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Theorem lgseisenlem3 15136

Description: Lemma for Eisenstein's lemma. (Contributed by Mario Carneiro, 17-Jun-2015.) (Proof shortened by AV, 28-Jul-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
lgseisenlem3	⊢ (𝜑 → (𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄)))) = (1_r‘𝑌))

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 lgseisenlem3 Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 5918	. . . . . . . . 9 ⊢ (𝑘 = 𝑥 → (2 · 𝑘) = (2 · 𝑥))
2	1	fveq2d 5550	. . . . . . . 8 ⊢ (𝑘 = 𝑥 → (𝐿‘(2 · 𝑘)) = (𝐿‘(2 · 𝑥)))
3	2	cbvmptv 4125	. . . . . . 7 ⊢ (𝑘 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑘))) = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥)))
4	3	oveq2i 5921	. . . . . 6 ⊢ (𝐺 Σ_g (𝑘 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑘)))) = (𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥))))
5		eqid 2193	. . . . . . 7 ⊢ (Base‘𝐺) = (Base‘𝐺)
6		eqid 2193	. . . . . . 7 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
7		lgseisen.1	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))
8	7	eldifad 3164	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ ℙ)
9		lgseisen.7	. . . . . . . . . . 11 ⊢ 𝑌 = (ℤ/nℤ‘𝑃)
10	9	znidom 14122	. . . . . . . . . 10 ⊢ (𝑃 ∈ ℙ → 𝑌 ∈ IDomn)
11	8, 10	syl 14	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ IDomn)
12	11	idomcringd 13758	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ CRing)
13		lgseisen.8	. . . . . . . . 9 ⊢ 𝐺 = (mulGrp‘𝑌)
14	13	crngmgp 13484	. . . . . . . 8 ⊢ (𝑌 ∈ CRing → 𝐺 ∈ CMnd)
15	12, 14	syl 14	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ CMnd)
16		1zzd 9334	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℤ)
17		oddn2prm 12389	. . . . . . . . 9 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ¬ 2 ∥ 𝑃)
18	7, 17	syl 14	. . . . . . . 8 ⊢ (𝜑 → ¬ 2 ∥ 𝑃)
19		prmz 12239	. . . . . . . . 9 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
20		oddm1d2 12023	. . . . . . . . 9 ⊢ (𝑃 ∈ ℤ → (¬ 2 ∥ 𝑃 ↔ ((𝑃 − 1) / 2) ∈ ℤ))
21	8, 19, 20	3syl 17	. . . . . . . 8 ⊢ (𝜑 → (¬ 2 ∥ 𝑃 ↔ ((𝑃 − 1) / 2) ∈ ℤ))
22	18, 21	mpbid 147	. . . . . . 7 ⊢ (𝜑 → ((𝑃 − 1) / 2) ∈ ℤ)
23	11	idomringd 13759	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑌 ∈ Ring)
24		lgseisen.9	. . . . . . . . . . . 12 ⊢ 𝐿 = (ℤRHom‘𝑌)
25	24	zrhrhm 14088	. . . . . . . . . . 11 ⊢ (𝑌 ∈ Ring → 𝐿 ∈ (ℤ_ring RingHom 𝑌))
26		zringbas 14062	. . . . . . . . . . . 12 ⊢ ℤ = (Base‘ℤ_ring)
27		eqid 2193	. . . . . . . . . . . 12 ⊢ (Base‘𝑌) = (Base‘𝑌)
28	26, 27	rhmf 13643	. . . . . . . . . . 11 ⊢ (𝐿 ∈ (ℤ_ring RingHom 𝑌) → 𝐿:ℤ⟶(Base‘𝑌))
29	23, 25, 28	3syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐿:ℤ⟶(Base‘𝑌))
30		2z 9335	. . . . . . . . . . 11 ⊢ 2 ∈ ℤ
31		elfzelz 10081	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (1...((𝑃 − 1) / 2)) → 𝑘 ∈ ℤ)
32		zmulcl 9360	. . . . . . . . . . 11 ⊢ ((2 ∈ ℤ ∧ 𝑘 ∈ ℤ) → (2 · 𝑘) ∈ ℤ)
33	30, 31, 32	sylancr 414	. . . . . . . . . 10 ⊢ (𝑘 ∈ (1...((𝑃 − 1) / 2)) → (2 · 𝑘) ∈ ℤ)
34		ffvelcdm 5683	. . . . . . . . . 10 ⊢ ((𝐿:ℤ⟶(Base‘𝑌) ∧ (2 · 𝑘) ∈ ℤ) → (𝐿‘(2 · 𝑘)) ∈ (Base‘𝑌))
35	29, 33, 34	syl2an 289	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘(2 · 𝑘)) ∈ (Base‘𝑌))
36	35	fmpttd 5705	. . . . . . . 8 ⊢ (𝜑 → (𝑘 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑘))):(1...((𝑃 − 1) / 2))⟶(Base‘𝑌))
37	13, 27	mgpbasg 13406	. . . . . . . . . 10 ⊢ (𝑌 ∈ CRing → (Base‘𝑌) = (Base‘𝐺))
38	12, 37	syl 14	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝑌) = (Base‘𝐺))
