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Theorem lgseisenlem3 26430

Description: Lemma for lgseisen 26432. (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 7263	. . . . . . . . 9 ⊢ (𝑘 = 𝑥 → (2 · 𝑘) = (2 · 𝑥))
2	1	fveq2d 6760	. . . . . . . 8 ⊢ (𝑘 = 𝑥 → (𝐿‘(2 · 𝑘)) = (𝐿‘(2 · 𝑥)))
3	2	cbvmptv 5183	. . . . . . 7 ⊢ (𝑘 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑘))) = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥)))
4	3	oveq2i 7266	. . . . . 6 ⊢ (𝐺 Σ_g (𝑘 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑘)))) = (𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥))))
5		lgseisen.8	. . . . . . . 8 ⊢ 𝐺 = (mulGrp‘𝑌)
6		eqid 2738	. . . . . . . 8 ⊢ (Base‘𝑌) = (Base‘𝑌)
7	5, 6	mgpbas 19641	. . . . . . 7 ⊢ (Base‘𝑌) = (Base‘𝐺)
8		eqid 2738	. . . . . . 7 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
9		lgseisen.1	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))
10	9	eldifad 3895	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ ℙ)
11		lgseisen.7	. . . . . . . . . . 11 ⊢ 𝑌 = (ℤ/nℤ‘𝑃)
12	11	znfld 20680	. . . . . . . . . 10 ⊢ (𝑃 ∈ ℙ → 𝑌 ∈ Field)
13	10, 12	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ Field)
14		isfld 19915	. . . . . . . . . 10 ⊢ (𝑌 ∈ Field ↔ (𝑌 ∈ DivRing ∧ 𝑌 ∈ CRing))
15	14	simprbi 496	. . . . . . . . 9 ⊢ (𝑌 ∈ Field → 𝑌 ∈ CRing)
16	13, 15	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ CRing)
17	5	crngmgp 19706	. . . . . . . 8 ⊢ (𝑌 ∈ CRing → 𝐺 ∈ CMnd)
18	16, 17	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ CMnd)
19		fzfid 13621	. . . . . . 7 ⊢ (𝜑 → (1...((𝑃 − 1) / 2)) ∈ Fin)
20		crngring 19710	. . . . . . . . . . . 12 ⊢ (𝑌 ∈ CRing → 𝑌 ∈ Ring)
21	16, 20	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑌 ∈ Ring)
22		lgseisen.9	. . . . . . . . . . . 12 ⊢ 𝐿 = (ℤRHom‘𝑌)
23	22	zrhrhm 20625	. . . . . . . . . . 11 ⊢ (𝑌 ∈ Ring → 𝐿 ∈ (ℤ_ring RingHom 𝑌))
24	21, 23	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐿 ∈ (ℤ_ring RingHom 𝑌))
25		zringbas 20588	. . . . . . . . . . 11 ⊢ ℤ = (Base‘ℤ_ring)
26	25, 6	rhmf 19885	. . . . . . . . . 10 ⊢ (𝐿 ∈ (ℤ_ring RingHom 𝑌) → 𝐿:ℤ⟶(Base‘𝑌))
27	24, 26	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐿:ℤ⟶(Base‘𝑌))
28		2z 12282	. . . . . . . . . 10 ⊢ 2 ∈ ℤ
29		elfzelz 13185	. . . . . . . . . 10 ⊢ (𝑘 ∈ (1...((𝑃 − 1) / 2)) → 𝑘 ∈ ℤ)
30		zmulcl 12299	. . . . . . . . . 10 ⊢ ((2 ∈ ℤ ∧ 𝑘 ∈ ℤ) → (2 · 𝑘) ∈ ℤ)
31	28, 29, 30	sylancr 586	. . . . . . . . 9 ⊢ (𝑘 ∈ (1...((𝑃 − 1) / 2)) → (2 · 𝑘) ∈ ℤ)
32		ffvelrn 6941	. . . . . . . . 9 ⊢ ((𝐿:ℤ⟶(Base‘𝑌) ∧ (2 · 𝑘) ∈ ℤ) → (𝐿‘(2 · 𝑘)) ∈ (Base‘𝑌))
33	27, 31, 32	syl2an 595	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘(2 · 𝑘)) ∈ (Base‘𝑌))
34	33	fmpttd 6971	. . . . . . 7 ⊢ (𝜑 → (𝑘 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑘))):(1...((𝑃 − 1) / 2))⟶(Base‘𝑌))
35		eqid 2738	. . . . . . . 8 ⊢ (𝑘 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑘))) = (𝑘 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑘)))
36		fvexd 6771	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘(2 · 𝑘)) ∈ V)
37		fvexd 6771	. . . . . . . 8 ⊢ (𝜑 → (0_g‘𝐺) ∈ V)
38	35, 19, 36, 37	fsuppmptdm 9069	. . . . . . 7 ⊢ (𝜑 → (𝑘 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑘))) finSupp (0_g‘𝐺))
39		lgseisen.2	. . . . . . . 8 ⊢ (𝜑 → 𝑄 ∈ (ℙ ∖ {2}))
40		lgseisen.3	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ≠ 𝑄)
