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Theorem lgseisenlem4 15795

Description: Lemma for lgseisen 15796. (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 variables 𝑘 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		zringbas 14603	. . . . 5 ⊢ ℤ = (Base‘ℤ_ring)
2		zring0 14607	. . . . 5 ⊢ 0 = (0_g‘ℤ_ring)
3		zringabl 14601	. . . . . 6 ⊢ ℤ_ring ∈ Abel
4		ablcmn 13871	. . . . . 6 ⊢ (ℤ_ring ∈ Abel → ℤ_ring ∈ CMnd)
5	3, 4	mp1i 10	. . . . 5 ⊢ (𝜑 → ℤ_ring ∈ CMnd)
6		lgseisen.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))
7	6	eldifad 3209	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ ℙ)
8		prmnn 12675	. . . . . . . . . 10 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
9	8	nnnn0d 9448	. . . . . . . . 9 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ₀)
10	7, 9	syl 14	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ ℕ₀)
11		lgseisen.7	. . . . . . . . 9 ⊢ 𝑌 = (ℤ/nℤ‘𝑃)
12	11	zncrng 14652	. . . . . . . 8 ⊢ (𝑃 ∈ ℕ₀ → 𝑌 ∈ CRing)
13	10, 12	syl 14	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ CRing)
14		lgseisen.8	. . . . . . . 8 ⊢ 𝐺 = (mulGrp‘𝑌)
15	14	crngmgp 14010	. . . . . . 7 ⊢ (𝑌 ∈ CRing → 𝐺 ∈ CMnd)
16	13, 15	syl 14	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ CMnd)
17	16	cmnmndd 13888	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ Mnd)
18		1zzd 9499	. . . . 5 ⊢ (𝜑 → 1 ∈ ℤ)
19		oddprm 12825	. . . . . . 7 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((𝑃 − 1) / 2) ∈ ℕ)
20	6, 19	syl 14	. . . . . 6 ⊢ (𝜑 → ((𝑃 − 1) / 2) ∈ ℕ)
21	20	nnzd 9594	. . . . 5 ⊢ (𝜑 → ((𝑃 − 1) / 2) ∈ ℤ)
22	13	crngringd 14015	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ Ring)
23		lgseisen.9	. . . . . . . . . 10 ⊢ 𝐿 = (ℤRHom‘𝑌)
24	23	zrhrhm 14630	. . . . . . . . 9 ⊢ (𝑌 ∈ Ring → 𝐿 ∈ (ℤ_ring RingHom 𝑌))
25	22, 24	syl 14	. . . . . . . 8 ⊢ (𝜑 → 𝐿 ∈ (ℤ_ring RingHom 𝑌))
26		eqid 2229	. . . . . . . . 9 ⊢ (Base‘𝑌) = (Base‘𝑌)
27	1, 26	rhmf 14170	. . . . . . . 8 ⊢ (𝐿 ∈ (ℤ_ring RingHom 𝑌) → 𝐿:ℤ⟶(Base‘𝑌))
28	25, 27	syl 14	. . . . . . 7 ⊢ (𝜑 → 𝐿:ℤ⟶(Base‘𝑌))
29		m1expcl 10817	. . . . . . . 8 ⊢ (𝑘 ∈ ℤ → (-1↑𝑘) ∈ ℤ)
30	29	adantl 277	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → (-1↑𝑘) ∈ ℤ)
31	28, 30	cofmpt 5812	. . . . . 6 ⊢ (𝜑 → (𝐿 ∘ (𝑘 ∈ ℤ ↦ (-1↑𝑘))) = (𝑘 ∈ ℤ ↦ (𝐿‘(-1↑𝑘))))
32		zringmpg 14613	. . . . . . . . 9 ⊢ ((mulGrp‘ℂ_fld) ↾_s ℤ) = (mulGrp‘ℤ_ring)
33	32, 14	rhmmhm 14166	. . . . . . . 8 ⊢ (𝐿 ∈ (ℤ_ring RingHom 𝑌) → 𝐿 ∈ (((mulGrp‘ℂ_fld) ↾_s ℤ) MndHom 𝐺))
34	25, 33	syl 14	. . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ (((mulGrp‘ℂ_fld) ↾_s ℤ) MndHom 𝐺))
35		neg1cn 9241	. . . . . . . . . . 11 ⊢ -1 ∈ ℂ
36		neg1ap0 9245	. . . . . . . . . . 11 ⊢ -1 # 0
37		eqid 2229	. . . . . . . . . . . 12 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
38		eqid 2229	. . . . . . . . . . . 12 ⊢ ((mulGrp‘ℂ_fld) ↾_s {𝑧 ∈ ℂ ∣ 𝑧 # 0}) = ((mulGrp‘ℂ_fld) ↾_s {𝑧 ∈ ℂ ∣ 𝑧 # 0})
39	37, 38	expghmap 14614	. . . . . . . . . . 11 ⊢ ((-1 ∈ ℂ ∧ -1 # 0) → (𝑘 ∈ ℤ ↦ (-1↑𝑘)) ∈ (ℤ_ring GrpHom ((mulGrp‘ℂ_fld) ↾_s {𝑧 ∈ ℂ ∣ 𝑧 # 0})))
40	35, 36, 39	mp2an 426	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℤ ↦ (-1↑𝑘)) ∈ (ℤ_ring GrpHom ((mulGrp‘ℂ_fld) ↾_s {𝑧 ∈ ℂ ∣ 𝑧 # 0}))
41		ghmmhm 13833	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℤ ↦ (-1↑𝑘)) ∈ (ℤ_ring GrpHom ((mulGrp‘ℂ_fld) ↾_s {𝑧 ∈ ℂ ∣ 𝑧 # 0})) → (𝑘 ∈ ℤ ↦ (-1↑𝑘)) ∈ (ℤ_ring MndHom ((mulGrp‘ℂ_fld) ↾_s {𝑧 ∈ ℂ ∣ 𝑧 # 0})))
42	40, 41	ax-mp 5	. . . . . . . . 9 ⊢ (𝑘 ∈ ℤ ↦ (-1↑𝑘)) ∈ (ℤ_ring MndHom ((mulGrp‘ℂ_fld) ↾_s {𝑧 ∈ ℂ ∣ 𝑧 # 0}))
43		cnring 14577	. . . . . . . . . 10 ⊢ ℂ_fld ∈ Ring
44		cnfldui 14596	. . . . . . . . . . 11 ⊢ {𝑧 ∈ ℂ ∣ 𝑧 # 0} = (Unit‘ℂ_fld)
45	44, 37	unitsubm 14126	. . . . . . . . . 10 ⊢ (ℂ_fld ∈ Ring → {𝑧 ∈ ℂ ∣ 𝑧 # 0} ∈ (SubMnd‘(mulGrp‘ℂ_fld)))
46	43, 45	ax-mp 5	. . . . . . . . 9 ⊢ {𝑧 ∈ ℂ ∣ 𝑧 # 0} ∈ (SubMnd‘(mulGrp‘ℂ_fld))
47	38	resmhm2 13564	. . . . . . . . 9 ⊢ (((𝑘 ∈ ℤ ↦ (-1↑𝑘)) ∈ (ℤ_ring MndHom ((mulGrp‘ℂ_fld) ↾_s {𝑧 ∈ ℂ ∣ 𝑧 # 0})) ∧ {𝑧 ∈ ℂ ∣ 𝑧 # 0} ∈ (SubMnd‘(mulGrp‘ℂ_fld))) → (𝑘 ∈ ℤ ↦ (-1↑𝑘)) ∈ (ℤ_ring MndHom (mulGrp‘ℂ_fld)))
48	42, 46, 47	mp2an 426	. . . . . . . 8 ⊢ (𝑘 ∈ ℤ ↦ (-1↑𝑘)) ∈ (ℤ_ring MndHom (mulGrp‘ℂ_fld))