39	38	feq3d 5384	. . . . . . . 8 ⊢ (𝜑 → ((𝑘 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑘))):(1...((𝑃 − 1) / 2))⟶(Base‘𝑌) ↔ (𝑘 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑘))):(1...((𝑃 − 1) / 2))⟶(Base‘𝐺)))
40	36, 39	mpbid 147	. . . . . . 7 ⊢ (𝜑 → (𝑘 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑘))):(1...((𝑃 − 1) / 2))⟶(Base‘𝐺))
41		lgseisen.2	. . . . . . . 8 ⊢ (𝜑 → 𝑄 ∈ (ℙ ∖ {2}))
42		lgseisen.3	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ≠ 𝑄)
43		lgseisen.4	. . . . . . . 8 ⊢ 𝑅 = ((𝑄 · (2 · 𝑥)) mod 𝑃)
44		lgseisen.5	. . . . . . . 8 ⊢ 𝑀 = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ ((((-1↑𝑅) · 𝑅) mod 𝑃) / 2))
45		lgseisen.6	. . . . . . . 8 ⊢ 𝑆 = ((𝑄 · (2 · 𝑦)) mod 𝑃)
46	7, 41, 42, 43, 44, 45	lgseisenlem2 15135	. . . . . . 7 ⊢ (𝜑 → 𝑀:(1...((𝑃 − 1) / 2))–1-1-onto→(1...((𝑃 − 1) / 2)))
47	5, 6, 15, 16, 22, 40, 46	gsumfzreidx 13396	. . . . . 6 ⊢ (𝜑 → (𝐺 Σ_g (𝑘 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑘)))) = (𝐺 Σ_g ((𝑘 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑘))) ∘ 𝑀)))
48	4, 47	eqtr3id 2240	. . . . 5 ⊢ (𝜑 → (𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥)))) = (𝐺 Σ_g ((𝑘 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑘))) ∘ 𝑀)))
49	7, 41, 42, 43, 44	lgseisenlem1 15134	. . . . . . . 8 ⊢ (𝜑 → 𝑀:(1...((𝑃 − 1) / 2))⟶(1...((𝑃 − 1) / 2)))
50	44	fmpt 5700	. . . . . . . 8 ⊢ (∀𝑥 ∈ (1...((𝑃 − 1) / 2))((((-1↑𝑅) · 𝑅) mod 𝑃) / 2) ∈ (1...((𝑃 − 1) / 2)) ↔ 𝑀:(1...((𝑃 − 1) / 2))⟶(1...((𝑃 − 1) / 2)))
51	49, 50	sylibr 134	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ (1...((𝑃 − 1) / 2))((((-1↑𝑅) · 𝑅) mod 𝑃) / 2) ∈ (1...((𝑃 − 1) / 2)))
52	44	a1i 9	. . . . . . 7 ⊢ (𝜑 → 𝑀 = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ ((((-1↑𝑅) · 𝑅) mod 𝑃) / 2)))
53		eqidd 2194	. . . . . . 7 ⊢ (𝜑 → (𝑘 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑘))) = (𝑘 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑘))))
54		oveq2 5918	. . . . . . . 8 ⊢ (𝑘 = ((((-1↑𝑅) · 𝑅) mod 𝑃) / 2) → (2 · 𝑘) = (2 · ((((-1↑𝑅) · 𝑅) mod 𝑃) / 2)))
55	54	fveq2d 5550	. . . . . . 7 ⊢ (𝑘 = ((((-1↑𝑅) · 𝑅) mod 𝑃) / 2) → (𝐿‘(2 · 𝑘)) = (𝐿‘(2 · ((((-1↑𝑅) · 𝑅) mod 𝑃) / 2))))
56	51, 52, 53, 55	fmptcof 5717	. . . . . 6 ⊢ (𝜑 → ((𝑘 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑘))) ∘ 𝑀) = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · ((((-1↑𝑅) · 𝑅) mod 𝑃) / 2)))))
57	56	oveq2d 5926	. . . . 5 ⊢ (𝜑 → (𝐺 Σ_g ((𝑘 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑘))) ∘ 𝑀)) = (𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · ((((-1↑𝑅) · 𝑅) mod 𝑃) / 2))))))
58	41	eldifad 3164	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑄 ∈ ℙ)
59	58	adantr 276	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑄 ∈ ℙ)
60		prmz 12239	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑄 ∈ ℙ → 𝑄 ∈ ℤ)
61	59, 60	syl 14	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑄 ∈ ℤ)
62		2nn 9133	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 2 ∈ ℕ
63		elfznn 10110	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ (1...((𝑃 − 1) / 2)) → 𝑥 ∈ ℕ)
64	63	adantl 277	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑥 ∈ ℕ)
65		nnmulcl 8993	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((2 ∈ ℕ ∧ 𝑥 ∈ ℕ) → (2 · 𝑥) ∈ ℕ)
66	62, 64, 65	sylancr 414	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (2 · 𝑥) ∈ ℕ)
67	66	nnzd 9428	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (2 · 𝑥) ∈ ℤ)
68	61, 67	zmulcld 9435	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑄 · (2 · 𝑥)) ∈ ℤ)
