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Theorem lgsqrlem1 27234

Description: Lemma for lgsqr 27239. (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}))
lgsqrlem1.3	⊢ (𝜑 → 𝐴 ∈ ℤ)
lgsqrlem1.4	⊢ (𝜑 → ((𝐴↑((𝑃 − 1) / 2)) mod 𝑃) = (1 mod 𝑃))

**Assertion**
Ref	Expression
lgsqrlem1	⊢ (𝜑 → ((𝑂‘𝑇)‘(𝐿‘𝐴)) = (0_g‘𝑌))

**Proof of Theorem lgsqrlem1**
Step	Hyp	Ref	Expression
1		lgsqr.t	. . . . 5 ⊢ 𝑇 = ((((𝑃 − 1) / 2) ↑ 𝑋) − 1 )
2	1	fveq2i 6888	. . . 4 ⊢ (𝑂‘𝑇) = (𝑂‘((((𝑃 − 1) / 2) ↑ 𝑋) − 1 ))
3	2	fveq1i 6886	. . 3 ⊢ ((𝑂‘𝑇)‘(𝐿‘𝐴)) = ((𝑂‘((((𝑃 − 1) / 2) ↑ 𝑋) − 1 ))‘(𝐿‘𝐴))
4		lgsqr.o	. . . . 5 ⊢ 𝑂 = (eval₁‘𝑌)
5		lgsqr.s	. . . . 5 ⊢ 𝑆 = (Poly₁‘𝑌)
6		eqid 2726	. . . . 5 ⊢ (Base‘𝑌) = (Base‘𝑌)
7		lgsqr.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑆)
8		lgsqr.1	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))
9	8	eldifad 3955	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ ℙ)
10		lgsqr.y	. . . . . . . . 9 ⊢ 𝑌 = (ℤ/nℤ‘𝑃)
11	10	znfld 21455	. . . . . . . 8 ⊢ (𝑃 ∈ ℙ → 𝑌 ∈ Field)
12	9, 11	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ Field)
13		fldidom 21219	. . . . . . 7 ⊢ (𝑌 ∈ Field → 𝑌 ∈ IDomn)
14	12, 13	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ IDomn)
15		isidom 21216	. . . . . . 7 ⊢ (𝑌 ∈ IDomn ↔ (𝑌 ∈ CRing ∧ 𝑌 ∈ Domn))
16	15	simplbi 497	. . . . . 6 ⊢ (𝑌 ∈ IDomn → 𝑌 ∈ CRing)
17	14, 16	syl 17	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ CRing)
18		crngring 20150	. . . . . . . . 9 ⊢ (𝑌 ∈ CRing → 𝑌 ∈ Ring)
19	17, 18	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ Ring)
20		lgsqr.l	. . . . . . . . 9 ⊢ 𝐿 = (ℤRHom‘𝑌)
21	20	zrhrhm 21398	. . . . . . . 8 ⊢ (𝑌 ∈ Ring → 𝐿 ∈ (ℤ_ring RingHom 𝑌))
22	19, 21	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ (ℤ_ring RingHom 𝑌))
23		zringbas 21340	. . . . . . . 8 ⊢ ℤ = (Base‘ℤ_ring)
24	23, 6	rhmf 20387	. . . . . . 7 ⊢ (𝐿 ∈ (ℤ_ring RingHom 𝑌) → 𝐿:ℤ⟶(Base‘𝑌))
25	22, 24	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐿:ℤ⟶(Base‘𝑌))
26		lgsqrlem1.3	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℤ)