41		lgseisen.4	. . . . . . . 8 ⊢ 𝑅 = ((𝑄 · (2 · 𝑥)) mod 𝑃)
42		lgseisen.5	. . . . . . . 8 ⊢ 𝑀 = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ ((((-1↑𝑅) · 𝑅) mod 𝑃) / 2))
43		lgseisen.6	. . . . . . . 8 ⊢ 𝑆 = ((𝑄 · (2 · 𝑦)) mod 𝑃)
44	9, 39, 40, 41, 42, 43	lgseisenlem2 26429	. . . . . . 7 ⊢ (𝜑 → 𝑀:(1...((𝑃 − 1) / 2))–1-1-onto→(1...((𝑃 − 1) / 2)))
45	7, 8, 18, 19, 34, 38, 44	gsumf1o 19432	. . . . . 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)))
46	4, 45	eqtr3id 2793	. . . . 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)))
47	9, 39, 40, 41, 42	lgseisenlem1 26428	. . . . . . . 8 ⊢ (𝜑 → 𝑀:(1...((𝑃 − 1) / 2))⟶(1...((𝑃 − 1) / 2)))
48	42	fmpt 6966	. . . . . . . 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)))
49	47, 48	sylibr 233	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ (1...((𝑃 − 1) / 2))((((-1↑𝑅) · 𝑅) mod 𝑃) / 2) ∈ (1...((𝑃 − 1) / 2)))
50	42	a1i 11	. . . . . . 7 ⊢ (𝜑 → 𝑀 = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ ((((-1↑𝑅) · 𝑅) mod 𝑃) / 2)))
51		eqidd 2739	. . . . . . 7 ⊢ (𝜑 → (𝑘 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑘))) = (𝑘 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑘))))
52		oveq2 7263	. . . . . . . 8 ⊢ (𝑘 = ((((-1↑𝑅) · 𝑅) mod 𝑃) / 2) → (2 · 𝑘) = (2 · ((((-1↑𝑅) · 𝑅) mod 𝑃) / 2)))
53	52	fveq2d 6760	. . . . . . 7 ⊢ (𝑘 = ((((-1↑𝑅) · 𝑅) mod 𝑃) / 2) → (𝐿‘(2 · 𝑘)) = (𝐿‘(2 · ((((-1↑𝑅) · 𝑅) mod 𝑃) / 2))))
54	49, 50, 51, 53	fmptcof 6984	. . . . . 6 ⊢ (𝜑 → ((𝑘 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑘))) ∘ 𝑀) = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · ((((-1↑𝑅) · 𝑅) mod 𝑃) / 2)))))
55	54	oveq2d 7271	. . . . 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))))))
56	39	eldifad 3895	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑄 ∈ ℙ)
57	56	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑄 ∈ ℙ)
58		prmz 16308	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑄 ∈ ℙ → 𝑄 ∈ ℤ)
59	57, 58	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑄 ∈ ℤ)
60		2nn 11976	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 2 ∈ ℕ
61		elfznn 13214	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ (1...((𝑃 − 1) / 2)) → 𝑥 ∈ ℕ)
62	61	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑥 ∈ ℕ)
63		nnmulcl 11927	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((2 ∈ ℕ ∧ 𝑥 ∈ ℕ) → (2 · 𝑥) ∈ ℕ)
64	60, 62, 63	sylancr 586	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (2 · 𝑥) ∈ ℕ)
65	64	nnzd 12354	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (2 · 𝑥) ∈ ℤ)
66	59, 65	zmulcld 12361	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑄 · (2 · 𝑥)) ∈ ℤ)
67	10	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 ∈ ℙ)
68		prmnn 16307	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
69	67, 68	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 ∈ ℕ)
70	66, 69	zmodcld 13540	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝑄 · (2 · 𝑥)) mod 𝑃) ∈ ℕ₀)
71	41, 70	eqeltrid 2843	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑅 ∈ ℕ₀)
72	71	nn0zd 12353	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑅 ∈ ℤ)
73		m1expcl 13733	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ ℤ → (-1↑𝑅) ∈ ℤ)
74	72, 73	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (-1↑𝑅) ∈ ℤ)
