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Theorem lgseisenlem4 27705

Description: Lemma for lgseisen 27706. (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 21759	. . . . 5 ⊢ ℤ = (Base‘ℤ_ring)
2		zring0 21764	. . . . 5 ⊢ 0 = (0_g‘ℤ_ring)
3		zringabl 21757	. . . . . 6 ⊢ ℤ_ring ∈ Abel
4		ablcmn 20001	. . . . . 6 ⊢ (ℤ_ring ∈ Abel → ℤ_ring ∈ CMnd)
5	3, 4	mp1i 14	. . . . 5 ⊢ (𝜑 → ℤ_ring ∈ CMnd)
6		lgseisen.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))
7	6	eldifad 4051	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ ℙ)
8		lgseisen.7	. . . . . . . . . 10 ⊢ 𝑌 = (ℤ/nℤ‘𝑃)
9	8	znfld 21866	. . . . . . . . 9 ⊢ (𝑃 ∈ ℙ → 𝑌 ∈ Field)
10	7, 9	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ Field)
11		isfld 20993	. . . . . . . . 9 ⊢ (𝑌 ∈ Field ↔ (𝑌 ∈ DivRing ∧ 𝑌 ∈ CRing))
12	11	simprbi 503	. . . . . . . 8 ⊢ (𝑌 ∈ Field → 𝑌 ∈ CRing)
13	10, 12	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ CRing)
14		lgseisen.8	. . . . . . . 8 ⊢ 𝐺 = (mulGrp‘𝑌)
15	14	crngmgp 20467	. . . . . . 7 ⊢ (𝑌 ∈ CRing → 𝐺 ∈ CMnd)
16	13, 15	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ CMnd)
17		cmnmnd 20011	. . . . . 6 ⊢ (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
18	16, 17	syl 18	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ Mnd)
19		fzfid 14116	. . . . 5 ⊢ (𝜑 → (1...((𝑃 − 1) / 2)) ∈ Fin)
20		crngring 20472	. . . . . . . . . 10 ⊢ (𝑌 ∈ CRing → 𝑌 ∈ Ring)
21	13, 20	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ Ring)
22		lgseisen.9	. . . . . . . . . 10 ⊢ 𝐿 = (ℤRHom‘𝑌)
23	22	zrhrhm 21817	. . . . . . . . 9 ⊢ (𝑌 ∈ Ring → 𝐿 ∈ (ℤ_ring RingHom 𝑌))
24	21, 23	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐿 ∈ (ℤ_ring RingHom 𝑌))
25		eqid 2760	. . . . . . . . 9 ⊢ (Base‘𝑌) = (Base‘𝑌)
26	1, 25	rhmf 20715	. . . . . . . 8 ⊢ (𝐿 ∈ (ℤ_ring RingHom 𝑌) → 𝐿:ℤ⟶(Base‘𝑌))
27	24, 26	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐿:ℤ⟶(Base‘𝑌))
28		m1expcl 14229	. . . . . . . 8 ⊢ (𝑘 ∈ ℤ → ( -1↑𝑘) ∈ ℤ)
29	28	adantl 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → ( -1↑𝑘) ∈ ℤ)
30	27, 29	cofmpt 7133	. . . . . 6 ⊢ (𝜑 → (𝐿 ∘ (𝑘 ∈ ℤ ↦ ( -1↑𝑘))) = (𝑘 ∈ ℤ ↦ (𝐿‘( -1↑𝑘))))
31		zringmpg 21777	. . . . . . . . 9 ⊢ ((mulGrp‘ℂ_fld) ↾_s ℤ) = (mulGrp‘ℤ_ring)
32	31, 14	rhmmhm 20710	. . . . . . . 8 ⊢ (𝐿 ∈ (ℤ_ring RingHom 𝑌) → 𝐿 ∈ (((mulGrp‘ℂ_fld) ↾_s ℤ) MndHom 𝐺))
33	24, 32	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ (((mulGrp‘ℂ_fld) ↾_s ℤ) MndHom 𝐺))
34		neg1cn 12305	. . . . . . . . . . 11 ⊢ -1 ∈ ℂ
35		neg1ne0 12307	. . . . . . . . . . 11 ⊢ -1 ≠ 0
36		eqid 2760	. . . . . . . . . . . 12 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
37		eqid 2760	. . . . . . . . . . . 12 ⊢ ((mulGrp‘ℂ_fld) ↾_s (ℂ ∖ {0})) = ((mulGrp‘ℂ_fld) ↾_s (ℂ ∖ {0}))
38	36, 37	expghm 21781	. . . . . . . . . . 11 ⊢ (( -1 ∈ ℂ ∧ -1 ≠ 0) → (𝑘 ∈ ℤ ↦ ( -1↑𝑘)) ∈ (ℤ_ring GrpHom ((mulGrp‘ℂ_fld) ↾_s (ℂ ∖ {0}))))
39	34, 35, 38	mp2an 705	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℤ ↦ ( -1↑𝑘)) ∈ (ℤ_ring GrpHom ((mulGrp‘ℂ_fld) ↾_s (ℂ ∖ {0})))
40		ghmmhm 19440	. . . . . . . . . 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}))))
41	39, 40	ax-mp 5	. . . . . . . . 9 ⊢ (𝑘 ∈ ℤ ↦ ( -1↑𝑘)) ∈ (ℤ_ring MndHom ((mulGrp‘ℂ_fld) ↾_s (ℂ ∖ {0})))
42		cnring 21700	. . . . . . . . . 10 ⊢ ℂ_fld ∈ Ring
43		cnfldbas 21682	. . . . . . . . . . . 12 ⊢ ℂ = (Base‘ℂ_fld)
44		cnfld0 21702	. . . . . . . . . . . 12 ⊢ 0 = (0_g‘ℂ_fld)
45		cndrng 21707	. . . . . . . . . . . 12 ⊢ ℂ_fld ∈ DivRing
46	43, 44, 45	drngui 20986	. . . . . . . . . . 11 ⊢ (ℂ ∖ {0}) = (Unit‘ℂ_fld)
47	46, 36	unitsubm 20616	. . . . . . . . . 10 ⊢ (ℂ_fld ∈ Ring → (ℂ ∖ {0}) ∈ (SubMnd‘(mulGrp‘ℂ_fld)))
48	42, 47	ax-mp 5	. . . . . . . . 9 ⊢ (ℂ ∖ {0}) ∈ (SubMnd‘(mulGrp‘ℂ_fld))
