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Theorem lgseisenlem4 16409

Description: Lemma for lgseisen 16410. (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 15041	. . . . 5 ⊢ ℤ = (Base‘ℤ_ring)
2		zring0 15045	. . . . 5 ⊢ 0 = (0_g‘ℤ_ring)
3		zringabl 15039	. . . . . 6 ⊢ ℤ_ring ∈ Abel
4		ablcmn 14185	. . . . . 6 ⊢ (ℤ_ring ∈ Abel → ℤ_ring ∈ CMnd)
5	3, 4	mp1i 10	. . . . 5 ⊢ (𝜑 → ℤ_ring ∈ CMnd)
6		lgseisen.1	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))
7	6	eldifad 3231	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ ℙ)
8		prmnn 12914	. . . . . . . . 9 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
9	8	nnnn0d 9632	. . . . . . . 8 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ₀)
10	7, 9	syl 14	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ ℕ₀)
11		lgseisen.7	. . . . . . . 8 ⊢ 𝑌 = (ℤ/nℤ‘𝑃)
12	11	zncrng 15090	. . . . . . 7 ⊢ (𝑃 ∈ ℕ₀ → 𝑌 ∈ CRing)
13	10, 12	syl 14	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ CRing)
14		lgseisen.8	. . . . . . 7 ⊢ 𝐺 = (mulGrp‘𝑌)
15	14	crngmgp 14416	. . . . . 6 ⊢ (𝑌 ∈ CRing → 𝐺 ∈ CMnd)
16	13, 15	syl 14	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ CMnd)
17		1zzd 9683	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℤ)
18		oddprm 13068	. . . . . . . 8 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((𝑃 − 1) / 2) ∈ ℕ)
19	6, 18	syl 14	. . . . . . 7 ⊢ (𝜑 → ((𝑃 − 1) / 2) ∈ ℕ)
20	19	nnzd 9779	. . . . . 6 ⊢ (𝜑 → ((𝑃 − 1) / 2) ∈ ℤ)
21	17, 20	fzfigd 10890	. . . . 5 ⊢ (𝜑 → (1...((𝑃 − 1) / 2)) ∈ Fin)
22	13	crngringd 14421	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ Ring)
23		lgseisen.9	. . . . . . . . . 10 ⊢ 𝐿 = (ℤRHom‘𝑌)
24	23	zrhrhm 15068	. . . . . . . . 9 ⊢ (𝑌 ∈ Ring → 𝐿 ∈ (ℤ_ring RingHom 𝑌))
25	22, 24	syl 14	. . . . . . . 8 ⊢ (𝜑 → 𝐿 ∈ (ℤ_ring RingHom 𝑌))
26		eqid 2238	. . . . . . . . 9 ⊢ (Base‘𝑌) = (Base‘𝑌)
27	1, 26	rhmf 14580	. . . . . . . 8 ⊢ (𝐿 ∈ (ℤ_ring RingHom 𝑌) → 𝐿:ℤ⟶(Base‘𝑌))
28	25, 27	syl 14	. . . . . . 7 ⊢ (𝜑 → 𝐿:ℤ⟶(Base‘𝑌))
29		m1expcl 11021	. . . . . . . 8 ⊢ (𝑘 ∈ ℤ → ( -1↑𝑘) ∈ ℤ)
30	29	adantl 277	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → ( -1↑𝑘) ∈ ℤ)
31	28, 30	cofmpt 5880	. . . . . 6 ⊢ (𝜑 → (𝐿 ∘ (𝑘 ∈ ℤ ↦ ( -1↑𝑘))) = (𝑘 ∈ ℤ ↦ (𝐿‘( -1↑𝑘))))
32		zringmpg 15051	. . . . . . . . 9 ⊢ ((mulGrp‘ℂ_fld) ↾_s ℤ) = (mulGrp‘ℤ_ring)
33	32, 14	rhmmhm 14576	. . . . . . . 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 9419	. . . . . . . . . . 11 ⊢ -1 ∈ ℂ
36		neg1ap0 9423	. . . . . . . . . . 11 ⊢ -1 # 0
37		eqid 2238	. . . . . . . . . . . 12 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
38		eqid 2238	. . . . . . . . . . . 12 ⊢ ((mulGrp‘ℂ_fld) ↾_s {𝑧 ∈ ℂ ∣ 𝑧 # 0}) = ((mulGrp‘ℂ_fld) ↾_s {𝑧 ∈ ℂ ∣ 𝑧 # 0})
39	37, 38	expghmap 15052	. . . . . . . . . . 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 430	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℤ ↦ ( -1↑𝑘)) ∈ (ℤ_ring GrpHom ((mulGrp‘ℂ_fld) ↾_s {𝑧 ∈ ℂ ∣ 𝑧 # 0}))
41		ghmmhm 14116	. . . . . . . . . 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 15017	. . . . . . . . . 10 ⊢ ℂ_fld ∈ Ring
44		cnfldui 15034	. . . . . . . . . . 11 ⊢ {𝑧 ∈ ℂ ∣ 𝑧 # 0} = (Unit‘ℂ_fld)
45	44, 37	unitsubm 14536	. . . . . . . . . 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 13855	. . . . . . . . 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 430	. . . . . . . 8 ⊢ (𝑘 ∈ ℤ ↦ ( -1↑𝑘)) ∈ (ℤ_ring MndHom (mulGrp‘ℂ_fld))
49		zsubrg 15028	. . . . . . . . . 10 ⊢ ℤ ∈ (SubRing‘ℂ_fld)
50	37	subrgsubm 14652	. . . . . . . . . 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 5866	. . . . . . . . . 10 ⊢ (𝜑 → (𝑘 ∈ ℤ ↦ ( -1↑𝑘)):ℤ⟶ℤ)
53	52	frnd 5546	. . . . . . . . 9 ⊢ (𝜑 → ran (𝑘 ∈ ℤ ↦ ( -1↑𝑘)) ⊆ ℤ)
54		eqid 2238	. . . . . . . . . 10 ⊢ ((mulGrp‘ℂ_fld) ↾_s ℤ) = ((mulGrp‘ℂ_fld) ↾_s ℤ)
55	54	resmhm2b 13856	. . . . . . . . 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 418	. . . . . . . 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 13857	. . . . . . 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 415	. . . . . 6 ⊢ (𝜑 → (𝐿 ∘ (𝑘 ∈ ℤ ↦ ( -1↑𝑘))) ∈ (ℤ_ring MndHom 𝐺))
