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Theorem lgsqrlem2 27391

Description: Lemma for lgsqr 27395. (Contributed by Mario Carneiro, 15-Jun-2015.)

**Hypotheses**
Ref	Expression
lgsqr.y	⊢ 𝑌 = (ℤ/nℤ‘𝑃)
lgsqr.s	⊢ 𝑆 = (Poly₁‘𝑌)
lgsqr.b	⊢ 𝐵 = (Base‘𝑆)
lgsqr.d	⊢ 𝐷 = (deg₁‘𝑌)
lgsqr.o	⊢ 𝑂 = (eval₁‘𝑌)
lgsqr.e	⊢ ↑ = (._g‘(mulGrp‘𝑆))
lgsqr.x	⊢ 𝑋 = (var₁‘𝑌)
lgsqr.m	⊢ − = (-_g‘𝑆)
lgsqr.u	⊢ 1 = (1_r‘𝑆)
lgsqr.t	⊢ 𝑇 = ((((𝑃 − 1) / 2) ↑ 𝑋) − 1 )
lgsqr.l	⊢ 𝐿 = (ℤRHom‘𝑌)
lgsqr.1	⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))
lgsqr.g	⊢ 𝐺 = (𝑦 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(𝑦↑2)))

**Assertion**
Ref	Expression
lgsqrlem2	⊢ (𝜑 → 𝐺:(1...((𝑃 − 1) / 2))–1-1→(^◡(𝑂‘𝑇) “ {(0_g‘𝑌)}))

Distinct variable groups: 𝑦,𝑂 𝑦,𝑃 𝜑,𝑦 𝑦,𝑇 𝑦,𝐿 𝑦,𝑌 Allowed substitution hints: 𝐵(𝑦) 𝐷(𝑦) 𝑆(𝑦) 1 (𝑦) ↑ (𝑦) 𝐺(𝑦) − (𝑦) 𝑋(𝑦)

Proof of Theorem lgsqrlem2 Dummy variables 𝑥 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lgsqr.1	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))
2	1	eldifad 3963	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑃 ∈ ℙ)
3		lgsqr.y	. . . . . . . . . . . . 13 ⊢ 𝑌 = (ℤ/nℤ‘𝑃)
4	3	znfld 21579	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ ℙ → 𝑌 ∈ Field)
5	2, 4	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑌 ∈ Field)
6		fldidom 20771	. . . . . . . . . . 11 ⊢ (𝑌 ∈ Field → 𝑌 ∈ IDomn)
7	5, 6	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ IDomn)
8		isidom 20725	. . . . . . . . . . 11 ⊢ (𝑌 ∈ IDomn ↔ (𝑌 ∈ CRing ∧ 𝑌 ∈ Domn))
9	8	simplbi 497	. . . . . . . . . 10 ⊢ (𝑌 ∈ IDomn → 𝑌 ∈ CRing)
10	7, 9	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ CRing)
11		crngring 20242	. . . . . . . . 9 ⊢ (𝑌 ∈ CRing → 𝑌 ∈ Ring)
12	10, 11	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ Ring)
13		lgsqr.l	. . . . . . . . 9 ⊢ 𝐿 = (ℤRHom‘𝑌)
14	13	zrhrhm 21522	. . . . . . . 8 ⊢ (𝑌 ∈ Ring → 𝐿 ∈ (ℤ_ring RingHom 𝑌))
15	12, 14	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ (ℤ_ring RingHom 𝑌))
16		zringbas 21464	. . . . . . . 8 ⊢ ℤ = (Base‘ℤ_ring)
17		eqid 2737	. . . . . . . 8 ⊢ (Base‘𝑌) = (Base‘𝑌)
18	16, 17	rhmf 20485	. . . . . . 7 ⊢ (𝐿 ∈ (ℤ_ring RingHom 𝑌) → 𝐿:ℤ⟶(Base‘𝑌))
19	15, 18	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐿:ℤ⟶(Base‘𝑌))
20	19	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...((𝑃 − 1) / 2))) → 𝐿:ℤ⟶(Base‘𝑌))
21		elfzelz 13564	. . . . . . 7 ⊢ (𝑦 ∈ (1...((𝑃 − 1) / 2)) → 𝑦 ∈ ℤ)
22	21	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...((𝑃 − 1) / 2))) → 𝑦 ∈ ℤ)
23		zsqcl 14169	. . . . . 6 ⊢ (𝑦 ∈ ℤ → (𝑦↑2) ∈ ℤ)
24	22, 23	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...((𝑃 − 1) / 2))) → (𝑦↑2) ∈ ℤ)
25	20, 24	ffvelcdmd 7105	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘(𝑦↑2)) ∈ (Base‘𝑌))
26		lgsqr.s	. . . . 5 ⊢ 𝑆 = (Poly₁‘𝑌)
27		lgsqr.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑆)
28		lgsqr.d	. . . . 5 ⊢ 𝐷 = (deg₁‘𝑌)
29		lgsqr.o	. . . . 5 ⊢ 𝑂 = (eval₁‘𝑌)
30		lgsqr.e	. . . . 5 ⊢ ↑ = (._g‘(mulGrp‘𝑆))
31		lgsqr.x	. . . . 5 ⊢ 𝑋 = (var₁‘𝑌)
32		lgsqr.m	. . . . 5 ⊢ − = (-_g‘𝑆)
33		lgsqr.u	. . . . 5 ⊢ 1 = (1_r‘𝑆)