69	8	adantr 276	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 ∈ ℙ)
70		prmnn 12238	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
71	69, 70	syl 14	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 ∈ ℕ)
72	68, 71	zmodcld 10406	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝑄 · (2 · 𝑥)) mod 𝑃) ∈ ℕ₀)
73	43, 72	eqeltrid 2280	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑅 ∈ ℕ₀)
74	73	nn0zd 9427	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑅 ∈ ℤ)
75		m1expcl 10623	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ ℤ → (-1↑𝑅) ∈ ℤ)
76	74, 75	syl 14	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (-1↑𝑅) ∈ ℤ)
77	76, 74	zmulcld 9435	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((-1↑𝑅) · 𝑅) ∈ ℤ)
78	77, 71	zmodcld 10406	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (((-1↑𝑅) · 𝑅) mod 𝑃) ∈ ℕ₀)
79	78	nn0cnd 9285	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (((-1↑𝑅) · 𝑅) mod 𝑃) ∈ ℂ)
80		2cnd 9045	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 2 ∈ ℂ)
81		2ap0 9065	. . . . . . . . . . . 12 ⊢ 2 # 0
82	81	a1i 9	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 2 # 0)
83	79, 80, 82	divcanap2d 8801	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (2 · ((((-1↑𝑅) · 𝑅) mod 𝑃) / 2)) = (((-1↑𝑅) · 𝑅) mod 𝑃))
84	83	fveq2d 5550	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘(2 · ((((-1↑𝑅) · 𝑅) mod 𝑃) / 2))) = (𝐿‘(((-1↑𝑅) · 𝑅) mod 𝑃)))
85		zq 9681	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ ℤ → 𝑃 ∈ ℚ)
86	8, 19, 85	3syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑃 ∈ ℚ)
87	86	adantr 276	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 ∈ ℚ)
88	71	nngt0d 9016	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 0 < 𝑃)
89		eqidd 2194	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((-1↑𝑅) mod 𝑃) = ((-1↑𝑅) mod 𝑃))
90	43	oveq1i 5920	. . . . . . . . . . . . . 14 ⊢ (𝑅 mod 𝑃) = (((𝑄 · (2 · 𝑥)) mod 𝑃) mod 𝑃)
91		zq 9681	. . . . . . . . . . . . . . . 16 ⊢ ((𝑄 · (2 · 𝑥)) ∈ ℤ → (𝑄 · (2 · 𝑥)) ∈ ℚ)
92	68, 91	syl 14	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑄 · (2 · 𝑥)) ∈ ℚ)
93		modqabs2 10419	. . . . . . . . . . . . . . 15 ⊢ (((𝑄 · (2 · 𝑥)) ∈ ℚ ∧ 𝑃 ∈ ℚ ∧ 0 < 𝑃) → (((𝑄 · (2 · 𝑥)) mod 𝑃) mod 𝑃) = ((𝑄 · (2 · 𝑥)) mod 𝑃))
94	92, 87, 88, 93	syl3anc 1249	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (((𝑄 · (2 · 𝑥)) mod 𝑃) mod 𝑃) = ((𝑄 · (2 · 𝑥)) mod 𝑃))
95	90, 94	eqtrid 2238	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑅 mod 𝑃) = ((𝑄 · (2 · 𝑥)) mod 𝑃))
96	76, 76, 74, 68, 87, 88, 89, 95	modqmul12d 10439	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (((-1↑𝑅) · 𝑅) mod 𝑃) = (((-1↑𝑅) · (𝑄 · (2 · 𝑥))) mod 𝑃))
97		zq 9681	. . . . . . . . . . . . . 14 ⊢ (((-1↑𝑅) · 𝑅) ∈ ℤ → ((-1↑𝑅) · 𝑅) ∈ ℚ)
98	77, 97	syl 14	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((-1↑𝑅) · 𝑅) ∈ ℚ)
99		modqabs2 10419	. . . . . . . . . . . . 13 ⊢ ((((-1↑𝑅) · 𝑅) ∈ ℚ ∧ 𝑃 ∈ ℚ ∧ 0 < 𝑃) → ((((-1↑𝑅) · 𝑅) mod 𝑃) mod 𝑃) = (((-1↑𝑅) · 𝑅) mod 𝑃))
100	98, 87, 88, 99	syl3anc 1249	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((((-1↑𝑅) · 𝑅) mod 𝑃) mod 𝑃) = (((-1↑𝑅) · 𝑅) mod 𝑃))
101	76	zcnd 9430	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (-1↑𝑅) ∈ ℂ)
102	61	zcnd 9430	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑄 ∈ ℂ)
103	67	zcnd 9430	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (2 · 𝑥) ∈ ℂ)
104	101, 102, 103	mulassd 8033	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (((-1↑𝑅) · 𝑄) · (2 · 𝑥)) = ((-1↑𝑅) · (𝑄 · (2 · 𝑥))))
