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Theorem lgseisenlem4 25302

Description: Lemma for lgseisen 25303. The function 𝑀 is an injection (and hence a bijection by the pigeonhole principle). (Contributed by Mario Carneiro, 18-Jun-2015.) (Proof shortened by AV, 15-Jun-2019.)

**Hypotheses**
Ref	Expression
lgseisen.1	⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))
lgseisen.2	⊢ (𝜑 → 𝑄 ∈ (ℙ ∖ {2}))
lgseisen.3	⊢ (𝜑 → 𝑃 ≠ 𝑄)
lgseisen.4	⊢ 𝑅 = ((𝑄 · (2 · 𝑥)) mod 𝑃)
lgseisen.5	⊢ 𝑀 = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ ((((-1↑𝑅) · 𝑅) mod 𝑃) / 2))
lgseisen.6	⊢ 𝑆 = ((𝑄 · (2 · 𝑦)) mod 𝑃)
lgseisen.7	⊢ 𝑌 = (ℤ/nℤ‘𝑃)
lgseisen.8	⊢ 𝐺 = (mulGrp‘𝑌)
lgseisen.9	⊢ 𝐿 = (ℤRHom‘𝑌)

**Assertion**
Ref	Expression
lgseisenlem4	⊢ (𝜑 → ((𝑄↑((𝑃 − 1) / 2)) mod 𝑃) = ((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) mod 𝑃))

Distinct variable groups: 𝑥,𝐺 𝑥,𝐿 𝑥,𝑦,𝑃 𝜑,𝑥,𝑦 𝑦,𝑀 𝑥,𝑄,𝑦 𝑥,𝑌 𝑥,𝑆 Allowed substitution hints: 𝑅(𝑥,𝑦) 𝑆(𝑦) 𝐺(𝑦) 𝐿(𝑦) 𝑀(𝑥) 𝑌(𝑦)

Proof of Theorem lgseisenlem4 Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		zringbas 20026	. . . . 5 ⊢ ℤ = (Base‘ℤ_ring)
2		zring0 20030	. . . . 5 ⊢ 0 = (0_g‘ℤ_ring)
3		zringabl 20024	. . . . . 6 ⊢ ℤ_ring ∈ Abel
4		ablcmn 18399	. . . . . 6 ⊢ (ℤ_ring ∈ Abel → ℤ_ring ∈ CMnd)
5	3, 4	mp1i 13	. . . . 5 ⊢ (𝜑 → ℤ_ring ∈ CMnd)
6		lgseisen.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))
7	6	eldifad 3727	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ ℙ)
8		lgseisen.7	. . . . . . . . . 10 ⊢ 𝑌 = (ℤ/nℤ‘𝑃)
9	8	znfld 20111	. . . . . . . . 9 ⊢ (𝑃 ∈ ℙ → 𝑌 ∈ Field)
10	7, 9	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ Field)
11		isfld 18958	. . . . . . . . 9 ⊢ (𝑌 ∈ Field ↔ (𝑌 ∈ DivRing ∧ 𝑌 ∈ CRing))
12	11	simprbi 483	. . . . . . . 8 ⊢ (𝑌 ∈ Field → 𝑌 ∈ CRing)
13	10, 12	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ CRing)
14		lgseisen.8	. . . . . . . 8 ⊢ 𝐺 = (mulGrp‘𝑌)
15	14	crngmgp 18755	. . . . . . 7 ⊢ (𝑌 ∈ CRing → 𝐺 ∈ CMnd)
16	13, 15	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ CMnd)
17		cmnmnd 18408	. . . . . 6 ⊢ (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
18	16, 17	syl 17	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ Mnd)
19		fzfid 12966	. . . . 5 ⊢ (𝜑 → (1...((𝑃 − 1) / 2)) ∈ Fin)
20		m1expcl 13077	. . . . . . . 8 ⊢ (𝑘 ∈ ℤ → (-1↑𝑘) ∈ ℤ)
21	20	adantl 473	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → (-1↑𝑘) ∈ ℤ)
22		eqidd 2761	. . . . . . 7 ⊢ (𝜑 → (𝑘 ∈ ℤ ↦ (-1↑𝑘)) = (𝑘 ∈ ℤ ↦ (-1↑𝑘)))
23		crngring 18758	. . . . . . . . . . 11 ⊢ (𝑌 ∈ CRing → 𝑌 ∈ Ring)
24	13, 23	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ Ring)
25		lgseisen.9	. . . . . . . . . . 11 ⊢ 𝐿 = (ℤRHom‘𝑌)
26	25	zrhrhm 20062	. . . . . . . . . 10 ⊢ (𝑌 ∈ Ring → 𝐿 ∈ (ℤ_ring RingHom 𝑌))
27	24, 26	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐿 ∈ (ℤ_ring RingHom 𝑌))
28		eqid 2760	. . . . . . . . . 10 ⊢ (Base‘𝑌) = (Base‘𝑌)
29	1, 28	rhmf 18928	. . . . . . . . 9 ⊢ (𝐿 ∈ (ℤ_ring RingHom 𝑌) → 𝐿:ℤ⟶(Base‘𝑌))
30	27, 29	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐿:ℤ⟶(Base‘𝑌))
31	30	feqmptd 6411	. . . . . . 7 ⊢ (𝜑 → 𝐿 = (𝑥 ∈ ℤ ↦ (𝐿‘𝑥)))
32		fveq2 6352	. . . . . . 7 ⊢ (𝑥 = (-1↑𝑘) → (𝐿‘𝑥) = (𝐿‘(-1↑𝑘)))
33	21, 22, 31, 32	fmptco 6559	. . . . . 6 ⊢ (𝜑 → (𝐿 ∘ (𝑘 ∈ ℤ ↦ (-1↑𝑘))) = (𝑘 ∈ ℤ ↦ (𝐿‘(-1↑𝑘))))
34		zringmpg 20042	. . . . . . . . 9 ⊢ ((mulGrp‘ℂ_fld) ↾_s ℤ) = (mulGrp‘ℤ_ring)
35	34, 14	rhmmhm 18924	. . . . . . . 8 ⊢ (𝐿 ∈ (ℤ_ring RingHom 𝑌) → 𝐿 ∈ (((mulGrp‘ℂ_fld) ↾_s ℤ) MndHom 𝐺))
36	27, 35	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ (((mulGrp‘ℂ_fld) ↾_s ℤ) MndHom 𝐺))
37		neg1cn 11316	. . . . . . . . . . 11 ⊢ -1 ∈ ℂ
38		neg1ne0 11318	. . . . . . . . . . 11 ⊢ -1 ≠ 0
39		eqid 2760	. . . . . . . . . . . 12 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
40		eqid 2760	. . . . . . . . . . . 12 ⊢ ((mulGrp‘ℂ_fld) ↾_s (ℂ ∖ {0})) = ((mulGrp‘ℂ_fld) ↾_s (ℂ ∖ {0}))
41	39, 40	expghm 20046	. . . . . . . . . . 11 ⊢ ((-1 ∈ ℂ ∧ -1 ≠ 0) → (𝑘 ∈ ℤ ↦ (-1↑𝑘)) ∈ (ℤ_ring GrpHom ((mulGrp‘ℂ_fld) ↾_s (ℂ ∖ {0}))))
42	37, 38, 41	mp2an 710	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℤ ↦ (-1↑𝑘)) ∈ (ℤ_ring GrpHom ((mulGrp‘ℂ_fld) ↾_s (ℂ ∖ {0})))
43		ghmmhm 17871	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℤ ↦ (-1↑𝑘)) ∈ (ℤ_ring GrpHom ((mulGrp‘ℂ_fld) ↾_s (ℂ ∖ {0}))) → (𝑘 ∈ ℤ ↦ (-1↑𝑘)) ∈ (ℤ_ring MndHom ((mulGrp‘ℂ_fld) ↾_s (ℂ ∖ {0}))))