49	37	resmhm2 19017	. . . . . . . . 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)))
50	41, 48, 49	mp2an 705	. . . . . . . 8 ⊢ (𝑘 ∈ ℤ ↦ ( -1↑𝑘)) ∈ (ℤ_ring MndHom (mulGrp‘ℂ_fld))
51		zsubrg 21726	. . . . . . . . . 10 ⊢ ℤ ∈ (SubRing‘ℂ_fld)
52	36	subrgsubm 20837	. . . . . . . . . 10 ⊢ (ℤ ∈ (SubRing‘ℂ_fld) → ℤ ∈ (SubMnd‘(mulGrp‘ℂ_fld)))
53	51, 52	ax-mp 5	. . . . . . . . 9 ⊢ ℤ ∈ (SubMnd‘(mulGrp‘ℂ_fld))
54	29	fmpttd 7115	. . . . . . . . . 10 ⊢ (𝜑 → (𝑘 ∈ ℤ ↦ ( -1↑𝑘)):ℤ⟶ℤ)
55	54	frnd 6718	. . . . . . . . 9 ⊢ (𝜑 → ran (𝑘 ∈ ℤ ↦ ( -1↑𝑘)) ⊆ ℤ)
56		eqid 2760	. . . . . . . . . 10 ⊢ ((mulGrp‘ℂ_fld) ↾_s ℤ) = ((mulGrp‘ℂ_fld) ↾_s ℤ)
57	56	resmhm2b 19018	. . . . . . . . 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))))
58	53, 55, 57	sylancr 599	. . . . . . . 8 ⊢ (𝜑 → ((𝑘 ∈ ℤ ↦ ( -1↑𝑘)) ∈ (ℤ_ring MndHom (mulGrp‘ℂ_fld)) ↔ (𝑘 ∈ ℤ ↦ ( -1↑𝑘)) ∈ (ℤ_ring MndHom ((mulGrp‘ℂ_fld) ↾_s ℤ))))
59	50, 58	mpbii 236	. . . . . . 7 ⊢ (𝜑 → (𝑘 ∈ ℤ ↦ ( -1↑𝑘)) ∈ (ℤ_ring MndHom ((mulGrp‘ℂ_fld) ↾_s ℤ)))
60		mhmco 19019	. . . . . . 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))
61	33, 59, 60	syl2anc 596	. . . . . 6 ⊢ (𝜑 → (𝐿 ∘ (𝑘 ∈ ℤ ↦ ( -1↑𝑘))) ∈ (ℤ_ring MndHom 𝐺))
62	30, 61	eqeltrrd 2861	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ ℤ ↦ (𝐿‘( -1↑𝑘))) ∈ (ℤ_ring MndHom 𝐺))
63		lgseisen.2	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑄 ∈ (ℙ ∖ {2}))
64	63	gausslemma2dlem0a 27683	. . . . . . . . . 10 ⊢ (𝜑 → 𝑄 ∈ ℕ)
65	64	nnred 12350	. . . . . . . . 9 ⊢ (𝜑 → 𝑄 ∈ ℝ)
66	6	gausslemma2dlem0a 27683	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ ℕ)
67	65, 66	nndivred 12392	. . . . . . . 8 ⊢ (𝜑 → (𝑄 / 𝑃) ∈ ℝ)
68	67	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑄 / 𝑃) ∈ ℝ)
69		2nn 12416	. . . . . . . . 9 ⊢ 2 ∈ ℕ
70		elfznn 13687	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1...((𝑃 − 1) / 2)) → 𝑥 ∈ ℕ)
71	70	adantl 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑥 ∈ ℕ)
72		nnmulcl 12359	. . . . . . . . 9 ⊢ ((2 ∈ ℕ ∧ 𝑥 ∈ ℕ) → (2 · 𝑥) ∈ ℕ)
73	69, 71, 72	sylancr 599	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (2 · 𝑥) ∈ ℕ)
74	73	nnred 12350	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (2 · 𝑥) ∈ ℝ)
75	68, 74	remulcld 11339	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝑄 / 𝑃) · (2 · 𝑥)) ∈ ℝ)
76	75	flcld 13938	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))) ∈ ℤ)
77		eqid 2760	. . . . . 6 ⊢ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))
78		fvexd 6900	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))) ∈ V)
79		c0ex 11300	. . . . . . 7 ⊢ 0 ∈ V
80	79	a1i 11	. . . . . 6 ⊢ (𝜑 → 0 ∈ V)
81	77, 19, 78, 80	fsuppmptdm 9368	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) finSupp 0)
82		oveq2 7428	. . . . . 6 ⊢ (𝑘 = (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))) → ( -1↑𝑘) = ( -1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))
83	82	fveq2d 6889	. . . . 5 ⊢ (𝑘 = (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))) → (𝐿‘( -1↑𝑘)) = (𝐿‘( -1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))))
84		oveq2 7428	. . . . . 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)))))))
85	84	fveq2d 6889	. . . . 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))))))))
86	1, 2, 5, 18, 19, 62, 76, 81, 83, 85	gsummhm2 20153	. . . 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))))))))
87	14, 25	mgpbas 20365	. . . . . . 7 ⊢ (Base‘𝑌) = (Base‘𝐺)
88		eqid 2760	. . . . . . . 8 ⊢ (._r‘𝑌) = (._r‘𝑌)
89	14, 88	mgpplusg 20364	. . . . . . 7 ⊢ (._r‘𝑌) = (+_g‘𝐺)
90	27	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝐿:ℤ⟶(Base‘𝑌))
91		m1expcl 14229	. . . . . . . . 9 ⊢ ((⌊‘((𝑄 / 𝑃) · (2 · 𝑥))) ∈ ℤ → ( -1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ∈ ℤ)