34		lgsqr.t	. . . . 5 ⊢ 𝑇 = ((((𝑃 − 1) / 2) ↑ 𝑋) − 1 )
35	1	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 ∈ (ℙ ∖ {2}))
36		elfznn 13593	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (1...((𝑃 − 1) / 2)) → 𝑦 ∈ ℕ)
37	36	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...((𝑃 − 1) / 2))) → 𝑦 ∈ ℕ)
38	37	nncnd 12282	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...((𝑃 − 1) / 2))) → 𝑦 ∈ ℂ)
39		oddprm 16848	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((𝑃 − 1) / 2) ∈ ℕ)
40	1, 39	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑃 − 1) / 2) ∈ ℕ)
41	40	nnnn0d 12587	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑃 − 1) / 2) ∈ ℕ₀)
42	41	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...((𝑃 − 1) / 2))) → ((𝑃 − 1) / 2) ∈ ℕ₀)
43		2nn0 12543	. . . . . . . . . 10 ⊢ 2 ∈ ℕ₀
44	43	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...((𝑃 − 1) / 2))) → 2 ∈ ℕ₀)
45	38, 42, 44	expmuld 14189	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...((𝑃 − 1) / 2))) → (𝑦↑(2 · ((𝑃 − 1) / 2))) = ((𝑦↑2)↑((𝑃 − 1) / 2)))
46		prmnn 16711	. . . . . . . . . . . . . . . 16 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
47	2, 46	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑃 ∈ ℕ)
48	47	nnred 12281	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑃 ∈ ℝ)
49		peano2rem 11576	. . . . . . . . . . . . . 14 ⊢ (𝑃 ∈ ℝ → (𝑃 − 1) ∈ ℝ)
50	48, 49	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑃 − 1) ∈ ℝ)
51	50	recnd 11289	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑃 − 1) ∈ ℂ)
52		2cnd 12344	. . . . . . . . . . . 12 ⊢ (𝜑 → 2 ∈ ℂ)
53		2ne0 12370	. . . . . . . . . . . . 13 ⊢ 2 ≠ 0
54	53	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 2 ≠ 0)
55	51, 52, 54	divcan2d 12045	. . . . . . . . . . 11 ⊢ (𝜑 → (2 · ((𝑃 − 1) / 2)) = (𝑃 − 1))
56		phiprm 16814	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ ℙ → (ϕ‘𝑃) = (𝑃 − 1))
57	2, 56	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (ϕ‘𝑃) = (𝑃 − 1))
58	55, 57	eqtr4d 2780	. . . . . . . . . 10 ⊢ (𝜑 → (2 · ((𝑃 − 1) / 2)) = (ϕ‘𝑃))
59	58	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...((𝑃 − 1) / 2))) → (2 · ((𝑃 − 1) / 2)) = (ϕ‘𝑃))
60	59	oveq2d 7447	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...((𝑃 − 1) / 2))) → (𝑦↑(2 · ((𝑃 − 1) / 2))) = (𝑦↑(ϕ‘𝑃)))
61	45, 60	eqtr3d 2779	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...((𝑃 − 1) / 2))) → ((𝑦↑2)↑((𝑃 − 1) / 2)) = (𝑦↑(ϕ‘𝑃)))
62	61	oveq1d 7446	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...((𝑃 − 1) / 2))) → (((𝑦↑2)↑((𝑃 − 1) / 2)) mod 𝑃) = ((𝑦↑(ϕ‘𝑃)) mod 𝑃))
63	2	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 ∈ ℙ)
64	63, 46	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 ∈ ℕ)
65	47	nnzd 12640	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ ℤ)
66	65	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 ∈ ℤ)
67	22, 66	gcdcomd 16551	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...((𝑃 − 1) / 2))) → (𝑦 gcd 𝑃) = (𝑃 gcd 𝑦))
68	37	nnred 12281	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...((𝑃 − 1) / 2))) → 𝑦 ∈ ℝ)