44	42, 43	ax-mp 5	. . . . . . . . 9 ⊢ (𝑘 ∈ ℤ ↦ (-1↑𝑘)) ∈ (ℤ_ring MndHom ((mulGrp‘ℂ_fld) ↾_s (ℂ ∖ {0})))
45		cnring 19970	. . . . . . . . . 10 ⊢ ℂ_fld ∈ Ring
46		cnfldbas 19952	. . . . . . . . . . . 12 ⊢ ℂ = (Base‘ℂ_fld)
47		cnfld0 19972	. . . . . . . . . . . 12 ⊢ 0 = (0_g‘ℂ_fld)
48		cndrng 19977	. . . . . . . . . . . 12 ⊢ ℂ_fld ∈ DivRing
49	46, 47, 48	drngui 18955	. . . . . . . . . . 11 ⊢ (ℂ ∖ {0}) = (Unit‘ℂ_fld)
50	49, 39	unitsubm 18870	. . . . . . . . . 10 ⊢ (ℂ_fld ∈ Ring → (ℂ ∖ {0}) ∈ (SubMnd‘(mulGrp‘ℂ_fld)))
51	45, 50	ax-mp 5	. . . . . . . . 9 ⊢ (ℂ ∖ {0}) ∈ (SubMnd‘(mulGrp‘ℂ_fld))
52	40	resmhm2 17561	. . . . . . . . 9 ⊢ (((𝑘 ∈ ℤ ↦ (-1↑𝑘)) ∈ (ℤ_ring MndHom ((mulGrp‘ℂ_fld) ↾_s (ℂ ∖ {0}))) ∧ (ℂ ∖ {0}) ∈ (SubMnd‘(mulGrp‘ℂ_fld))) → (𝑘 ∈ ℤ ↦ (-1↑𝑘)) ∈ (ℤ_ring MndHom (mulGrp‘ℂ_fld)))
53	44, 51, 52	mp2an 710	. . . . . . . 8 ⊢ (𝑘 ∈ ℤ ↦ (-1↑𝑘)) ∈ (ℤ_ring MndHom (mulGrp‘ℂ_fld))
54		zsubrg 20001	. . . . . . . . . 10 ⊢ ℤ ∈ (SubRing‘ℂ_fld)
55	39	subrgsubm 18995	. . . . . . . . . 10 ⊢ (ℤ ∈ (SubRing‘ℂ_fld) → ℤ ∈ (SubMnd‘(mulGrp‘ℂ_fld)))
56	54, 55	ax-mp 5	. . . . . . . . 9 ⊢ ℤ ∈ (SubMnd‘(mulGrp‘ℂ_fld))
57		eqid 2760	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℤ ↦ (-1↑𝑘)) = (𝑘 ∈ ℤ ↦ (-1↑𝑘))
58	21, 57	fmptd 6548	. . . . . . . . . 10 ⊢ (𝜑 → (𝑘 ∈ ℤ ↦ (-1↑𝑘)):ℤ⟶ℤ)
59		frn 6214	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℤ ↦ (-1↑𝑘)):ℤ⟶ℤ → ran (𝑘 ∈ ℤ ↦ (-1↑𝑘)) ⊆ ℤ)
60	58, 59	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ran (𝑘 ∈ ℤ ↦ (-1↑𝑘)) ⊆ ℤ)
61		eqid 2760	. . . . . . . . . 10 ⊢ ((mulGrp‘ℂ_fld) ↾_s ℤ) = ((mulGrp‘ℂ_fld) ↾_s ℤ)
62	61	resmhm2b 17562	. . . . . . . . 9 ⊢ ((ℤ ∈ (SubMnd‘(mulGrp‘ℂ_fld)) ∧ ran (𝑘 ∈ ℤ ↦ (-1↑𝑘)) ⊆ ℤ) → ((𝑘 ∈ ℤ ↦ (-1↑𝑘)) ∈ (ℤ_ring MndHom (mulGrp‘ℂ_fld)) ↔ (𝑘 ∈ ℤ ↦ (-1↑𝑘)) ∈ (ℤ_ring MndHom ((mulGrp‘ℂ_fld) ↾_s ℤ))))
63	56, 60, 62	sylancr 698	. . . . . . . 8 ⊢ (𝜑 → ((𝑘 ∈ ℤ ↦ (-1↑𝑘)) ∈ (ℤ_ring MndHom (mulGrp‘ℂ_fld)) ↔ (𝑘 ∈ ℤ ↦ (-1↑𝑘)) ∈ (ℤ_ring MndHom ((mulGrp‘ℂ_fld) ↾_s ℤ))))
64	53, 63	mpbii 223	. . . . . . 7 ⊢ (𝜑 → (𝑘 ∈ ℤ ↦ (-1↑𝑘)) ∈ (ℤ_ring MndHom ((mulGrp‘ℂ_fld) ↾_s ℤ)))
65		mhmco 17563	. . . . . . 7 ⊢ ((𝐿 ∈ (((mulGrp‘ℂ_fld) ↾_s ℤ) MndHom 𝐺) ∧ (𝑘 ∈ ℤ ↦ (-1↑𝑘)) ∈ (ℤ_ring MndHom ((mulGrp‘ℂ_fld) ↾_s ℤ))) → (𝐿 ∘ (𝑘 ∈ ℤ ↦ (-1↑𝑘))) ∈ (ℤ_ring MndHom 𝐺))
66	36, 64, 65	syl2anc 696	. . . . . 6 ⊢ (𝜑 → (𝐿 ∘ (𝑘 ∈ ℤ ↦ (-1↑𝑘))) ∈ (ℤ_ring MndHom 𝐺))
67	33, 66	eqeltrrd 2840	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ ℤ ↦ (𝐿‘(-1↑𝑘))) ∈ (ℤ_ring MndHom 𝐺))
68		lgseisen.2	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑄 ∈ (ℙ ∖ {2}))
69	68	eldifad 3727	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑄 ∈ ℙ)
70		prmnn 15590	. . . . . . . . . . 11 ⊢ (𝑄 ∈ ℙ → 𝑄 ∈ ℕ)
71	69, 70	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑄 ∈ ℕ)
72	71	nnred 11227	. . . . . . . . 9 ⊢ (𝜑 → 𝑄 ∈ ℝ)
73		prmnn 15590	. . . . . . . . . 10 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
74	7, 73	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ ℕ)
75	72, 74	nndivred 11261	. . . . . . . 8 ⊢ (𝜑 → (𝑄 / 𝑃) ∈ ℝ)
76	75	adantr 472	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑄 / 𝑃) ∈ ℝ)
77		2nn 11377	. . . . . . . . 9 ⊢ 2 ∈ ℕ
78		elfznn 12563	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1...((𝑃 − 1) / 2)) → 𝑥 ∈ ℕ)
79	78	adantl 473	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑥 ∈ ℕ)
80		nnmulcl 11235	. . . . . . . . 9 ⊢ ((2 ∈ ℕ ∧ 𝑥 ∈ ℕ) → (2 · 𝑥) ∈ ℕ)
81	77, 79, 80	sylancr 698	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (2 · 𝑥) ∈ ℕ)
82	81	nnred 11227	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (2 · 𝑥) ∈ ℝ)
83	76, 82	remulcld 10262	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝑄 / 𝑃) · (2 · 𝑥)) ∈ ℝ)
84	83	flcld 12793	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))) ∈ ℤ)
85		eqid 2760	. . . . . 6 ⊢ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))
86		fvexd 6364	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))) ∈ V)
87		c0ex 10226	. . . . . . 7 ⊢ 0 ∈ V
88	87	a1i 11	. . . . . 6 ⊢ (𝜑 → 0 ∈ V)
89	85, 19, 86, 88	fsuppmptdm 8451	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) finSupp 0)
90		oveq2 6821	. . . . . 6 ⊢ (𝑘 = (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))) → (-1↑𝑘) = (-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))
91	90	fveq2d 6356	. . . . 5 ⊢ (𝑘 = (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))) → (𝐿‘(-1↑𝑘)) = (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))))