92	76, 91	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ( -1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ∈ ℤ)
93	90, 92	ffvelcdmd 7085	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘( -1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) ∈ (Base‘𝑌))
94		neg1z 12732	. . . . . . . . . 10 ⊢ -1 ∈ ℤ
95		lgseisen.4	. . . . . . . . . . 11 ⊢ 𝑅 = ((𝑄 · (2 · 𝑥)) mod 𝑃)
96	63	eldifad 4051	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑄 ∈ ℙ)
97	96	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑄 ∈ ℙ)
98		prmz 16850	. . . . . . . . . . . . . 14 ⊢ (𝑄 ∈ ℙ → 𝑄 ∈ ℤ)
99	97, 98	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑄 ∈ ℤ)
100	73	nnzd 12719	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (2 · 𝑥) ∈ ℤ)
101	99, 100	zmulcld 12809	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑄 · (2 · 𝑥)) ∈ ℤ)
102	7	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 ∈ ℙ)
103		prmnn 16849	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
104	102, 103	syl 18	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 ∈ ℕ)
105	101, 104	zmodcld 14032	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝑄 · (2 · 𝑥)) mod 𝑃) ∈ ℕ₀)
106	95, 105	eqeltrid 2864	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑅 ∈ ℕ₀)
107		zexpcl 14219	. . . . . . . . . 10 ⊢ (( -1 ∈ ℤ ∧ 𝑅 ∈ ℕ₀) → ( -1↑𝑅) ∈ ℤ)
108	94, 106, 107	sylancr 599	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ( -1↑𝑅) ∈ ℤ)
109	108, 99	zmulcld 12809	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (( -1↑𝑅) · 𝑄) ∈ ℤ)
110	90, 109	ffvelcdmd 7085	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘(( -1↑𝑅) · 𝑄)) ∈ (Base‘𝑌))
111		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))))))
112		eqid 2760	. . . . . . 7 ⊢ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(( -1↑𝑅) · 𝑄))) = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(( -1↑𝑅) · 𝑄)))
113	87, 89, 16, 19, 93, 110, 111, 112	gsummptfidmadd2 20140	. . . . . 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))))))
114		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)))))))
115		eqidd 2761	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(( -1↑𝑅) · 𝑄))) = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(( -1↑𝑅) · 𝑄))))
116	19, 93, 110, 114, 115	offval2 7713	. . . . . . . 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)))))
117	24	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝐿 ∈ (ℤ_ring RingHom 𝑌))
118		zringmulr 21763	. . . . . . . . . . . 12 ⊢ · = (._r‘ℤ_ring)
119	1, 118, 88	rhmmul 20720	. . . . . . . . . . 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))))
120	117, 92, 109, 119	syl3anc 1398	. . . . . . . . . 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))))
121	101	zred 12803	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑄 · (2 · 𝑥)) ∈ ℝ)
122	104	nnrpd 13162	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 ∈ ℝ⁺)
123		modval 14011	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑄 · (2 · 𝑥)) ∈ ℝ ∧ 𝑃 ∈ ℝ⁺) → ((𝑄 · (2 · 𝑥)) mod 𝑃) = ((𝑄 · (2 · 𝑥)) − (𝑃 · (⌊‘((𝑄 · (2 · 𝑥)) / 𝑃)))))
124	121, 122, 123	syl2anc 596	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝑄 · (2 · 𝑥)) mod 𝑃) = ((𝑄 · (2 · 𝑥)) − (𝑃 · (⌊‘((𝑄 · (2 · 𝑥)) / 𝑃)))))
125	95, 124	eqtrid 2807	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑅 = ((𝑄 · (2 · 𝑥)) − (𝑃 · (⌊‘((𝑄 · (2 · 𝑥)) / 𝑃)))))
126	99	zcnd 12804	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑄 ∈ ℂ)
127	73	nncnd 12351	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (2 · 𝑥) ∈ ℂ)
128	104	nncnd 12351	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 ∈ ℂ)
129	104	nnne0d 12388	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 ≠ 0)
