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Theorem lgsqrlem1 26838

Description: Lemma for lgsqr 26843. (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 6891	. . . 4 ⊢ (𝑂‘𝑇) = (𝑂‘((((𝑃 − 1) / 2) ↑ 𝑋) − 1 ))
3	2	fveq1i 6889	. . 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 2732	. . . . 5 ⊢ (Base‘𝑌) = (Base‘𝑌)
7		lgsqr.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑆)
8		lgsqr.1	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))
9	8	eldifad 3959	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ ℙ)
10		lgsqr.y	. . . . . . . . 9 ⊢ 𝑌 = (ℤ/nℤ‘𝑃)
11	10	znfld 21107	. . . . . . . 8 ⊢ (𝑃 ∈ ℙ → 𝑌 ∈ Field)
12	9, 11	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ Field)
13		fldidom 20915	. . . . . . 7 ⊢ (𝑌 ∈ Field → 𝑌 ∈ IDomn)
14	12, 13	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ IDomn)
15		isidom 20914	. . . . . . 7 ⊢ (𝑌 ∈ IDomn ↔ (𝑌 ∈ CRing ∧ 𝑌 ∈ Domn))
16	15	simplbi 498	. . . . . 6 ⊢ (𝑌 ∈ IDomn → 𝑌 ∈ CRing)
17	14, 16	syl 17	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ CRing)
18		crngring 20061	. . . . . . . . 9 ⊢ (𝑌 ∈ CRing → 𝑌 ∈ Ring)
19	17, 18	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ Ring)
20		lgsqr.l	. . . . . . . . 9 ⊢ 𝐿 = (ℤRHom‘𝑌)
21	20	zrhrhm 21052	. . . . . . . 8 ⊢ (𝑌 ∈ Ring → 𝐿 ∈ (ℤ_ring RingHom 𝑌))
22	19, 21	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ (ℤ_ring RingHom 𝑌))
23		zringbas 21015	. . . . . . . 8 ⊢ ℤ = (Base‘ℤ_ring)
24	23, 6	rhmf 20255	. . . . . . 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 7084	. . . . 5 ⊢ (𝜑 → (𝐿‘𝐴) ∈ (Base‘𝑌))
28		lgsqr.x	. . . . . . . 8 ⊢ 𝑋 = (var₁‘𝑌)
29	4, 28, 6, 5, 7, 17, 27	evl1vard 21847	. . . . . . 7 ⊢ (𝜑 → (𝑋 ∈ 𝐵 ∧ ((𝑂‘𝑋)‘(𝐿‘𝐴)) = (𝐿‘𝐴)))
30		lgsqr.e	. . . . . . 7 ⊢ ↑ = (._g‘(mulGrp‘𝑆))
31		eqid 2732	. . . . . . 7 ⊢ (._g‘(mulGrp‘𝑌)) = (._g‘(mulGrp‘𝑌))
32		oddprm 16739	. . . . . . . . 9 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((𝑃 − 1) / 2) ∈ ℕ)
33	8, 32	syl 17	. . . . . . . 8 ⊢ (𝜑 → ((𝑃 − 1) / 2) ∈ ℕ)
34	33	nnnn0d 12528	. . . . . . 7 ⊢ (𝜑 → ((𝑃 − 1) / 2) ∈ ℕ₀)
35	4, 5, 6, 7, 17, 27, 29, 30, 31, 34	evl1expd 21855	. . . . . 6 ⊢ (𝜑 → ((((𝑃 − 1) / 2) ↑ 𝑋) ∈ 𝐵 ∧ ((𝑂‘(((𝑃 − 1) / 2) ↑ 𝑋))‘(𝐿‘𝐴)) = (((𝑃 − 1) / 2)(._g‘(mulGrp‘𝑌))(𝐿‘𝐴))))
36		zringmpg 21032	. . . . . . . . . . . 12 ⊢ ((mulGrp‘ℂ_fld) ↾_s ℤ) = (mulGrp‘ℤ_ring)
37		eqid 2732	. . . . . . . . . . . 12 ⊢ (mulGrp‘𝑌) = (mulGrp‘𝑌)
38	36, 37	rhmmhm 20250	. . . . . . . . . . 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 19987	. . . . . . . . . . 11 ⊢ ℤ = (Base‘((mulGrp‘ℂ_fld) ↾_s ℤ))
41		eqid 2732	. . . . . . . . . . 11 ⊢ (._g‘((mulGrp‘ℂ_fld) ↾_s ℤ)) = (._g‘((mulGrp‘ℂ_fld) ↾_s ℤ))
42	40, 41, 31	mhmmulg 18989	. . . . . . . . . 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 1371	. . . . . . . . 9 ⊢ (𝜑 → (𝐿‘(((𝑃 − 1) / 2)(._g‘((mulGrp‘ℂ_fld) ↾_s ℤ))𝐴)) = (((𝑃 − 1) / 2)(._g‘(mulGrp‘𝑌))(𝐿‘𝐴)))
44		zsubrg 20990	. . . . . . . . . . . . . 14 ⊢ ℤ ∈ (SubRing‘ℂ_fld)