105	104	oveq1d 5925	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((((-1↑𝑅) · 𝑄) · (2 · 𝑥)) mod 𝑃) = (((-1↑𝑅) · (𝑄 · (2 · 𝑥))) mod 𝑃))
106	96, 100, 105	3eqtr4d 2236	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((((-1↑𝑅) · 𝑅) mod 𝑃) mod 𝑃) = ((((-1↑𝑅) · 𝑄) · (2 · 𝑥)) mod 𝑃))
107	8, 70	syl 14	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑃 ∈ ℕ)
108	107	adantr 276	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 ∈ ℕ)
109	78	nn0zd 9427	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (((-1↑𝑅) · 𝑅) mod 𝑃) ∈ ℤ)
110	76, 61	zmulcld 9435	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((-1↑𝑅) · 𝑄) ∈ ℤ)
111	110, 67	zmulcld 9435	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (((-1↑𝑅) · 𝑄) · (2 · 𝑥)) ∈ ℤ)
112		moddvds 11932	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ ℕ ∧ (((-1↑𝑅) · 𝑅) mod 𝑃) ∈ ℤ ∧ (((-1↑𝑅) · 𝑄) · (2 · 𝑥)) ∈ ℤ) → (((((-1↑𝑅) · 𝑅) mod 𝑃) mod 𝑃) = ((((-1↑𝑅) · 𝑄) · (2 · 𝑥)) mod 𝑃) ↔ 𝑃 ∥ ((((-1↑𝑅) · 𝑅) mod 𝑃) − (((-1↑𝑅) · 𝑄) · (2 · 𝑥)))))
113	108, 109, 111, 112	syl3anc 1249	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (((((-1↑𝑅) · 𝑅) mod 𝑃) mod 𝑃) = ((((-1↑𝑅) · 𝑄) · (2 · 𝑥)) mod 𝑃) ↔ 𝑃 ∥ ((((-1↑𝑅) · 𝑅) mod 𝑃) − (((-1↑𝑅) · 𝑄) · (2 · 𝑥)))))
114	106, 113	mpbid 147	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 ∥ ((((-1↑𝑅) · 𝑅) mod 𝑃) − (((-1↑𝑅) · 𝑄) · (2 · 𝑥))))
115	71	nnnn0d 9283	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 ∈ ℕ₀)
116	9, 24	zndvds 14114	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ ℕ₀ ∧ (((-1↑𝑅) · 𝑅) mod 𝑃) ∈ ℤ ∧ (((-1↑𝑅) · 𝑄) · (2 · 𝑥)) ∈ ℤ) → ((𝐿‘(((-1↑𝑅) · 𝑅) mod 𝑃)) = (𝐿‘(((-1↑𝑅) · 𝑄) · (2 · 𝑥))) ↔ 𝑃 ∥ ((((-1↑𝑅) · 𝑅) mod 𝑃) − (((-1↑𝑅) · 𝑄) · (2 · 𝑥)))))
117	115, 109, 111, 116	syl3anc 1249	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝐿‘(((-1↑𝑅) · 𝑅) mod 𝑃)) = (𝐿‘(((-1↑𝑅) · 𝑄) · (2 · 𝑥))) ↔ 𝑃 ∥ ((((-1↑𝑅) · 𝑅) mod 𝑃) − (((-1↑𝑅) · 𝑄) · (2 · 𝑥)))))
118	114, 117	mpbird 167	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘(((-1↑𝑅) · 𝑅) mod 𝑃)) = (𝐿‘(((-1↑𝑅) · 𝑄) · (2 · 𝑥))))
119	23, 25	syl 14	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐿 ∈ (ℤ_ring RingHom 𝑌))
120	119	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝐿 ∈ (ℤ_ring RingHom 𝑌))
121		zringmulr 14065	. . . . . . . . . . 11 ⊢ · = (._r‘ℤ_ring)
122		eqid 2193	. . . . . . . . . . 11 ⊢ (._r‘𝑌) = (._r‘𝑌)
123	26, 121, 122	rhmmul 13644	. . . . . . . . . 10 ⊢ ((𝐿 ∈ (ℤ_ring RingHom 𝑌) ∧ ((-1↑𝑅) · 𝑄) ∈ ℤ ∧ (2 · 𝑥) ∈ ℤ) → (𝐿‘(((-1↑𝑅) · 𝑄) · (2 · 𝑥))) = ((𝐿‘((-1↑𝑅) · 𝑄))(._r‘𝑌)(𝐿‘(2 · 𝑥))))
124	120, 110, 67, 123	syl3anc 1249	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘(((-1↑𝑅) · 𝑄) · (2 · 𝑥))) = ((𝐿‘((-1↑𝑅) · 𝑄))(._r‘𝑌)(𝐿‘(2 · 𝑥))))
125	84, 118, 124	3eqtrd 2230	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘(2 · ((((-1↑𝑅) · 𝑅) mod 𝑃) / 2))) = ((𝐿‘((-1↑𝑅) · 𝑄))(._r‘𝑌)(𝐿‘(2 · 𝑥))))
126	125	mpteq2dva 4119	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · ((((-1↑𝑅) · 𝑅) mod 𝑃) / 2)))) = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ ((𝐿‘((-1↑𝑅) · 𝑄))(._r‘𝑌)(𝐿‘(2 · 𝑥)))))
127	16, 22	fzfigd 10492	. . . . . . . 8 ⊢ (𝜑 → (1...((𝑃 − 1) / 2)) ∈ Fin)
128	29	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝐿:ℤ⟶(Base‘𝑌))
129	128, 110	ffvelcdmd 5686	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘((-1↑𝑅) · 𝑄)) ∈ (Base‘𝑌))
130	128, 67	ffvelcdmd 5686	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘(2 · 𝑥)) ∈ (Base‘𝑌))
131		eqidd 2194	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))) = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))))