130	126, 127, 128, 129	div23d 12130	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝑄 · (2 · 𝑥)) / 𝑃) = ((𝑄 / 𝑃) · (2 · 𝑥)))
131	130	fveq2d 6889	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (⌊‘((𝑄 · (2 · 𝑥)) / 𝑃)) = (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))
132	131	oveq2d 7436	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑃 · (⌊‘((𝑄 · (2 · 𝑥)) / 𝑃))) = (𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))
133	132	oveq2d 7436	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝑄 · (2 · 𝑥)) − (𝑃 · (⌊‘((𝑄 · (2 · 𝑥)) / 𝑃)))) = ((𝑄 · (2 · 𝑥)) − (𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))))
134	125, 133	eqtrd 2795	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑅 = ((𝑄 · (2 · 𝑥)) − (𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))))
135	134	oveq2d 7436	. . . . . . . . . . . . . . . 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)))))))
136		prmz 16850	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
137	102, 136	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 ∈ ℤ)
138	137, 76	zmulcld 12809	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ∈ ℤ)
139	138	zcnd 12804	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ∈ ℂ)
140	101	zcnd 12804	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑄 · (2 · 𝑥)) ∈ ℂ)
141	139, 140	pncan3d 11672	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + ((𝑄 · (2 · 𝑥)) − (𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))) = (𝑄 · (2 · 𝑥)))
142		2cnd 12421	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 2 ∈ ℂ)
143	71	nncnd 12351	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑥 ∈ ℂ)
144	126, 142, 143	mul12d 11519	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑄 · (2 · 𝑥)) = (2 · (𝑄 · 𝑥)))
145	135, 141, 144	3eqtrd 2799	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 𝑅) = (2 · (𝑄 · 𝑥)))
146	145	oveq2d 7436	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ( -1↑((𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 𝑅)) = ( -1↑(2 · (𝑄 · 𝑥))))
147	34	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → -1 ∈ ℂ)
148	35	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → -1 ≠ 0)
149	106	nn0zd 12718	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑅 ∈ ℤ)
150		expaddz 14249	. . . . . . . . . . . . . . . 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)))
151	147, 148, 138, 149, 150	syl22anc 852	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ( -1↑((𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 𝑅)) = (( -1↑(𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) · ( -1↑𝑅)))
152		expmulz 14251	. . . . . . . . . . . . . . . . . 18 ⊢ ((( -1 ∈ ℂ ∧ -1 ≠ 0) ∧ (𝑃 ∈ ℤ ∧ (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))) ∈ ℤ)) → ( -1↑(𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) = (( -1↑𝑃)↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))
153	147, 148, 137, 76, 152	syl22anc 852	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ( -1↑(𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) = (( -1↑𝑃)↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))
154		1cnd 11302	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 1 ∈ ℂ)
155		eldifsni 4752	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ≠ 2)
156	6, 155	syl 18	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝑃 ≠ 2)
157	156	necomd 3010	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 2 ≠ 𝑃)
158	157	neneqd 2960	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ¬ 2 = 𝑃)
159	158	adantr 486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ¬ 2 = 𝑃)
160		2z 12728	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 2 ∈ ℤ