69	50	rehalfcld 12513	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑃 − 1) / 2) ∈ ℝ)
70	69	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...((𝑃 − 1) / 2))) → ((𝑃 − 1) / 2) ∈ ℝ)
71	48	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...((𝑃 − 1) / 2))) → 𝑃 ∈ ℝ)
72		elfzle2 13568	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (1...((𝑃 − 1) / 2)) → 𝑦 ≤ ((𝑃 − 1) / 2))
73	72	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...((𝑃 − 1) / 2))) → 𝑦 ≤ ((𝑃 − 1) / 2))
74		prmuz2 16733	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ_≥‘2))
75	2, 74	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑃 ∈ (ℤ_≥‘2))
76		uz2m1nn 12965	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃 ∈ (ℤ_≥‘2) → (𝑃 − 1) ∈ ℕ)
77	75, 76	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑃 − 1) ∈ ℕ)
78	77	nnrpd 13075	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑃 − 1) ∈ ℝ⁺)
79		rphalflt 13064	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 − 1) ∈ ℝ⁺ → ((𝑃 − 1) / 2) < (𝑃 − 1))
80	78, 79	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑃 − 1) / 2) < (𝑃 − 1))
81	48	ltm1d 12200	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑃 − 1) < 𝑃)
82	69, 50, 48, 80, 81	lttrd 11422	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑃 − 1) / 2) < 𝑃)
83	82	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...((𝑃 − 1) / 2))) → ((𝑃 − 1) / 2) < 𝑃)
84	68, 70, 71, 73, 83	lelttrd 11419	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...((𝑃 − 1) / 2))) → 𝑦 < 𝑃)
85	68, 71	ltnled 11408	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...((𝑃 − 1) / 2))) → (𝑦 < 𝑃 ↔ ¬ 𝑃 ≤ 𝑦))
86	84, 85	mpbid 232	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...((𝑃 − 1) / 2))) → ¬ 𝑃 ≤ 𝑦)
87		dvdsle 16347	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ ℤ ∧ 𝑦 ∈ ℕ) → (𝑃 ∥ 𝑦 → 𝑃 ≤ 𝑦))
88	66, 37, 87	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...((𝑃 − 1) / 2))) → (𝑃 ∥ 𝑦 → 𝑃 ≤ 𝑦))
89	86, 88	mtod 198	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...((𝑃 − 1) / 2))) → ¬ 𝑃 ∥ 𝑦)
90		coprm 16748	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ 𝑦 ∈ ℤ) → (¬ 𝑃 ∥ 𝑦 ↔ (𝑃 gcd 𝑦) = 1))
91	63, 22, 90	syl2anc 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...((𝑃 − 1) / 2))) → (¬ 𝑃 ∥ 𝑦 ↔ (𝑃 gcd 𝑦) = 1))
92	89, 91	mpbid 232	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...((𝑃 − 1) / 2))) → (𝑃 gcd 𝑦) = 1)
93	67, 92	eqtrd 2777	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...((𝑃 − 1) / 2))) → (𝑦 gcd 𝑃) = 1)
94		eulerth 16820	. . . . . . 7 ⊢ ((𝑃 ∈ ℕ ∧ 𝑦 ∈ ℤ ∧ (𝑦 gcd 𝑃) = 1) → ((𝑦↑(ϕ‘𝑃)) mod 𝑃) = (1 mod 𝑃))
95	64, 22, 93, 94	syl3anc 1373	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...((𝑃 − 1) / 2))) → ((𝑦↑(ϕ‘𝑃)) mod 𝑃) = (1 mod 𝑃))
96	62, 95	eqtrd 2777	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...((𝑃 − 1) / 2))) → (((𝑦↑2)↑((𝑃 − 1) / 2)) mod 𝑃) = (1 mod 𝑃))
97	3, 26, 27, 28, 29, 30, 31, 32, 33, 34, 13, 35, 24, 96	lgsqrlem1 27390	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...((𝑃 − 1) / 2))) → ((𝑂‘𝑇)‘(𝐿‘(𝑦↑2))) = (0_g‘𝑌))
98		eqid 2737	. . . . . . . 8 ⊢ (𝑌 ↑_s (Base‘𝑌)) = (𝑌 ↑_s (Base‘𝑌))
