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Theorem lgseisenlem3 25102

Description: Lemma for lgseisen 25104. (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 6658	. . . . . . . . 9 ⊢ (𝑘 = 𝑥 → (2 · 𝑘) = (2 · 𝑥))
2	1	fveq2d 6195	. . . . . . . 8 ⊢ (𝑘 = 𝑥 → (𝐿‘(2 · 𝑘)) = (𝐿‘(2 · 𝑥)))
3	2	cbvmptv 4750	. . . . . . 7 ⊢ (𝑘 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑘))) = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥)))
4	3	oveq2i 6661	. . . . . 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 2622	. . . . . . . 8 ⊢ (Base‘𝑌) = (Base‘𝑌)
7	5, 6	mgpbas 18495	. . . . . . 7 ⊢ (Base‘𝑌) = (Base‘𝐺)
8		eqid 2622	. . . . . . 7 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
9		lgseisen.1	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))
10	9	eldifad 3586	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ ℙ)
11		lgseisen.7	. . . . . . . . . . 11 ⊢ 𝑌 = (ℤ/nℤ‘𝑃)
12	11	znfld 19909	. . . . . . . . . 10 ⊢ (𝑃 ∈ ℙ → 𝑌 ∈ Field)
13	10, 12	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ Field)
14		isfld 18756	. . . . . . . . . 10 ⊢ (𝑌 ∈ Field ↔ (𝑌 ∈ DivRing ∧ 𝑌 ∈ CRing))
15	14	simprbi 480	. . . . . . . . 9 ⊢ (𝑌 ∈ Field → 𝑌 ∈ CRing)
16	13, 15	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ CRing)
17	5	crngmgp 18555	. . . . . . . 8 ⊢ (𝑌 ∈ CRing → 𝐺 ∈ CMnd)
18	16, 17	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ CMnd)
19		fzfid 12772	. . . . . . 7 ⊢ (𝜑 → (1...((𝑃 − 1) / 2)) ∈ Fin)
20		crngring 18558	. . . . . . . . . . . 12 ⊢ (𝑌 ∈ CRing → 𝑌 ∈ Ring)
21	16, 20	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑌 ∈ Ring)
22		lgseisen.9	. . . . . . . . . . . 12 ⊢ 𝐿 = (ℤRHom‘𝑌)
23	22	zrhrhm 19860	. . . . . . . . . . 11 ⊢ (𝑌 ∈ Ring → 𝐿 ∈ (ℤ_ring RingHom 𝑌))
24	21, 23	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐿 ∈ (ℤ_ring RingHom 𝑌))
25		zringbas 19824	. . . . . . . . . . 11 ⊢ ℤ = (Base‘ℤ_ring)
26	25, 6	rhmf 18726	. . . . . . . . . 10 ⊢ (𝐿 ∈ (ℤ_ring RingHom 𝑌) → 𝐿:ℤ⟶(Base‘𝑌))
27	24, 26	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐿:ℤ⟶(Base‘𝑌))
28		2z 11409	. . . . . . . . . 10 ⊢ 2 ∈ ℤ
29		elfzelz 12342	. . . . . . . . . 10 ⊢ (𝑘 ∈ (1...((𝑃 − 1) / 2)) → 𝑘 ∈ ℤ)
30		zmulcl 11426	. . . . . . . . . 10 ⊢ ((2 ∈ ℤ ∧ 𝑘 ∈ ℤ) → (2 · 𝑘) ∈ ℤ)
31	28, 29, 30	sylancr 695	. . . . . . . . 9 ⊢ (𝑘 ∈ (1...((𝑃 − 1) / 2)) → (2 · 𝑘) ∈ ℤ)
32		ffvelrn 6357	. . . . . . . . 9 ⊢ ((𝐿:ℤ⟶(Base‘𝑌) ∧ (2 · 𝑘) ∈ ℤ) → (𝐿‘(2 · 𝑘)) ∈ (Base‘𝑌))
33	27, 31, 32	syl2an 494	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘(2 · 𝑘)) ∈ (Base‘𝑌))
34		eqid 2622	. . . . . . . 8 ⊢ (𝑘 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑘))) = (𝑘 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑘)))
35	33, 34	fmptd 6385	. . . . . . 7 ⊢ (𝜑 → (𝑘 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑘))):(1...((𝑃 − 1) / 2))⟶(Base‘𝑌))
36		fvexd 6203	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘(2 · 𝑘)) ∈ V)
37		fvexd 6203	. . . . . . . 8 ⊢ (𝜑 → (0_g‘𝐺) ∈ V)
38	34, 19, 36, 37	fsuppmptdm 8286	. . . . . . 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 25101	. . . . . . 7 ⊢ (𝜑 → 𝑀:(1...((𝑃 − 1) / 2))–1-1-onto→(1...((𝑃 − 1) / 2)))
45	7, 8, 18, 19, 35, 38, 44	gsumf1o 18317	. . . . . 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	syl5eqr 2670	. . . . 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 25100	. . . . . . . 8 ⊢ (𝜑 → 𝑀:(1...((𝑃 − 1) / 2))⟶(1...((𝑃 − 1) / 2)))