161		uzid 12980	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (2 ∈ ℤ → 2 ∈ (ℤ_≥‘2))
162	160, 161	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 2 ∈ (ℤ_≥‘2)
163		dvdsprm 16879	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((2 ∈ (ℤ_≥‘2) ∧ 𝑃 ∈ ℙ) → (2 ∥ 𝑃 ↔ 2 = 𝑃))
164	162, 102, 163	sylancr 599	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (2 ∥ 𝑃 ↔ 2 = 𝑃))
165	159, 164	mtbird 328	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ¬ 2 ∥ 𝑃)
166		oexpneg 16515	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1 ∈ ℂ ∧ 𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃) → ( -1↑𝑃) = -(1↑𝑃))
167	154, 104, 165, 166	syl3anc 1398	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ( -1↑𝑃) = -(1↑𝑃))
168		1exp 14234	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑃 ∈ ℤ → (1↑𝑃) = 1)
169	137, 168	syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (1↑𝑃) = 1)
170	169	negeqd 11551	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → -(1↑𝑃) = -1)
171	167, 170	eqtrd 2795	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ( -1↑𝑃) = -1)
172	171	oveq1d 7435	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (( -1↑𝑃)↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) = ( -1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))
173	153, 172	eqtrd 2795	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ( -1↑(𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) = ( -1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))
174	173	oveq1d 7435	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (( -1↑(𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) · ( -1↑𝑅)) = (( -1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) · ( -1↑𝑅)))
175	151, 174	eqtrd 2795	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ( -1↑((𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 𝑅)) = (( -1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) · ( -1↑𝑅)))
176		nnmulcl 12359	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑄 ∈ ℕ ∧ 𝑥 ∈ ℕ) → (𝑄 · 𝑥) ∈ ℕ)
177	64, 70, 176	syl2an 608	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑄 · 𝑥) ∈ ℕ)
178	177	nnnn0d 12667	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑄 · 𝑥) ∈ ℕ₀)
179		2nn0 12623	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℕ₀
180	179	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 2 ∈ ℕ₀)
181	147, 178, 180	expmuld 14292	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ( -1↑(2 · (𝑄 · 𝑥))) = (( -1↑2)↑(𝑄 · 𝑥)))
182		neg1sqe1 14339	. . . . . . . . . . . . . . . . 17 ⊢ ( -1↑2) = 1
183	182	oveq1i 7430	. . . . . . . . . . . . . . . 16 ⊢ (( -1↑2)↑(𝑄 · 𝑥)) = (1↑(𝑄 · 𝑥))
184	177	nnzd 12719	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑄 · 𝑥) ∈ ℤ)
185		1exp 14234	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑄 · 𝑥) ∈ ℤ → (1↑(𝑄 · 𝑥)) = 1)
186	184, 185	syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (1↑(𝑄 · 𝑥)) = 1)
187	183, 186	eqtrid 2807	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (( -1↑2)↑(𝑄 · 𝑥)) = 1)
188	181, 187	eqtrd 2795	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ( -1↑(2 · (𝑄 · 𝑥))) = 1)
189	146, 175, 188	3eqtr3d 2803	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (( -1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) · ( -1↑𝑅)) = 1)
190	189	oveq1d 7435	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((( -1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) · ( -1↑𝑅)) · 𝑄) = (1 · 𝑄))
191	92	zcnd 12804	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ( -1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ∈ ℂ)
192	108	zcnd 12804	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ( -1↑𝑅) ∈ ℂ)