99		eqid 2737	. . . . . . . 8 ⊢ (Base‘(𝑌 ↑_s (Base‘𝑌))) = (Base‘(𝑌 ↑_s (Base‘𝑌)))
100		fvexd 6921	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝑌) ∈ V)
101	29, 26, 98, 17	evl1rhm 22336	. . . . . . . . . . 11 ⊢ (𝑌 ∈ CRing → 𝑂 ∈ (𝑆 RingHom (𝑌 ↑_s (Base‘𝑌))))
102	10, 101	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑂 ∈ (𝑆 RingHom (𝑌 ↑_s (Base‘𝑌))))
103	27, 99	rhmf 20485	. . . . . . . . . 10 ⊢ (𝑂 ∈ (𝑆 RingHom (𝑌 ↑_s (Base‘𝑌))) → 𝑂:𝐵⟶(Base‘(𝑌 ↑_s (Base‘𝑌))))
104	102, 103	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑂:𝐵⟶(Base‘(𝑌 ↑_s (Base‘𝑌))))
105	26	ply1ring 22249	. . . . . . . . . . . . 13 ⊢ (𝑌 ∈ Ring → 𝑆 ∈ Ring)
106	12, 105	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ∈ Ring)
107		ringgrp 20235	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ Ring → 𝑆 ∈ Grp)
108	106, 107	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 ∈ Grp)
109		eqid 2737	. . . . . . . . . . . . 13 ⊢ (mulGrp‘𝑆) = (mulGrp‘𝑆)
110	109, 27	mgpbas 20142	. . . . . . . . . . . 12 ⊢ 𝐵 = (Base‘(mulGrp‘𝑆))
111	109	ringmgp 20236	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ Ring → (mulGrp‘𝑆) ∈ Mnd)
112	106, 111	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (mulGrp‘𝑆) ∈ Mnd)
113	31, 26, 27	vr1cl 22219	. . . . . . . . . . . . 13 ⊢ (𝑌 ∈ Ring → 𝑋 ∈ 𝐵)
114	12, 113	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
115	110, 30, 112, 41, 114	mulgnn0cld 19113	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝑃 − 1) / 2) ↑ 𝑋) ∈ 𝐵)
116	27, 33	ringidcl 20262	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ Ring → 1 ∈ 𝐵)
117	106, 116	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 1 ∈ 𝐵)
118	27, 32	grpsubcl 19038	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Grp ∧ (((𝑃 − 1) / 2) ↑ 𝑋) ∈ 𝐵 ∧ 1 ∈ 𝐵) → ((((𝑃 − 1) / 2) ↑ 𝑋) − 1 ) ∈ 𝐵)
119	108, 115, 117, 118	syl3anc 1373	. . . . . . . . . 10 ⊢ (𝜑 → ((((𝑃 − 1) / 2) ↑ 𝑋) − 1 ) ∈ 𝐵)
120	34, 119	eqeltrid 2845	. . . . . . . . 9 ⊢ (𝜑 → 𝑇 ∈ 𝐵)
121	104, 120	ffvelcdmd 7105	. . . . . . . 8 ⊢ (𝜑 → (𝑂‘𝑇) ∈ (Base‘(𝑌 ↑_s (Base‘𝑌))))
122	98, 17, 99, 5, 100, 121	pwselbas 17534	. . . . . . 7 ⊢ (𝜑 → (𝑂‘𝑇):(Base‘𝑌)⟶(Base‘𝑌))
123	122	ffnd 6737	. . . . . 6 ⊢ (𝜑 → (𝑂‘𝑇) Fn (Base‘𝑌))
124	123	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...((𝑃 − 1) / 2))) → (𝑂‘𝑇) Fn (Base‘𝑌))
125		fniniseg 7080	. . . . 5 ⊢ ((𝑂‘𝑇) Fn (Base‘𝑌) → ((𝐿‘(𝑦↑2)) ∈ (^◡(𝑂‘𝑇) “ {(0_g‘𝑌)}) ↔ ((𝐿‘(𝑦↑2)) ∈ (Base‘𝑌) ∧ ((𝑂‘𝑇)‘(𝐿‘(𝑦↑2))) = (0_g‘𝑌))))
126	124, 125	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...((𝑃 − 1) / 2))) → ((𝐿‘(𝑦↑2)) ∈ (^◡(𝑂‘𝑇) “ {(0_g‘𝑌)}) ↔ ((𝐿‘(𝑦↑2)) ∈ (Base‘𝑌) ∧ ((𝑂‘𝑇)‘(𝐿‘(𝑦↑2))) = (0_g‘𝑌))))
127	25, 97, 126	mpbir2and 713	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...((𝑃 − 1) / 2))) → (𝐿‘(𝑦↑2)) ∈ (^◡(𝑂‘𝑇) “ {(0_g‘𝑌)}))
128		lgsqr.g	. . 3 ⊢ 𝐺 = (𝑦 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(𝑦↑2)))
129	127, 128	fmptd 7134	. 2 ⊢ (𝜑 → 𝐺:(1...((𝑃 − 1) / 2))⟶(^◡(𝑂‘𝑇) “ {(0_g‘𝑌)}))
130		fvoveq1 7454	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝐿‘(𝑦↑2)) = (𝐿‘(𝑥↑2)))
131		fvex 6919	. . . . . . . 8 ⊢ (𝐿‘(𝑥↑2)) ∈ V