60	31, 59	eqeltrrd 2316	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ ℤ ↦ (𝐿‘( -1↑𝑘))) ∈ (ℤ_ring MndHom 𝐺))
61		lgseisen.2	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑄 ∈ (ℙ ∖ {2}))
62	61	gausslemma2dlem0a 16385	. . . . . . . . . 10 ⊢ (𝜑 → 𝑄 ∈ ℕ)
63	62	nnzd 9779	. . . . . . . . 9 ⊢ (𝜑 → 𝑄 ∈ ℤ)
64	63	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑄 ∈ ℤ)
65	6	gausslemma2dlem0a 16385	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ ℕ)
66	65	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 ∈ ℕ)
67		znq 10041	. . . . . . . 8 ⊢ ((𝑄 ∈ ℤ ∧ 𝑃 ∈ ℕ) → (𝑄 / 𝑃) ∈ ℚ)
68	64, 66, 67	syl2anc 415	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑄 / 𝑃) ∈ ℚ)
69		2nn 9478	. . . . . . . . . 10 ⊢ 2 ∈ ℕ
70		elfznn 10478	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (1...((𝑃 − 1) / 2)) → 𝑥 ∈ ℕ)
71	70	adantl 277	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑥 ∈ ℕ)
72		nnmulcl 9335	. . . . . . . . . 10 ⊢ ((2 ∈ ℕ ∧ 𝑥 ∈ ℕ) → (2 · 𝑥) ∈ ℕ)
73	69, 71, 72	sylancr 418	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (2 · 𝑥) ∈ ℕ)
74	73	nnzd 9779	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (2 · 𝑥) ∈ ℤ)
75		zq 10043	. . . . . . . 8 ⊢ ((2 · 𝑥) ∈ ℤ → (2 · 𝑥) ∈ ℚ)
76	74, 75	syl 14	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (2 · 𝑥) ∈ ℚ)
77		qmulcl 10054	. . . . . . 7 ⊢ (((𝑄 / 𝑃) ∈ ℚ ∧ (2 · 𝑥) ∈ ℚ) → ((𝑄 / 𝑃) · (2 · 𝑥)) ∈ ℚ)
78	68, 76, 77	syl2anc 415	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝑄 / 𝑃) · (2 · 𝑥)) ∈ ℚ)
79	78	flqcld 10732	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))) ∈ ℤ)
80		oveq2 6096	. . . . . 6 ⊢ (𝑘 = (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))) → ( -1↑𝑘) = ( -1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))
81	80	fveq2d 5702	. . . . 5 ⊢ (𝑘 = (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))) → (𝐿‘( -1↑𝑘)) = (𝐿‘( -1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))))
82		oveq2 6096	. . . . . 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 5702	. . . . 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, 16, 21, 60, 79, 81, 83	gsummhm2fi 14266	. . . 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 2238	. . . . . . . 8 ⊢ (Base‘𝐺) = (Base‘𝐺)
86		eqid 2238	. . . . . . . 8 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
87	28	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝐿:ℤ⟶(Base‘𝑌))
88		m1expcl 11021	. . . . . . . . . . 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 5847	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘( -1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) ∈ (Base‘𝑌))
91	14, 26	mgpbasg 14332	. . . . . . . . . . 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 2317	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘( -1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) ∈ (Base‘𝐺))
95		neg1z 9688	. . . . . . . . . . . 12 ⊢ -1 ∈ ℤ
96		lgseisen.4	. . . . . . . . . . . . 13 ⊢ 𝑅 = ((𝑄 · (2 · 𝑥)) mod 𝑃)
97	61	eldifad 3231	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑄 ∈ ℙ)
98	97	adantr 276	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑄 ∈ ℙ)
99		prmz 12915	. . . . . . . . . . . . . . . 16 ⊢ (𝑄 ∈ ℙ → 𝑄 ∈ ℤ)
100	98, 99	syl 14	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑄 ∈ ℤ)
101	100, 74	zmulcld 9786	. . . . . . . . . . . . . 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 10804	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝑄 · (2 · 𝑥)) mod 𝑃) ∈ ℕ₀)
105	96, 104	eqeltrid 2325	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑅 ∈ ℕ₀)
106		zexpcl 11013	. . . . . . . . . . . 12 ⊢ (( -1 ∈ ℤ ∧ 𝑅 ∈ ℕ₀) → ( -1↑𝑅) ∈ ℤ)
107	95, 105, 106	sylancr 418	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ( -1↑𝑅) ∈ ℤ)
108	107, 100	zmulcld 9786	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (( -1↑𝑅) · 𝑄) ∈ ℤ)
109	87, 108	ffvelcdmd 5847	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘(( -1↑𝑅) · 𝑄)) ∈ (Base‘𝑌))