75	74, 72	zmulcld 12361	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((-1↑𝑅) · 𝑅) ∈ ℤ)
76	75, 69	zmodcld 13540	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (((-1↑𝑅) · 𝑅) mod 𝑃) ∈ ℕ₀)
77	76	nn0cnd 12225	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (((-1↑𝑅) · 𝑅) mod 𝑃) ∈ ℂ)
78		2cnd 11981	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 2 ∈ ℂ)
79		2ne0 12007	. . . . . . . . . . . 12 ⊢ 2 ≠ 0
80	79	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 2 ≠ 0)
81	77, 78, 80	divcan2d 11683	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (2 · ((((-1↑𝑅) · 𝑅) mod 𝑃) / 2)) = (((-1↑𝑅) · 𝑅) mod 𝑃))
82	81	fveq2d 6760	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘(2 · ((((-1↑𝑅) · 𝑅) mod 𝑃) / 2))) = (𝐿‘(((-1↑𝑅) · 𝑅) mod 𝑃)))
83	69	nnrpd 12699	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 ∈ ℝ⁺)
84		eqidd 2739	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((-1↑𝑅) mod 𝑃) = ((-1↑𝑅) mod 𝑃))
85	41	oveq1i 7265	. . . . . . . . . . . . . 14 ⊢ (𝑅 mod 𝑃) = (((𝑄 · (2 · 𝑥)) mod 𝑃) mod 𝑃)
86	66	zred 12355	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑄 · (2 · 𝑥)) ∈ ℝ)
87		modabs2 13553	. . . . . . . . . . . . . . 15 ⊢ (((𝑄 · (2 · 𝑥)) ∈ ℝ ∧ 𝑃 ∈ ℝ⁺) → (((𝑄 · (2 · 𝑥)) mod 𝑃) mod 𝑃) = ((𝑄 · (2 · 𝑥)) mod 𝑃))
88	86, 83, 87	syl2anc 583	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (((𝑄 · (2 · 𝑥)) mod 𝑃) mod 𝑃) = ((𝑄 · (2 · 𝑥)) mod 𝑃))
89	85, 88	syl5eq 2791	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑅 mod 𝑃) = ((𝑄 · (2 · 𝑥)) mod 𝑃))
90	74, 74, 72, 66, 83, 84, 89	modmul12d 13573	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (((-1↑𝑅) · 𝑅) mod 𝑃) = (((-1↑𝑅) · (𝑄 · (2 · 𝑥))) mod 𝑃))
91	75	zred 12355	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((-1↑𝑅) · 𝑅) ∈ ℝ)
92		modabs2 13553	. . . . . . . . . . . . 13 ⊢ ((((-1↑𝑅) · 𝑅) ∈ ℝ ∧ 𝑃 ∈ ℝ⁺) → ((((-1↑𝑅) · 𝑅) mod 𝑃) mod 𝑃) = (((-1↑𝑅) · 𝑅) mod 𝑃))
93	91, 83, 92	syl2anc 583	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((((-1↑𝑅) · 𝑅) mod 𝑃) mod 𝑃) = (((-1↑𝑅) · 𝑅) mod 𝑃))
94	74	zcnd 12356	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (-1↑𝑅) ∈ ℂ)
95	59	zcnd 12356	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑄 ∈ ℂ)
96	65	zcnd 12356	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (2 · 𝑥) ∈ ℂ)
97	94, 95, 96	mulassd 10929	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (((-1↑𝑅) · 𝑄) · (2 · 𝑥)) = ((-1↑𝑅) · (𝑄 · (2 · 𝑥))))
98	97	oveq1d 7270	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((((-1↑𝑅) · 𝑄) · (2 · 𝑥)) mod 𝑃) = (((-1↑𝑅) · (𝑄 · (2 · 𝑥))) mod 𝑃))
99	90, 93, 98	3eqtr4d 2788	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((((-1↑𝑅) · 𝑅) mod 𝑃) mod 𝑃) = ((((-1↑𝑅) · 𝑄) · (2 · 𝑥)) mod 𝑃))
100	10, 68	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑃 ∈ ℕ)
101	100	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 ∈ ℕ)
102	76	nn0zd 12353	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (((-1↑𝑅) · 𝑅) mod 𝑃) ∈ ℤ)
103	74, 59	zmulcld 12361	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((-1↑𝑅) · 𝑄) ∈ ℤ)
104	103, 65	zmulcld 12361	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (((-1↑𝑅) · 𝑄) · (2 · 𝑥)) ∈ ℤ)
105		moddvds 15902	. . . . . . . . . . . 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)))))
106	101, 102, 104, 105	syl3anc 1369	. . . . . . . . . . 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)))))