193	191, 192, 126	mulassd 11332	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((( -1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) · ( -1↑𝑅)) · 𝑄) = (( -1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) · (( -1↑𝑅) · 𝑄)))
194	126	mullidd 11327	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (1 · 𝑄) = 𝑄)
195	190, 193, 194	3eqtr3d 2803	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (( -1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) · (( -1↑𝑅) · 𝑄)) = 𝑄)
196	195	fveq2d 6889	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘(( -1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) · (( -1↑𝑅) · 𝑄))) = (𝐿‘𝑄))
197	120, 196	eqtr3d 2797	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝐿‘( -1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))(._r‘𝑌)(𝐿‘(( -1↑𝑅) · 𝑄))) = (𝐿‘𝑄))
198	197	mpteq2dva 5197	. . . . . . . 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)))
199	116, 198	eqtrd 2795	. . . . . . 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)))
200	199	oveq2d 7436	. . . . . 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))))
201		lgseisen.3	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ≠ 𝑄)
202		lgseisen.5	. . . . . . . 8 ⊢ 𝑀 = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (((( -1↑𝑅) · 𝑅) mod 𝑃) / 2))
203		lgseisen.6	. . . . . . . 8 ⊢ 𝑆 = ((𝑄 · (2 · 𝑦)) mod 𝑃)
204	6, 63, 201, 95, 202, 203, 8, 14, 22	lgseisenlem3 27704	. . . . . . 7 ⊢ (𝜑 → (𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(( -1↑𝑅) · 𝑄)))) = (1_r‘𝑌))
205	204	oveq2d 7436	. . . . . 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)))
206	113, 200, 205	3eqtr3rd 2804	. . . . 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))))
207		eqid 2760	. . . . . . 7 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
208	93	fmpttd 7115	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘( -1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))):(1...((𝑃 − 1) / 2))⟶(Base‘𝑌))
209		fvexd 6900	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘( -1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) ∈ V)
210		fvexd 6900	. . . . . . . 8 ⊢ (𝜑 → (0_g‘𝐺) ∈ V)
211	111, 19, 209, 210	fsuppmptdm 9368	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘( -1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))) finSupp (0_g‘𝐺))
212	87, 207, 16, 19, 208, 211	gsumcl 20129	. . . . . 6 ⊢ (𝜑 → (𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘( -1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))))) ∈ (Base‘𝑌))
213		eqid 2760	. . . . . . 7 ⊢ (1_r‘𝑌) = (1_r‘𝑌)
214	25, 88, 213	ringridm 20499	. . . . . 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))))))))
215	21, 212, 214	syl2anc 596	. . . . 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))))))))
216	96, 98	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝑄 ∈ ℤ)
217	27, 216	ffvelcdmd 7085	. . . . . . 7 ⊢ (𝜑 → (𝐿‘𝑄) ∈ (Base‘𝑌))
218		eqid 2760	. . . . . . . 8 ⊢ (._g‘𝐺) = (._g‘𝐺)
219	87, 218	gsumconst 20148	. . . . . . 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)))
220	18, 19, 217, 219	syl3anc 1398	. . . . . 6 ⊢ (𝜑 → (𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘𝑄))) = ((♯‘(1...((𝑃 − 1) / 2)))(._g‘𝐺)(𝐿‘𝑄)))
221		oddprm 16988	. . . . . . . . . 10 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((𝑃 − 1) / 2) ∈ ℕ)
222	6, 221	syl 18	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃 − 1) / 2) ∈ ℕ)
223	222	nnnn0d 12667	. . . . . . . 8 ⊢ (𝜑 → ((𝑃 − 1) / 2) ∈ ℕ₀)
224		hashfz1 14490	. . . . . . . 8 ⊢ (((𝑃 − 1) / 2) ∈ ℕ₀ → (♯‘(1...((𝑃 − 1) / 2))) = ((𝑃 − 1) / 2))
225	223, 224	syl 18	. . . . . . 7 ⊢ (𝜑 → (♯‘(1...((𝑃 − 1) / 2))) = ((𝑃 − 1) / 2))