110	109, 93	eleqtrd 2317	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘(( -1↑𝑅) · 𝑄)) ∈ (Base‘𝐺))
111		eqid 2238	. . . . . . . 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 2238	. . . . . . . 8 ⊢ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(( -1↑𝑅) · 𝑄))) = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(( -1↑𝑅) · 𝑄)))
113	85, 86, 16, 21, 94, 110, 111, 112	gsummptfidmadd2 14253	. . . . . . 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 2238	. . . . . . . . . . . 12 ⊢ (._r‘𝑌) = (._r‘𝑌)
115	14, 114	mgpplusgg 14329	. . . . . . . . . . 11 ⊢ (𝑌 ∈ CRing → (._r‘𝑌) = (+_g‘𝐺))
116	13, 115	syl 14	. . . . . . . . . 10 ⊢ (𝜑 → (._r‘𝑌) = (+_g‘𝐺))
117	116	ofeqd 6307	. . . . . . . . 9 ⊢ (𝜑 → ∘_𝑓 (._r‘𝑌) = ∘_𝑓 (+_g‘𝐺))
118	117	oveqd 6105	. . . . . . . 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 6104	. . . . . . 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 6105	. . . . . . 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 2281	. . . . . 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		eqidd 2239	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘( -1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))) = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘( -1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))))
123		eqidd 2239	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(( -1↑𝑅) · 𝑄))) = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(( -1↑𝑅) · 𝑄))))
124	21, 90, 109, 122, 123	offval2 6322	. . . . . . . 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)))))
125	25	adantr 276	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝐿 ∈ (ℤ_ring RingHom 𝑌))
126		zringmulr 15044	. . . . . . . . . . . 12 ⊢ · = (._r‘ℤ_ring)
127	1, 126, 114	rhmmul 14581	. . . . . . . . . . 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))))
128	125, 89, 108, 127	syl3anc 1278	. . . . . . . . . 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))))
129	62	adantr 276	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑄 ∈ ℕ)
130	129, 73	nnmulcld 9363	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑄 · (2 · 𝑥)) ∈ ℕ)
131		nnq 10050	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑄 · (2 · 𝑥)) ∈ ℕ → (𝑄 · (2 · 𝑥)) ∈ ℚ)
132	130, 131	syl 14	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑄 · (2 · 𝑥)) ∈ ℚ)
133		nnq 10050	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℚ)
134	65, 133	syl 14	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑃 ∈ ℚ)
135	134	adantr 276	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 ∈ ℚ)
136	66	nngt0d 9358	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 0 < 𝑃)
137		modqval 10783	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑄 · (2 · 𝑥)) ∈ ℚ ∧ 𝑃 ∈ ℚ ∧ 0 < 𝑃) → ((𝑄 · (2 · 𝑥)) mod 𝑃) = ((𝑄 · (2 · 𝑥)) − (𝑃 · (⌊‘((𝑄 · (2 · 𝑥)) / 𝑃)))))
138	132, 135, 136, 137	syl3anc 1278	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝑄 · (2 · 𝑥)) mod 𝑃) = ((𝑄 · (2 · 𝑥)) − (𝑃 · (⌊‘((𝑄 · (2 · 𝑥)) / 𝑃)))))
139	96, 138	eqtrid 2283	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑅 = ((𝑄 · (2 · 𝑥)) − (𝑃 · (⌊‘((𝑄 · (2 · 𝑥)) / 𝑃)))))
140	100	zcnd 9781	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑄 ∈ ℂ)
141	73	nncnd 9328	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (2 · 𝑥) ∈ ℂ)
142	103	nncnd 9328	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 ∈ ℂ)
143	103	nnap0d 9360	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 # 0)
144	140, 141, 142, 143	div23apd 9167	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝑄 · (2 · 𝑥)) / 𝑃) = ((𝑄 / 𝑃) · (2 · 𝑥)))
145	144	fveq2d 5702	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (⌊‘((𝑄 · (2 · 𝑥)) / 𝑃)) = (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))
146	145	oveq2d 6104	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑃 · (⌊‘((𝑄 · (2 · 𝑥)) / 𝑃))) = (𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))