132		eqidd 2194	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥))) = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥))))
133	127, 129, 130, 131, 132	offval2 6138	. . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))) ∘_𝑓 (._r‘𝑌)(𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥)))) = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ ((𝐿‘((-1↑𝑅) · 𝑄))(._r‘𝑌)(𝐿‘(2 · 𝑥)))))
134	126, 133	eqtr4d 2229	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · ((((-1↑𝑅) · 𝑅) mod 𝑃) / 2)))) = ((𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))) ∘_𝑓 (._r‘𝑌)(𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥)))))
135	134	oveq2d 5926	. . . . 5 ⊢ (𝜑 → (𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · ((((-1↑𝑅) · 𝑅) mod 𝑃) / 2))))) = (𝐺 Σ_g ((𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))) ∘_𝑓 (._r‘𝑌)(𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥))))))
136	48, 57, 135	3eqtrd 2230	. . . 4 ⊢ (𝜑 → (𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥)))) = (𝐺 Σ_g ((𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))) ∘_𝑓 (._r‘𝑌)(𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥))))))
137		eqid 2193	. . . . . 6 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
138	38	eleq2d 2263	. . . . . . . 8 ⊢ (𝜑 → ((𝐿‘((-1↑𝑅) · 𝑄)) ∈ (Base‘𝑌) ↔ (𝐿‘((-1↑𝑅) · 𝑄)) ∈ (Base‘𝐺)))
139	138	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝐿‘((-1↑𝑅) · 𝑄)) ∈ (Base‘𝑌) ↔ (𝐿‘((-1↑𝑅) · 𝑄)) ∈ (Base‘𝐺)))
140	129, 139	mpbid 147	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘((-1↑𝑅) · 𝑄)) ∈ (Base‘𝐺))
141	38	eleq2d 2263	. . . . . . . 8 ⊢ (𝜑 → ((𝐿‘(2 · 𝑥)) ∈ (Base‘𝑌) ↔ (𝐿‘(2 · 𝑥)) ∈ (Base‘𝐺)))
142	141	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝐿‘(2 · 𝑥)) ∈ (Base‘𝑌) ↔ (𝐿‘(2 · 𝑥)) ∈ (Base‘𝐺)))
143	130, 142	mpbid 147	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘(2 · 𝑥)) ∈ (Base‘𝐺))
144		eqid 2193	. . . . . 6 ⊢ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))) = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄)))
145		eqid 2193	. . . . . 6 ⊢ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥))) = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥)))
146	5, 137, 15, 16, 22, 140, 143, 144, 145	gsumfzmptfidmadd2 13399	. . . . 5 ⊢ (𝜑 → (𝐺 Σ_g ((𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))) ∘_𝑓 (+_g‘𝐺)(𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥))))) = ((𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))))(+_g‘𝐺)(𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥))))))
147	13, 122	mgpplusgg 13404	. . . . . . . . 9 ⊢ (𝑌 ∈ CRing → (._r‘𝑌) = (+_g‘𝐺))
148	12, 147	syl 14	. . . . . . . 8 ⊢ (𝜑 → (._r‘𝑌) = (+_g‘𝐺))
149	148	ofeqd 6124	. . . . . . 7 ⊢ (𝜑 → ∘_𝑓 (._r‘𝑌) = ∘_𝑓 (+_g‘𝐺))
150	149	oveqd 5927	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))) ∘_𝑓 (._r‘𝑌)(𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥)))) = ((𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))) ∘_𝑓 (+_g‘𝐺)(𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥)))))
151	150	oveq2d 5926	. . . . 5 ⊢ (𝜑 → (𝐺 Σ_g ((𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))) ∘_𝑓 (._r‘𝑌)(𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥))))) = (𝐺 Σ_g ((𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))) ∘_𝑓 (+_g‘𝐺)(𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥))))))
152	148	oveqd 5927	. . . . 5 ⊢ (𝜑 → ((𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))))(._r‘𝑌)(𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥))))) = ((𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))))(+_g‘𝐺)(𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥))))))