27	25, 26	ffvelcdmd 7081	. . . . 5 ⊢ (𝜑 → (𝐿‘𝐴) ∈ (Base‘𝑌))
28		lgsqr.x	. . . . . . . 8 ⊢ 𝑋 = (var₁‘𝑌)
29	4, 28, 6, 5, 7, 17, 27	evl1vard 22211	. . . . . . 7 ⊢ (𝜑 → (𝑋 ∈ 𝐵 ∧ ((𝑂‘𝑋)‘(𝐿‘𝐴)) = (𝐿‘𝐴)))
30		lgsqr.e	. . . . . . 7 ⊢ ↑ = (._g‘(mulGrp‘𝑆))
31		eqid 2726	. . . . . . 7 ⊢ (._g‘(mulGrp‘𝑌)) = (._g‘(mulGrp‘𝑌))
32		oddprm 16752	. . . . . . . . 9 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((𝑃 − 1) / 2) ∈ ℕ)
33	8, 32	syl 17	. . . . . . . 8 ⊢ (𝜑 → ((𝑃 − 1) / 2) ∈ ℕ)
34	33	nnnn0d 12536	. . . . . . 7 ⊢ (𝜑 → ((𝑃 − 1) / 2) ∈ ℕ₀)
35	4, 5, 6, 7, 17, 27, 29, 30, 31, 34	evl1expd 22219	. . . . . 6 ⊢ (𝜑 → ((((𝑃 − 1) / 2) ↑ 𝑋) ∈ 𝐵 ∧ ((𝑂‘(((𝑃 − 1) / 2) ↑ 𝑋))‘(𝐿‘𝐴)) = (((𝑃 − 1) / 2)(._g‘(mulGrp‘𝑌))(𝐿‘𝐴))))
36		zringmpg 21358	. . . . . . . . . . . 12 ⊢ ((mulGrp‘ℂ_fld) ↾_s ℤ) = (mulGrp‘ℤ_ring)
37		eqid 2726	. . . . . . . . . . . 12 ⊢ (mulGrp‘𝑌) = (mulGrp‘𝑌)
38	36, 37	rhmmhm 20381	. . . . . . . . . . 11 ⊢ (𝐿 ∈ (ℤ_ring RingHom 𝑌) → 𝐿 ∈ (((mulGrp‘ℂ_fld) ↾_s ℤ) MndHom (mulGrp‘𝑌)))
39	22, 38	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐿 ∈ (((mulGrp‘ℂ_fld) ↾_s ℤ) MndHom (mulGrp‘𝑌)))
40	36, 23	mgpbas 20045	. . . . . . . . . . 11 ⊢ ℤ = (Base‘((mulGrp‘ℂ_fld) ↾_s ℤ))
41		eqid 2726	. . . . . . . . . . 11 ⊢ (._g‘((mulGrp‘ℂ_fld) ↾_s ℤ)) = (._g‘((mulGrp‘ℂ_fld) ↾_s ℤ))
42	40, 41, 31	mhmmulg 19042	. . . . . . . . . 10 ⊢ ((𝐿 ∈ (((mulGrp‘ℂ_fld) ↾_s ℤ) MndHom (mulGrp‘𝑌)) ∧ ((𝑃 − 1) / 2) ∈ ℕ₀ ∧ 𝐴 ∈ ℤ) → (𝐿‘(((𝑃 − 1) / 2)(._g‘((mulGrp‘ℂ_fld) ↾_s ℤ))𝐴)) = (((𝑃 − 1) / 2)(._g‘(mulGrp‘𝑌))(𝐿‘𝐴)))
43	39, 34, 26, 42	syl3anc 1368	. . . . . . . . 9 ⊢ (𝜑 → (𝐿‘(((𝑃 − 1) / 2)(._g‘((mulGrp‘ℂ_fld) ↾_s ℤ))𝐴)) = (((𝑃 − 1) / 2)(._g‘(mulGrp‘𝑌))(𝐿‘𝐴)))
44		zsubrg 21314	. . . . . . . . . . . . . 14 ⊢ ℤ ∈ (SubRing‘ℂ_fld)
45		eqid 2726	. . . . . . . . . . . . . . 15 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
46	45	subrgsubm 20487	. . . . . . . . . . . . . 14 ⊢ (ℤ ∈ (SubRing‘ℂ_fld) → ℤ ∈ (SubMnd‘(mulGrp‘ℂ_fld)))
47	44, 46	mp1i 13	. . . . . . . . . . . . 13 ⊢ (𝜑 → ℤ ∈ (SubMnd‘(mulGrp‘ℂ_fld)))