132	130, 128, 131	fvmpt 7016	. . . . . . 7 ⊢ (𝑥 ∈ (1...((𝑃 − 1) / 2)) → (𝐺‘𝑥) = (𝐿‘(𝑥↑2)))
133	132	ad2antrl 728	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ∧ 𝑧 ∈ (1...((𝑃 − 1) / 2)))) → (𝐺‘𝑥) = (𝐿‘(𝑥↑2)))
134		fvoveq1 7454	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → (𝐿‘(𝑦↑2)) = (𝐿‘(𝑧↑2)))
135		fvex 6919	. . . . . . . 8 ⊢ (𝐿‘(𝑧↑2)) ∈ V
136	134, 128, 135	fvmpt 7016	. . . . . . 7 ⊢ (𝑧 ∈ (1...((𝑃 − 1) / 2)) → (𝐺‘𝑧) = (𝐿‘(𝑧↑2)))
137	136	ad2antll 729	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ∧ 𝑧 ∈ (1...((𝑃 − 1) / 2)))) → (𝐺‘𝑧) = (𝐿‘(𝑧↑2)))
138	133, 137	eqeq12d 2753	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ∧ 𝑧 ∈ (1...((𝑃 − 1) / 2)))) → ((𝐺‘𝑥) = (𝐺‘𝑧) ↔ (𝐿‘(𝑥↑2)) = (𝐿‘(𝑧↑2))))
139	47	nnnn0d 12587	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ ℕ₀)
140	139	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ∧ 𝑧 ∈ (1...((𝑃 − 1) / 2)))) → 𝑃 ∈ ℕ₀)
141		elfzelz 13564	. . . . . . . 8 ⊢ (𝑥 ∈ (1...((𝑃 − 1) / 2)) → 𝑥 ∈ ℤ)
142	141	ad2antrl 728	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ∧ 𝑧 ∈ (1...((𝑃 − 1) / 2)))) → 𝑥 ∈ ℤ)
143		zsqcl 14169	. . . . . . 7 ⊢ (𝑥 ∈ ℤ → (𝑥↑2) ∈ ℤ)
144	142, 143	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ∧ 𝑧 ∈ (1...((𝑃 − 1) / 2)))) → (𝑥↑2) ∈ ℤ)
145		elfzelz 13564	. . . . . . . 8 ⊢ (𝑧 ∈ (1...((𝑃 − 1) / 2)) → 𝑧 ∈ ℤ)
146	145	ad2antll 729	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ∧ 𝑧 ∈ (1...((𝑃 − 1) / 2)))) → 𝑧 ∈ ℤ)
147		zsqcl 14169	. . . . . . 7 ⊢ (𝑧 ∈ ℤ → (𝑧↑2) ∈ ℤ)
148	146, 147	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ∧ 𝑧 ∈ (1...((𝑃 − 1) / 2)))) → (𝑧↑2) ∈ ℤ)
149	3, 13	zndvds 21568	. . . . . 6 ⊢ ((𝑃 ∈ ℕ₀ ∧ (𝑥↑2) ∈ ℤ ∧ (𝑧↑2) ∈ ℤ) → ((𝐿‘(𝑥↑2)) = (𝐿‘(𝑧↑2)) ↔ 𝑃 ∥ ((𝑥↑2) − (𝑧↑2))))
150	140, 144, 148, 149	syl3anc 1373	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ∧ 𝑧 ∈ (1...((𝑃 − 1) / 2)))) → ((𝐿‘(𝑥↑2)) = (𝐿‘(𝑧↑2)) ↔ 𝑃 ∥ ((𝑥↑2) − (𝑧↑2))))
151		elfznn 13593	. . . . . . . . 9 ⊢ (𝑥 ∈ (1...((𝑃 − 1) / 2)) → 𝑥 ∈ ℕ)
152	151	ad2antrl 728	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ∧ 𝑧 ∈ (1...((𝑃 − 1) / 2)))) → 𝑥 ∈ ℕ)
153	152	nncnd 12282	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ∧ 𝑧 ∈ (1...((𝑃 − 1) / 2)))) → 𝑥 ∈ ℂ)
154		elfznn 13593	. . . . . . . . 9 ⊢ (𝑧 ∈ (1...((𝑃 − 1) / 2)) → 𝑧 ∈ ℕ)
155	154	ad2antll 729	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ∧ 𝑧 ∈ (1...((𝑃 − 1) / 2)))) → 𝑧 ∈ ℕ)
156	155	nncnd 12282	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ∧ 𝑧 ∈ (1...((𝑃 − 1) / 2)))) → 𝑧 ∈ ℂ)
157		subsq 14249	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((𝑥↑2) − (𝑧↑2)) = ((𝑥 + 𝑧) · (𝑥 − 𝑧)))
158	153, 156, 157	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ∧ 𝑧 ∈ (1...((𝑃 − 1) / 2)))) → ((𝑥↑2) − (𝑧↑2)) = ((𝑥 + 𝑧) · (𝑥 − 𝑧)))
159	158	breq2d 5155	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ∧ 𝑧 ∈ (1...((𝑃 − 1) / 2)))) → (𝑃 ∥ ((𝑥↑2) − (𝑧↑2)) ↔ 𝑃 ∥ ((𝑥 + 𝑧) · (𝑥 − 𝑧))))
160	138, 150, 159	3bitrd 305	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ∧ 𝑧 ∈ (1...((𝑃 − 1) / 2)))) → ((𝐺‘𝑥) = (𝐺‘𝑧) ↔ 𝑃 ∥ ((𝑥 + 𝑧) · (𝑥 − 𝑧))))