153	146, 151, 152	3eqtr4d 2236	. . . 4 ⊢ (𝜑 → (𝐺 Σ_g ((𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))) ∘_𝑓 (._r‘𝑌)(𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥))))) = ((𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))))(._r‘𝑌)(𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥))))))
154	136, 153	eqtrd 2226	. . 3 ⊢ (𝜑 → (𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥)))) = ((𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))))(._r‘𝑌)(𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥))))))
155	154	oveq1d 5925	. 2 ⊢ (𝜑 → ((𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥))))(/_r‘𝑌)(𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥))))) = (((𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))))(._r‘𝑌)(𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥)))))(/_r‘𝑌)(𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥))))))
156	15	cmnmndd 13367	. . . 4 ⊢ (𝜑 → 𝐺 ∈ Mnd)
157		eqid 2193	. . . . . 6 ⊢ (Unit‘𝑌) = (Unit‘𝑌)
158	157, 13	unitsubm 13599	. . . . 5 ⊢ (𝑌 ∈ Ring → (Unit‘𝑌) ∈ (SubMnd‘𝐺))
159	23, 158	syl 14	. . . 4 ⊢ (𝜑 → (Unit‘𝑌) ∈ (SubMnd‘𝐺))
160		elfzle2 10084	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1...((𝑃 − 1) / 2)) → 𝑥 ≤ ((𝑃 − 1) / 2))
161	160	adantl 277	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑥 ≤ ((𝑃 − 1) / 2))
162	64	nnred 8985	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑥 ∈ ℝ)
163		prmuz2 12259	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ_≥‘2))
164		uz2m1nn 9660	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ (ℤ_≥‘2) → (𝑃 − 1) ∈ ℕ)
165	69, 163, 164	3syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑃 − 1) ∈ ℕ)
166	165	nnred 8985	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑃 − 1) ∈ ℝ)
167		2re 9042	. . . . . . . . . . 11 ⊢ 2 ∈ ℝ
168	167	a1i 9	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 2 ∈ ℝ)
169		2pos 9063	. . . . . . . . . . 11 ⊢ 0 < 2
170	169	a1i 9	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 0 < 2)
171		lemuldiv2 8891	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ (𝑃 − 1) ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → ((2 · 𝑥) ≤ (𝑃 − 1) ↔ 𝑥 ≤ ((𝑃 − 1) / 2)))
172	162, 166, 168, 170, 171	syl112anc 1253	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((2 · 𝑥) ≤ (𝑃 − 1) ↔ 𝑥 ≤ ((𝑃 − 1) / 2)))
173	161, 172	mpbird 167	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (2 · 𝑥) ≤ (𝑃 − 1))
174	69, 19	syl 14	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 ∈ ℤ)
175		peano2zm 9345	. . . . . . . . 9 ⊢ (𝑃 ∈ ℤ → (𝑃 − 1) ∈ ℤ)
176		fznn 10145	. . . . . . . . 9 ⊢ ((𝑃 − 1) ∈ ℤ → ((2 · 𝑥) ∈ (1...(𝑃 − 1)) ↔ ((2 · 𝑥) ∈ ℕ ∧ (2 · 𝑥) ≤ (𝑃 − 1))))
177	174, 175, 176	3syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((2 · 𝑥) ∈ (1...(𝑃 − 1)) ↔ ((2 · 𝑥) ∈ ℕ ∧ (2 · 𝑥) ≤ (𝑃 − 1))))
178	66, 173, 177	mpbir2and 946	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (2 · 𝑥) ∈ (1...(𝑃 − 1)))
179		fzm1ndvds 11988	. . . . . . 7 ⊢ ((𝑃 ∈ ℕ ∧ (2 · 𝑥) ∈ (1...(𝑃 − 1))) → ¬ 𝑃 ∥ (2 · 𝑥))
180	71, 178, 179	syl2anc 411	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ¬ 𝑃 ∥ (2 · 𝑥))
181	9, 157, 24	znunit 14124	. . . . . . . 8 ⊢ ((𝑃 ∈ ℕ₀ ∧ (2 · 𝑥) ∈ ℤ) → ((𝐿‘(2 · 𝑥)) ∈ (Unit‘𝑌) ↔ ((2 · 𝑥) gcd 𝑃) = 1))
182	115, 67, 181	syl2anc 411	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝐿‘(2 · 𝑥)) ∈ (Unit‘𝑌) ↔ ((2 · 𝑥) gcd 𝑃) = 1))