107	99, 106	mpbid 231	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 ∥ ((((-1↑𝑅) · 𝑅) mod 𝑃) − (((-1↑𝑅) · 𝑄) · (2 · 𝑥))))
108	69	nnnn0d 12223	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 ∈ ℕ₀)
109	11, 22	zndvds 20669	. . . . . . . . . . 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)))))
110	108, 102, 104, 109	syl3anc 1369	. . . . . . . . . 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)))))
111	107, 110	mpbird 256	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘(((-1↑𝑅) · 𝑅) mod 𝑃)) = (𝐿‘(((-1↑𝑅) · 𝑄) · (2 · 𝑥))))
112	24	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝐿 ∈ (ℤ_ring RingHom 𝑌))
113		zringmulr 20591	. . . . . . . . . . 11 ⊢ · = (._r‘ℤ_ring)
114		eqid 2738	. . . . . . . . . . 11 ⊢ (._r‘𝑌) = (._r‘𝑌)
115	25, 113, 114	rhmmul 19886	. . . . . . . . . 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))))
116	112, 103, 65, 115	syl3anc 1369	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘(((-1↑𝑅) · 𝑄) · (2 · 𝑥))) = ((𝐿‘((-1↑𝑅) · 𝑄))(._r‘𝑌)(𝐿‘(2 · 𝑥))))
117	82, 111, 116	3eqtrd 2782	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘(2 · ((((-1↑𝑅) · 𝑅) mod 𝑃) / 2))) = ((𝐿‘((-1↑𝑅) · 𝑄))(._r‘𝑌)(𝐿‘(2 · 𝑥))))
118	117	mpteq2dva 5170	. . . . . . 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)))))
119	27	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝐿:ℤ⟶(Base‘𝑌))
120	119, 103	ffvelrnd 6944	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘((-1↑𝑅) · 𝑄)) ∈ (Base‘𝑌))
121	119, 65	ffvelrnd 6944	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘(2 · 𝑥)) ∈ (Base‘𝑌))
122		eqidd 2739	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))) = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))))
123		eqidd 2739	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥))) = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥))))
124	19, 120, 121, 122, 123	offval2 7531	. . . . . . 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)))))
125	118, 124	eqtr4d 2781	. . . . . 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)))))
126	125	oveq2d 7271	. . . . 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))))))
127	46, 55, 126	3eqtrd 2782	. . . 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))))))
128	5, 114	mgpplusg 19639	. . . . 5 ⊢ (._r‘𝑌) = (+_g‘𝐺)
129		eqid 2738	. . . . 5 ⊢ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))) = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄)))
130		eqid 2738	. . . . 5 ⊢ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥))) = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥)))
131	7, 128, 18, 19, 120, 121, 129, 130	gsummptfidmadd2 19442	. . . 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))))))
132	127, 131	eqtrd 2778	. . 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))))))
133	132	oveq1d 7270	. 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))))))
134		eqid 2738	. . . . . 6 ⊢ (Unit‘𝑌) = (Unit‘𝑌)
135	134, 5	unitsubm 19827	. . . . 5 ⊢ (𝑌 ∈ Ring → (Unit‘𝑌) ∈ (SubMnd‘𝐺))
136	21, 135	syl 17	. . . 4 ⊢ (𝜑 → (Unit‘𝑌) ∈ (SubMnd‘𝐺))
137		elfzle2 13189	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (1...((𝑃 − 1) / 2)) → 𝑥 ≤ ((𝑃 − 1) / 2))
138	137	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑥 ≤ ((𝑃 − 1) / 2))
139	62	nnred 11918	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑥 ∈ ℝ)