48		eqid 2726	. . . . . . . . . . . . . 14 ⊢ (._g‘(mulGrp‘ℂ_fld)) = (._g‘(mulGrp‘ℂ_fld))
49		eqid 2726	. . . . . . . . . . . . . 14 ⊢ ((mulGrp‘ℂ_fld) ↾_s ℤ) = ((mulGrp‘ℂ_fld) ↾_s ℤ)
50	48, 49, 41	submmulg 19045	. . . . . . . . . . . . 13 ⊢ ((ℤ ∈ (SubMnd‘(mulGrp‘ℂ_fld)) ∧ ((𝑃 − 1) / 2) ∈ ℕ₀ ∧ 𝐴 ∈ ℤ) → (((𝑃 − 1) / 2)(._g‘(mulGrp‘ℂ_fld))𝐴) = (((𝑃 − 1) / 2)(._g‘((mulGrp‘ℂ_fld) ↾_s ℤ))𝐴))
51	47, 34, 26, 50	syl3anc 1368	. . . . . . . . . . . 12 ⊢ (𝜑 → (((𝑃 − 1) / 2)(._g‘(mulGrp‘ℂ_fld))𝐴) = (((𝑃 − 1) / 2)(._g‘((mulGrp‘ℂ_fld) ↾_s ℤ))𝐴))
52	26	zcnd 12671	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ∈ ℂ)
53		cnfldexp 21293	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ ((𝑃 − 1) / 2) ∈ ℕ₀) → (((𝑃 − 1) / 2)(._g‘(mulGrp‘ℂ_fld))𝐴) = (𝐴↑((𝑃 − 1) / 2)))
54	52, 34, 53	syl2anc 583	. . . . . . . . . . . 12 ⊢ (𝜑 → (((𝑃 − 1) / 2)(._g‘(mulGrp‘ℂ_fld))𝐴) = (𝐴↑((𝑃 − 1) / 2)))
55	51, 54	eqtr3d 2768	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝑃 − 1) / 2)(._g‘((mulGrp‘ℂ_fld) ↾_s ℤ))𝐴) = (𝐴↑((𝑃 − 1) / 2)))
56	55	fveq2d 6889	. . . . . . . . . 10 ⊢ (𝜑 → (𝐿‘(((𝑃 − 1) / 2)(._g‘((mulGrp‘ℂ_fld) ↾_s ℤ))𝐴)) = (𝐿‘(𝐴↑((𝑃 − 1) / 2))))
57		lgsqrlem1.4	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐴↑((𝑃 − 1) / 2)) mod 𝑃) = (1 mod 𝑃))
58		prmnn 16618	. . . . . . . . . . . . . 14 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
59	9, 58	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑃 ∈ ℕ)
60		zexpcl 14047	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℤ ∧ ((𝑃 − 1) / 2) ∈ ℕ₀) → (𝐴↑((𝑃 − 1) / 2)) ∈ ℤ)
61	26, 34, 60	syl2anc 583	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴↑((𝑃 − 1) / 2)) ∈ ℤ)
62		1zzd 12597	. . . . . . . . . . . . 13 ⊢ (𝜑 → 1 ∈ ℤ)
63		moddvds 16215	. . . . . . . . . . . . 13 ⊢ ((𝑃 ∈ ℕ ∧ (𝐴↑((𝑃 − 1) / 2)) ∈ ℤ ∧ 1 ∈ ℤ) → (((𝐴↑((𝑃 − 1) / 2)) mod 𝑃) = (1 mod 𝑃) ↔ 𝑃 ∥ ((𝐴↑((𝑃 − 1) / 2)) − 1)))
64	59, 61, 62, 63	syl3anc 1368	. . . . . . . . . . . 12 ⊢ (𝜑 → (((𝐴↑((𝑃 − 1) / 2)) mod 𝑃) = (1 mod 𝑃) ↔ 𝑃 ∥ ((𝐴↑((𝑃 − 1) / 2)) − 1)))