49		zsubrg 14588	. . . . . . . . . 10 ⊢ ℤ ∈ (SubRing‘ℂ_fld)
50	37	subrgsubm 14241	. . . . . . . . . 10 ⊢ (ℤ ∈ (SubRing‘ℂ_fld) → ℤ ∈ (SubMnd‘(mulGrp‘ℂ_fld)))
51	49, 50	ax-mp 5	. . . . . . . . 9 ⊢ ℤ ∈ (SubMnd‘(mulGrp‘ℂ_fld))
52	30	fmpttd 5798	. . . . . . . . . 10 ⊢ (𝜑 → (𝑘 ∈ ℤ ↦ (-1↑𝑘)):ℤ⟶ℤ)
53	52	frnd 5489	. . . . . . . . 9 ⊢ (𝜑 → ran (𝑘 ∈ ℤ ↦ (-1↑𝑘)) ⊆ ℤ)
54		eqid 2229	. . . . . . . . . 10 ⊢ ((mulGrp‘ℂ_fld) ↾_s ℤ) = ((mulGrp‘ℂ_fld) ↾_s ℤ)
55	54	resmhm2b 13565	. . . . . . . . 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))))
56	51, 53, 55	sylancr 414	. . . . . . . 8 ⊢ (𝜑 → ((𝑘 ∈ ℤ ↦ (-1↑𝑘)) ∈ (ℤ_ring MndHom (mulGrp‘ℂ_fld)) ↔ (𝑘 ∈ ℤ ↦ (-1↑𝑘)) ∈ (ℤ_ring MndHom ((mulGrp‘ℂ_fld) ↾_s ℤ))))
57	48, 56	mpbii 148	. . . . . . 7 ⊢ (𝜑 → (𝑘 ∈ ℤ ↦ (-1↑𝑘)) ∈ (ℤ_ring MndHom ((mulGrp‘ℂ_fld) ↾_s ℤ)))
58		mhmco 13566	. . . . . . 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))
59	34, 57, 58	syl2anc 411	. . . . . 6 ⊢ (𝜑 → (𝐿 ∘ (𝑘 ∈ ℤ ↦ (-1↑𝑘))) ∈ (ℤ_ring MndHom 𝐺))
60	31, 59	eqeltrrd 2307	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ ℤ ↦ (𝐿‘(-1↑𝑘))) ∈ (ℤ_ring MndHom 𝐺))
61		lgseisen.2	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑄 ∈ (ℙ ∖ {2}))
62	61	gausslemma2dlem0a 15771	. . . . . . . . . 10 ⊢ (𝜑 → 𝑄 ∈ ℕ)
63	62	nnzd 9594	. . . . . . . . 9 ⊢ (𝜑 → 𝑄 ∈ ℤ)
64	63	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑄 ∈ ℤ)
65	6	gausslemma2dlem0a 15771	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ ℕ)
66	65	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 ∈ ℕ)
67		znq 9851	. . . . . . . 8 ⊢ ((𝑄 ∈ ℤ ∧ 𝑃 ∈ ℕ) → (𝑄 / 𝑃) ∈ ℚ)
68	64, 66, 67	syl2anc 411	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑄 / 𝑃) ∈ ℚ)
69		2nn 9298	. . . . . . . . . 10 ⊢ 2 ∈ ℕ
70		elfznn 10282	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (1...((𝑃 − 1) / 2)) → 𝑥 ∈ ℕ)
71	70	adantl 277	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑥 ∈ ℕ)
72		nnmulcl 9157	. . . . . . . . . 10 ⊢ ((2 ∈ ℕ ∧ 𝑥 ∈ ℕ) → (2 · 𝑥) ∈ ℕ)
73	69, 71, 72	sylancr 414	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (2 · 𝑥) ∈ ℕ)
74	73	nnzd 9594	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (2 · 𝑥) ∈ ℤ)
75		zq 9853	. . . . . . . 8 ⊢ ((2 · 𝑥) ∈ ℤ → (2 · 𝑥) ∈ ℚ)
76	74, 75	syl 14	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (2 · 𝑥) ∈ ℚ)
77		qmulcl 9864	. . . . . . 7 ⊢ (((𝑄 / 𝑃) ∈ ℚ ∧ (2 · 𝑥) ∈ ℚ) → ((𝑄 / 𝑃) · (2 · 𝑥)) ∈ ℚ)
78	68, 76, 77	syl2anc 411	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝑄 / 𝑃) · (2 · 𝑥)) ∈ ℚ)
79	78	flqcld 10530	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))) ∈ ℤ)
80		oveq2 6021	. . . . . 6 ⊢ (𝑘 = (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))) → (-1↑𝑘) = (-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))
81	80	fveq2d 5639	. . . . 5 ⊢ (𝑘 = (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))) → (𝐿‘(-1↑𝑘)) = (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))))
82		oveq2 6021	. . . . . 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)))))))
83	82	fveq2d 5639	. . . . 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))))))))
84	1, 2, 5, 17, 18, 21, 60, 79, 81, 83	gsumfzmhm2 13924	. . . 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))))))))
85		eqid 2229	. . . . . . . 8 ⊢ (Base‘𝐺) = (Base‘𝐺)
86		eqid 2229	. . . . . . . 8 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
87	28	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝐿:ℤ⟶(Base‘𝑌))
88		m1expcl 10817	. . . . . . . . . . 11 ⊢ ((⌊‘((𝑄 / 𝑃) · (2 · 𝑥))) ∈ ℤ → (-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ∈ ℤ)
89	79, 88	syl 14	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ∈ ℤ)
90	87, 89	ffvelcdmd 5779	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) ∈ (Base‘𝑌))
91	14, 26	mgpbasg 13932	. . . . . . . . . . 11 ⊢ (𝑌 ∈ CRing → (Base‘𝑌) = (Base‘𝐺))
92	13, 91	syl 14	. . . . . . . . . 10 ⊢ (𝜑 → (Base‘𝑌) = (Base‘𝐺))
93	92	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (Base‘𝑌) = (Base‘𝐺))
94	90, 93	eleqtrd 2308	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) ∈ (Base‘𝐺))
95		neg1z 9504	. . . . . . . . . . . 12 ⊢ -1 ∈ ℤ