140		prmuz2 16329	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ_≥‘2))
141		uz2m1nn 12592	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ (ℤ_≥‘2) → (𝑃 − 1) ∈ ℕ)
142	67, 140, 141	3syl 18	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑃 − 1) ∈ ℕ)
143	142	nnred 11918	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑃 − 1) ∈ ℝ)
144		2re 11977	. . . . . . . . . . . 12 ⊢ 2 ∈ ℝ
145	144	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 2 ∈ ℝ)
146		2pos 12006	. . . . . . . . . . . 12 ⊢ 0 < 2
147	146	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 0 < 2)
148		lemuldiv2 11786	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ ∧ (𝑃 − 1) ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → ((2 · 𝑥) ≤ (𝑃 − 1) ↔ 𝑥 ≤ ((𝑃 − 1) / 2)))
149	139, 143, 145, 147, 148	syl112anc 1372	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((2 · 𝑥) ≤ (𝑃 − 1) ↔ 𝑥 ≤ ((𝑃 − 1) / 2)))
150	138, 149	mpbird 256	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (2 · 𝑥) ≤ (𝑃 − 1))
151		prmz 16308	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
152	67, 151	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 ∈ ℤ)
153		peano2zm 12293	. . . . . . . . . . 11 ⊢ (𝑃 ∈ ℤ → (𝑃 − 1) ∈ ℤ)
154	152, 153	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑃 − 1) ∈ ℤ)
155		fznn 13253	. . . . . . . . . 10 ⊢ ((𝑃 − 1) ∈ ℤ → ((2 · 𝑥) ∈ (1...(𝑃 − 1)) ↔ ((2 · 𝑥) ∈ ℕ ∧ (2 · 𝑥) ≤ (𝑃 − 1))))
156	154, 155	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((2 · 𝑥) ∈ (1...(𝑃 − 1)) ↔ ((2 · 𝑥) ∈ ℕ ∧ (2 · 𝑥) ≤ (𝑃 − 1))))
157	64, 150, 156	mpbir2and 709	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (2 · 𝑥) ∈ (1...(𝑃 − 1)))
158		fzm1ndvds 15959	. . . . . . . 8 ⊢ ((𝑃 ∈ ℕ ∧ (2 · 𝑥) ∈ (1...(𝑃 − 1))) → ¬ 𝑃 ∥ (2 · 𝑥))
159	69, 157, 158	syl2anc 583	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ¬ 𝑃 ∥ (2 · 𝑥))
160		eqid 2738	. . . . . . . . . 10 ⊢ (0_g‘𝑌) = (0_g‘𝑌)
161	11, 22, 160	zndvds0 20670	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℕ₀ ∧ (2 · 𝑥) ∈ ℤ) → ((𝐿‘(2 · 𝑥)) = (0_g‘𝑌) ↔ 𝑃 ∥ (2 · 𝑥)))
162	108, 65, 161	syl2anc 583	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝐿‘(2 · 𝑥)) = (0_g‘𝑌) ↔ 𝑃 ∥ (2 · 𝑥)))
163	162	necon3abid 2979	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝐿‘(2 · 𝑥)) ≠ (0_g‘𝑌) ↔ ¬ 𝑃 ∥ (2 · 𝑥)))
164	159, 163	mpbird 256	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘(2 · 𝑥)) ≠ (0_g‘𝑌))
165	14	simplbi 497	. . . . . . . . 9 ⊢ (𝑌 ∈ Field → 𝑌 ∈ DivRing)
166	13, 165	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ DivRing)
167	166	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑌 ∈ DivRing)
168	6, 134, 160	drngunit 19911	. . . . . . 7 ⊢ (𝑌 ∈ DivRing → ((𝐿‘(2 · 𝑥)) ∈ (Unit‘𝑌) ↔ ((𝐿‘(2 · 𝑥)) ∈ (Base‘𝑌) ∧ (𝐿‘(2 · 𝑥)) ≠ (0_g‘𝑌))))
169	167, 168	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝐿‘(2 · 𝑥)) ∈ (Unit‘𝑌) ↔ ((𝐿‘(2 · 𝑥)) ∈ (Base‘𝑌) ∧ (𝐿‘(2 · 𝑥)) ≠ (0_g‘𝑌))))
170	121, 164, 169	mpbir2and 709	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘(2 · 𝑥)) ∈ (Unit‘𝑌))
171	170	fmpttd 6971	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥))):(1...((𝑃 − 1) / 2))⟶(Unit‘𝑌))
172		fvexd 6771	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘(2 · 𝑥)) ∈ V)
173	130, 19, 172, 37	fsuppmptdm 9069	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥))) finSupp (0_g‘𝐺))