161	2	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ∧ 𝑧 ∈ (1...((𝑃 − 1) / 2)))) → 𝑃 ∈ ℙ)
162	142, 146	zaddcld 12726	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ∧ 𝑧 ∈ (1...((𝑃 − 1) / 2)))) → (𝑥 + 𝑧) ∈ ℤ)
163	142, 146	zsubcld 12727	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ∧ 𝑧 ∈ (1...((𝑃 − 1) / 2)))) → (𝑥 − 𝑧) ∈ ℤ)
164		euclemma 16750	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ (𝑥 + 𝑧) ∈ ℤ ∧ (𝑥 − 𝑧) ∈ ℤ) → (𝑃 ∥ ((𝑥 + 𝑧) · (𝑥 − 𝑧)) ↔ (𝑃 ∥ (𝑥 + 𝑧) ∨ 𝑃 ∥ (𝑥 − 𝑧))))
165	161, 162, 163, 164	syl3anc 1373	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ∧ 𝑧 ∈ (1...((𝑃 − 1) / 2)))) → (𝑃 ∥ ((𝑥 + 𝑧) · (𝑥 − 𝑧)) ↔ (𝑃 ∥ (𝑥 + 𝑧) ∨ 𝑃 ∥ (𝑥 − 𝑧))))
166	161, 46	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ∧ 𝑧 ∈ (1...((𝑃 − 1) / 2)))) → 𝑃 ∈ ℕ)
167	166	nnzd 12640	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ∧ 𝑧 ∈ (1...((𝑃 − 1) / 2)))) → 𝑃 ∈ ℤ)
168	152, 155	nnaddcld 12318	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ∧ 𝑧 ∈ (1...((𝑃 − 1) / 2)))) → (𝑥 + 𝑧) ∈ ℕ)
169		dvdsle 16347	. . . . . . . 8 ⊢ ((𝑃 ∈ ℤ ∧ (𝑥 + 𝑧) ∈ ℕ) → (𝑃 ∥ (𝑥 + 𝑧) → 𝑃 ≤ (𝑥 + 𝑧)))
170	167, 168, 169	syl2anc 584	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ∧ 𝑧 ∈ (1...((𝑃 − 1) / 2)))) → (𝑃 ∥ (𝑥 + 𝑧) → 𝑃 ≤ (𝑥 + 𝑧)))
171	168	nnred 12281	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ∧ 𝑧 ∈ (1...((𝑃 − 1) / 2)))) → (𝑥 + 𝑧) ∈ ℝ)
172	166	nnred 12281	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ∧ 𝑧 ∈ (1...((𝑃 − 1) / 2)))) → 𝑃 ∈ ℝ)
173	172, 49	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ∧ 𝑧 ∈ (1...((𝑃 − 1) / 2)))) → (𝑃 − 1) ∈ ℝ)
174	152	nnred 12281	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ∧ 𝑧 ∈ (1...((𝑃 − 1) / 2)))) → 𝑥 ∈ ℝ)
175	155	nnred 12281	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ∧ 𝑧 ∈ (1...((𝑃 − 1) / 2)))) → 𝑧 ∈ ℝ)
176	69	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ∧ 𝑧 ∈ (1...((𝑃 − 1) / 2)))) → ((𝑃 − 1) / 2) ∈ ℝ)
177		elfzle2 13568	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (1...((𝑃 − 1) / 2)) → 𝑥 ≤ ((𝑃 − 1) / 2))
178	177	ad2antrl 728	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ∧ 𝑧 ∈ (1...((𝑃 − 1) / 2)))) → 𝑥 ≤ ((𝑃 − 1) / 2))
179		elfzle2 13568	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ (1...((𝑃 − 1) / 2)) → 𝑧 ≤ ((𝑃 − 1) / 2))
180	179	ad2antll 729	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ∧ 𝑧 ∈ (1...((𝑃 − 1) / 2)))) → 𝑧 ≤ ((𝑃 − 1) / 2))
181	174, 175, 176, 176, 178, 180	le2addd 11882	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ∧ 𝑧 ∈ (1...((𝑃 − 1) / 2)))) → (𝑥 + 𝑧) ≤ (((𝑃 − 1) / 2) + ((𝑃 − 1) / 2)))
182	51	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ∧ 𝑧 ∈ (1...((𝑃 − 1) / 2)))) → (𝑃 − 1) ∈ ℂ)
183	182	2halvesd 12512	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ∧ 𝑧 ∈ (1...((𝑃 − 1) / 2)))) → (((𝑃 − 1) / 2) + ((𝑃 − 1) / 2)) = (𝑃 − 1))
184	181, 183	breqtrd 5169	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ∧ 𝑧 ∈ (1...((𝑃 − 1) / 2)))) → (𝑥 + 𝑧) ≤ (𝑃 − 1))
185	172	ltm1d 12200	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ∧ 𝑧 ∈ (1...((𝑃 − 1) / 2)))) → (𝑃 − 1) < 𝑃)