96		lgseisen.4	. . . . . . . . . . . . 13 ⊢ 𝑅 = ((𝑄 · (2 · 𝑥)) mod 𝑃)
97	61	eldifad 3209	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑄 ∈ ℙ)
98	97	adantr 276	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑄 ∈ ℙ)
99		prmz 12676	. . . . . . . . . . . . . . . 16 ⊢ (𝑄 ∈ ℙ → 𝑄 ∈ ℤ)
100	98, 99	syl 14	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑄 ∈ ℤ)
101	100, 74	zmulcld 9601	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑄 · (2 · 𝑥)) ∈ ℤ)
102	7	adantr 276	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 ∈ ℙ)
103	102, 8	syl 14	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 ∈ ℕ)
104	101, 103	zmodcld 10600	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝑄 · (2 · 𝑥)) mod 𝑃) ∈ ℕ₀)
105	96, 104	eqeltrid 2316	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑅 ∈ ℕ₀)
106		zexpcl 10809	. . . . . . . . . . . 12 ⊢ ((-1 ∈ ℤ ∧ 𝑅 ∈ ℕ₀) → (-1↑𝑅) ∈ ℤ)
107	95, 105, 106	sylancr 414	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (-1↑𝑅) ∈ ℤ)
108	107, 100	zmulcld 9601	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((-1↑𝑅) · 𝑄) ∈ ℤ)
109	87, 108	ffvelcdmd 5779	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘((-1↑𝑅) · 𝑄)) ∈ (Base‘𝑌))
110	109, 93	eleqtrd 2308	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘((-1↑𝑅) · 𝑄)) ∈ (Base‘𝐺))
111		eqid 2229	. . . . . . . 8 ⊢ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))) = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))))
112		eqid 2229	. . . . . . . 8 ⊢ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))) = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄)))
113	85, 86, 16, 18, 21, 94, 110, 111, 112	gsumfzmptfidmadd2 13920	. . . . . . 7 ⊢ (𝜑 → (𝐺 Σ_g ((𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))) ∘_𝑓 (+_g‘𝐺)(𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))))) = ((𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))))(+_g‘𝐺)(𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))))))
114		eqid 2229	. . . . . . . . . . . 12 ⊢ (._r‘𝑌) = (._r‘𝑌)
115	14, 114	mgpplusgg 13930	. . . . . . . . . . 11 ⊢ (𝑌 ∈ CRing → (._r‘𝑌) = (+_g‘𝐺))
116	13, 115	syl 14	. . . . . . . . . 10 ⊢ (𝜑 → (._r‘𝑌) = (+_g‘𝐺))
117	116	ofeqd 6232	. . . . . . . . 9 ⊢ (𝜑 → ∘_𝑓 (._r‘𝑌) = ∘_𝑓 (+_g‘𝐺))
118	117	oveqd 6030	. . . . . . . 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)))))) ∘_𝑓 (+_g‘𝐺)(𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄)))))
119	118	oveq2d 6029	. . . . . . 7 ⊢ (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)))))) ∘_𝑓 (+_g‘𝐺)(𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))))))
120	116	oveqd 6030	. . . . . . 7 ⊢ (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)))))))(+_g‘𝐺)(𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))))))
121	113, 119, 120	3eqtr4d 2272	. . . . . 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))))))
122	18, 21	fzfigd 10686	. . . . . . . . 9 ⊢ (𝜑 → (1...((𝑃 − 1) / 2)) ∈ Fin)
123		eqidd 2230	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))) = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))))
124		eqidd 2230	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))) = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))))
125	122, 90, 109, 123, 124	offval2 6246	. . . . . . . 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)))))
126	25	adantr 276	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝐿 ∈ (ℤ_ring RingHom 𝑌))
127		zringmulr 14606	. . . . . . . . . . . 12 ⊢ · = (._r‘ℤ_ring)
128	1, 127, 114	rhmmul 14171	. . . . . . . . . . 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))))
129	126, 89, 108, 128	syl3anc 1271	. . . . . . . . . 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))))
130	62	adantr 276	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑄 ∈ ℕ)
131	130, 73	nnmulcld 9185	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑄 · (2 · 𝑥)) ∈ ℕ)
132		nnq 9860	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑄 · (2 · 𝑥)) ∈ ℕ → (𝑄 · (2 · 𝑥)) ∈ ℚ)
133	131, 132	syl 14	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑄 · (2 · 𝑥)) ∈ ℚ)