48	42	fmpt 6381	. . . . . . . 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 224	. . . . . . 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 2623	. . . . . . 7 ⊢ (𝜑 → (𝑘 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑘))) = (𝑘 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑘))))
52		oveq2 6658	. . . . . . . 8 ⊢ (𝑘 = ((((-1↑𝑅) · 𝑅) mod 𝑃) / 2) → (2 · 𝑘) = (2 · ((((-1↑𝑅) · 𝑅) mod 𝑃) / 2)))
53	52	fveq2d 6195	. . . . . . 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 6397	. . . . . 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 6666	. . . . 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 3586	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑄 ∈ ℙ)
57	56	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑄 ∈ ℙ)
58		prmz 15389	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑄 ∈ ℙ → 𝑄 ∈ ℤ)
59	57, 58	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑄 ∈ ℤ)
60		2nn 11185	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 2 ∈ ℕ
61		elfznn 12370	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ (1...((𝑃 − 1) / 2)) → 𝑥 ∈ ℕ)
62	61	adantl 482	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑥 ∈ ℕ)
63		nnmulcl 11043	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((2 ∈ ℕ ∧ 𝑥 ∈ ℕ) → (2 · 𝑥) ∈ ℕ)
64	60, 62, 63	sylancr 695	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (2 · 𝑥) ∈ ℕ)
65	64	nnzd 11481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (2 · 𝑥) ∈ ℤ)
66	59, 65	zmulcld 11488	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑄 · (2 · 𝑥)) ∈ ℤ)
67	10	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 ∈ ℙ)
68		prmnn 15388	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
69	67, 68	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 ∈ ℕ)
70	66, 69	zmodcld 12691	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝑄 · (2 · 𝑥)) mod 𝑃) ∈ ℕ₀)
71	41, 70	syl5eqel 2705	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑅 ∈ ℕ₀)
72	71	nn0zd 11480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑅 ∈ ℤ)
73		m1expcl 12883	. . . . . . . . . . . . . . 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 11488	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((-1↑𝑅) · 𝑅) ∈ ℤ)
76	75, 69	zmodcld 12691	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (((-1↑𝑅) · 𝑅) mod 𝑃) ∈ ℕ₀)
77	76	nn0cnd 11353	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (((-1↑𝑅) · 𝑅) mod 𝑃) ∈ ℂ)
78		2cnd 11093	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 2 ∈ ℂ)
79		2ne0 11113	. . . . . . . . . . . 12 ⊢ 2 ≠ 0
80	79	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 2 ≠ 0)
81	77, 78, 80	divcan2d 10803	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (2 · ((((-1↑𝑅) · 𝑅) mod 𝑃) / 2)) = (((-1↑𝑅) · 𝑅) mod 𝑃))
82	81	fveq2d 6195	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘(2 · ((((-1↑𝑅) · 𝑅) mod 𝑃) / 2))) = (𝐿‘(((-1↑𝑅) · 𝑅) mod 𝑃)))
83	69	nnrpd 11870	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 ∈ ℝ⁺)
84		eqidd 2623	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((-1↑𝑅) mod 𝑃) = ((-1↑𝑅) mod 𝑃))
85	41	oveq1i 6660	. . . . . . . . . . . . . 14 ⊢ (𝑅 mod 𝑃) = (((𝑄 · (2 · 𝑥)) mod 𝑃) mod 𝑃)
86	66	zred 11482	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑄 · (2 · 𝑥)) ∈ ℝ)
87		modabs2 12704	. . . . . . . . . . . . . . 15 ⊢ (((𝑄 · (2 · 𝑥)) ∈ ℝ ∧ 𝑃 ∈ ℝ⁺) → (((𝑄 · (2 · 𝑥)) mod 𝑃) mod 𝑃) = ((𝑄 · (2 · 𝑥)) mod 𝑃))