92		oveq2 6821	. . . . . 6 ⊢ (𝑘 = (ℤ_ring Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) → (-1↑𝑘) = (-1↑(ℤ_ring Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))))
93	92	fveq2d 6356	. . . . 5 ⊢ (𝑘 = (ℤ_ring Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) → (𝐿‘(-1↑𝑘)) = (𝐿‘(-1↑(ℤ_ring Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))))))
94	1, 2, 5, 18, 19, 67, 84, 89, 91, 93	gsummhm2 18539	. . . 4 ⊢ (𝜑 → (𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))))) = (𝐿‘(-1↑(ℤ_ring Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))))))
95	14, 28	mgpbas 18695	. . . . . . 7 ⊢ (Base‘𝑌) = (Base‘𝐺)
96		eqid 2760	. . . . . . . 8 ⊢ (._r‘𝑌) = (._r‘𝑌)
97	14, 96	mgpplusg 18693	. . . . . . 7 ⊢ (._r‘𝑌) = (+_g‘𝐺)
98	30	adantr 472	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝐿:ℤ⟶(Base‘𝑌))
99		m1expcl 13077	. . . . . . . . 9 ⊢ ((⌊‘((𝑄 / 𝑃) · (2 · 𝑥))) ∈ ℤ → (-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ∈ ℤ)
100	84, 99	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ∈ ℤ)
101	98, 100	ffvelrnd 6523	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) ∈ (Base‘𝑌))
102		neg1z 11605	. . . . . . . . . 10 ⊢ -1 ∈ ℤ
103		lgseisen.4	. . . . . . . . . . 11 ⊢ 𝑅 = ((𝑄 · (2 · 𝑥)) mod 𝑃)
104	69	adantr 472	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑄 ∈ ℙ)
105		prmz 15591	. . . . . . . . . . . . . 14 ⊢ (𝑄 ∈ ℙ → 𝑄 ∈ ℤ)
106	104, 105	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑄 ∈ ℤ)
107	81	nnzd 11673	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (2 · 𝑥) ∈ ℤ)
108	106, 107	zmulcld 11680	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑄 · (2 · 𝑥)) ∈ ℤ)
109	7	adantr 472	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 ∈ ℙ)
110	109, 73	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 ∈ ℕ)
111	108, 110	zmodcld 12885	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝑄 · (2 · 𝑥)) mod 𝑃) ∈ ℕ₀)
112	103, 111	syl5eqel 2843	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑅 ∈ ℕ₀)
113		zexpcl 13069	. . . . . . . . . 10 ⊢ ((-1 ∈ ℤ ∧ 𝑅 ∈ ℕ₀) → (-1↑𝑅) ∈ ℤ)
114	102, 112, 113	sylancr 698	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (-1↑𝑅) ∈ ℤ)
115	114, 106	zmulcld 11680	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((-1↑𝑅) · 𝑄) ∈ ℤ)
116	98, 115	ffvelrnd 6523	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘((-1↑𝑅) · 𝑄)) ∈ (Base‘𝑌))
117		eqid 2760	. . . . . . 7 ⊢ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))) = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))))
118		eqid 2760	. . . . . . 7 ⊢ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))) = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄)))
119	95, 97, 16, 19, 101, 116, 117, 118	gsummptfidmadd2 18526	. . . . . 6 ⊢ (𝜑 → (𝐺 Σ_g ((𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))) ∘_𝑓 (._r‘𝑌)(𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))))) = ((𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))))(._r‘𝑌)(𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))))))
120		eqidd 2761	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))) = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))))
121		eqidd 2761	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))) = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))))
122	19, 101, 116, 120, 121	offval2 7079	. . . . . . . 8 ⊢ (𝜑 → ((𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))) ∘_𝑓 (._r‘𝑌)(𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄)))) = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ ((𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))(._r‘𝑌)(𝐿‘((-1↑𝑅) · 𝑄)))))
123	27	adantr 472	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝐿 ∈ (ℤ_ring RingHom 𝑌))
124		zringmulr 20029	. . . . . . . . . . . 12 ⊢ · = (._r‘ℤ_ring)
125	1, 124, 96	rhmmul 18929	. . . . . . . . . . 11 ⊢ ((𝐿 ∈ (ℤ_ring RingHom 𝑌) ∧ (-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ∈ ℤ ∧ ((-1↑𝑅) · 𝑄) ∈ ℤ) → (𝐿‘((-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) · ((-1↑𝑅) · 𝑄))) = ((𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))(._r‘𝑌)(𝐿‘((-1↑𝑅) · 𝑄))))
126	123, 100, 115, 125	syl3anc 1477	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘((-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) · ((-1↑𝑅) · 𝑄))) = ((𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))(._r‘𝑌)(𝐿‘((-1↑𝑅) · 𝑄))))