134		nnq 9860	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℚ)
135	65, 134	syl 14	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑃 ∈ ℚ)
136	135	adantr 276	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 ∈ ℚ)
137	66	nngt0d 9180	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 0 < 𝑃)
138		modqval 10579	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑄 · (2 · 𝑥)) ∈ ℚ ∧ 𝑃 ∈ ℚ ∧ 0 < 𝑃) → ((𝑄 · (2 · 𝑥)) mod 𝑃) = ((𝑄 · (2 · 𝑥)) − (𝑃 · (⌊‘((𝑄 · (2 · 𝑥)) / 𝑃)))))
139	133, 136, 137, 138	syl3anc 1271	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝑄 · (2 · 𝑥)) mod 𝑃) = ((𝑄 · (2 · 𝑥)) − (𝑃 · (⌊‘((𝑄 · (2 · 𝑥)) / 𝑃)))))
140	96, 139	eqtrid 2274	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑅 = ((𝑄 · (2 · 𝑥)) − (𝑃 · (⌊‘((𝑄 · (2 · 𝑥)) / 𝑃)))))
141	100	zcnd 9596	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑄 ∈ ℂ)
142	73	nncnd 9150	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (2 · 𝑥) ∈ ℂ)
143	103	nncnd 9150	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 ∈ ℂ)
144	103	nnap0d 9182	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 # 0)
145	141, 142, 143, 144	div23apd 9001	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝑄 · (2 · 𝑥)) / 𝑃) = ((𝑄 / 𝑃) · (2 · 𝑥)))
146	145	fveq2d 5639	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (⌊‘((𝑄 · (2 · 𝑥)) / 𝑃)) = (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))
147	146	oveq2d 6029	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑃 · (⌊‘((𝑄 · (2 · 𝑥)) / 𝑃))) = (𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))
148	147	oveq2d 6029	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝑄 · (2 · 𝑥)) − (𝑃 · (⌊‘((𝑄 · (2 · 𝑥)) / 𝑃)))) = ((𝑄 · (2 · 𝑥)) − (𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))))
149	140, 148	eqtrd 2262	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑅 = ((𝑄 · (2 · 𝑥)) − (𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))))
150	149	oveq2d 6029	. . . . . . . . . . . . . . . 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)))))))
151		prmz 12676	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
152	102, 151	syl 14	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 ∈ ℤ)
153	152, 79	zmulcld 9601	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ∈ ℤ)
154	153	zcnd 9596	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ∈ ℂ)
155	101	zcnd 9596	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑄 · (2 · 𝑥)) ∈ ℂ)
156	154, 155	pncan3d 8486	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + ((𝑄 · (2 · 𝑥)) − (𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))) = (𝑄 · (2 · 𝑥)))
157		2cnd 9209	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 2 ∈ ℂ)
158	71	nncnd 9150	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑥 ∈ ℂ)
159	141, 157, 158	mul12d 8324	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑄 · (2 · 𝑥)) = (2 · (𝑄 · 𝑥)))
160	150, 156, 159	3eqtrd 2266	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 𝑅) = (2 · (𝑄 · 𝑥)))
161	160	oveq2d 6029	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (-1↑((𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 𝑅)) = (-1↑(2 · (𝑄 · 𝑥))))
162	35	a1i 9	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → -1 ∈ ℂ)
163	36	a1i 9	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → -1 # 0)
164	105	nn0zd 9593	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑅 ∈ ℤ)
165		expaddzap 10838	. . . . . . . . . . . . . . . 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)))
166	162, 163, 153, 164, 165	syl22anc 1272	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (-1↑((𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 𝑅)) = ((-1↑(𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) · (-1↑𝑅)))
167		expmulzap 10840	. . . . . . . . . . . . . . . . . 18 ⊢ (((-1 ∈ ℂ ∧ -1 # 0) ∧ (𝑃 ∈ ℤ ∧ (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))) ∈ ℤ)) → (-1↑(𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) = ((-1↑𝑃)↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))