147	146	oveq2d 6104	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝑄 · (2 · 𝑥)) − (𝑃 · (⌊‘((𝑄 · (2 · 𝑥)) / 𝑃)))) = ((𝑄 · (2 · 𝑥)) − (𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))))
148	139, 147	eqtrd 2271	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑅 = ((𝑄 · (2 · 𝑥)) − (𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))))
149	148	oveq2d 6104	. . . . . . . . . . . . . . . 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)))))))
150		prmz 12915	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
151	102, 150	syl 14	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 ∈ ℤ)
152	151, 79	zmulcld 9786	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ∈ ℤ)
153	152	zcnd 9781	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ∈ ℂ)
154	101	zcnd 9781	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑄 · (2 · 𝑥)) ∈ ℂ)
155	153, 154	pncan3d 8648	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + ((𝑄 · (2 · 𝑥)) − (𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))) = (𝑄 · (2 · 𝑥)))
156		2cnd 9387	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 2 ∈ ℂ)
157	71	nncnd 9328	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑥 ∈ ℂ)
158	140, 156, 157	mul12d 8486	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑄 · (2 · 𝑥)) = (2 · (𝑄 · 𝑥)))
159	149, 155, 158	3eqtrd 2275	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 𝑅) = (2 · (𝑄 · 𝑥)))
160	159	oveq2d 6104	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ( -1↑((𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 𝑅)) = ( -1↑(2 · (𝑄 · 𝑥))))
161	35	a1i 9	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → -1 ∈ ℂ)
162	36	a1i 9	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → -1 # 0)
163	105	nn0zd 9778	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑅 ∈ ℤ)
164		expaddzap 11042	. . . . . . . . . . . . . . . 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)))
165	161, 162, 152, 163, 164	syl22anc 1279	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ( -1↑((𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 𝑅)) = (( -1↑(𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) · ( -1↑𝑅)))
166		expmulzap 11044	. . . . . . . . . . . . . . . . . 18 ⊢ ((( -1 ∈ ℂ ∧ -1 # 0) ∧ (𝑃 ∈ ℤ ∧ (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))) ∈ ℤ)) → ( -1↑(𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) = (( -1↑𝑃)↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))
167	161, 162, 151, 79, 166	syl22anc 1279	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ( -1↑(𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) = (( -1↑𝑃)↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))
168		1cnd 8349	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 1 ∈ ℂ)
169		eldifsni 3843	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ≠ 2)
170	6, 169	syl 14	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝑃 ≠ 2)
171	170	necomd 2506	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 2 ≠ 𝑃)
172	171	neneqd 2441	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ¬ 2 = 𝑃)
173	172	adantr 276	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ¬ 2 = 𝑃)
174		2z 9684	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 2 ∈ ℤ
175		uzid 9953	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (2 ∈ ℤ → 2 ∈ (ℤ_≥‘2))
176	174, 175	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 2 ∈ (ℤ_≥‘2)
177		dvdsprm 12942	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((2 ∈ (ℤ_≥‘2) ∧ 𝑃 ∈ ℙ) → (2 ∥ 𝑃 ↔ 2 = 𝑃))
178	176, 102, 177	sylancr 418	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (2 ∥ 𝑃 ↔ 2 = 𝑃))
179	173, 178	mtbird 684	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ¬ 2 ∥ 𝑃)
180		oexpneg 12670	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1 ∈ ℂ ∧ 𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃) → ( -1↑𝑃) = -(1↑𝑃))
181	168, 103, 179, 180	syl3anc 1278	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ( -1↑𝑃) = -(1↑𝑃))