127	108	zred 11674	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑄 · (2 · 𝑥)) ∈ ℝ)
128	110	nnrpd 12063	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 ∈ ℝ⁺)
129		modval 12864	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑄 · (2 · 𝑥)) ∈ ℝ ∧ 𝑃 ∈ ℝ⁺) → ((𝑄 · (2 · 𝑥)) mod 𝑃) = ((𝑄 · (2 · 𝑥)) − (𝑃 · (⌊‘((𝑄 · (2 · 𝑥)) / 𝑃)))))
130	127, 128, 129	syl2anc 696	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝑄 · (2 · 𝑥)) mod 𝑃) = ((𝑄 · (2 · 𝑥)) − (𝑃 · (⌊‘((𝑄 · (2 · 𝑥)) / 𝑃)))))
131	103, 130	syl5eq 2806	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑅 = ((𝑄 · (2 · 𝑥)) − (𝑃 · (⌊‘((𝑄 · (2 · 𝑥)) / 𝑃)))))
132	106	zcnd 11675	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑄 ∈ ℂ)
133	81	nncnd 11228	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (2 · 𝑥) ∈ ℂ)
134	110	nncnd 11228	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 ∈ ℂ)
135	110	nnne0d 11257	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 ≠ 0)
136	132, 133, 134, 135	div23d 11030	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝑄 · (2 · 𝑥)) / 𝑃) = ((𝑄 / 𝑃) · (2 · 𝑥)))
137	136	fveq2d 6356	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (⌊‘((𝑄 · (2 · 𝑥)) / 𝑃)) = (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))
138	137	oveq2d 6829	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑃 · (⌊‘((𝑄 · (2 · 𝑥)) / 𝑃))) = (𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))
139	138	oveq2d 6829	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝑄 · (2 · 𝑥)) − (𝑃 · (⌊‘((𝑄 · (2 · 𝑥)) / 𝑃)))) = ((𝑄 · (2 · 𝑥)) − (𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))))
140	131, 139	eqtrd 2794	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑅 = ((𝑄 · (2 · 𝑥)) − (𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))))
141	140	oveq2d 6829	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 𝑅) = ((𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + ((𝑄 · (2 · 𝑥)) − (𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))))
142		prmz 15591	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
143	109, 142	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 ∈ ℤ)
144	143, 84	zmulcld 11680	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ∈ ℤ)
145	144	zcnd 11675	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ∈ ℂ)
146	108	zcnd 11675	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑄 · (2 · 𝑥)) ∈ ℂ)
147	145, 146	pncan3d 10587	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + ((𝑄 · (2 · 𝑥)) − (𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))) = (𝑄 · (2 · 𝑥)))
148		2cnd 11285	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 2 ∈ ℂ)
149	79	nncnd 11228	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑥 ∈ ℂ)
150	132, 148, 149	mul12d 10437	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑄 · (2 · 𝑥)) = (2 · (𝑄 · 𝑥)))
151	141, 147, 150	3eqtrd 2798	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 𝑅) = (2 · (𝑄 · 𝑥)))
152	151	oveq2d 6829	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (-1↑((𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 𝑅)) = (-1↑(2 · (𝑄 · 𝑥))))
153	37	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → -1 ∈ ℂ)
154	38	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → -1 ≠ 0)
155	112	nn0zd 11672	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑅 ∈ ℤ)
156		expaddz 13098	. . . . . . . . . . . . . . . 16 ⊢ (((-1 ∈ ℂ ∧ -1 ≠ 0) ∧ ((𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ∈ ℤ ∧ 𝑅 ∈ ℤ)) → (-1↑((𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 𝑅)) = ((-1↑(𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) · (-1↑𝑅)))
157	153, 154, 144, 155, 156	syl22anc 1478	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (-1↑((𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 𝑅)) = ((-1↑(𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) · (-1↑𝑅)))
158		expmulz 13100	. . . . . . . . . . . . . . . . . 18 ⊢ (((-1 ∈ ℂ ∧ -1 ≠ 0) ∧ (𝑃 ∈ ℤ ∧ (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))) ∈ ℤ)) → (-1↑(𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) = ((-1↑𝑃)↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))
159	153, 154, 143, 84, 158	syl22anc 1478	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (-1↑(𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) = ((-1↑𝑃)↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))