226	225	oveq1d 7435	. . . . . 6 ⊢ (𝜑 → ((♯‘(1...((𝑃 − 1) / 2)))(._g‘𝐺)(𝐿‘𝑄)) = (((𝑃 − 1) / 2)(._g‘𝐺)(𝐿‘𝑄)))
227	31, 1	mgpbas 20365	. . . . . . . . 9 ⊢ ℤ = (Base‘((mulGrp‘ℂ_fld) ↾_s ℤ))
228		eqid 2760	. . . . . . . . 9 ⊢ (._g‘((mulGrp‘ℂ_fld) ↾_s ℤ)) = (._g‘((mulGrp‘ℂ_fld) ↾_s ℤ))
229	227, 228, 218	mhmmulg 19325	. . . . . . . 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)))
230	33, 223, 216, 229	syl3anc 1398	. . . . . . 7 ⊢ (𝜑 → (𝐿‘(((𝑃 − 1) / 2)(._g‘((mulGrp‘ℂ_fld) ↾_s ℤ))𝑄)) = (((𝑃 − 1) / 2)(._g‘𝐺)(𝐿‘𝑄)))
231	53	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ℤ ∈ (SubMnd‘(mulGrp‘ℂ_fld)))
232		eqid 2760	. . . . . . . . . . 11 ⊢ (._g‘(mulGrp‘ℂ_fld)) = (._g‘(mulGrp‘ℂ_fld))
233	232, 56, 228	submmulg 19328	. . . . . . . . . 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))
234	231, 223, 216, 233	syl3anc 1398	. . . . . . . . 9 ⊢ (𝜑 → (((𝑃 − 1) / 2)(._g‘(mulGrp‘ℂ_fld))𝑄) = (((𝑃 − 1) / 2)(._g‘((mulGrp‘ℂ_fld) ↾_s ℤ))𝑄))
235	216	zcnd 12804	. . . . . . . . . 10 ⊢ (𝜑 → 𝑄 ∈ ℂ)
236		cnfldexp 21711	. . . . . . . . . 10 ⊢ ((𝑄 ∈ ℂ ∧ ((𝑃 − 1) / 2) ∈ ℕ₀) → (((𝑃 − 1) / 2)(._g‘(mulGrp‘ℂ_fld))𝑄) = (𝑄↑((𝑃 − 1) / 2)))
237	235, 223, 236	syl2anc 596	. . . . . . . . 9 ⊢ (𝜑 → (((𝑃 − 1) / 2)(._g‘(mulGrp‘ℂ_fld))𝑄) = (𝑄↑((𝑃 − 1) / 2)))
238	234, 237	eqtr3d 2797	. . . . . . . 8 ⊢ (𝜑 → (((𝑃 − 1) / 2)(._g‘((mulGrp‘ℂ_fld) ↾_s ℤ))𝑄) = (𝑄↑((𝑃 − 1) / 2)))
239	238	fveq2d 6889	. . . . . . 7 ⊢ (𝜑 → (𝐿‘(((𝑃 − 1) / 2)(._g‘((mulGrp‘ℂ_fld) ↾_s ℤ))𝑄)) = (𝐿‘(𝑄↑((𝑃 − 1) / 2))))
240	230, 239	eqtr3d 2797	. . . . . 6 ⊢ (𝜑 → (((𝑃 − 1) / 2)(._g‘𝐺)(𝐿‘𝑄)) = (𝐿‘(𝑄↑((𝑃 − 1) / 2))))
241	220, 226, 240	3eqtrd 2799	. . . . 5 ⊢ (𝜑 → (𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘𝑄))) = (𝐿‘(𝑄↑((𝑃 − 1) / 2))))
242	206, 215, 241	3eqtr3d 2803	. . . 4 ⊢ (𝜑 → (𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘( -1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))))) = (𝐿‘(𝑄↑((𝑃 − 1) / 2))))
243		subrgsubg 20829	. . . . . . . . . 10 ⊢ (ℤ ∈ (SubRing‘ℂ_fld) → ℤ ∈ (SubGrp‘ℂ_fld))
244	51, 243	ax-mp 5	. . . . . . . . 9 ⊢ ℤ ∈ (SubGrp‘ℂ_fld)
245		subgsubm 19359	. . . . . . . . 9 ⊢ (ℤ ∈ (SubGrp‘ℂ_fld) → ℤ ∈ (SubMnd‘ℂ_fld))
246	244, 245	mp1i 14	. . . . . . . 8 ⊢ (𝜑 → ℤ ∈ (SubMnd‘ℂ_fld))
247	76	fmpttd 7115	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))):(1...((𝑃 − 1) / 2))⟶ℤ)
248		df-zring 21753	. . . . . . . 8 ⊢ ℤ_ring = (ℂ_fld ↾_s ℤ)
249	19, 246, 247, 248	gsumsubm 19031	. . . . . . 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))))))
250	76	zcnd 12804	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))) ∈ ℂ)
251	19, 250	gsumfsum 21740	. . . . . . 7 ⊢ (𝜑 → (ℂ_fld Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) = Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))
252	249, 251	eqtr3d 2797	. . . . . 6 ⊢ (𝜑 → (ℤ_ring Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) = Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))
253	252	oveq2d 7436	. . . . 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)))))
254	253	fveq2d 6889	. . . 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))))))
255	86, 242, 254	3eqtr3d 2803	. . 3 ⊢ (𝜑 → (𝐿‘(𝑄↑((𝑃 − 1) / 2))) = (𝐿‘( -1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))))
256	66	nnnn0d 12667	. . . 4 ⊢ (𝜑 → 𝑃 ∈ ℕ₀)
257		zexpcl 14219	. . . . 5 ⊢ ((𝑄 ∈ ℤ ∧ ((𝑃 − 1) / 2) ∈ ℕ₀) → (𝑄↑((𝑃 − 1) / 2)) ∈ ℤ)
258	216, 223, 257	syl2anc 596	. . . 4 ⊢ (𝜑 → (𝑄↑((𝑃 − 1) / 2)) ∈ ℤ)
259	19, 76	fsumzcl 15901	. . . . 5 ⊢ (𝜑 → Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))) ∈ ℤ)
260		m1expcl 14229	. . . . 5 ⊢ (Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))) ∈ ℤ → ( -1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ∈ ℤ)
261	259, 260	syl 18	. . . 4 ⊢ (𝜑 → ( -1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ∈ ℤ)