186	171, 173, 172, 184, 185	lelttrd 11419	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ∧ 𝑧 ∈ (1...((𝑃 − 1) / 2)))) → (𝑥 + 𝑧) < 𝑃)
187	171, 172	ltnled 11408	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ∧ 𝑧 ∈ (1...((𝑃 − 1) / 2)))) → ((𝑥 + 𝑧) < 𝑃 ↔ ¬ 𝑃 ≤ (𝑥 + 𝑧)))
188	186, 187	mpbid 232	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ∧ 𝑧 ∈ (1...((𝑃 − 1) / 2)))) → ¬ 𝑃 ≤ (𝑥 + 𝑧))
189	188	pm2.21d 121	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ∧ 𝑧 ∈ (1...((𝑃 − 1) / 2)))) → (𝑃 ≤ (𝑥 + 𝑧) → 𝑥 = 𝑧))
190	170, 189	syld 47	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ∧ 𝑧 ∈ (1...((𝑃 − 1) / 2)))) → (𝑃 ∥ (𝑥 + 𝑧) → 𝑥 = 𝑧))
191		moddvds 16301	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℕ ∧ 𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ) → ((𝑥 mod 𝑃) = (𝑧 mod 𝑃) ↔ 𝑃 ∥ (𝑥 − 𝑧)))
192	166, 142, 146, 191	syl3anc 1373	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ∧ 𝑧 ∈ (1...((𝑃 − 1) / 2)))) → ((𝑥 mod 𝑃) = (𝑧 mod 𝑃) ↔ 𝑃 ∥ (𝑥 − 𝑧)))
193	166	nnrpd 13075	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ∧ 𝑧 ∈ (1...((𝑃 − 1) / 2)))) → 𝑃 ∈ ℝ⁺)
194	152	nnnn0d 12587	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ∧ 𝑧 ∈ (1...((𝑃 − 1) / 2)))) → 𝑥 ∈ ℕ₀)
195	194	nn0ge0d 12590	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ∧ 𝑧 ∈ (1...((𝑃 − 1) / 2)))) → 0 ≤ 𝑥)
196	82	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ∧ 𝑧 ∈ (1...((𝑃 − 1) / 2)))) → ((𝑃 − 1) / 2) < 𝑃)
197	174, 176, 172, 178, 196	lelttrd 11419	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ∧ 𝑧 ∈ (1...((𝑃 − 1) / 2)))) → 𝑥 < 𝑃)
198		modid 13936	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℝ ∧ 𝑃 ∈ ℝ⁺) ∧ (0 ≤ 𝑥 ∧ 𝑥 < 𝑃)) → (𝑥 mod 𝑃) = 𝑥)
199	174, 193, 195, 197, 198	syl22anc 839	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ∧ 𝑧 ∈ (1...((𝑃 − 1) / 2)))) → (𝑥 mod 𝑃) = 𝑥)
200	155	nnnn0d 12587	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ∧ 𝑧 ∈ (1...((𝑃 − 1) / 2)))) → 𝑧 ∈ ℕ₀)
201	200	nn0ge0d 12590	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ∧ 𝑧 ∈ (1...((𝑃 − 1) / 2)))) → 0 ≤ 𝑧)
202	175, 176, 172, 180, 196	lelttrd 11419	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ∧ 𝑧 ∈ (1...((𝑃 − 1) / 2)))) → 𝑧 < 𝑃)
203		modid 13936	. . . . . . . . . 10 ⊢ (((𝑧 ∈ ℝ ∧ 𝑃 ∈ ℝ⁺) ∧ (0 ≤ 𝑧 ∧ 𝑧 < 𝑃)) → (𝑧 mod 𝑃) = 𝑧)
204	175, 193, 201, 202, 203	syl22anc 839	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ∧ 𝑧 ∈ (1...((𝑃 − 1) / 2)))) → (𝑧 mod 𝑃) = 𝑧)
205	199, 204	eqeq12d 2753	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ∧ 𝑧 ∈ (1...((𝑃 − 1) / 2)))) → ((𝑥 mod 𝑃) = (𝑧 mod 𝑃) ↔ 𝑥 = 𝑧))
206	192, 205	bitr3d 281	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ∧ 𝑧 ∈ (1...((𝑃 − 1) / 2)))) → (𝑃 ∥ (𝑥 − 𝑧) ↔ 𝑥 = 𝑧))
207	206	biimpd 229	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ∧ 𝑧 ∈ (1...((𝑃 − 1) / 2)))) → (𝑃 ∥ (𝑥 − 𝑧) → 𝑥 = 𝑧))
208	190, 207	jaod 860	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ∧ 𝑧 ∈ (1...((𝑃 − 1) / 2)))) → ((𝑃 ∥ (𝑥 + 𝑧) ∨ 𝑃 ∥ (𝑥 − 𝑧)) → 𝑥 = 𝑧))
209	165, 208	sylbid 240	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ∧ 𝑧 ∈ (1...((𝑃 − 1) / 2)))) → (𝑃 ∥ ((𝑥 + 𝑧) · (𝑥 − 𝑧)) → 𝑥 = 𝑧))