160		1cnd 10248	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 1 ∈ ℂ)
161		eldifsni 4466	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ≠ 2)
162	6, 161	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝑃 ≠ 2)
163	162	necomd 2987	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 2 ≠ 𝑃)
164	163	neneqd 2937	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ¬ 2 = 𝑃)
165	164	adantr 472	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ¬ 2 = 𝑃)
166		2z 11601	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 2 ∈ ℤ
167		uzid 11894	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (2 ∈ ℤ → 2 ∈ (ℤ_≥‘2))
168	166, 167	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 2 ∈ (ℤ_≥‘2)
169		dvdsprm 15617	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((2 ∈ (ℤ_≥‘2) ∧ 𝑃 ∈ ℙ) → (2 ∥ 𝑃 ↔ 2 = 𝑃))
170	168, 109, 169	sylancr 698	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (2 ∥ 𝑃 ↔ 2 = 𝑃))
171	165, 170	mtbird 314	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ¬ 2 ∥ 𝑃)
172		oexpneg 15271	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1 ∈ ℂ ∧ 𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃) → (-1↑𝑃) = -(1↑𝑃))
173	160, 110, 171, 172	syl3anc 1477	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (-1↑𝑃) = -(1↑𝑃))
174		1exp 13083	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑃 ∈ ℤ → (1↑𝑃) = 1)
175	143, 174	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (1↑𝑃) = 1)
176	175	negeqd 10467	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → -(1↑𝑃) = -1)
177	173, 176	eqtrd 2794	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (-1↑𝑃) = -1)
178	177	oveq1d 6828	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((-1↑𝑃)↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) = (-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))
179	159, 178	eqtrd 2794	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (-1↑(𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) = (-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))
180	179	oveq1d 6828	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((-1↑(𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) · (-1↑𝑅)) = ((-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) · (-1↑𝑅)))
181	157, 180	eqtrd 2794	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (-1↑((𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 𝑅)) = ((-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) · (-1↑𝑅)))
182		nnmulcl 11235	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑄 ∈ ℕ ∧ 𝑥 ∈ ℕ) → (𝑄 · 𝑥) ∈ ℕ)
183	71, 78, 182	syl2an 495	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑄 · 𝑥) ∈ ℕ)
184	183	nnnn0d 11543	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑄 · 𝑥) ∈ ℕ₀)
185		2nn0 11501	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℕ₀
186	185	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 2 ∈ ℕ₀)
187	153, 184, 186	expmuld 13205	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (-1↑(2 · (𝑄 · 𝑥))) = ((-1↑2)↑(𝑄 · 𝑥)))
188		neg1sqe1 13153	. . . . . . . . . . . . . . . . 17 ⊢ (-1↑2) = 1
189	188	oveq1i 6823	. . . . . . . . . . . . . . . 16 ⊢ ((-1↑2)↑(𝑄 · 𝑥)) = (1↑(𝑄 · 𝑥))
190	183	nnzd 11673	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑄 · 𝑥) ∈ ℤ)
191		1exp 13083	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑄 · 𝑥) ∈ ℤ → (1↑(𝑄 · 𝑥)) = 1)
192	190, 191	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (1↑(𝑄 · 𝑥)) = 1)
193	189, 192	syl5eq 2806	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((-1↑2)↑(𝑄 · 𝑥)) = 1)
194	187, 193	eqtrd 2794	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (-1↑(2 · (𝑄 · 𝑥))) = 1)
195	152, 181, 194	3eqtr3d 2802	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) · (-1↑𝑅)) = 1)
196	195	oveq1d 6828	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (((-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) · (-1↑𝑅)) · 𝑄) = (1 · 𝑄))
197	100	zcnd 11675	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ∈ ℂ)
198	114	zcnd 11675	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (-1↑𝑅) ∈ ℂ)
199	197, 198, 132	mulassd 10255	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (((-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) · (-1↑𝑅)) · 𝑄) = ((-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) · ((-1↑𝑅) · 𝑄)))