174	8, 18, 19, 136, 171, 173	gsumsubmcl 19435	. . 3 ⊢ (𝜑 → (𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥)))) ∈ (Unit‘𝑌))
175		eqid 2738	. . . 4 ⊢ (/_r‘𝑌) = (/_r‘𝑌)
176		eqid 2738	. . . 4 ⊢ (1_r‘𝑌) = (1_r‘𝑌)
177	134, 175, 176	dvrid 19845	. . 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))
178	21, 174, 177	syl2anc 583	. 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))
179	120	fmpttd 6971	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))):(1...((𝑃 − 1) / 2))⟶(Base‘𝑌))
180		fvexd 6771	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘((-1↑𝑅) · 𝑄)) ∈ V)
181	129, 19, 180, 37	fsuppmptdm 9069	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))) finSupp (0_g‘𝐺))
182	7, 8, 18, 19, 179, 181	gsumcl 19431	. . 3 ⊢ (𝜑 → (𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄)))) ∈ (Base‘𝑌))
183	6, 134, 175, 114	dvrcan3 19849	. . 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)))))
184	21, 182, 174, 183	syl3anc 1369	. 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)))))
185	133, 178, 184	3eqtr3rd 2787	1 ⊢ (𝜑 → (𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄)))) = (1_r‘𝑌))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 ∀wral 3063 Vcvv 3422 ∖ cdif 3880 {csn 4558 class class class wbr 5070 ↦ cmpt 5153 ∘ ccom 5584 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 ∘_f cof 7509 Fincfn 8691 ℝcr 10801 0cc0 10802 1c1 10803 · cmul 10807 < clt 10940 ≤ cle 10941 − cmin 11135 -cneg 11136 / cdiv 11562 ℕcn 11903 2c2 11958 ℕ₀cn0 12163 ℤcz 12249 ℤ_≥cuz 12511 ℝ⁺crp 12659 ...cfz 13168 mod cmo 13517 ↑cexp 13710 ∥ cdvds 15891 ℙcprime 16304 Basecbs 16840 ._rcmulr 16889 0_gc0g 17067 Σ_g cgsu 17068 SubMndcsubmnd 18344 CMndccmn 19301 mulGrpcmgp 19635 1_rcur 19652 Ringcrg 19698 CRingccrg 19699 Unitcui 19796 /_rcdvr 19839 RingHom crh 19871 DivRingcdr 19906 Fieldcfield 19907 ℤ_ringzring 20582 ℤRHomczrh 20613 ℤ/nℤczn 20616

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 ax-addf 10881 ax-mulf 10882

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-tpos 8013 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-oadd 8271 df-er 8456 df-ec 8458 df-qs 8462 df-map 8575 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-sup 9131 df-inf 9132 df-oi 9199 df-dju 9590 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-xnn0 12236 df-z 12250 df-dec 12367 df-uz 12512 df-rp 12660 df-fz 13169 df-fzo 13312 df-fl 13440 df-mod 13518 df-seq 13650 df-exp 13711 df-hash 13973 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-dvds 15892 df-gcd 16130 df-prm 16305 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-0g 17069 df-gsum 17070 df-imas 17136 df-qus 17137 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-mhm 18345 df-submnd 18346 df-grp 18495 df-minusg 18496 df-sbg 18497 df-mulg 18616 df-subg 18667 df-nsg 18668 df-eqg 18669 df-ghm 18747 df-cntz 18838 df-cmn 19303 df-abl 19304 df-mgp 19636 df-ur 19653 df-ring 19700 df-cring 19701 df-oppr 19777 df-dvdsr 19798 df-unit 19799 df-invr 19829 df-dvr 19840 df-rnghom 19874 df-drng 19908 df-field 19909 df-subrg 19937 df-lmod 20040 df-lss 20109 df-lsp 20149 df-sra 20349 df-rgmod 20350 df-lidl 20351 df-rsp 20352 df-2idl 20416 df-nzr 20442 df-rlreg 20467 df-domn 20468 df-idom 20469 df-cnfld 20511 df-zring 20583 df-zrh 20617 df-zn 20620

This theorem is referenced by: lgseisenlem4 26431

W3C validator