168	162, 163, 152, 79, 167	syl22anc 1272	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (-1↑(𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) = ((-1↑𝑃)↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))
169		1cnd 8188	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 1 ∈ ℂ)
170		eldifsni 3800	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ≠ 2)
171	6, 170	syl 14	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝑃 ≠ 2)
172	171	necomd 2486	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 2 ≠ 𝑃)
173	172	neneqd 2421	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ¬ 2 = 𝑃)
174	173	adantr 276	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ¬ 2 = 𝑃)
175		2z 9500	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 2 ∈ ℤ
176		uzid 9763	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (2 ∈ ℤ → 2 ∈ (ℤ_≥‘2))
177	175, 176	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 2 ∈ (ℤ_≥‘2)
178		dvdsprm 12702	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((2 ∈ (ℤ_≥‘2) ∧ 𝑃 ∈ ℙ) → (2 ∥ 𝑃 ↔ 2 = 𝑃))
179	177, 102, 178	sylancr 414	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (2 ∥ 𝑃 ↔ 2 = 𝑃))
180	174, 179	mtbird 677	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ¬ 2 ∥ 𝑃)
181		oexpneg 12431	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1 ∈ ℂ ∧ 𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃) → (-1↑𝑃) = -(1↑𝑃))
182	169, 103, 180, 181	syl3anc 1271	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (-1↑𝑃) = -(1↑𝑃))
183		1exp 10823	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑃 ∈ ℤ → (1↑𝑃) = 1)
184	152, 183	syl 14	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (1↑𝑃) = 1)
185	184	negeqd 8367	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → -(1↑𝑃) = -1)
186	182, 185	eqtrd 2262	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (-1↑𝑃) = -1)
187	186	oveq1d 6028	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((-1↑𝑃)↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) = (-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))
188	168, 187	eqtrd 2262	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (-1↑(𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) = (-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))
189	188	oveq1d 6028	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((-1↑(𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) · (-1↑𝑅)) = ((-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) · (-1↑𝑅)))
190	166, 189	eqtrd 2262	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (-1↑((𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 𝑅)) = ((-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) · (-1↑𝑅)))
191		nnmulcl 9157	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑄 ∈ ℕ ∧ 𝑥 ∈ ℕ) → (𝑄 · 𝑥) ∈ ℕ)
192	62, 70, 191	syl2an 289	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑄 · 𝑥) ∈ ℕ)
193	192	nnnn0d 9448	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑄 · 𝑥) ∈ ℕ₀)
194		2nn0 9412	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℕ₀
195	194	a1i 9	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 2 ∈ ℕ₀)
196	162, 193, 195	expmuld 10931	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (-1↑(2 · (𝑄 · 𝑥))) = ((-1↑2)↑(𝑄 · 𝑥)))
197		neg1sqe1 10889	. . . . . . . . . . . . . . . . 17 ⊢ (-1↑2) = 1
198	197	oveq1i 6023	. . . . . . . . . . . . . . . 16 ⊢ ((-1↑2)↑(𝑄 · 𝑥)) = (1↑(𝑄 · 𝑥))
199	192	nnzd 9594	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑄 · 𝑥) ∈ ℤ)
200		1exp 10823	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑄 · 𝑥) ∈ ℤ → (1↑(𝑄 · 𝑥)) = 1)
201	199, 200	syl 14	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (1↑(𝑄 · 𝑥)) = 1)
202	198, 201	eqtrid 2274	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((-1↑2)↑(𝑄 · 𝑥)) = 1)
203	196, 202	eqtrd 2262	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (-1↑(2 · (𝑄 · 𝑥))) = 1)
204	161, 190, 203	3eqtr3d 2270	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) · (-1↑𝑅)) = 1)
205	204	oveq1d 6028	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (((-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) · (-1↑𝑅)) · 𝑄) = (1 · 𝑄))
206	89	zcnd 9596	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ∈ ℂ)