65	57, 64	mpbid 231	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ∥ ((𝐴↑((𝑃 − 1) / 2)) − 1))
66	59	nnnn0d 12536	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑃 ∈ ℕ₀)
67	10, 20	zndvds 21444	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ ℕ₀ ∧ (𝐴↑((𝑃 − 1) / 2)) ∈ ℤ ∧ 1 ∈ ℤ) → ((𝐿‘(𝐴↑((𝑃 − 1) / 2))) = (𝐿‘1) ↔ 𝑃 ∥ ((𝐴↑((𝑃 − 1) / 2)) − 1)))
68	66, 61, 62, 67	syl3anc 1368	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐿‘(𝐴↑((𝑃 − 1) / 2))) = (𝐿‘1) ↔ 𝑃 ∥ ((𝐴↑((𝑃 − 1) / 2)) − 1)))
69	65, 68	mpbird 257	. . . . . . . . . 10 ⊢ (𝜑 → (𝐿‘(𝐴↑((𝑃 − 1) / 2))) = (𝐿‘1))
70		zring1 21346	. . . . . . . . . . . 12 ⊢ 1 = (1_r‘ℤ_ring)
71		eqid 2726	. . . . . . . . . . . 12 ⊢ (1_r‘𝑌) = (1_r‘𝑌)
72	70, 71	rhm1 20391	. . . . . . . . . . 11 ⊢ (𝐿 ∈ (ℤ_ring RingHom 𝑌) → (𝐿‘1) = (1_r‘𝑌))
73	22, 72	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝐿‘1) = (1_r‘𝑌))
74	56, 69, 73	3eqtrd 2770	. . . . . . . . 9 ⊢ (𝜑 → (𝐿‘(((𝑃 − 1) / 2)(._g‘((mulGrp‘ℂ_fld) ↾_s ℤ))𝐴)) = (1_r‘𝑌))
75	43, 74	eqtr3d 2768	. . . . . . . 8 ⊢ (𝜑 → (((𝑃 − 1) / 2)(._g‘(mulGrp‘𝑌))(𝐿‘𝐴)) = (1_r‘𝑌))
76	75	eqeq2d 2737	. . . . . . 7 ⊢ (𝜑 → (((𝑂‘(((𝑃 − 1) / 2) ↑ 𝑋))‘(𝐿‘𝐴)) = (((𝑃 − 1) / 2)(._g‘(mulGrp‘𝑌))(𝐿‘𝐴)) ↔ ((𝑂‘(((𝑃 − 1) / 2) ↑ 𝑋))‘(𝐿‘𝐴)) = (1_r‘𝑌)))
77	76	anbi2d 628	. . . . . 6 ⊢ (𝜑 → (((((𝑃 − 1) / 2) ↑ 𝑋) ∈ 𝐵 ∧ ((𝑂‘(((𝑃 − 1) / 2) ↑ 𝑋))‘(𝐿‘𝐴)) = (((𝑃 − 1) / 2)(._g‘(mulGrp‘𝑌))(𝐿‘𝐴))) ↔ ((((𝑃 − 1) / 2) ↑ 𝑋) ∈ 𝐵 ∧ ((𝑂‘(((𝑃 − 1) / 2) ↑ 𝑋))‘(𝐿‘𝐴)) = (1_r‘𝑌))))
78	35, 77	mpbid 231	. . . . 5 ⊢ (𝜑 → ((((𝑃 − 1) / 2) ↑ 𝑋) ∈ 𝐵 ∧ ((𝑂‘(((𝑃 − 1) / 2) ↑ 𝑋))‘(𝐿‘𝐴)) = (1_r‘𝑌)))
79		eqid 2726	. . . . . . 7 ⊢ (algSc‘𝑆) = (algSc‘𝑆)
80	6, 71	ringidcl 20165	. . . . . . . 8 ⊢ (𝑌 ∈ Ring → (1_r‘𝑌) ∈ (Base‘𝑌))
81	19, 80	syl 17	. . . . . . 7 ⊢ (𝜑 → (1_r‘𝑌) ∈ (Base‘𝑌))
82	4, 5, 6, 79, 7, 17, 81, 27	evl1scad 22209	. . . . . 6 ⊢ (𝜑 → (((algSc‘𝑆)‘(1_r‘𝑌)) ∈ 𝐵 ∧ ((𝑂‘((algSc‘𝑆)‘(1_r‘𝑌)))‘(𝐿‘𝐴)) = (1_r‘𝑌)))
83		lgsqr.u	. . . . . . . . . 10 ⊢ 1 = (1_r‘𝑆)
84	5, 79, 71, 83	ply1scl1 22167	. . . . . . . . 9 ⊢ (𝑌 ∈ Ring → ((algSc‘𝑆)‘(1_r‘𝑌)) = 1 )