182		1exp 11027	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑃 ∈ ℤ → (1↑𝑃) = 1)
183	151, 182	syl 14	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (1↑𝑃) = 1)
184	183	negeqd 8529	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → -(1↑𝑃) = -1)
185	181, 184	eqtrd 2271	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ( -1↑𝑃) = -1)
186	185	oveq1d 6103	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (( -1↑𝑃)↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) = ( -1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))
187	167, 186	eqtrd 2271	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ( -1↑(𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) = ( -1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))
188	187	oveq1d 6103	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (( -1↑(𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) · ( -1↑𝑅)) = (( -1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) · ( -1↑𝑅)))
189	165, 188	eqtrd 2271	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ( -1↑((𝑃 · (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) + 𝑅)) = (( -1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) · ( -1↑𝑅)))
190		nnmulcl 9335	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑄 ∈ ℕ ∧ 𝑥 ∈ ℕ) → (𝑄 · 𝑥) ∈ ℕ)
191	62, 70, 190	syl2an 289	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑄 · 𝑥) ∈ ℕ)
192	191	nnnn0d 9632	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑄 · 𝑥) ∈ ℕ₀)
193		2nn0 9592	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℕ₀
194	193	a1i 9	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 2 ∈ ℕ₀)
195	161, 192, 194	expmuld 11136	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ( -1↑(2 · (𝑄 · 𝑥))) = (( -1↑2)↑(𝑄 · 𝑥)))
196		neg1sqe1 11093	. . . . . . . . . . . . . . . . 17 ⊢ ( -1↑2) = 1
197	196	oveq1i 6098	. . . . . . . . . . . . . . . 16 ⊢ (( -1↑2)↑(𝑄 · 𝑥)) = (1↑(𝑄 · 𝑥))
198	191	nnzd 9779	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑄 · 𝑥) ∈ ℤ)
199		1exp 11027	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑄 · 𝑥) ∈ ℤ → (1↑(𝑄 · 𝑥)) = 1)
200	198, 199	syl 14	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (1↑(𝑄 · 𝑥)) = 1)
201	197, 200	eqtrid 2283	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (( -1↑2)↑(𝑄 · 𝑥)) = 1)
202	195, 201	eqtrd 2271	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ( -1↑(2 · (𝑄 · 𝑥))) = 1)
203	160, 189, 202	3eqtr3d 2279	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (( -1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) · ( -1↑𝑅)) = 1)
204	203	oveq1d 6103	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((( -1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) · ( -1↑𝑅)) · 𝑄) = (1 · 𝑄))
205	89	zcnd 9781	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ( -1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ∈ ℂ)
206	107	zcnd 9781	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ( -1↑𝑅) ∈ ℂ)
207	205, 206, 140	mulassd 8356	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((( -1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) · ( -1↑𝑅)) · 𝑄) = (( -1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) · (( -1↑𝑅) · 𝑄)))
208	140	mullidd 8351	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (1 · 𝑄) = 𝑄)
209	204, 207, 208	3eqtr3d 2279	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (( -1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) · (( -1↑𝑅) · 𝑄)) = 𝑄)
210	209	fveq2d 5702	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘(( -1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) · (( -1↑𝑅) · 𝑄))) = (𝐿‘𝑄))
211	128, 210	eqtr3d 2273	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝐿‘( -1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))(._r‘𝑌)(𝐿‘(( -1↑𝑅) · 𝑄))) = (𝐿‘𝑄))
212	211	mpteq2dva 4223	. . . . . . . 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)))
213	124, 212	eqtrd 2271	. . . . . . 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)))