207	107	zcnd 9596	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (-1↑𝑅) ∈ ℂ)
208	206, 207, 141	mulassd 8196	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (((-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) · (-1↑𝑅)) · 𝑄) = ((-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) · ((-1↑𝑅) · 𝑄)))
209	141	mullidd 8190	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (1 · 𝑄) = 𝑄)
210	205, 208, 209	3eqtr3d 2270	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) · ((-1↑𝑅) · 𝑄)) = 𝑄)
211	210	fveq2d 5639	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘((-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) · ((-1↑𝑅) · 𝑄))) = (𝐿‘𝑄))
212	129, 211	eqtr3d 2264	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))(._r‘𝑌)(𝐿‘((-1↑𝑅) · 𝑄))) = (𝐿‘𝑄))
213	212	mpteq2dva 4177	. . . . . . . 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)))
214	125, 213	eqtrd 2262	. . . . . . 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)))
215	214	oveq2d 6029	. . . . . 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))))
216		lgseisen.3	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ≠ 𝑄)
217		lgseisen.5	. . . . . . . 8 ⊢ 𝑀 = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ ((((-1↑𝑅) · 𝑅) mod 𝑃) / 2))
218		lgseisen.6	. . . . . . . 8 ⊢ 𝑆 = ((𝑄 · (2 · 𝑦)) mod 𝑃)
219	6, 61, 216, 96, 217, 218, 11, 14, 23	lgseisenlem3 15794	. . . . . . 7 ⊢ (𝜑 → (𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄)))) = (1_r‘𝑌))
220	219	oveq2d 6029	. . . . . 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)))
221	121, 215, 220	3eqtr3rd 2271	. . . . 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))))
222		eqid 2229	. . . . . . . 8 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
223	90	fmpttd 5798	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))):(1...((𝑃 − 1) / 2))⟶(Base‘𝑌))
224	92	feq3d 5468	. . . . . . . . 9 ⊢ (𝜑 → ((𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))):(1...((𝑃 − 1) / 2))⟶(Base‘𝑌) ↔ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))):(1...((𝑃 − 1) / 2))⟶(Base‘𝐺)))
225	223, 224	mpbid 147	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))):(1...((𝑃 − 1) / 2))⟶(Base‘𝐺))
226	85, 222, 17, 18, 21, 225	gsumfzcl 13575	. . . . . . 7 ⊢ (𝜑 → (𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))))) ∈ (Base‘𝐺))
227	226, 92	eleqtrrd 2309	. . . . . 6 ⊢ (𝜑 → (𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))))) ∈ (Base‘𝑌))
228		eqid 2229	. . . . . . 7 ⊢ (1_r‘𝑌) = (1_r‘𝑌)
229	26, 114, 228	ringridm 14030	. . . . . 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))))))))
230	22, 227, 229	syl2anc 411	. . . . 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))))))))
231		nnuz 9785	. . . . . . . 8 ⊢ ℕ = (ℤ_≥‘1)
232	20, 231	eleqtrdi 2322	. . . . . . 7 ⊢ (𝜑 → ((𝑃 − 1) / 2) ∈ (ℤ_≥‘1))
233	97, 99	syl 14	. . . . . . . . 9 ⊢ (𝜑 → 𝑄 ∈ ℤ)
234	28, 233	ffvelcdmd 5779	. . . . . . . 8 ⊢ (𝜑 → (𝐿‘𝑄) ∈ (Base‘𝑌))
235	234, 92	eleqtrd 2308	. . . . . . 7 ⊢ (𝜑 → (𝐿‘𝑄) ∈ (Base‘𝐺))
236		eqid 2229	. . . . . . . 8 ⊢ (._g‘𝐺) = (._g‘𝐺)
237	85, 236	gsumfzconst 13921	. . . . . . 7 ⊢ ((𝐺 ∈ Mnd ∧ ((𝑃 − 1) / 2) ∈ (ℤ_≥‘1) ∧ (𝐿‘𝑄) ∈ (Base‘𝐺)) → (𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘𝑄))) = (((((𝑃 − 1) / 2) − 1) + 1)(._g‘𝐺)(𝐿‘𝑄)))
238	17, 232, 235, 237	syl3anc 1271	. . . . . 6 ⊢ (𝜑 → (𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘𝑄))) = (((((𝑃 − 1) / 2) − 1) + 1)(._g‘𝐺)(𝐿‘𝑄)))
239	20	nncnd 9150	. . . . . . . 8 ⊢ (𝜑 → ((𝑃 − 1) / 2) ∈ ℂ)
240		1cnd 8188	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℂ)
241	239, 240	npcand 8487	. . . . . . 7 ⊢ (𝜑 → ((((𝑃 − 1) / 2) − 1) + 1) = ((𝑃 − 1) / 2))
242	241	oveq1d 6028	. . . . . 6 ⊢ (𝜑 → (((((𝑃 − 1) / 2) − 1) + 1)(._g‘𝐺)(𝐿‘𝑄)) = (((𝑃 − 1) / 2)(._g‘𝐺)(𝐿‘𝑄)))
243	20	nnnn0d 9448	. . . . . . . 8 ⊢ (𝜑 → ((𝑃 − 1) / 2) ∈ ℕ₀)
244		zringring 14600	. . . . . . . . . 10 ⊢ ℤ_ring ∈ Ring
245	32, 1	mgpbasg 13932	. . . . . . . . . 10 ⊢ (ℤ_ring ∈ Ring → ℤ = (Base‘((mulGrp‘ℂ_fld) ↾_s ℤ)))
246	244, 245	ax-mp 5	. . . . . . . . 9 ⊢ ℤ = (Base‘((mulGrp‘ℂ_fld) ↾_s ℤ))
