝜑 → (𝑄↑((𝑃 − 1) / 2)) ∈ ℕ)
13		nnq 10042	. . . . . 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 9675	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℤ)
16	15	znegcld 9774	. . . . . . 7 ⊢ (𝜑 → -1 ∈ ℤ)
17		zq 10035	. . . . . . 7 ⊢ (-1 ∈ ℤ → -1 ∈ ℚ)
18	16, 17	syl 14	. . . . . 6 ⊢ (𝜑 → -1 ∈ ℚ)
19		neg1ne0 9413	. . . . . . 7 ⊢ -1 ≠ 0
20	19	a1i 9	. . . . . 6 ⊢ (𝜑 → -1 ≠ 0)
21	10	nnzd 9771	. . . . . . . 8 ⊢ (𝜑 → ((𝑃 − 1) / 2) ∈ ℤ)
22	15, 21	fzfigd 10881	. . . . . . 7 ⊢ (𝜑 → (1...((𝑃 − 1) / 2)) ∈ Fin)
23	5	gausslemma2dlem0a 16287	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ ℕ)
24		znq 10033	. . . . . . . . . 10 ⊢ ((𝑄 ∈ ℤ ∧ 𝑃 ∈ ℕ) → (𝑄 / 𝑃) ∈ ℚ)
25	4, 23, 24	syl2anc 415	. . . . . . . . 9 ⊢ (𝜑 → (𝑄 / 𝑃) ∈ ℚ)
26		2z 9676	. . . . . . . . . . . 12 ⊢ 2 ∈ ℤ
27	26	a1i 9	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 2 ∈ ℤ)
28		elfznn 10470	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (1...((𝑃 − 1) / 2)) → 𝑥 ∈ ℕ)
29	28	adantl 277	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑥 ∈ ℕ)
30	29	nnzd 9771	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑥 ∈ ℤ)
31	27, 30	zmulcld 9778	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (2 · 𝑥) ∈ ℤ)
32		zq 10035	. . . . . . . . . 10 ⊢ ((2 · 𝑥) ∈ ℤ → (2 · 𝑥) ∈ ℚ)
33	31, 32	syl 14	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (2 · 𝑥) ∈ ℚ)
34		qmulcl 10046	. . . . . . . . 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 10724	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))) ∈ ℤ)
37	22, 36	fsumzcl 12185	. . . . . 6 ⊢ (𝜑 → Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))) ∈ ℤ)
38		qexpclz 11010	. . . . . 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 9674	. . . . . 6 ⊢ 1 ∈ ℤ
41		zq 10035	. . . . . 6 ⊢ (1 ∈ ℤ → 1 ∈ ℚ)
42	40, 41	mp1i 10	. . . . 5 ⊢ (𝜑 → 1 ∈ ℚ)
43		nnq 10042	. . . . . 6 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℚ)
44	23, 43	syl 14	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℚ)
45	23	nngt0d 9350	. . . . 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 16311	. . . . 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 10811	. . . 4 ⊢ (𝜑 → (((𝑄↑((𝑃 − 1) / 2)) + 1) mod 𝑃) = (((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 1) mod 𝑃))
55		qaddcl 10044	. . . . . 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 8501	. . . . . . 7 ⊢ -1 = (0 − 1)
58		neg1cn 9411	. . . . . . . . . . . 12 ⊢ -1 ∈ ℂ
59		neg1ap0 9415	. . . . . . . . . . . 12 ⊢ -1 # 0
60		absexpzap 11861	. . . . . . . . . . . 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 8272	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℂ
63	62	absnegi 11928	. . . . . . . . . . . . . 14 ⊢ (abs‘-1) = (abs‘1)
64		abs1 11852	. . . . . . . . . . . . . 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 11018	. . . . . . . . . . . . 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 8902	. . . . . . . . . 10 ⊢ 1 ≤ 1
72	70, 71	eqbrtrdi 4169	. . . . . . . . 9 ⊢ (𝜑 → (abs‘(-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) ≤ 1)
73		neg1rr 9412	. . . . . . . . . . . 12 ⊢ -1 ∈ ℝ
74	73	a1i 9	. . . . . . . . . . 11 ⊢ (𝜑 → -1 ∈ ℝ)
75	59	a1i 9	. . . . . . . . . . 11 ⊢ (𝜑 → -1 # 0)
76	74, 75, 37	reexpclzapd 11149	. . . . . . . . . 10 ⊢ (𝜑 → (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ∈ ℝ)
77		1re 8325	. . . . . . . . . 10 ⊢ 1 ∈ ℝ
78		absle 11870	. . . . . . . . . 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 8327	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℝ)
84		1red 8341	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℝ)
85	83, 84, 76	lesubaddd 8871	. . . . . 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 9319	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ ℝ)
88		peano2rem 8594	. . . . . . . 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 9365	. . . . . . . . 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 9771	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑃 ∈ ℤ)
95		zapne 9723	. . . . . . . . . . . 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 9376	. . . . . . . . . . . 12 ⊢ 2 ∈ ℝ
99	98	a1i 9	. . . . . . . . . . 11 ⊢ (𝜑 → 2 ∈ ℝ)
100	5	eldifad 3231	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑃 ∈ ℙ)
101		prmuz2 12926	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ_≥‘2))
102		eluzle 9943	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ (ℤ_≥‘2) → 2 ≤ 𝑃)
103	100, 101, 102	3syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 2 ≤ 𝑃)
104	99, 87, 103	leltapd 8969	. . . . . . . . . 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 8874	. . . . . . . 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 8452	. . . . . 6 ⊢ (𝜑 → (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) < (𝑃 − 1))
110	76, 84, 87	ltaddsubd 8874	. . . . . 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 10799	. . . . 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 8354	. . 3 ⊢ (𝜑 → (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ∈ ℂ)
117		pncan 8533	. . 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 8177 ℝcr 8178 0cc0 8179 1c1 8180 + caddc 8182 · cmul 8184 < clt 8360 ≤ cle 8361 − cmin 8498 -cneg 8499 # cap 8911 / cdiv 9004 ℕcn 9306 2c2 9357 ℤcz 9648 ℤ_≥cuz 9930 ℚcq 10028 ...cfz 10421 ⌊cfl 10713 mod cmo 10772 ↑cexp 10988 abscabs 11777 Σcsu 12135 ℙcprime 12901 mulGrpcmgp 14266 ℤRHomczrh 14995 ℤ/nℤczn 14997 /_L clgs 16235

