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Theorem lgseisenlem3 15791

Description: Lemma for lgseisen 15793. (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 6021	. . . . . . . . 9 ⊢ (𝑘 = 𝑥 → (2 · 𝑘) = (2 · 𝑥))
2	1	fveq2d 5639	. . . . . . . 8 ⊢ (𝑘 = 𝑥 → (𝐿‘(2 · 𝑘)) = (𝐿‘(2 · 𝑥)))
3	2	cbvmptv 4183	. . . . . . 7 ⊢ (𝑘 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑘))) = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥)))
4	3	oveq2i 6024	. . . . . 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 2229	. . . . . . 7 ⊢ (Base‘𝐺) = (Base‘𝐺)
6		eqid 2229	. . . . . . 7 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
7		lgseisen.1	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))
8	7	eldifad 3209	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ ℙ)
9		lgseisen.7	. . . . . . . . . . 11 ⊢ 𝑌 = (ℤ/nℤ‘𝑃)
10	9	znidom 14661	. . . . . . . . . 10 ⊢ (𝑃 ∈ ℙ → 𝑌 ∈ IDomn)
11	8, 10	syl 14	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ IDomn)
12	11	idomcringd 14282	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ CRing)
13		lgseisen.8	. . . . . . . . 9 ⊢ 𝐺 = (mulGrp‘𝑌)
14	13	crngmgp 14007	. . . . . . . 8 ⊢ (𝑌 ∈ CRing → 𝐺 ∈ CMnd)
15	12, 14	syl 14	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ CMnd)
16		1zzd 9496	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℤ)
17		oddn2prm 12824	. . . . . . . . 9 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ¬ 2 ∥ 𝑃)
18	7, 17	syl 14	. . . . . . . 8 ⊢ (𝜑 → ¬ 2 ∥ 𝑃)
19		prmz 12673	. . . . . . . . 9 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
20		oddm1d2 12443	. . . . . . . . 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 14283	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑌 ∈ Ring)
24		lgseisen.9	. . . . . . . . . . . 12 ⊢ 𝐿 = (ℤRHom‘𝑌)
25	24	zrhrhm 14627	. . . . . . . . . . 11 ⊢ (𝑌 ∈ Ring → 𝐿 ∈ (ℤ_ring RingHom 𝑌))
26		zringbas 14600	. . . . . . . . . . . 12 ⊢ ℤ = (Base‘ℤ_ring)
27		eqid 2229	. . . . . . . . . . . 12 ⊢ (Base‘𝑌) = (Base‘𝑌)
28	26, 27	rhmf 14167	. . . . . . . . . . 11 ⊢ (𝐿 ∈ (ℤ_ring RingHom 𝑌) → 𝐿:ℤ⟶(Base‘𝑌))
29	23, 25, 28	3syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐿:ℤ⟶(Base‘𝑌))
30		2z 9497	. . . . . . . . . . 11 ⊢ 2 ∈ ℤ
31		elfzelz 10250	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (1...((𝑃 − 1) / 2)) → 𝑘 ∈ ℤ)
32		zmulcl 9523	. . . . . . . . . . 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 5776	. . . . . . . . . 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 5798	. . . . . . . 8 ⊢ (𝜑 → (𝑘 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑘))):(1...((𝑃 − 1) / 2))⟶(Base‘𝑌))
37	13, 27	mgpbasg 13929	. . . . . . . . . 10 ⊢ (𝑌 ∈ CRing → (Base‘𝑌) = (Base‘𝐺))
38	12, 37	syl 14	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝑌) = (Base‘𝐺))
39	38	feq3d 5468	. . . . . . . 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 15790	. . . . . . 7 ⊢ (𝜑 → 𝑀:(1...((𝑃 − 1) / 2))–1-1-onto→(1...((𝑃 − 1) / 2)))
47	5, 6, 15, 16, 22, 40, 46	gsumfzreidx 13914	. . . . . 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 2276	. . . . 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 15789	. . . . . . . 8 ⊢ (𝜑 → 𝑀:(1...((𝑃 − 1) / 2))⟶(1...((𝑃 − 1) / 2)))
50	44	fmpt 5793	. . . . . . . 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 2230	. . . . . . 7 ⊢ (𝜑 → (𝑘 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑘))) = (𝑘 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑘))))
54		oveq2 6021	. . . . . . . 8 ⊢ (𝑘 = ((((-1↑𝑅) · 𝑅) mod 𝑃) / 2) → (2 · 𝑘) = (2 · ((((-1↑𝑅) · 𝑅) mod 𝑃) / 2)))
55	54	fveq2d 5639	. . . . . . 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 5810	. . . . . 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 6029	. . . . 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 3209	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑄 ∈ ℙ)
59	58	adantr 276	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑄 ∈ ℙ)
60		prmz 12673	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑄 ∈ ℙ → 𝑄 ∈ ℤ)
61	59, 60	syl 14	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑄 ∈ ℤ)