262	8, 22	zndvds 21855	. . . 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)))))))
263	256, 258, 261, 262	syl3anc 1398	. . 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)))))))
264	255, 263	mpbid 235	. 2 ⊢ (𝜑 → 𝑃 ∥ ((𝑄↑((𝑃 − 1) / 2)) − ( -1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))))
265		moddvds 16433	. . 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)))))))
266	66, 258, 261, 265	syl3anc 1398	. 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)))))))
267	264, 266	mpbird 260	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

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1569 ∈ wcel 2144 ≠ wne 2955 Vcvv 3450 ⊆ wss 3895 ∖ cdif 4045 {csn 4583 class class class wbr 5102 ↦ cmpt 5185 ran crn 5651 ∘ ccom 5654 ⟶wf 6534 ‘cfv 6538 (class class class)co 7420 ∘_f cof 7691 Fincfn 8973 ℂcc 11198 ℝcr 11199 0cc0 11200 1c1 11201 + caddc 11203 · cmul 11205 − cmin 11541 -cneg 11542 / cdiv 11973 ℕcn 12335 2c2 12397 ℕ₀cn0 12606 ℤcz 12693 ℤ_≥cuz 12965 ℝ⁺crp 13120 ...cfz 13639 ⌊cfl 13930 mod cmo 14009 ↑cexp 14204 ♯chash 14474 Σcsu 15853 ∥ cdvds 16422 ℙcprime 16846 Basecbs 17387 ↾_s cress 17408 ._rcmulr 17429 0_gc0g 17610 Σ_g cgsu 17611 Mndcmnd 18923 MndHom cmhm 18976 SubMndcsubmnd 18977 ._gcmg 19277 SubGrpcsubg 19330 GrpHom cghm 19427 CMndccmn 19994 Abelcabl 19995 mulGrpcmgp 20360 1_rcur 20407 Ringcrg 20459 CRingccrg 20460 RingHom crh 20699 SubRingcsubrg 20821 DivRingcdr 20980 Fieldcfield 20981 ℂ_fldccnfld 21678 ℤ_ringczring 21752 ℤRHomczrh 21805 ℤ/nℤczn 21808

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-inf2 9642 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277 ax-pre-sup 11278 ax-addf 11279 ax-mulf 11280

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-tp 4588 df-op 4590 df-uni 4867 df-int 4907 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-se 5604 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7693 df-om 7878 df-1st 8001 df-2nd 8002 df-supp 8178 df-tpos 8243 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-1o 8476 df-2o 8477 df-oadd 8480 df-er 8717 df-ec 8719 df-qs 8723 df-map 8849 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-fsupp 9354 df-sup 9434 df-inf 9435 df-oi 9504 df-dju 9982 df-card 10020 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-div 11974 df-nn 12336 df-2 12405 df-3 12406 df-4 12407 df-5 12408 df-6 12409 df-7 12410 df-8 12411 df-9 12412 df-n0 12607 df-xnn0 12680 df-z 12694 df-dec 12815 df-uz 12966 df-rp 13121 df-fz 13640 df-fzo 13789 df-fl 13932 df-mod 14010 df-seq 14145 df-exp 14205 df-hash 14475 df-cj 15266 df-re 15267 df-im 15268 df-sqrt 15402 df-abs 15403 df-clim 15655 df-sum 15854 df-dvds 16423 df-gcd 16665 df-prm 16847 df-struct 17325 df-sets 17342 df-slot 17360 df-ndx 17372 df-base 17388 df-ress 17409 df-plusg 17441 df-mulr 17442 df-starv 17443 df-sca 17444 df-vsca 17445 df-ip 17446 df-tset 17447 df-ple 17448 df-ds 17450 df-unif 17451 df-0g 17612 df-gsum 17613 df-imas 17680 df-qus 17681 df-mgm 18816 df-sgrp 18908 df-mnd 18924 df-mhm 18978 df-submnd 18979 df-grp 19147 df-minusg 19148 df-sbg 19149 df-mulg 19278 df-subg 19333 df-nsg 19334 df-eqg 19335 df-ghm 19428 df-cntz 19531 df-cmn 19996 df-abl 19997 df-mgp 20361 df-rng 20375 df-ur 20408 df-ring 20461 df-cring 20462 df-oppr 20567 df-dvdsr 20587 df-unit 20588 df-invr 20618 df-dvr 20631 df-rhm 20702 df-nzr 20763 df-subrng 20798 df-subrg 20822 df-rlreg 20946 df-domn 20947 df-idom 20948 df-drng 20982 df-field 20983 df-lmod 21137 df-lss 21207 df-lsp 21247 df-sra 21448 df-rgmod 21449 df-lidl 21486 df-rsp 21487 df-2idl 21543 df-cnfld 21679 df-zring 21753 df-zrh 21809 df-zn 21812

This theorem is used by: lgseisen 27706

W3C validator