85	19, 84	syl 17	. . . . . . . 8 ⊢ (𝜑 → ((algSc‘𝑆)‘(1_r‘𝑌)) = 1 )
86	85	eleq1d 2812	. . . . . . 7 ⊢ (𝜑 → (((algSc‘𝑆)‘(1_r‘𝑌)) ∈ 𝐵 ↔ 1 ∈ 𝐵))
87	85	fveq2d 6889	. . . . . . . . 9 ⊢ (𝜑 → (𝑂‘((algSc‘𝑆)‘(1_r‘𝑌))) = (𝑂‘ 1 ))
88	87	fveq1d 6887	. . . . . . . 8 ⊢ (𝜑 → ((𝑂‘((algSc‘𝑆)‘(1_r‘𝑌)))‘(𝐿‘𝐴)) = ((𝑂‘ 1 )‘(𝐿‘𝐴)))
89	88	eqeq1d 2728	. . . . . . 7 ⊢ (𝜑 → (((𝑂‘((algSc‘𝑆)‘(1_r‘𝑌)))‘(𝐿‘𝐴)) = (1_r‘𝑌) ↔ ((𝑂‘ 1 )‘(𝐿‘𝐴)) = (1_r‘𝑌)))
90	86, 89	anbi12d 630	. . . . . 6 ⊢ (𝜑 → ((((algSc‘𝑆)‘(1_r‘𝑌)) ∈ 𝐵 ∧ ((𝑂‘((algSc‘𝑆)‘(1_r‘𝑌)))‘(𝐿‘𝐴)) = (1_r‘𝑌)) ↔ ( 1 ∈ 𝐵 ∧ ((𝑂‘ 1 )‘(𝐿‘𝐴)) = (1_r‘𝑌))))
91	82, 90	mpbid 231	. . . . 5 ⊢ (𝜑 → ( 1 ∈ 𝐵 ∧ ((𝑂‘ 1 )‘(𝐿‘𝐴)) = (1_r‘𝑌)))
92		lgsqr.m	. . . . 5 ⊢ − = (-_g‘𝑆)
93		eqid 2726	. . . . 5 ⊢ (-_g‘𝑌) = (-_g‘𝑌)
94	4, 5, 6, 7, 17, 27, 78, 91, 92, 93	evl1subd 22216	. . . 4 ⊢ (𝜑 → (((((𝑃 − 1) / 2) ↑ 𝑋) − 1 ) ∈ 𝐵 ∧ ((𝑂‘((((𝑃 − 1) / 2) ↑ 𝑋) − 1 ))‘(𝐿‘𝐴)) = ((1_r‘𝑌)(-_g‘𝑌)(1_r‘𝑌))))
95	94	simprd 495	. . 3 ⊢ (𝜑 → ((𝑂‘((((𝑃 − 1) / 2) ↑ 𝑋) − 1 ))‘(𝐿‘𝐴)) = ((1_r‘𝑌)(-_g‘𝑌)(1_r‘𝑌)))
96	3, 95	eqtrid 2778	. 2 ⊢ (𝜑 → ((𝑂‘𝑇)‘(𝐿‘𝐴)) = ((1_r‘𝑌)(-_g‘𝑌)(1_r‘𝑌)))
97		ringgrp 20143	. . . 4 ⊢ (𝑌 ∈ Ring → 𝑌 ∈ Grp)
98	19, 97	syl 17	. . 3 ⊢ (𝜑 → 𝑌 ∈ Grp)
99		eqid 2726	. . . 4 ⊢ (0_g‘𝑌) = (0_g‘𝑌)
100	6, 99, 93	grpsubid 18952	. . 3 ⊢ ((𝑌 ∈ Grp ∧ (1_r‘𝑌) ∈ (Base‘𝑌)) → ((1_r‘𝑌)(-_g‘𝑌)(1_r‘𝑌)) = (0_g‘𝑌))
101	98, 81, 100	syl2anc 583	. 2 ⊢ (𝜑 → ((1_r‘𝑌)(-_g‘𝑌)(1_r‘𝑌)) = (0_g‘𝑌))
102	96, 101	eqtrd 2766	1 ⊢ (𝜑 → ((𝑂‘𝑇)‘(𝐿‘𝐴)) = (0_g‘𝑌))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∖ cdif 3940 {csn 4623 class class class wbr 5141 ⟶wf 6533 ‘cfv 6537 (class class class)co 7405 ℂcc 11110 1c1 11113 − cmin 11448 / cdiv 11875 ℕcn 12216 2c2 12271 ℕ₀cn0 12476 ℤcz 12562 mod cmo 13840 ↑cexp 14032 ∥ cdvds 16204 ℙcprime 16615 Basecbs 17153 ↾_s cress 17182 0_gc0g 17394 MndHom cmhm 18711 SubMndcsubmnd 18712 Grpcgrp 18863 -_gcsg 18865 ._gcmg 18995 mulGrpcmgp 20039 1_rcur 20086 Ringcrg 20138 CRingccrg 20139 RingHom crh 20371 SubRingcsubrg 20469 Fieldcfield 20588 Domncdomn 21190 IDomncidom 21191 ℂ_fldccnfld 21240 ℤ_ringczring 21333 ℤRHomczrh 21386 ℤ/nℤczn 21389 algSccascl 21747 var₁cv1 22050 Poly₁cpl1 22051 eval₁ce1 22188 deg₁ cdg1 25942