45		eqid 2732	. . . . . . . . . . . . . . 15 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
46	45	subrgsubm 20368	. . . . . . . . . . . . . 14 ⊢ (ℤ ∈ (SubRing‘ℂ_fld) → ℤ ∈ (SubMnd‘(mulGrp‘ℂ_fld)))
47	44, 46	mp1i 13	. . . . . . . . . . . . 13 ⊢ (𝜑 → ℤ ∈ (SubMnd‘(mulGrp‘ℂ_fld)))
48		eqid 2732	. . . . . . . . . . . . . 14 ⊢ (._g‘(mulGrp‘ℂ_fld)) = (._g‘(mulGrp‘ℂ_fld))
49		eqid 2732	. . . . . . . . . . . . . 14 ⊢ ((mulGrp‘ℂ_fld) ↾_s ℤ) = ((mulGrp‘ℂ_fld) ↾_s ℤ)
50	48, 49, 41	submmulg 18992	. . . . . . . . . . . . 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 1371	. . . . . . . . . . . 12 ⊢ (𝜑 → (((𝑃 − 1) / 2)(._g‘(mulGrp‘ℂ_fld))𝐴) = (((𝑃 − 1) / 2)(._g‘((mulGrp‘ℂ_fld) ↾_s ℤ))𝐴))
52	26	zcnd 12663	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ∈ ℂ)
53		cnfldexp 20970	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ ((𝑃 − 1) / 2) ∈ ℕ₀) → (((𝑃 − 1) / 2)(._g‘(mulGrp‘ℂ_fld))𝐴) = (𝐴↑((𝑃 − 1) / 2)))
54	52, 34, 53	syl2anc 584	. . . . . . . . . . . 12 ⊢ (𝜑 → (((𝑃 − 1) / 2)(._g‘(mulGrp‘ℂ_fld))𝐴) = (𝐴↑((𝑃 − 1) / 2)))
55	51, 54	eqtr3d 2774	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝑃 − 1) / 2)(._g‘((mulGrp‘ℂ_fld) ↾_s ℤ))𝐴) = (𝐴↑((𝑃 − 1) / 2)))
56	55	fveq2d 6892	. . . . . . . . . 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 16607	. . . . . . . . . . . . . 14 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
59	9, 58	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑃 ∈ ℕ)
60		zexpcl 14038	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℤ ∧ ((𝑃 − 1) / 2) ∈ ℕ₀) → (𝐴↑((𝑃 − 1) / 2)) ∈ ℤ)
61	26, 34, 60	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴↑((𝑃 − 1) / 2)) ∈ ℤ)
62		1zzd 12589	. . . . . . . . . . . . 13 ⊢ (𝜑 → 1 ∈ ℤ)
63		moddvds 16204	. . . . . . . . . . . . 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 1371	. . . . . . . . . . . 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 12528	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑃 ∈ ℕ₀)
67	10, 20	zndvds 21096	. . . . . . . . . . . 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 1371	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐿‘(𝐴↑((𝑃 − 1) / 2))) = (𝐿‘1) ↔ 𝑃 ∥ ((𝐴↑((𝑃 − 1) / 2)) − 1)))
69	65, 68	mpbird 256	. . . . . . . . . 10 ⊢ (𝜑 → (𝐿‘(𝐴↑((𝑃 − 1) / 2))) = (𝐿‘1))
70		zring1 21020	. . . . . . . . . . . 12 ⊢ 1 = (1_r‘ℤ_ring)
71		eqid 2732	. . . . . . . . . . . 12 ⊢ (1_r‘𝑌) = (1_r‘𝑌)
72	70, 71	rhm1 20259	. . . . . . . . . . 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 2776	. . . . . . . . 9 ⊢ (𝜑 → (𝐿‘(((𝑃 − 1) / 2)(._g‘((mulGrp‘ℂ_fld) ↾_s ℤ))𝐴)) = (1_r‘𝑌))
75	43, 74	eqtr3d 2774	. . . . . . . 8 ⊢ (𝜑 → (((𝑃 − 1) / 2)(._g‘(mulGrp‘𝑌))(𝐿‘𝐴)) = (1_r‘𝑌))
76	75	eqeq2d 2743	. . . . . . 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 629	. . . . . 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 2732	. . . . . . 7 ⊢ (algSc‘𝑆) = (algSc‘𝑆)