200	132	mulid2d 10250	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (1 · 𝑄) = 𝑄)
201	196, 199, 200	3eqtr3d 2802	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) · ((-1↑𝑅) · 𝑄)) = 𝑄)
202	201	fveq2d 6356	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘((-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) · ((-1↑𝑅) · 𝑄))) = (𝐿‘𝑄))
203	126, 202	eqtr3d 2796	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))(._r‘𝑌)(𝐿‘((-1↑𝑅) · 𝑄))) = (𝐿‘𝑄))
204	203	mpteq2dva 4896	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ ((𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))(._r‘𝑌)(𝐿‘((-1↑𝑅) · 𝑄)))) = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘𝑄)))
205	122, 204	eqtrd 2794	. . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))) ∘_𝑓 (._r‘𝑌)(𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄)))) = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘𝑄)))
206	205	oveq2d 6829	. . . . . 6 ⊢ (𝜑 → (𝐺 Σ_g ((𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))) ∘_𝑓 (._r‘𝑌)(𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))))) = (𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘𝑄))))
207		lgseisen.3	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ≠ 𝑄)
208		lgseisen.5	. . . . . . . 8 ⊢ 𝑀 = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ ((((-1↑𝑅) · 𝑅) mod 𝑃) / 2))
209		lgseisen.6	. . . . . . . 8 ⊢ 𝑆 = ((𝑄 · (2 · 𝑦)) mod 𝑃)
210	6, 68, 207, 103, 208, 209, 8, 14, 25	lgseisenlem3 25301	. . . . . . 7 ⊢ (𝜑 → (𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄)))) = (1_r‘𝑌))
211	210	oveq2d 6829	. . . . . 6 ⊢ (𝜑 → ((𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))))(._r‘𝑌)(𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))))) = ((𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))))(._r‘𝑌)(1_r‘𝑌)))
212	119, 206, 211	3eqtr3rd 2803	. . . . 5 ⊢ (𝜑 → ((𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))))(._r‘𝑌)(1_r‘𝑌)) = (𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘𝑄))))
213		eqid 2760	. . . . . . 7 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
214	101, 117	fmptd 6548	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))):(1...((𝑃 − 1) / 2))⟶(Base‘𝑌))
215		fvexd 6364	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) ∈ V)
216		fvexd 6364	. . . . . . . 8 ⊢ (𝜑 → (0_g‘𝐺) ∈ V)
217	117, 19, 215, 216	fsuppmptdm 8451	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))) finSupp (0_g‘𝐺))
218	95, 213, 16, 19, 214, 217	gsumcl 18516	. . . . . 6 ⊢ (𝜑 → (𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))))) ∈ (Base‘𝑌))
219		eqid 2760	. . . . . . 7 ⊢ (1_r‘𝑌) = (1_r‘𝑌)
220	28, 96, 219	ringridm 18772	. . . . . 6 ⊢ ((𝑌 ∈ Ring ∧ (𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))))) ∈ (Base‘𝑌)) → ((𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))))(._r‘𝑌)(1_r‘𝑌)) = (𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))))))
221	24, 218, 220	syl2anc 696	. . . . 5 ⊢ (𝜑 → ((𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))))(._r‘𝑌)(1_r‘𝑌)) = (𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))))))
222	69, 105	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑄 ∈ ℤ)
223	30, 222	ffvelrnd 6523	. . . . . . 7 ⊢ (𝜑 → (𝐿‘𝑄) ∈ (Base‘𝑌))
224		eqid 2760	. . . . . . . 8 ⊢ (._g‘𝐺) = (._g‘𝐺)
225	95, 224	gsumconst 18534	. . . . . . 7 ⊢ ((𝐺 ∈ Mnd ∧ (1...((𝑃 − 1) / 2)) ∈ Fin ∧ (𝐿‘𝑄) ∈ (Base‘𝑌)) → (𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘𝑄))) = ((♯‘(1...((𝑃 − 1) / 2)))(._g‘𝐺)(𝐿‘𝑄)))
226	18, 19, 223, 225	syl3anc 1477	. . . . . 6 ⊢ (𝜑 → (𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘𝑄))) = ((♯‘(1...((𝑃 − 1) / 2)))(._g‘𝐺)(𝐿‘𝑄)))
227		oddprm 15717	. . . . . . . . . 10 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((𝑃 − 1) / 2) ∈ ℕ)
228	6, 227	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃 − 1) / 2) ∈ ℕ)
229	228	nnnn0d 11543	. . . . . . . 8 ⊢ (𝜑 → ((𝑃 − 1) / 2) ∈ ℕ₀)
230		hashfz1 13328	. . . . . . . 8 ⊢ (((𝑃 − 1) / 2) ∈ ℕ₀ → (♯‘(1...((𝑃 − 1) / 2))) = ((𝑃 − 1) / 2))
231	229, 230	syl 17	. . . . . . 7 ⊢ (𝜑 → (♯‘(1...((𝑃 − 1) / 2))) = ((𝑃 − 1) / 2))
232	231	oveq1d 6828	. . . . . 6 ⊢ (𝜑 → ((♯‘(1...((𝑃 − 1) / 2)))(._g‘𝐺)(𝐿‘𝑄)) = (((𝑃 − 1) / 2)(._g‘𝐺)(𝐿‘𝑄)))
233	34, 1	mgpbas 18695	. . . . . . . . 9 ⊢ ℤ = (Base‘((mulGrp‘ℂ_fld) ↾_s ℤ))
234		eqid 2760	. . . . . . . . 9 ⊢ (._g‘((mulGrp‘ℂ_fld) ↾_s ℤ)) = (._g‘((mulGrp‘ℂ_fld) ↾_s ℤ))