214	213	oveq2d 6104	. . . . . 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))))
215		lgseisen.3	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ≠ 𝑄)
216		lgseisen.5	. . . . . . . 8 ⊢ 𝑀 = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (((( -1↑𝑅) · 𝑅) mod 𝑃) / 2))
217		lgseisen.6	. . . . . . . 8 ⊢ 𝑆 = ((𝑄 · (2 · 𝑦)) mod 𝑃)
218	6, 61, 215, 96, 216, 217, 11, 14, 23	lgseisenlem3 16408	. . . . . . 7 ⊢ (𝜑 → (𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(( -1↑𝑅) · 𝑄)))) = (1_r‘𝑌))
219	218	oveq2d 6104	. . . . . 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)))
220	121, 214, 219	3eqtr3rd 2280	. . . . 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))))
221		eqid 2238	. . . . . . . 8 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
222	90	fmpttd 5866	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘( -1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))):(1...((𝑃 − 1) / 2))⟶(Base‘𝑌))
223	92	feq3d 5525	. . . . . . . . 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)))
224	222, 223	mpbid 147	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘( -1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))):(1...((𝑃 − 1) / 2))⟶(Base‘𝐺))
225	85, 221, 16, 21, 224	gsumclfi 14250	. . . . . . 7 ⊢ (𝜑 → (𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘( -1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))))) ∈ (Base‘𝐺))
226	225, 92	eleqtrrd 2318	. . . . . 6 ⊢ (𝜑 → (𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘( -1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))))) ∈ (Base‘𝑌))
227		eqid 2238	. . . . . . 7 ⊢ (1_r‘𝑌) = (1_r‘𝑌)
228	26, 114, 227	ringridm 14437	. . . . . 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))))))))
229	22, 226, 228	syl2anc 415	. . . . 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))))))))
230	97, 99	syl 14	. . . . . . . . . 10 ⊢ (𝜑 → 𝑄 ∈ ℤ)
231	28, 230	ffvelcdmd 5847	. . . . . . . . 9 ⊢ (𝜑 → (𝐿‘𝑄) ∈ (Base‘𝑌))
232	231, 92	eleqtrd 2317	. . . . . . . 8 ⊢ (𝜑 → (𝐿‘𝑄) ∈ (Base‘𝐺))
233		eqid 2238	. . . . . . . . 9 ⊢ (._g‘𝐺) = (._g‘𝐺)
234	85, 233	gsumconstcmn 14267	. . . . . . . 8 ⊢ ((𝐺 ∈ CMnd ∧ (1...((𝑃 − 1) / 2)) ∈ Fin ∧ (𝐿‘𝑄) ∈ (Base‘𝐺)) → (𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘𝑄))) = ((♯‘(1...((𝑃 − 1) / 2)))(._g‘𝐺)(𝐿‘𝑄)))
235	16, 21, 232, 234	syl3anc 1278	. . . . . . 7 ⊢ (𝜑 → (𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘𝑄))) = ((♯‘(1...((𝑃 − 1) / 2)))(._g‘𝐺)(𝐿‘𝑄)))
236		nnuz 9975	. . . . . . . . . 10 ⊢ ℕ = (ℤ_≥‘1)
237	19, 236	eleqtrdi 2331	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃 − 1) / 2) ∈ (ℤ_≥‘1))
238		hashfz 11285	. . . . . . . . 9 ⊢ (((𝑃 − 1) / 2) ∈ (ℤ_≥‘1) → (♯‘(1...((𝑃 − 1) / 2))) = ((((𝑃 − 1) / 2) − 1) + 1))
239	237, 238	syl 14	. . . . . . . 8 ⊢ (𝜑 → (♯‘(1...((𝑃 − 1) / 2))) = ((((𝑃 − 1) / 2) − 1) + 1))
240	239	oveq1d 6103	. . . . . . 7 ⊢ (𝜑 → ((♯‘(1...((𝑃 − 1) / 2)))(._g‘𝐺)(𝐿‘𝑄)) = (((((𝑃 − 1) / 2) − 1) + 1)(._g‘𝐺)(𝐿‘𝑄)))
241	235, 240	eqtrd 2271	. . . . . 6 ⊢ (𝜑 → (𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘𝑄))) = (((((𝑃 − 1) / 2) − 1) + 1)(._g‘𝐺)(𝐿‘𝑄)))
242	19	nncnd 9328	. . . . . . . 8 ⊢ (𝜑 → ((𝑃 − 1) / 2) ∈ ℂ)
243		1cnd 8349	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℂ)
244	242, 243	npcand 8649	. . . . . . 7 ⊢ (𝜑 → ((((𝑃 − 1) / 2) − 1) + 1) = ((𝑃 − 1) / 2))
245	244	oveq1d 6103	. . . . . 6 ⊢ (𝜑 → (((((𝑃 − 1) / 2) − 1) + 1)(._g‘𝐺)(𝐿‘𝑄)) = (((𝑃 − 1) / 2)(._g‘𝐺)(𝐿‘𝑄)))
246	19	nnnn0d 9632	. . . . . . . 8 ⊢ (𝜑 → ((𝑃 − 1) / 2) ∈ ℕ₀)
247		zringring 15038	. . . . . . . . . 10 ⊢ ℤ_ring ∈ Ring
248	32, 1	mgpbasg 14332	. . . . . . . . . 10 ⊢ (ℤ_ring ∈ Ring → ℤ = (Base‘((mulGrp‘ℂ_fld) ↾_s ℤ)))
249	247, 248	ax-mp 5	. . . . . . . . 9 ⊢ ℤ = (Base‘((mulGrp‘ℂ_fld) ↾_s ℤ))
250		eqid 2238	. . . . . . . . 9 ⊢ (._g‘((mulGrp‘ℂ_fld) ↾_s ℤ)) = (._g‘((mulGrp‘ℂ_fld) ↾_s ℤ))
251	249, 250, 233	mhmmulg 14026	. . . . . . . 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)))