62		2nn 9295	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 2 ∈ ℕ
63		elfznn 10279	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ (1...((𝑃 − 1) / 2)) → 𝑥 ∈ ℕ)
64	63	adantl 277	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑥 ∈ ℕ)
65		nnmulcl 9154	. . . . . . . . . . . . . . . . . . . . 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 9591	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (2 · 𝑥) ∈ ℤ)
68	61, 67	zmulcld 9598	. . . . . . . . . . . . . . . . . 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 12672	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
71	69, 70	syl 14	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 ∈ ℕ)
72	68, 71	zmodcld 10597	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝑄 · (2 · 𝑥)) mod 𝑃) ∈ ℕ₀)
73	43, 72	eqeltrid 2316	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑅 ∈ ℕ₀)
74	73	nn0zd 9590	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑅 ∈ ℤ)
75		m1expcl 10814	. . . . . . . . . . . . . . 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 9598	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((-1↑𝑅) · 𝑅) ∈ ℤ)
78	77, 71	zmodcld 10597	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (((-1↑𝑅) · 𝑅) mod 𝑃) ∈ ℕ₀)
79	78	nn0cnd 9447	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (((-1↑𝑅) · 𝑅) mod 𝑃) ∈ ℂ)
80		2cnd 9206	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 2 ∈ ℂ)
81		2ap0 9226	. . . . . . . . . . . 12 ⊢ 2 # 0
82	81	a1i 9	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 2 # 0)
83	79, 80, 82	divcanap2d 8962	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (2 · ((((-1↑𝑅) · 𝑅) mod 𝑃) / 2)) = (((-1↑𝑅) · 𝑅) mod 𝑃))
84	83	fveq2d 5639	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘(2 · ((((-1↑𝑅) · 𝑅) mod 𝑃) / 2))) = (𝐿‘(((-1↑𝑅) · 𝑅) mod 𝑃)))
85		zq 9850	. . . . . . . . . . . . . . 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 9177	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 0 < 𝑃)
89		eqidd 2230	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((-1↑𝑅) mod 𝑃) = ((-1↑𝑅) mod 𝑃))
90	43	oveq1i 6023	. . . . . . . . . . . . . 14 ⊢ (𝑅 mod 𝑃) = (((𝑄 · (2 · 𝑥)) mod 𝑃) mod 𝑃)
91		zq 9850	. . . . . . . . . . . . . . . 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 10610	. . . . . . . . . . . . . . 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 1271	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (((𝑄 · (2 · 𝑥)) mod 𝑃) mod 𝑃) = ((𝑄 · (2 · 𝑥)) mod 𝑃))
95	90, 94	eqtrid 2274	. . . . . . . . . . . . 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 10630	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (((-1↑𝑅) · 𝑅) mod 𝑃) = (((-1↑𝑅) · (𝑄 · (2 · 𝑥))) mod 𝑃))
97		zq 9850	. . . . . . . . . . . . . 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 10610	. . . . . . . . . . . . 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 1271	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((((-1↑𝑅) · 𝑅) mod 𝑃) mod 𝑃) = (((-1↑𝑅) · 𝑅) mod 𝑃))
101	76	zcnd 9593	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (-1↑𝑅) ∈ ℂ)
102	61	zcnd 9593	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑄 ∈ ℂ)
103	67	zcnd 9593	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (2 · 𝑥) ∈ ℂ)
104	101, 102, 103	mulassd 8193	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (((-1↑𝑅) · 𝑄) · (2 · 𝑥)) = ((-1↑𝑅) · (𝑄 · (2 · 𝑥))))
105	104	oveq1d 6028	. . . . . . . . . . . 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 2272	. . . . . . . . . . 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 9590	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (((-1↑𝑅) · 𝑅) mod 𝑃) ∈ ℤ)
110	76, 61	zmulcld 9598	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((-1↑𝑅) · 𝑄) ∈ ℤ)
111	110, 67	zmulcld 9598	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (((-1↑𝑅) · 𝑄) · (2 · 𝑥)) ∈ ℤ)
112		moddvds 12350	. . . . . . . . . . . 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 1271	. . . . . . . . . . 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 9445	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 ∈ ℕ₀)