235	233, 234, 224	mhmmulg 17784	. . . . . . . 8 ⊢ ((𝐿 ∈ (((mulGrp‘ℂ_fld) ↾_s ℤ) MndHom 𝐺) ∧ ((𝑃 − 1) / 2) ∈ ℕ₀ ∧ 𝑄 ∈ ℤ) → (𝐿‘(((𝑃 − 1) / 2)(._g‘((mulGrp‘ℂ_fld) ↾_s ℤ))𝑄)) = (((𝑃 − 1) / 2)(._g‘𝐺)(𝐿‘𝑄)))
236	36, 229, 222, 235	syl3anc 1477	. . . . . . 7 ⊢ (𝜑 → (𝐿‘(((𝑃 − 1) / 2)(._g‘((mulGrp‘ℂ_fld) ↾_s ℤ))𝑄)) = (((𝑃 − 1) / 2)(._g‘𝐺)(𝐿‘𝑄)))
237	56	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ℤ ∈ (SubMnd‘(mulGrp‘ℂ_fld)))
238		eqid 2760	. . . . . . . . . . 11 ⊢ (._g‘(mulGrp‘ℂ_fld)) = (._g‘(mulGrp‘ℂ_fld))
239	238, 61, 234	submmulg 17787	. . . . . . . . . 10 ⊢ ((ℤ ∈ (SubMnd‘(mulGrp‘ℂ_fld)) ∧ ((𝑃 − 1) / 2) ∈ ℕ₀ ∧ 𝑄 ∈ ℤ) → (((𝑃 − 1) / 2)(._g‘(mulGrp‘ℂ_fld))𝑄) = (((𝑃 − 1) / 2)(._g‘((mulGrp‘ℂ_fld) ↾_s ℤ))𝑄))
240	237, 229, 222, 239	syl3anc 1477	. . . . . . . . 9 ⊢ (𝜑 → (((𝑃 − 1) / 2)(._g‘(mulGrp‘ℂ_fld))𝑄) = (((𝑃 − 1) / 2)(._g‘((mulGrp‘ℂ_fld) ↾_s ℤ))𝑄))
241	222	zcnd 11675	. . . . . . . . . 10 ⊢ (𝜑 → 𝑄 ∈ ℂ)
242		cnfldexp 19981	. . . . . . . . . 10 ⊢ ((𝑄 ∈ ℂ ∧ ((𝑃 − 1) / 2) ∈ ℕ₀) → (((𝑃 − 1) / 2)(._g‘(mulGrp‘ℂ_fld))𝑄) = (𝑄↑((𝑃 − 1) / 2)))
243	241, 229, 242	syl2anc 696	. . . . . . . . 9 ⊢ (𝜑 → (((𝑃 − 1) / 2)(._g‘(mulGrp‘ℂ_fld))𝑄) = (𝑄↑((𝑃 − 1) / 2)))
244	240, 243	eqtr3d 2796	. . . . . . . 8 ⊢ (𝜑 → (((𝑃 − 1) / 2)(._g‘((mulGrp‘ℂ_fld) ↾_s ℤ))𝑄) = (𝑄↑((𝑃 − 1) / 2)))
245	244	fveq2d 6356	. . . . . . 7 ⊢ (𝜑 → (𝐿‘(((𝑃 − 1) / 2)(._g‘((mulGrp‘ℂ_fld) ↾_s ℤ))𝑄)) = (𝐿‘(𝑄↑((𝑃 − 1) / 2))))
246	236, 245	eqtr3d 2796	. . . . . 6 ⊢ (𝜑 → (((𝑃 − 1) / 2)(._g‘𝐺)(𝐿‘𝑄)) = (𝐿‘(𝑄↑((𝑃 − 1) / 2))))
247	226, 232, 246	3eqtrd 2798	. . . . 5 ⊢ (𝜑 → (𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘𝑄))) = (𝐿‘(𝑄↑((𝑃 − 1) / 2))))
248	212, 221, 247	3eqtr3d 2802	. . . 4 ⊢ (𝜑 → (𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))))) = (𝐿‘(𝑄↑((𝑃 − 1) / 2))))
249		subrgsubg 18988	. . . . . . . . . 10 ⊢ (ℤ ∈ (SubRing‘ℂ_fld) → ℤ ∈ (SubGrp‘ℂ_fld))
250	54, 249	ax-mp 5	. . . . . . . . 9 ⊢ ℤ ∈ (SubGrp‘ℂ_fld)
251		subgsubm 17817	. . . . . . . . 9 ⊢ (ℤ ∈ (SubGrp‘ℂ_fld) → ℤ ∈ (SubMnd‘ℂ_fld))
252	250, 251	mp1i 13	. . . . . . . 8 ⊢ (𝜑 → ℤ ∈ (SubMnd‘ℂ_fld))
253	84, 85	fmptd 6548	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))):(1...((𝑃 − 1) / 2))⟶ℤ)
254		df-zring 20021	. . . . . . . 8 ⊢ ℤ_ring = (ℂ_fld ↾_s ℤ)
255	19, 252, 253, 254	gsumsubm 17574	. . . . . . 7 ⊢ (𝜑 → (ℂ_fld Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) = (ℤ_ring Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))))
256	84	zcnd 11675	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))) ∈ ℂ)
257	19, 256	gsumfsum 20015	. . . . . . 7 ⊢ (𝜑 → (ℂ_fld Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) = Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))
258	255, 257	eqtr3d 2796	. . . . . 6 ⊢ (𝜑 → (ℤ_ring Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) = Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))
259	258	oveq2d 6829	. . . . 5 ⊢ (𝜑 → (-1↑(ℤ_ring Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))) = (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))
260	259	fveq2d 6356	. . . 4 ⊢ (𝜑 → (𝐿‘(-1↑(ℤ_ring Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))))) = (𝐿‘(-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))))
261	94, 248, 260	3eqtr3d 2802	. . 3 ⊢ (𝜑 → (𝐿‘(𝑄↑((𝑃 − 1) / 2))) = (𝐿‘(-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))))
262	74	nnnn0d 11543	. . . 4 ⊢ (𝜑 → 𝑃 ∈ ℕ₀)
263		zexpcl 13069	. . . . 5 ⊢ ((𝑄 ∈ ℤ ∧ ((𝑃 − 1) / 2) ∈ ℕ₀) → (𝑄↑((𝑃 − 1) / 2)) ∈ ℤ)
264	222, 229, 263	syl2anc 696	. . . 4 ⊢ (𝜑 → (𝑄↑((𝑃 − 1) / 2)) ∈ ℤ)
265	19, 84	fsumzcl 14665	. . . . 5 ⊢ (𝜑 → Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))) ∈ ℤ)
266		m1expcl 13077	. . . . 5 ⊢ (Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))) ∈ ℤ → (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ∈ ℤ)
267	265, 266	syl 17	. . . 4 ⊢ (𝜑 → (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ∈ ℤ)
268	8, 25	zndvds 20100	. . . 4 ⊢ ((𝑃 ∈ ℕ₀ ∧ (𝑄↑((𝑃 − 1) / 2)) ∈ ℤ ∧ (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ∈ ℤ) → ((𝐿‘(𝑄↑((𝑃 − 1) / 2))) = (𝐿‘(-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) ↔ 𝑃 ∥ ((𝑄↑((𝑃 − 1) / 2)) − (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))))
269	262, 264, 267, 268	syl3anc 1477	. . 3 ⊢ (𝜑 → ((𝐿‘(𝑄↑((𝑃 − 1) / 2))) = (𝐿‘(-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) ↔ 𝑃 ∥ ((𝑄↑((𝑃 − 1) / 2)) − (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))))