88	86, 83, 87	syl2anc 693	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (((𝑄 · (2 · 𝑥)) mod 𝑃) mod 𝑃) = ((𝑄 · (2 · 𝑥)) mod 𝑃))
89	85, 88	syl5eq 2668	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑅 mod 𝑃) = ((𝑄 · (2 · 𝑥)) mod 𝑃))
90	74, 74, 72, 66, 83, 84, 89	modmul12d 12724	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (((-1↑𝑅) · 𝑅) mod 𝑃) = (((-1↑𝑅) · (𝑄 · (2 · 𝑥))) mod 𝑃))
91	75	zred 11482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((-1↑𝑅) · 𝑅) ∈ ℝ)
92		modabs2 12704	. . . . . . . . . . . . 13 ⊢ ((((-1↑𝑅) · 𝑅) ∈ ℝ ∧ 𝑃 ∈ ℝ⁺) → ((((-1↑𝑅) · 𝑅) mod 𝑃) mod 𝑃) = (((-1↑𝑅) · 𝑅) mod 𝑃))
93	91, 83, 92	syl2anc 693	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((((-1↑𝑅) · 𝑅) mod 𝑃) mod 𝑃) = (((-1↑𝑅) · 𝑅) mod 𝑃))
94	74	zcnd 11483	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (-1↑𝑅) ∈ ℂ)
95	59	zcnd 11483	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑄 ∈ ℂ)
96	65	zcnd 11483	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (2 · 𝑥) ∈ ℂ)
97	94, 95, 96	mulassd 10063	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (((-1↑𝑅) · 𝑄) · (2 · 𝑥)) = ((-1↑𝑅) · (𝑄 · (2 · 𝑥))))
98	97	oveq1d 6665	. . . . . . . . . . . 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 2666	. . . . . . . . . . 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 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 ∈ ℕ)
102	76	nn0zd 11480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (((-1↑𝑅) · 𝑅) mod 𝑃) ∈ ℤ)
103	74, 59	zmulcld 11488	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((-1↑𝑅) · 𝑄) ∈ ℤ)
104	103, 65	zmulcld 11488	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (((-1↑𝑅) · 𝑄) · (2 · 𝑥)) ∈ ℤ)
105		moddvds 14991	. . . . . . . . . . . 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 1326	. . . . . . . . . . 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 222	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 ∥ ((((-1↑𝑅) · 𝑅) mod 𝑃) − (((-1↑𝑅) · 𝑄) · (2 · 𝑥))))
108	69	nnnn0d 11351	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 ∈ ℕ₀)
109	11, 22	zndvds 19898	. . . . . . . . . . 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 1326	. . . . . . . . . 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 247	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘(((-1↑𝑅) · 𝑅) mod 𝑃)) = (𝐿‘(((-1↑𝑅) · 𝑄) · (2 · 𝑥))))
112	24	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝐿 ∈ (ℤ_ring RingHom 𝑌))
113		zringmulr 19827	. . . . . . . . . . 11 ⊢ · = (._r‘ℤ_ring)
114		eqid 2622	. . . . . . . . . . 11 ⊢ (._r‘𝑌) = (._r‘𝑌)
115	25, 113, 114	rhmmul 18727	. . . . . . . . . 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 1326	. . . . . . . . 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 2660	. . . . . . . 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 4744	. . . . . . 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 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝐿:ℤ⟶(Base‘𝑌))
120	119, 103	ffvelrnd 6360	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘((-1↑𝑅) · 𝑄)) ∈ (Base‘𝑌))
121	119, 65	ffvelrnd 6360	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘(2 · 𝑥)) ∈ (Base‘𝑌))
122		eqidd 2623	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))) = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))))