183		coprm 12272	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℙ ∧ (2 · 𝑥) ∈ ℤ) → (¬ 𝑃 ∥ (2 · 𝑥) ↔ (𝑃 gcd (2 · 𝑥)) = 1))
184	19	adantr 276	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ ℙ ∧ (2 · 𝑥) ∈ ℤ) → 𝑃 ∈ ℤ)
185		simpr 110	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ ℙ ∧ (2 · 𝑥) ∈ ℤ) → (2 · 𝑥) ∈ ℤ)
186	184, 185	gcdcomd 12101	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ (2 · 𝑥) ∈ ℤ) → (𝑃 gcd (2 · 𝑥)) = ((2 · 𝑥) gcd 𝑃))
187	186	eqeq1d 2202	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℙ ∧ (2 · 𝑥) ∈ ℤ) → ((𝑃 gcd (2 · 𝑥)) = 1 ↔ ((2 · 𝑥) gcd 𝑃) = 1))
188	183, 187	bitrd 188	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ (2 · 𝑥) ∈ ℤ) → (¬ 𝑃 ∥ (2 · 𝑥) ↔ ((2 · 𝑥) gcd 𝑃) = 1))
189	69, 67, 188	syl2anc 411	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (¬ 𝑃 ∥ (2 · 𝑥) ↔ ((2 · 𝑥) gcd 𝑃) = 1))
190	182, 189	bitr4d 191	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝐿‘(2 · 𝑥)) ∈ (Unit‘𝑌) ↔ ¬ 𝑃 ∥ (2 · 𝑥)))
191	180, 190	mpbird 167	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘(2 · 𝑥)) ∈ (Unit‘𝑌))
192	191	fmpttd 5705	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥))):(1...((𝑃 − 1) / 2))⟶(Unit‘𝑌))
193	156, 16, 22, 159, 192	gsumfzsubmcl 13397	. . 3 ⊢ (𝜑 → (𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥)))) ∈ (Unit‘𝑌))
194		eqid 2193	. . . 4 ⊢ (/_r‘𝑌) = (/_r‘𝑌)
195		eqid 2193	. . . 4 ⊢ (1_r‘𝑌) = (1_r‘𝑌)
196	157, 194, 195	dvrid 13617	. . 3 ⊢ ((𝑌 ∈ Ring ∧ (𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥)))) ∈ (Unit‘𝑌)) → ((𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥))))(/_r‘𝑌)(𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥))))) = (1_r‘𝑌))
197	23, 193, 196	syl2anc 411	. 2 ⊢ (𝜑 → ((𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥))))(/_r‘𝑌)(𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥))))) = (1_r‘𝑌))
198	129	fmpttd 5705	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))):(1...((𝑃 − 1) / 2))⟶(Base‘𝑌))
199	38	feq3d 5384	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))):(1...((𝑃 − 1) / 2))⟶(Base‘𝑌) ↔ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))):(1...((𝑃 − 1) / 2))⟶(Base‘𝐺)))
200	198, 199	mpbid 147	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))):(1...((𝑃 − 1) / 2))⟶(Base‘𝐺))
201	5, 6, 156, 16, 22, 200	gsumfzcl 13061	. . . 4 ⊢ (𝜑 → (𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄)))) ∈ (Base‘𝐺))
202	201, 38	eleqtrrd 2273	. . 3 ⊢ (𝜑 → (𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄)))) ∈ (Base‘𝑌))
203	27, 157, 194, 122	dvrcan3 13621	. . 3 ⊢ ((𝑌 ∈ Ring ∧ (𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄)))) ∈ (Base‘𝑌) ∧ (𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥)))) ∈ (Unit‘𝑌)) → (((𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))))(._r‘𝑌)(𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥)))))(/_r‘𝑌)(𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥))))) = (𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄)))))
204	23, 202, 193, 203	syl3anc 1249	. 2 ⊢ (𝜑 → (((𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))))(._r‘𝑌)(𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥)))))(/_r‘𝑌)(𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥))))) = (𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄)))))