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 8270 ax-resscn 8271 ax-1cn 8272 ax-1re 8273 ax-icn 8274 ax-addcl 8275 ax-addrcl 8276 ax-mulcl 8277 ax-mulrcl 8278 ax-addcom 8279 ax-mulcom 8280 ax-addass 8281 ax-mulass 8282 ax-distr 8283 ax-i2m1 8284 ax-0lt1 8285 ax-1rid 8286 ax-0id 8287 ax-rnegex 8288 ax-precex 8289 ax-cnre 8290 ax-pre-ltirr 8291 ax-pre-ltwlin 8292 ax-pre-lttrn 8293 ax-pre-apti 8294 ax-pre-ltadd 8295 ax-pre-mulgt0 8296 ax-pre-mulext 8297 ax-arch 8298 ax-caucvg 8299 ax-addf 8301 ax-mulf 8302

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 7324 df-inf 7325 df-pnf 8362 df-mnf 8363 df-xr 8364 df-ltxr 8365 df-le 8366 df-sub 8500 df-neg 8501 df-reap 8905 df-ap 8912 df-div 9005 df-inn 9307 df-2 9365 df-3 9366 df-4 9367 df-5 9368 df-6 9369 df-7 9370 df-8 9371 df-9 9372 df-n0 9568 df-z 9649 df-dec 9782 df-uz 9931 df-q 10029 df-rp 10065 df-fz 10422 df-fzo 10560 df-fl 10715 df-mod 10773 df-seqfrec 10898 df-exp 10989 df-ihash 11229 df-cj 11621 df-re 11622 df-im 11623 df-rsqrt 11778 df-abs 11779 df-clim 12061 df-sumdc 12136 df-proddc 12334 df-dvds 12571 df-gcd 12747 df-prm 12902 df-phi 13009 df-pc 13084 df-struct 13403 df-ndx 13404 df-slot 13405 df-base 13407 df-sets 13408 df-iress 13409 df-plusg 13493 df-mulr 13494 df-starv 13495 df-sca 13496 df-vsca 13497 df-ip 13498 df-tset 13499 df-ple 13500 df-ds 13502 df-unif 13503 df-0g 13661 df-gzsum 13662 df-topgen 13663 df-iimas 13673 df-qus 13674 df-mgm 13725 df-sgrp 13766 df-mnd 13779 df-mhm 13815 df-submnd 13816 df-grp 13857 df-minusg 13858 df-sbg 13859 df-mulg 13972 df-subg 14022 df-nsg 14023 df-eqg 14024 df-ghm 14093 df-cmn 14138 df-abl 14139 df-gsumfi 14200 df-mgp 14267 df-rng 14281 df-ur 14312 df-srg 14317 df-ring 14351 df-cring 14352 df-oppr 14422 df-dvdsr 14444 df-unit 14445 df-invr 14477 df-dvr 14488 df-rhm 14508 df-nzr 14536 df-subrg 14576 df-domn 14616 df-idom 14617 df-lmod 14674 df-lssm 14739 df-lsp 14773 df-sra 14821 df-rgmod 14822 df-lidl 14855 df-rsp 14856 df-2idl 14886 df-bl 14932 df-mopn 14933 df-fg 14935 df-metu 14936 df-cnfld 14943 df-zring 14975 df-zrh 14998 df-zn 15000 df-lgs 16236

This theorem is used by: lgsquadlem2 16316

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