252	34, 246, 230, 251	syl3anc 1278	. . . . . . 7 ⊢ (𝜑 → (𝐿‘(((𝑃 − 1) / 2)(._g‘((mulGrp‘ℂ_fld) ↾_s ℤ))𝑄)) = (((𝑃 − 1) / 2)(._g‘𝐺)(𝐿‘𝑄)))
253	49, 50	mp1i 10	. . . . . . . . . 10 ⊢ (𝜑 → ℤ ∈ (SubMnd‘(mulGrp‘ℂ_fld)))
254		eqid 2238	. . . . . . . . . . 11 ⊢ (._g‘(mulGrp‘ℂ_fld)) = (._g‘(mulGrp‘ℂ_fld))
255	254, 54, 250	submmulg 14029	. . . . . . . . . 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))
256	253, 246, 230, 255	syl3anc 1278	. . . . . . . . 9 ⊢ (𝜑 → (((𝑃 − 1) / 2)(._g‘(mulGrp‘ℂ_fld))𝑄) = (((𝑃 − 1) / 2)(._g‘((mulGrp‘ℂ_fld) ↾_s ℤ))𝑄))
257	230	zcnd 9781	. . . . . . . . . 10 ⊢ (𝜑 → 𝑄 ∈ ℂ)
258		cnfldexp 15024	. . . . . . . . . 10 ⊢ ((𝑄 ∈ ℂ ∧ ((𝑃 − 1) / 2) ∈ ℕ₀) → (((𝑃 − 1) / 2)(._g‘(mulGrp‘ℂ_fld))𝑄) = (𝑄↑((𝑃 − 1) / 2)))
259	257, 246, 258	syl2anc 415	. . . . . . . . 9 ⊢ (𝜑 → (((𝑃 − 1) / 2)(._g‘(mulGrp‘ℂ_fld))𝑄) = (𝑄↑((𝑃 − 1) / 2)))
260	256, 259	eqtr3d 2273	. . . . . . . 8 ⊢ (𝜑 → (((𝑃 − 1) / 2)(._g‘((mulGrp‘ℂ_fld) ↾_s ℤ))𝑄) = (𝑄↑((𝑃 − 1) / 2)))
261	260	fveq2d 5702	. . . . . . 7 ⊢ (𝜑 → (𝐿‘(((𝑃 − 1) / 2)(._g‘((mulGrp‘ℂ_fld) ↾_s ℤ))𝑄)) = (𝐿‘(𝑄↑((𝑃 − 1) / 2))))
262	252, 261	eqtr3d 2273	. . . . . 6 ⊢ (𝜑 → (((𝑃 − 1) / 2)(._g‘𝐺)(𝐿‘𝑄)) = (𝐿‘(𝑄↑((𝑃 − 1) / 2))))
263	241, 245, 262	3eqtrd 2275	. . . . 5 ⊢ (𝜑 → (𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘𝑄))) = (𝐿‘(𝑄↑((𝑃 − 1) / 2))))
264	220, 229, 263	3eqtr3d 2279	. . . 4 ⊢ (𝜑 → (𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘( -1↑(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))))) = (𝐿‘(𝑄↑((𝑃 − 1) / 2))))
265		subrgsubg 14645	. . . . . . . . . 10 ⊢ (ℤ ∈ (SubRing‘ℂ_fld) → ℤ ∈ (SubGrp‘ℂ_fld))
266	49, 265	ax-mp 5	. . . . . . . . 9 ⊢ ℤ ∈ (SubGrp‘ℂ_fld)
267		subgsubm 14059	. . . . . . . . 9 ⊢ (ℤ ∈ (SubGrp‘ℂ_fld) → ℤ ∈ (SubMnd‘ℂ_fld))
268	266, 267	mp1i 10	. . . . . . . 8 ⊢ (𝜑 → ℤ ∈ (SubMnd‘ℂ_fld))
269		ringcmn 14446	. . . . . . . . 9 ⊢ (ℂ_fld ∈ Ring → ℂ_fld ∈ CMnd)
270	43, 269	mp1i 10	. . . . . . . 8 ⊢ (𝜑 → ℂ_fld ∈ CMnd)
271	79	fmpttd 5866	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))):(1...((𝑃 − 1) / 2))⟶ℤ)
272		df-zring 15036	. . . . . . . 8 ⊢ ℤ_ring = (ℂ_fld ↾_s ℤ)
273	21, 268, 270, 271, 272	gsumsubmfi 14269	. . . . . . 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))))))
274	79	zcnd 9781	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))) ∈ ℂ)
275	21, 274	gsumfsum 15033	. . . . . . 7 ⊢ (𝜑 → (ℂ_fld Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) = Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))
276	273, 275	eqtr3d 2273	. . . . . 6 ⊢ (𝜑 → (ℤ_ring Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))) = Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))
277	276	oveq2d 6104	. . . . 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)))))
278	277	fveq2d 5702	. . . 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))))))
279	84, 264, 278	3eqtr3d 2279	. . 3 ⊢ (𝜑 → (𝐿‘(𝑄↑((𝑃 − 1) / 2))) = (𝐿‘( -1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))))
280	65	nnnn0d 9632	. . . 4 ⊢ (𝜑 → 𝑃 ∈ ℕ₀)
281		zexpcl 11013	. . . . 5 ⊢ ((𝑄 ∈ ℤ ∧ ((𝑃 − 1) / 2) ∈ ℕ₀) → (𝑄↑((𝑃 − 1) / 2)) ∈ ℤ)
282	230, 246, 281	syl2anc 415	. . . 4 ⊢ (𝜑 → (𝑄↑((𝑃 − 1) / 2)) ∈ ℤ)
283	21, 79	fsumzcl 12195	. . . . 5 ⊢ (𝜑 → Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))) ∈ ℤ)
284		m1expcl 11021	. . . . 5 ⊢ (Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))) ∈ ℤ → ( -1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ∈ ℤ)
285	283, 284	syl 14	. . . 4 ⊢ (𝜑 → ( -1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) ∈ ℤ)
286	11, 23	zndvds 15094	. . . 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)))))))
287	280, 282, 285, 286	syl3anc 1278	. . 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)))))))
288	279, 287	mpbid 147	. 2 ⊢ (𝜑 → 𝑃 ∥ ((𝑄↑((𝑃 − 1) / 2)) − ( -1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥))))))
289		moddvds 12592	. . 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)))))))
290	65, 282, 285, 289	syl3anc 1278	. 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)))))))