116	9, 24	zndvds 14653	. . . . . . . . . . 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 1271	. . . . . . . . . 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 14603	. . . . . . . . . . 11 ⊢ · = (._r‘ℤ_ring)
122		eqid 2229	. . . . . . . . . . 11 ⊢ (._r‘𝑌) = (._r‘𝑌)
123	26, 121, 122	rhmmul 14168	. . . . . . . . . 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 1271	. . . . . . . . 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 2266	. . . . . . . 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 4177	. . . . . . 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 10683	. . . . . . . 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 5779	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘((-1↑𝑅) · 𝑄)) ∈ (Base‘𝑌))
130	128, 67	ffvelcdmd 5779	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘(2 · 𝑥)) ∈ (Base‘𝑌))
131		eqidd 2230	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))) = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))))
132		eqidd 2230	. . . . . . . 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 6246	. . . . . . 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 2265	. . . . . 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 6029	. . . . 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 2266	. . . 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 2229	. . . . . 6 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
138	38	eleq2d 2299	. . . . . . . 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 2299	. . . . . . . 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 2229	. . . . . 6 ⊢ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))) = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄)))
145		eqid 2229	. . . . . 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 13917	. . . . 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 13927	. . . . . . . . 9 ⊢ (𝑌 ∈ CRing → (._r‘𝑌) = (+_g‘𝐺))
148	12, 147	syl 14	. . . . . . . 8 ⊢ (𝜑 → (._r‘𝑌) = (+_g‘𝐺))
149	148	ofeqd 6232	. . . . . . 7 ⊢ (𝜑 → ∘_𝑓 (._r‘𝑌) = ∘_𝑓 (+_g‘𝐺))
150	149	oveqd 6030	. . . . . 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 6029	. . . . 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 6030	. . . . 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 2272	. . . 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 2262	. . 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 6028	. 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 13885	. . . 4 ⊢ (𝜑 → 𝐺 ∈ Mnd)
157		eqid 2229	. . . . . 6 ⊢ (Unit‘𝑌) = (Unit‘𝑌)
158	157, 13	unitsubm 14123	. . . . 5 ⊢ (𝑌 ∈ Ring → (Unit‘𝑌) ∈ (SubMnd‘𝐺))
159	23, 158	syl 14	. . . 4 ⊢ (𝜑 → (Unit‘𝑌) ∈ (SubMnd‘𝐺))
160		elfzle2 10253	. . . . . . . . . 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 9146	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑥 ∈ ℝ)
163		prmuz2 12693	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ_≥‘2))
164		uz2m1nn 9829	. . . . . . . . . . . 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 9146	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑃 − 1) ∈ ℝ)
167		2re 9203	. . . . . . . . . . 11 ⊢ 2 ∈ ℝ
168	167	a1i 9	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 2 ∈ ℝ)
169		2pos 9224	. . . . . . . . . . 11 ⊢ 0 < 2
170	169	a1i 9	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 0 < 2)
171		lemuldiv2 9052	. . . . . . . . . 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 1275	. . . . . . . . 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 9507	. . . . . . . . 9 ⊢ (𝑃 ∈ ℤ → (𝑃 − 1) ∈ ℤ)
176		fznn 10314	. . . . . . . . 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 950	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (2 · 𝑥) ∈ (1...(𝑃 − 1)))
179		fzm1ndvds 12407	. . . . . . 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 14663	. . . . . . . 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 12706	. . . . . . . . 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 12535	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ (2 · 𝑥) ∈ ℤ) → (𝑃 gcd (2 · 𝑥)) = ((2 · 𝑥) gcd 𝑃))