205	155, 197, 204	3eqtr3rd 2235	1 ⊢ (𝜑 → (𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄)))) = (1_r‘𝑌))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1364 ∈ wcel 2164 ≠ wne 2364 ∀wral 2472 ∖ cdif 3150 {csn 3618 class class class wbr 4029 ↦ cmpt 4090 ∘ ccom 4659 ⟶wf 5242 ‘cfv 5246 (class class class)co 5910 ∘_𝑓 cof 6120 Fincfn 6785 ℝcr 7861 0cc0 7862 1c1 7863 · cmul 7867 < clt 8044 ≤ cle 8045 − cmin 8180 -cneg 8181 # cap 8590 / cdiv 8681 ℕcn 8972 2c2 9023 ℕ₀cn0 9230 ℤcz 9307 ℤ_≥cuz 9582 ℚcq 9674 ...cfz 10064 mod cmo 10383 ↑cexp 10599 ∥ cdvds 11920 gcd cgcd 12069 ℙcprime 12235 Basecbs 12608 +_gcplusg 12685 ._rcmulr 12686 0_gc0g 12857 Σ_g cgsu 12858 SubMndcsubmnd 13020 CMndccmn 13343 mulGrpcmgp 13400 1_rcur 13439 Ringcrg 13476 CRingccrg 13477 Unitcui 13567 /_rcdvr 13611 RingHom crh 13630 IDomncidom 13737 ℤ_ringczring 14056 ℤRHomczrh 14076 ℤ/nℤczn 14078

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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4144 ax-sep 4147 ax-nul 4155 ax-pow 4203 ax-pr 4238 ax-un 4462 ax-setind 4565 ax-iinf 4616 ax-cnex 7953 ax-resscn 7954 ax-1cn 7955 ax-1re 7956 ax-icn 7957 ax-addcl 7958 ax-addrcl 7959 ax-mulcl 7960 ax-mulrcl 7961 ax-addcom 7962 ax-mulcom 7963 ax-addass 7964 ax-mulass 7965 ax-distr 7966 ax-i2m1 7967 ax-0lt1 7968 ax-1rid 7969 ax-0id 7970 ax-rnegex 7971 ax-precex 7972 ax-cnre 7973 ax-pre-ltirr 7974 ax-pre-ltwlin 7975 ax-pre-lttrn 7976 ax-pre-apti 7977 ax-pre-ltadd 7978 ax-pre-mulgt0 7979 ax-pre-mulext 7980 ax-arch 7981 ax-caucvg 7982 ax-addf 7984 ax-mulf 7985

This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-xor 1387 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-nul 3447 df-if 3558 df-pw 3603 df-sn 3624 df-pr 3625 df-tp 3626 df-op 3627 df-uni 3836 df-int 3871 df-iun 3914 df-br 4030 df-opab 4091 df-mpt 4092 df-tr 4128 df-id 4322 df-po 4325 df-iso 4326 df-iord 4395 df-on 4397 df-ilim 4398 df-suc 4400 df-iom 4619 df-xp 4661 df-rel 4662 df-cnv 4663 df-co 4664 df-dm 4665 df-rn 4666 df-res 4667 df-ima 4668 df-iota 5207 df-fun 5248 df-fn 5249 df-f 5250 df-f1 5251 df-fo 5252 df-f1o 5253 df-fv 5254 df-riota 5865 df-ov 5913 df-oprab 5914 df-mpo 5915 df-of 6122 df-1st 6184 df-2nd 6185 df-tpos 6289 df-recs 6349 df-irdg 6414 df-frec 6435 df-1o 6460 df-2o 6461 df-oadd 6464 df-er 6578 df-ec 6580 df-qs 6584 df-map 6695 df-en 6786 df-dom 6787 df-fin 6788 df-sup 7033 df-pnf 8046 df-mnf 8047 df-xr 8048 df-ltxr 8049 df-le 8050 df-sub 8182 df-neg 8183 df-reap 8584 df-ap 8591 df-div 8682 df-inn 8973 df-2 9031 df-3 9032 df-4 9033 df-5 9034 df-6 9035 df-7 9036 df-8 9037 df-9 9038 df-n0 9231 df-z 9308 df-dec 9439 df-uz 9583 df-q 9675 df-rp 9710 df-fz 10065 df-fzo 10199 df-fl 10329 df-mod 10384 df-seqfrec 10509 df-exp 10600 df-ihash 10837 df-cj 10976 df-re 10977 df-im 10978 df-rsqrt 11132 df-abs 11133 df-dvds 11921 df-gcd 12070 df-prm 12236 df-struct 12610 df-ndx 12611 df-slot 12612 df-base 12614 df-sets 12615 df-iress 12616 df-plusg 12698 df-mulr 12699 df-starv 12700 df-sca 12701 df-vsca 12702 df-ip 12703 df-ple 12705 df-0g 12859 df-igsum 12860 df-iimas 12875 df-qus 12876 df-mgm 12929 df-sgrp 12975 df-mnd 12988 df-mhm 13021 df-submnd 13022 df-grp 13065 df-minusg 13066 df-sbg 13067 df-mulg 13180 df-subg 13229 df-nsg 13230 df-eqg 13231 df-ghm 13300 df-cmn 13345 df-abl 13346 df-mgp 13401 df-rng 13413 df-ur 13440 df-srg 13444 df-ring 13478 df-cring 13479 df-oppr 13548 df-dvdsr 13569 df-unit 13570 df-invr 13601 df-dvr 13612 df-rhm 13632 df-nzr 13660 df-subrg 13699 df-domn 13739 df-idom 13740 df-lmod 13769 df-lssm 13833 df-lsp 13867 df-sra 13915 df-rgmod 13916 df-lidl 13949 df-rsp 13950 df-2idl 13980 df-icnfld 14032 df-zring 14057 df-zrh 14079 df-zn 14081

This theorem is referenced by: (None)

W3C validator