291	288, 290	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

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1402 ∈ wcel 2209 ≠ wne 2420 {crab 2532 ∖ cdif 3217 ⊆ wss 3220 {csn 3709 class class class wbr 4131 ↦ cmpt 4193 ran crn 4777 ∘ ccom 4780 ⟶wf 5376 ‘cfv 5380 (class class class)co 6088 ∘_𝑓 cof 6303 Fincfn 7027 ℂcc 8184 0cc0 8186 1c1 8187 + caddc 8189 · cmul 8191 < clt 8367 − cmin 8505 -cneg 8506 # cap 8918 / cdiv 9011 ℕcn 9314 2c2 9365 ℕ₀cn0 9575 ℤcz 9656 ℤ_≥cuz 9938 ℚcq 10036 ...cfz 10429 ⌊cfl 10721 mod cmo 10781 ↑cexp 10997 ♯chash 11237 Σcsu 12145 ∥ cdvds 12580 ℙcprime 12911 Basecbs 13411 ↾_s cress 13412 +_gcplusg 13491 ._rcmulr 13492 0_gc0g 13670 MndHom cmhm 13824 SubMndcsubmnd 13825 ._gcmg 13982 SubGrpcsubg 14030 GrpHom cghm 14103 CMndccmn 14178 Abelcabl 14179 Σ_g cgsu 14241 mulGrpcmgp 14325 1_rcur 14370 Ringcrg 14408 CRingccrg 14409 RingHom crh 14567 SubRingcsubrg 14635 ℂ_fldccnfld 15003 ℤ_ringczring 15035 ℤRHomczrh 15056 ℤ/nℤczn 15058

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4248 ax-sep 4251 ax-nul 4261 ax-pow 4313 ax-pr 4348 ax-un 4580 ax-setind 4686 ax-iinf 4737 ax-cnex 8277 ax-resscn 8278 ax-1cn 8279 ax-1re 8280 ax-icn 8281 ax-addcl 8282 ax-addrcl 8283 ax-mulcl 8284 ax-mulrcl 8285 ax-addcom 8286 ax-mulcom 8287 ax-addass 8288 ax-mulass 8289 ax-distr 8290 ax-i2m1 8291 ax-0lt1 8292 ax-1rid 8293 ax-0id 8294 ax-rnegex 8295 ax-precex 8296 ax-cnre 8297 ax-pre-ltirr 8298 ax-pre-ltwlin 8299 ax-pre-lttrn 8300 ax-pre-apti 8301 ax-pre-ltadd 8302 ax-pre-mulgt0 8303 ax-pre-mulext 8304 ax-arch 8305 ax-caucvg 8306 ax-addf 8308 ax-mulf 8309

This proof depends on definitions: df-bi 117 df-stab 843 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-xor 1425 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-tp 3717 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4132 df-opab 4194 df-mpt 4195 df-tr 4232 df-id 4440 df-po 4443 df-iso 4444 df-iord 4513 df-on 4515 df-ilim 4516 df-suc 4518 df-iom 4740 df-xp 4782 df-rel 4783 df-cnv 4784 df-co 4785 df-dm 4786 df-rn 4787 df-res 4788 df-ima 4789 df-iota 5340 df-fun 5382 df-fn 5383 df-f 5384 df-f1 5385 df-fo 5386 df-f1o 5387 df-fv 5388 df-isom 5389 df-riota 6041 df-ov 6091 df-oprab 6092 df-mpo 6093 df-of 6305 df-1st 6378 df-2nd 6379 df-tpos 6520 df-recs 6580 df-irdg 6645 df-frec 6666 df-1o 6691 df-2o 6692 df-oadd 6695 df-er 6811 df-ec 6813 df-qs 6817 df-map 6928 df-en 7028 df-dom 7029 df-fin 7030 df-sup 7331 df-pnf 8369 df-mnf 8370 df-xr 8371 df-ltxr 8372 df-le 8373 df-sub 8507 df-neg 8508 df-reap 8912 df-ap 8919 df-div 9012 df-inn 9315 df-2 9373 df-3 9374 df-4 9375 df-5 9376 df-6 9377 df-7 9378 df-8 9379 df-9 9380 df-n0 9576 df-z 9657 df-dec 9790 df-uz 9939 df-q 10037 df-rp 10073 df-fz 10430 df-fzo 10568 df-fl 10723 df-mod 10782 df-seqfrec 10907 df-exp 10998 df-ihash 11238 df-cj 11630 df-re 11631 df-im 11632 df-rsqrt 11787 df-abs 11788 df-clim 12071 df-sumdc 12146 df-dvds 12581 df-gcd 12757 df-prm 12912 df-struct 13413 df-ndx 13414 df-slot 13415 df-base 13417 df-sets 13418 df-iress 13419 df-plusg 13504 df-mulr 13505 df-starv 13506 df-sca 13507 df-vsca 13508 df-ip 13509 df-tset 13510 df-ple 13511 df-ds 13513 df-unif 13514 df-0g 13672 df-gzsum 13673 df-topgen 13674 df-iimas 13684 df-qus 13685 df-mgm 13736 df-sgrp 13777 df-mnd 13790 df-mhm 13826 df-submnd 13827 df-grp 13868 df-minusg 13869 df-sbg 13870 df-mulg 13983 df-subg 14033 df-nsg 14034 df-eqg 14035 df-ghm 14104 df-cmn 14180 df-abl 14181 df-gsumfi 14242 df-mgp 14326 df-rng 14340 df-ur 14371 df-srg 14376 df-ring 14410 df-cring 14411 df-oppr 14483 df-dvdsr 14505 df-unit 14506 df-invr 14538 df-dvr 14549 df-rhm 14569 df-nzr 14597 df-subrg 14637 df-domn 14677 df-idom 14678 df-lmod 14735 df-lssm 14800 df-lsp 14834 df-sra 14882 df-rgmod 14883 df-lidl 14916 df-rsp 14917 df-2idl 14947 df-bl 14993 df-mopn 14994 df-fg 14996 df-metu 14997 df-cnfld 15004 df-zring 15036 df-zrh 15059 df-zn 15061

This theorem is used by: lgseisen 16410

W3C validator