247		eqid 2229	. . . . . . . . 9 ⊢ (._g‘((mulGrp‘ℂ_fld) ↾_s ℤ)) = (._g‘((mulGrp‘ℂ_fld) ↾_s ℤ))
248	246, 247, 236	mhmmulg 13743	. . . . . . . 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)))
249	34, 243, 233, 248	syl3anc 1271	. . . . . . 7 ⊢ (𝜑 → (𝐿‘(((𝑃 − 1) / 2)(._g‘((mulGrp‘ℂ_fld) ↾_s ℤ))𝑄)) = (((𝑃 − 1) / 2)(._g‘𝐺)(𝐿‘𝑄)))
250	51	a1i 9	. . . . . . . . . 10 ⊢ (𝜑 → ℤ ∈ (SubMnd‘(mulGrp‘ℂ_fld)))
251		eqid 2229	. . . . . . . . . . 11 ⊢ (._g‘(mulGrp‘ℂ_fld)) = (._g‘(mulGrp‘ℂ_fld))
252	251, 54, 247	submmulg 13746	. . . . . . . . . 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))
253	250, 243, 233, 252	syl3anc 1271	. . . . . . . . 9 ⊢ (𝜑 → (((𝑃 − 1) / 2)(._g‘(mulGrp‘ℂ_fld))𝑄) = (((𝑃 − 1) / 2)(._g‘((mulGrp‘ℂ_fld) ↾_s ℤ))𝑄))
254	233	zcnd 9596	. . . . . . . . . 10 ⊢ (𝜑 → 𝑄 ∈ ℂ)
255		cnfldexp 14584	. . . . . . . . . 10 ⊢ ((𝑄 ∈ ℂ ∧ ((𝑃 − 1) / 2) ∈ ℕ₀) → (((𝑃 − 1) / 2)(._g‘(mulGrp‘ℂ_fld))𝑄) = (𝑄↑((𝑃 − 1) / 2)))
256	254, 243, 255	syl2anc 411	. . . . . . . . 9 ⊢ (𝜑 → (((𝑃 − 1) / 2)(._g‘(mulGrp‘ℂ_fld))𝑄) = (𝑄↑((𝑃 − 1) / 2)))
257	253, 256	eqtr3d 2264	. . . . . . . 8 ⊢ (𝜑 → (((𝑃 − 1) / 2)(._g‘((mulGrp‘ℂ_fld) ↾_s ℤ))𝑄) = (𝑄↑((𝑃 − 1) / 2)))
258	257	fveq2d 5639	. . . . . . 7 ⊢ (𝜑 → (𝐿‘(((𝑃 − 1) / 2)(._g‘((mulGrp‘ℂ_fld) ↾_s ℤ))𝑄)) = (𝐿‘(𝑄↑((𝑃 − 1) / 2))))
259	249, 258	eqtr3d 2264	. . . . . 6 ⊢ (𝜑 → (((𝑃 − 1) / 2)(._g‘𝐺)(𝐿‘𝑄)) = (𝐿‘(𝑄↑((𝑃 − 1) / 2))))
260	238, 242, 259	3eqtrd 2266	. . . . 5 ⊢ (𝜑 → (𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘𝑄))) = (𝐿‘(𝑄↑((𝑃 − 1) / 2))))
261	221, 230, 260	3eqtr3d 2270	. . . 4 ⊢ (𝜑 → (𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(-1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))))) = (𝐿‘(𝑄↑((𝑃 − 1) / 2))))
262		subrgsubg 14234	. . . . . . . . . 10 ⊢ (ℤ ∈ (SubRing‘ℂ_fld) → ℤ ∈ (SubGrp‘ℂ_fld))
263	49, 262	ax-mp 5	. . . . . . . . 9 ⊢ ℤ ∈ (SubGrp‘ℂ_fld)
264		subgsubm 13776	. . . . . . . . 9 ⊢ (ℤ ∈ (SubGrp‘ℂ_fld) → ℤ ∈ (SubMnd‘ℂ_fld))
265	263, 264	mp1i 10	. . . . . . . 8 ⊢ (𝜑 → ℤ ∈ (SubMnd‘ℂ_fld))
266	79	fmpttd 5798	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))):(1...((𝑃 − 1) / 2))⟶ℤ)
267		df-zring 14598	. . . . . . . 8 ⊢ ℤ_ring = (ℂ_fld ↾_s ℤ)
268	122, 265, 266, 267	gsumsubm 13570	. . . . . . 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))))))
269	79	zcnd 9596	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))) ∈ ℂ)
270	18, 21, 269	gsumfzfsum 14595	. . . . . . 7 ⊢ (𝜑 → (ℂ_fld Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) = Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))
271	268, 270	eqtr3d 2264	. . . . . 6 ⊢ (𝜑 → (ℤ_ring Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) = Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))
272	271	oveq2d 6029	. . . . 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)))))
273	272	fveq2d 5639	. . . 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))))))
274	84, 261, 273	3eqtr3d 2270	. . 3 ⊢ (𝜑 → (𝐿‘(𝑄↑((𝑃 − 1) / 2))) = (𝐿‘(-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))))
275	65	nnnn0d 9448	. . . 4 ⊢ (𝜑 → 𝑃 ∈ ℕ₀)
276		zexpcl 10809	. . . . 5 ⊢ ((𝑄 ∈ ℤ ∧ ((𝑃 − 1) / 2) ∈ ℕ₀) → (𝑄↑((𝑃 − 1) / 2)) ∈ ℤ)
277	233, 243, 276	syl2anc 411	. . . 4 ⊢ (𝜑 → (𝑄↑((𝑃 − 1) / 2)) ∈ ℤ)
278	122, 79	fsumzcl 11956	. . . . 5 ⊢ (𝜑 → Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))) ∈ ℤ)
279		m1expcl 10817	. . . . 5 ⊢ (Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))) ∈ ℤ → (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ∈ ℤ)
280	278, 279	syl 14	. . . 4 ⊢ (𝜑 → (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ∈ ℤ)
281	11, 23	zndvds 14656	. . . 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)))))))
282	275, 277, 280, 281	syl3anc 1271	. . 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)))))))
283	274, 282	mpbid 147	. 2 ⊢ (𝜑 → 𝑃 ∥ ((𝑄↑((𝑃 − 1) / 2)) − (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))))
284		moddvds 12353	. . 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)))))))
285	65, 277, 280, 284	syl3anc 1271	. 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)))))))