210	160, 209	sylbid 240	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...((𝑃 − 1) / 2)) ∧ 𝑧 ∈ (1...((𝑃 − 1) / 2)))) → ((𝐺‘𝑥) = (𝐺‘𝑧) → 𝑥 = 𝑧))
211	210	ralrimivva 3202	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (1...((𝑃 − 1) / 2))∀𝑧 ∈ (1...((𝑃 − 1) / 2))((𝐺‘𝑥) = (𝐺‘𝑧) → 𝑥 = 𝑧))
212		dff13 7275	. 2 ⊢ (𝐺:(1...((𝑃 − 1) / 2))–1-1→(^◡(𝑂‘𝑇) “ {(0_g‘𝑌)}) ↔ (𝐺:(1...((𝑃 − 1) / 2))⟶(^◡(𝑂‘𝑇) “ {(0_g‘𝑌)}) ∧ ∀𝑥 ∈ (1...((𝑃 − 1) / 2))∀𝑧 ∈ (1...((𝑃 − 1) / 2))((𝐺‘𝑥) = (𝐺‘𝑧) → 𝑥 = 𝑧)))
213	129, 211, 212	sylanbrc 583	1 ⊢ (𝜑 → 𝐺:(1...((𝑃 − 1) / 2))–1-1→(^◡(𝑂‘𝑇) “ {(0_g‘𝑌)}))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 Vcvv 3480 ∖ cdif 3948 {csn 4626 class class class wbr 5143 ↦ cmpt 5225 ^◡ccnv 5684 “ cima 5688 Fn wfn 6556 ⟶wf 6557 –1-1→wf1 6558 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 ℝcr 11154 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 < clt 11295 ≤ cle 11296 − cmin 11492 / cdiv 11920 ℕcn 12266 2c2 12321 ℕ₀cn0 12526 ℤcz 12613 ℤ_≥cuz 12878 ℝ⁺crp 13034 ...cfz 13547 mod cmo 13909 ↑cexp 14102 ∥ cdvds 16290 gcd cgcd 16531 ℙcprime 16708 ϕcphi 16801 Basecbs 17247 0_gc0g 17484 ↑_s cpws 17491 Mndcmnd 18747 Grpcgrp 18951 -_gcsg 18953 ._gcmg 19085 mulGrpcmgp 20137 1_rcur 20178 Ringcrg 20230 CRingccrg 20231 RingHom crh 20469 Domncdomn 20692 IDomncidom 20693 Fieldcfield 20730 ℤ_ringczring 21457 ℤRHomczrh 21510 ℤ/nℤczn 21513 var₁cv1 22177 Poly₁cpl1 22178 eval₁ce1 22318 deg₁cdg1 26093

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 ax-mulf 11235

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofr 7698 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-oadd 8510 df-er 8745 df-ec 8747 df-qs 8751 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-sup 9482 df-inf 9483 df-oi 9550 df-dju 9941 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-xnn0 12600 df-z 12614 df-dec 12734 df-uz 12879 df-rp 13035 df-fz 13548 df-fzo 13695 df-fl 13832 df-mod 13910 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-dvds 16291 df-gcd 16532 df-prm 16709 df-phi 16803 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-0g 17486 df-gsum 17487 df-prds 17492 df-pws 17494 df-imas 17553 df-qus 17554 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mhm 18796 df-submnd 18797 df-grp 18954 df-minusg 18955 df-sbg 18956 df-mulg 19086 df-subg 19141 df-nsg 19142 df-eqg 19143 df-ghm 19231 df-cntz 19335 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-srg 20184 df-ring 20232 df-cring 20233 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-invr 20388 df-dvr 20401 df-rhm 20472 df-nzr 20513 df-subrng 20546 df-subrg 20570 df-rlreg 20694 df-domn 20695 df-idom 20696 df-drng 20731 df-field 20732 df-lmod 20860 df-lss 20930 df-lsp 20970 df-sra 21172 df-rgmod 21173 df-lidl 21218 df-rsp 21219 df-2idl 21260 df-cnfld 21365 df-zring 21458 df-zrh 21514 df-zn 21517 df-assa 21873 df-asp 21874 df-ascl 21875 df-psr 21929 df-mvr 21930 df-mpl 21931 df-opsr 21933 df-evls 22098 df-evl 22099 df-psr1 22181 df-vr1 22182 df-ply1 22183 df-evl1 22320

This theorem is referenced by: lgsqrlem4 27393

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