123		eqidd 2623	. . . . . . . 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 6914	. . . . . . 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 2659	. . . . . 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 6666	. . . . 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 2660	. . . 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 18493	. . . . 5 ⊢ (._r‘𝑌) = (+_g‘𝐺)
129		eqid 2622	. . . . 5 ⊢ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))) = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄)))
130		eqid 2622	. . . . 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 18326	. . . 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 2656	. . 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 6665	. 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 2622	. . . . . 6 ⊢ (Unit‘𝑌) = (Unit‘𝑌)
135	134, 5	unitsubm 18670	. . . . 5 ⊢ (𝑌 ∈ Ring → (Unit‘𝑌) ∈ (SubMnd‘𝐺))
136	21, 135	syl 17	. . . 4 ⊢ (𝜑 → (Unit‘𝑌) ∈ (SubMnd‘𝐺))
137		elfzle2 12345	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (1...((𝑃 − 1) / 2)) → 𝑥 ≤ ((𝑃 − 1) / 2))
138	137	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑥 ≤ ((𝑃 − 1) / 2))
139	62	nnred 11035	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑥 ∈ ℝ)
140		prmuz2 15408	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ_≥‘2))
141		uz2m1nn 11763	. . . . . . . . . . . . 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 11035	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑃 − 1) ∈ ℝ)
144		2re 11090	. . . . . . . . . . . 12 ⊢ 2 ∈ ℝ
145	144	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 2 ∈ ℝ)
146		2pos 11112	. . . . . . . . . . . 12 ⊢ 0 < 2
147	146	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 0 < 2)
148		lemuldiv2 10904	. . . . . . . . . . 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 1330	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((2 · 𝑥) ≤ (𝑃 − 1) ↔ 𝑥 ≤ ((𝑃 − 1) / 2)))
150	138, 149	mpbird 247	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (2 · 𝑥) ≤ (𝑃 − 1))
151		prmz 15389	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
152	67, 151	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 ∈ ℤ)
153		peano2zm 11420	. . . . . . . . . . 11 ⊢ (𝑃 ∈ ℤ → (𝑃 − 1) ∈ ℤ)
154	152, 153	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝑃 − 1) ∈ ℤ)
155		fznn 12408	. . . . . . . . . 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 957	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (2 · 𝑥) ∈ (1...(𝑃 − 1)))
158		fzm1ndvds 15044	. . . . . . . 8 ⊢ ((𝑃 ∈ ℕ ∧ (2 · 𝑥) ∈ (1...(𝑃 − 1))) → ¬ 𝑃 ∥ (2 · 𝑥))
159	69, 157, 158	syl2anc 693	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ¬ 𝑃 ∥ (2 · 𝑥))
160		eqid 2622	. . . . . . . . . 10 ⊢ (0_g‘𝑌) = (0_g‘𝑌)
161	11, 22, 160	zndvds0 19899	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℕ₀ ∧ (2 · 𝑥) ∈ ℤ) → ((𝐿‘(2 · 𝑥)) = (0_g‘𝑌) ↔ 𝑃 ∥ (2 · 𝑥)))
162	108, 65, 161	syl2anc 693	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝐿‘(2 · 𝑥)) = (0_g‘𝑌) ↔ 𝑃 ∥ (2 · 𝑥)))
163	162	necon3abid 2830	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → ((𝐿‘(2 · 𝑥)) ≠ (0_g‘𝑌) ↔ ¬ 𝑃 ∥ (2 · 𝑥)))
164	159, 163	mpbird 247	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘(2 · 𝑥)) ≠ (0_g‘𝑌))
165	14	simplbi 476	. . . . . . . . 9 ⊢ (𝑌 ∈ Field → 𝑌 ∈ DivRing)
166	13, 165	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ DivRing)
167	166	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → 𝑌 ∈ DivRing)