286	283, 285	mpbird 167	1 ⊢ (𝜑 → ((𝑄↑((𝑃 − 1) / 2)) mod 𝑃) = ((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) mod 𝑃))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1395 ∈ wcel 2200 ≠ wne 2400 {crab 2512 ∖ cdif 3195 ⊆ wss 3198 {csn 3667 class class class wbr 4086 ↦ cmpt 4148 ran crn 4724 ∘ ccom 4727 ⟶wf 5320 ‘cfv 5324 (class class class)co 6013 ∘_𝑓 cof 6228 Fincfn 6904 ℂcc 8023 0cc0 8025 1c1 8026 + caddc 8028 · cmul 8030 < clt 8207 − cmin 8343 -cneg 8344 # cap 8754 / cdiv 8845 ℕcn 9136 2c2 9187 ℕ₀cn0 9395 ℤcz 9472 ℤ_≥cuz 9748 ℚcq 9846 ...cfz 10236 ⌊cfl 10521 mod cmo 10577 ↑cexp 10793 Σcsu 11907 ∥ cdvds 12341 ℙcprime 12672 Basecbs 13075 ↾_s cress 13076 +_gcplusg 13153 ._rcmulr 13154 0_gc0g 13332 Σ_g cgsu 13333 Mndcmnd 13492 MndHom cmhm 13533 SubMndcsubmnd 13534 ._gcmg 13699 SubGrpcsubg 13747 GrpHom cghm 13820 CMndccmn 13864 Abelcabl 13865 mulGrpcmgp 13926 1_rcur 13965 Ringcrg 14002 CRingccrg 14003 RingHom crh 14157 SubRingcsubrg 14224 ℂ_fldccnfld 14563 ℤ_ringczring 14597 ℤRHomczrh 14618 ℤ/nℤczn 14620

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4202 ax-sep 4205 ax-nul 4213 ax-pow 4262 ax-pr 4297 ax-un 4528 ax-setind 4633 ax-iinf 4684 ax-cnex 8116 ax-resscn 8117 ax-1cn 8118 ax-1re 8119 ax-icn 8120 ax-addcl 8121 ax-addrcl 8122 ax-mulcl 8123 ax-mulrcl 8124 ax-addcom 8125 ax-mulcom 8126 ax-addass 8127 ax-mulass 8128 ax-distr 8129 ax-i2m1 8130 ax-0lt1 8131 ax-1rid 8132 ax-0id 8133 ax-rnegex 8134 ax-precex 8135 ax-cnre 8136 ax-pre-ltirr 8137 ax-pre-ltwlin 8138 ax-pre-lttrn 8139 ax-pre-apti 8140 ax-pre-ltadd 8141 ax-pre-mulgt0 8142 ax-pre-mulext 8143 ax-arch 8144 ax-caucvg 8145 ax-addf 8147 ax-mulf 8148

This theorem depends on definitions: df-bi 117 df-stab 836 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-xor 1418 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2802 df-sbc 3030 df-csb 3126 df-dif 3200 df-un 3202 df-in 3204 df-ss 3211 df-nul 3493 df-if 3604 df-pw 3652 df-sn 3673 df-pr 3674 df-tp 3675 df-op 3676 df-uni 3892 df-int 3927 df-iun 3970 df-br 4087 df-opab 4149 df-mpt 4150 df-tr 4186 df-id 4388 df-po 4391 df-iso 4392 df-iord 4461 df-on 4463 df-ilim 4464 df-suc 4466 df-iom 4687 df-xp 4729 df-rel 4730 df-cnv 4731 df-co 4732 df-dm 4733 df-rn 4734 df-res 4735 df-ima 4736 df-iota 5284 df-fun 5326 df-fn 5327 df-f 5328 df-f1 5329 df-fo 5330 df-f1o 5331 df-fv 5332 df-isom 5333 df-riota 5966 df-ov 6016 df-oprab 6017 df-mpo 6018 df-of 6230 df-1st 6298 df-2nd 6299 df-tpos 6406 df-recs 6466 df-irdg 6531 df-frec 6552 df-1o 6577 df-2o 6578 df-oadd 6581 df-er 6697 df-ec 6699 df-qs 6703 df-map 6814 df-en 6905 df-dom 6906 df-fin 6907 df-sup 7177 df-pnf 8209 df-mnf 8210 df-xr 8211 df-ltxr 8212 df-le 8213 df-sub 8345 df-neg 8346 df-reap 8748 df-ap 8755 df-div 8846 df-inn 9137 df-2 9195 df-3 9196 df-4 9197 df-5 9198 df-6 9199 df-7 9200 df-8 9201 df-9 9202 df-n0 9396 df-z 9473 df-dec 9605 df-uz 9749 df-q 9847 df-rp 9882 df-fz 10237 df-fzo 10371 df-fl 10523 df-mod 10578 df-seqfrec 10703 df-exp 10794 df-ihash 11031 df-cj 11396 df-re 11397 df-im 11398 df-rsqrt 11552 df-abs 11553 df-clim 11833 df-sumdc 11908 df-dvds 12342 df-gcd 12518 df-prm 12673 df-struct 13077 df-ndx 13078 df-slot 13079 df-base 13081 df-sets 13082 df-iress 13083 df-plusg 13166 df-mulr 13167 df-starv 13168 df-sca 13169 df-vsca 13170 df-ip 13171 df-tset 13172 df-ple 13173 df-ds 13175 df-unif 13176 df-0g 13334 df-igsum 13335 df-topgen 13336 df-iimas 13378 df-qus 13379 df-mgm 13432 df-sgrp 13478 df-mnd 13493 df-mhm 13535 df-submnd 13536 df-grp 13579 df-minusg 13580 df-sbg 13581 df-mulg 13700 df-subg 13750 df-nsg 13751 df-eqg 13752 df-ghm 13821 df-cmn 13866 df-abl 13867 df-mgp 13927 df-rng 13939 df-ur 13966 df-srg 13970 df-ring 14004 df-cring 14005 df-oppr 14074 df-dvdsr 14095 df-unit 14096 df-invr 14128 df-dvr 14139 df-rhm 14159 df-nzr 14187 df-subrg 14226 df-domn 14266 df-idom 14267 df-lmod 14296 df-lssm 14360 df-lsp 14394 df-sra 14442 df-rgmod 14443 df-lidl 14476 df-rsp 14477 df-2idl 14507 df-bl 14553 df-mopn 14554 df-fg 14556 df-metu 14557 df-cnfld 14564 df-zring 14598 df-zrh 14621 df-zn 14623

This theorem is referenced by: lgseisen 15796

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