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-addf 11191 ax-mulf 11192

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7667 df-ofr 7668 df-om 7853 df-1st 7974 df-2nd 7975 df-supp 8147 df-tpos 8212 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-2o 8468 df-oadd 8471 df-er 8705 df-ec 8707 df-qs 8711 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-sup 9439 df-inf 9440 df-oi 9507 df-dju 9898 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-xnn0 12549 df-z 12563 df-dec 12682 df-uz 12827 df-rp 12981 df-fz 13491 df-fzo 13634 df-fl 13763 df-mod 13841 df-seq 13973 df-exp 14033 df-hash 14296 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-dvds 16205 df-gcd 16443 df-prm 16616 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-starv 17221 df-sca 17222 df-vsca 17223 df-ip 17224 df-tset 17225 df-ple 17226 df-ds 17228 df-unif 17229 df-hom 17230 df-cco 17231 df-0g 17396 df-gsum 17397 df-prds 17402 df-pws 17404 df-imas 17463 df-qus 17464 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-mhm 18713 df-submnd 18714 df-grp 18866 df-minusg 18867 df-sbg 18868 df-mulg 18996 df-subg 19050 df-nsg 19051 df-eqg 19052 df-ghm 19139 df-cntz 19233 df-cmn 19702 df-abl 19703 df-mgp 20040 df-rng 20058 df-ur 20087 df-srg 20092 df-ring 20140 df-cring 20141 df-oppr 20236 df-dvdsr 20259 df-unit 20260 df-invr 20290 df-dvr 20303 df-rhm 20374 df-nzr 20415 df-subrng 20446 df-subrg 20471 df-drng 20589 df-field 20590 df-lmod 20708 df-lss 20779 df-lsp 20819 df-sra 21021 df-rgmod 21022 df-lidl 21067 df-rsp 21068 df-2idl 21107 df-rlreg 21193 df-domn 21194 df-idom 21195 df-cnfld 21241 df-zring 21334 df-zrh 21390 df-zn 21393 df-assa 21748 df-asp 21749 df-ascl 21750 df-psr 21803 df-mvr 21804 df-mpl 21805 df-opsr 21807 df-evls 21977 df-evl 21978 df-psr1 22054 df-vr1 22055 df-ply1 22056 df-evl1 22190

This theorem is referenced by: lgsqrlem2 27235 lgsqrlem3 27236

W3C validator