270	261, 269	mpbid 222	. 2 ⊢ (𝜑 → 𝑃 ∥ ((𝑄↑((𝑃 − 1) / 2)) − (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))))
271		moddvds 15193	. . 3 ⊢ ((𝑃 ∈ ℕ ∧ (𝑄↑((𝑃 − 1) / 2)) ∈ ℤ ∧ (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ∈ ℤ) → (((𝑄↑((𝑃 − 1) / 2)) mod 𝑃) = ((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) mod 𝑃) ↔ 𝑃 ∥ ((𝑄↑((𝑃 − 1) / 2)) − (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))))
272	74, 264, 267, 271	syl3anc 1477	. 2 ⊢ (𝜑 → (((𝑄↑((𝑃 − 1) / 2)) mod 𝑃) = ((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) mod 𝑃) ↔ 𝑃 ∥ ((𝑄↑((𝑃 − 1) / 2)) − (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))))
273	270, 272	mpbird 247	1 ⊢ (𝜑 → ((𝑄↑((𝑃 − 1) / 2)) mod 𝑃) = ((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) mod 𝑃))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 383 = wceq 1632 ∈ wcel 2139 ≠ wne 2932 Vcvv 3340 ∖ cdif 3712 ⊆ wss 3715 {csn 4321 class class class wbr 4804 ↦ cmpt 4881 ran crn 5267 ∘ ccom 5270 ⟶wf 6045 ‘cfv 6049 (class class class)co 6813 ∘_𝑓 cof 7060 Fincfn 8121 ℂcc 10126 ℝcr 10127 0cc0 10128 1c1 10129 + caddc 10131 · cmul 10133 − cmin 10458 -cneg 10459 / cdiv 10876 ℕcn 11212 2c2 11262 ℕ₀cn0 11484 ℤcz 11569 ℤ_≥cuz 11879 ℝ⁺crp 12025 ...cfz 12519 ⌊cfl 12785 mod cmo 12862 ↑cexp 13054 ♯chash 13311 Σcsu 14615 ∥ cdvds 15182 ℙcprime 15587 Basecbs 16059 ↾_s cress 16060 ._rcmulr 16144 0_gc0g 16302 Σ_g cgsu 16303 Mndcmnd 17495 MndHom cmhm 17534 SubMndcsubmnd 17535 ._gcmg 17741 SubGrpcsubg 17789 GrpHom cghm 17858 CMndccmn 18393 Abelcabl 18394 mulGrpcmgp 18689 1_rcur 18701 Ringcrg 18747 CRingccrg 18748 RingHom crh 18914 DivRingcdr 18949 Fieldcfield 18950 SubRingcsubrg 18978 ℂ_fldccnfld 19948 ℤ_ringzring 20020 ℤRHomczrh 20050 ℤ/nℤczn 20053

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-inf2 8711 ax-cnex 10184 ax-resscn 10185 ax-1cn 10186 ax-icn 10187 ax-addcl 10188 ax-addrcl 10189 ax-mulcl 10190 ax-mulrcl 10191 ax-mulcom 10192 ax-addass 10193 ax-mulass 10194 ax-distr 10195 ax-i2m1 10196 ax-1ne0 10197 ax-1rid 10198 ax-rnegex 10199 ax-rrecex 10200 ax-cnre 10201 ax-pre-lttri 10202 ax-pre-lttrn 10203 ax-pre-ltadd 10204 ax-pre-mulgt0 10205 ax-pre-sup 10206 ax-addf 10207 ax-mulf 10208

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-fal 1638 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-se 5226 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-isom 6058 df-riota 6774 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-of 7062 df-om 7231 df-1st 7333 df-2nd 7334 df-supp 7464 df-tpos 7521 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-1o 7729 df-2o 7730 df-oadd 7733 df-er 7911 df-ec 7913 df-qs 7917 df-map 8025 df-en 8122 df-dom 8123 df-sdom 8124 df-fin 8125 df-fsupp 8441 df-sup 8513 df-inf 8514 df-oi 8580 df-card 8955 df-cda 9182 df-pnf 10268 df-mnf 10269 df-xr 10270 df-ltxr 10271 df-le 10272 df-sub 10460 df-neg 10461 df-div 10877 df-nn 11213 df-2 11271 df-3 11272 df-4 11273 df-5 11274 df-6 11275 df-7 11276 df-8 11277 df-9 11278 df-n0 11485 df-xnn0 11556 df-z 11570 df-dec 11686 df-uz 11880 df-rp 12026 df-fz 12520 df-fzo 12660 df-fl 12787 df-mod 12863 df-seq 12996 df-exp 13055 df-hash 13312 df-cj 14038 df-re 14039 df-im 14040 df-sqrt 14174 df-abs 14175 df-clim 14418 df-sum 14616 df-dvds 15183 df-gcd 15419 df-prm 15588 df-struct 16061 df-ndx 16062 df-slot 16063 df-base 16065 df-sets 16066 df-ress 16067 df-plusg 16156 df-mulr 16157 df-starv 16158 df-sca 16159 df-vsca 16160 df-ip 16161 df-tset 16162 df-ple 16163 df-ds 16166 df-unif 16167 df-0g 16304 df-gsum 16305 df-imas 16370 df-qus 16371 df-mgm 17443 df-sgrp 17485 df-mnd 17496 df-mhm 17536 df-submnd 17537 df-grp 17626 df-minusg 17627 df-sbg 17628 df-mulg 17742 df-subg 17792 df-nsg 17793 df-eqg 17794 df-ghm 17859 df-cntz 17950 df-cmn 18395 df-abl 18396 df-mgp 18690 df-ur 18702 df-ring 18749 df-cring 18750 df-oppr 18823 df-dvdsr 18841 df-unit 18842 df-invr 18872 df-dvr 18883 df-rnghom 18917 df-drng 18951 df-field 18952 df-subrg 18980 df-lmod 19067 df-lss 19135 df-lsp 19174 df-sra 19374 df-rgmod 19375 df-lidl 19376 df-rsp 19377 df-2idl 19434 df-nzr 19460 df-rlreg 19485 df-domn 19486 df-idom 19487 df-cnfld 19949 df-zring 20021 df-zrh 20054 df-zn 20057

This theorem is referenced by: lgseisen 25303

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