168	6, 134, 160	drngunit 18752	. . . . . . 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 957	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘(2 · 𝑥)) ∈ (Unit‘𝑌))
171	170, 130	fmptd 6385	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥))):(1...((𝑃 − 1) / 2))⟶(Unit‘𝑌))
172		fvexd 6203	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘(2 · 𝑥)) ∈ V)
173	130, 19, 172, 37	fsuppmptdm 8286	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥))) finSupp (0_g‘𝐺))
174	8, 18, 19, 136, 171, 173	gsumsubmcl 18319	. . 3 ⊢ (𝜑 → (𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(2 · 𝑥)))) ∈ (Unit‘𝑌))
175		eqid 2622	. . . 4 ⊢ (/_r‘𝑌) = (/_r‘𝑌)
176		eqid 2622	. . . 4 ⊢ (1_r‘𝑌) = (1_r‘𝑌)
177	134, 175, 176	dvrid 18688	. . 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 693	. 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, 129	fmptd 6385	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))):(1...((𝑃 − 1) / 2))⟶(Base‘𝑌))
180		fvexd 6203	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘((-1↑𝑅) · 𝑄)) ∈ V)
181	129, 19, 180, 37	fsuppmptdm 8286	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄))) finSupp (0_g‘𝐺))
182	7, 8, 18, 19, 179, 181	gsumcl 18316	. . 3 ⊢ (𝜑 → (𝐺 Σ_g (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄)))) ∈ (Base‘𝑌))
183	6, 134, 175, 114	dvrcan3 18692	. . 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 1326	. 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 2665	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 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 Vcvv 3200 ∖ cdif 3571 {csn 4177 class class class wbr 4653 ↦ cmpt 4729 ∘ ccom 5118 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ∘_𝑓 cof 6895 Fincfn 7955 ℝcr 9935 0cc0 9936 1c1 9937 · cmul 9941 < clt 10074 ≤ cle 10075 − cmin 10266 -cneg 10267 / cdiv 10684 ℕcn 11020 2c2 11070 ℕ₀cn0 11292 ℤcz 11377 ℤ_≥cuz 11687 ℝ⁺crp 11832 ...cfz 12326 mod cmo 12668 ↑cexp 12860 ∥ cdvds 14983 ℙcprime 15385 Basecbs 15857 ._rcmulr 15942 0_gc0g 16100 Σ_g cgsu 16101 SubMndcsubmnd 17334 CMndccmn 18193 mulGrpcmgp 18489 1_rcur 18501 Ringcrg 18547 CRingccrg 18548 Unitcui 18639 /_rcdvr 18682 RingHom crh 18712 DivRingcdr 18747 Fieldcfield 18748 ℤ_ringzring 19818 ℤRHomczrh 19848 ℤ/nℤczn 19851

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-tpos 7352 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-ec 7744 df-qs 7748 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-xnn0 11364 df-z 11378 df-dec 11494 df-uz 11688 df-rp 11833 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-dvds 14984 df-gcd 15217 df-prm 15386 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-0g 16102 df-gsum 16103 df-imas 16168 df-qus 16169 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mhm 17335 df-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-mulg 17541 df-subg 17591 df-nsg 17592 df-eqg 17593 df-ghm 17658 df-cntz 17750 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-cring 18550 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-invr 18672 df-dvr 18683 df-rnghom 18715 df-drng 18749 df-field 18750 df-subrg 18778 df-lmod 18865 df-lss 18933 df-lsp 18972 df-sra 19172 df-rgmod 19173 df-lidl 19174 df-rsp 19175 df-2idl 19232 df-nzr 19258 df-rlreg 19283 df-domn 19284 df-idom 19285 df-cnfld 19747 df-zring 19819 df-zrh 19852 df-zn 19855

This theorem is referenced by: lgseisenlem4 25103

W3C validator