187	186	eqeq1d 2238	. . . . . . . . 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 5798	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥))):(1...((𝑃 − 1) / 2))⟶(Unit‘𝑌))
193	156, 16, 22, 159, 192	gsumfzsubmcl 13915	. . 3 ⊢ (𝜑 → (𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥)))) ∈ (Unit‘𝑌))
194		eqid 2229	. . . 4 ⊢ (/_r‘𝑌) = (/_r‘𝑌)
195		eqid 2229	. . . 4 ⊢ (1_r‘𝑌) = (1_r‘𝑌)
196	157, 194, 195	dvrid 14141	. . 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 5798	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))):(1...((𝑃 − 1) / 2))⟶(Base‘𝑌))
199	38	feq3d 5468	. . . . . 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 13572	. . . 4 ⊢ (𝜑 → (𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄)))) ∈ (Base‘𝐺))
202	201, 38	eleqtrrd 2309	. . 3 ⊢ (𝜑 → (𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄)))) ∈ (Base‘𝑌))
203	27, 157, 194, 122	dvrcan3 14145	. . 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 1271	. 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 2271	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 1395 ∈ wcel 2200 ≠ wne 2400 ∀wral 2508 ∖ cdif 3195 {csn 3667 class class class wbr 4086 ↦ cmpt 4148 ∘ ccom 4727 ⟶wf 5320 ‘cfv 5324 (class class class)co 6013 ∘_𝑓 cof 6228 Fincfn 6904 ℝcr 8021 0cc0 8022 1c1 8023 · cmul 8027 < clt 8204 ≤ cle 8205 − cmin 8340 -cneg 8341 # cap 8751 / cdiv 8842 ℕcn 9133 2c2 9184 ℕ₀cn0 9392 ℤcz 9469 ℤ_≥cuz 9745 ℚcq 9843 ...cfz 10233 mod cmo 10574 ↑cexp 10790 ∥ cdvds 12338 gcd cgcd 12514 ℙcprime 12669 Basecbs 13072 +_gcplusg 13150 ._rcmulr 13151 0_gc0g 13329 Σ_g cgsu 13330 SubMndcsubmnd 13531 CMndccmn 13861 mulGrpcmgp 13923 1_rcur 13962 Ringcrg 13999 CRingccrg 14000 Unitcui 14090 /_rcdvr 14135 RingHom crh 14154 IDomncidom 14261 ℤ_ringczring 14594 ℤRHomczrh 14615 ℤ/nℤczn 14617

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 8113 ax-resscn 8114 ax-1cn 8115 ax-1re 8116 ax-icn 8117 ax-addcl 8118 ax-addrcl 8119 ax-mulcl 8120 ax-mulrcl 8121 ax-addcom 8122 ax-mulcom 8123 ax-addass 8124 ax-mulass 8125 ax-distr 8126 ax-i2m1 8127 ax-0lt1 8128 ax-1rid 8129 ax-0id 8130 ax-rnegex 8131 ax-precex 8132 ax-cnre 8133 ax-pre-ltirr 8134 ax-pre-ltwlin 8135 ax-pre-lttrn 8136 ax-pre-apti 8137 ax-pre-ltadd 8138 ax-pre-mulgt0 8139 ax-pre-mulext 8140 ax-arch 8141 ax-caucvg 8142 ax-addf 8144 ax-mulf 8145

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-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 7174 df-pnf 8206 df-mnf 8207 df-xr 8208 df-ltxr 8209 df-le 8210 df-sub 8342 df-neg 8343 df-reap 8745 df-ap 8752 df-div 8843 df-inn 9134 df-2 9192 df-3 9193 df-4 9194 df-5 9195 df-6 9196 df-7 9197 df-8 9198 df-9 9199 df-n0 9393 df-z 9470 df-dec 9602 df-uz 9746 df-q 9844 df-rp 9879 df-fz 10234 df-fzo 10368 df-fl 10520 df-mod 10575 df-seqfrec 10700 df-exp 10791 df-ihash 11028 df-cj 11393 df-re 11394 df-im 11395 df-rsqrt 11549 df-abs 11550 df-dvds 12339 df-gcd 12515 df-prm 12670 df-struct 13074 df-ndx 13075 df-slot 13076 df-base 13078 df-sets 13079 df-iress 13080 df-plusg 13163 df-mulr 13164 df-starv 13165 df-sca 13166 df-vsca 13167 df-ip 13168 df-tset 13169 df-ple 13170 df-ds 13172 df-unif 13173 df-0g 13331 df-igsum 13332 df-topgen 13333 df-iimas 13375 df-qus 13376 df-mgm 13429 df-sgrp 13475 df-mnd 13490 df-mhm 13532 df-submnd 13533 df-grp 13576 df-minusg 13577 df-sbg 13578 df-mulg 13697 df-subg 13747 df-nsg 13748 df-eqg 13749 df-ghm 13818 df-cmn 13863 df-abl 13864 df-mgp 13924 df-rng 13936 df-ur 13963 df-srg 13967 df-ring 14001 df-cring 14002 df-oppr 14071 df-dvdsr 14092 df-unit 14093 df-invr 14125 df-dvr 14136 df-rhm 14156 df-nzr 14184 df-subrg 14223 df-domn 14263 df-idom 14264 df-lmod 14293 df-lssm 14357 df-lsp 14391 df-sra 14439 df-rgmod 14440 df-lidl 14473 df-rsp 14474 df-2idl 14504 df-bl 14550 df-mopn 14551 df-fg 14553 df-metu 14554 df-cnfld 14561 df-zring 14595 df-zrh 14618 df-zn 14620

This theorem is referenced by: lgseisenlem4 15792

W3C validator