80	6, 71	ringidcl 20076	. . . . . . . 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 21845	. . . . . 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 21806	. . . . . . . . 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 2818	. . . . . . 7 ⊢ (𝜑 → (((algSc‘𝑆)‘(1_r‘𝑌)) ∈ 𝐵 ↔ 1 ∈ 𝐵))
87	85	fveq2d 6892	. . . . . . . . 9 ⊢ (𝜑 → (𝑂‘((algSc‘𝑆)‘(1_r‘𝑌))) = (𝑂‘ 1 ))
88	87	fveq1d 6890	. . . . . . . 8 ⊢ (𝜑 → ((𝑂‘((algSc‘𝑆)‘(1_r‘𝑌)))‘(𝐿‘𝐴)) = ((𝑂‘ 1 )‘(𝐿‘𝐴)))
89	88	eqeq1d 2734	. . . . . . 7 ⊢ (𝜑 → (((𝑂‘((algSc‘𝑆)‘(1_r‘𝑌)))‘(𝐿‘𝐴)) = (1_r‘𝑌) ↔ ((𝑂‘ 1 )‘(𝐿‘𝐴)) = (1_r‘𝑌)))
90	86, 89	anbi12d 631	. . . . . 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 2732	. . . . 5 ⊢ (-_g‘𝑌) = (-_g‘𝑌)
94	4, 5, 6, 7, 17, 27, 78, 91, 92, 93	evl1subd 21852	. . . 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 496	. . 3 ⊢ (𝜑 → ((𝑂‘((((𝑃 − 1) / 2) ↑ 𝑋) − 1 ))‘(𝐿‘𝐴)) = ((1_r‘𝑌)(-_g‘𝑌)(1_r‘𝑌)))
96	3, 95	eqtrid 2784	. 2 ⊢ (𝜑 → ((𝑂‘𝑇)‘(𝐿‘𝐴)) = ((1_r‘𝑌)(-_g‘𝑌)(1_r‘𝑌)))
97		ringgrp 20054	. . . 4 ⊢ (𝑌 ∈ Ring → 𝑌 ∈ Grp)
98	19, 97	syl 17	. . 3 ⊢ (𝜑 → 𝑌 ∈ Grp)
99		eqid 2732	. . . 4 ⊢ (0_g‘𝑌) = (0_g‘𝑌)
100	6, 99, 93	grpsubid 18903	. . 3 ⊢ ((𝑌 ∈ Grp ∧ (1_r‘𝑌) ∈ (Base‘𝑌)) → ((1_r‘𝑌)(-_g‘𝑌)(1_r‘𝑌)) = (0_g‘𝑌))
101	98, 81, 100	syl2anc 584	. 2 ⊢ (𝜑 → ((1_r‘𝑌)(-_g‘𝑌)(1_r‘𝑌)) = (0_g‘𝑌))
102	96, 101	eqtrd 2772	1 ⊢ (𝜑 → ((𝑂‘𝑇)‘(𝐿‘𝐴)) = (0_g‘𝑌))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∖ cdif 3944 {csn 4627 class class class wbr 5147 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 ℂcc 11104 1c1 11107 − cmin 11440 / cdiv 11867 ℕcn 12208 2c2 12263 ℕ₀cn0 12468 ℤcz 12554 mod cmo 13830 ↑cexp 14023 ∥ cdvds 16193 ℙcprime 16604 Basecbs 17140 ↾_s cress 17169 0_gc0g 17381 MndHom cmhm 18665 SubMndcsubmnd 18666 Grpcgrp 18815 -_gcsg 18817 ._gcmg 18944 mulGrpcmgp 19981 1_rcur 19998 Ringcrg 20049 CRingccrg 20050 RingHom crh 20240 Fieldcfield 20308 SubRingcsubrg 20351 Domncdomn 20888 IDomncidom 20889 ℂ_fldccnfld 20936 ℤ_ringczring 21009 ℤRHomczrh 21040 ℤ/nℤczn 21043 algSccascl 21398 var₁cv1 21691 Poly₁cpl1 21692 eval₁ce1 21824 deg₁ cdg1 25560

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-ofr 7667 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8143 df-tpos 8207 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-oadd 8466 df-er 8699 df-ec 8701 df-qs 8705 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-sup 9433 df-inf 9434 df-oi 9501 df-dju 9892 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-xnn0 12541 df-z 12555 df-dec 12674 df-uz 12819 df-rp 12971 df-fz 13481 df-fzo 13624 df-fl 13753 df-mod 13831 df-seq 13963 df-exp 14024 df-hash 14287 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-dvds 16194 df-gcd 16432 df-prm 16605 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-hom 17217 df-cco 17218 df-0g 17383 df-gsum 17384 df-prds 17389 df-pws 17391 df-imas 17450 df-qus 17451 df-mre 17526 df-mrc 17527 df-acs 17529 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-mhm 18667 df-submnd 18668 df-grp 18818 df-minusg 18819 df-sbg 18820 df-mulg 18945 df-subg 18997 df-nsg 18998 df-eqg 18999 df-ghm 19084 df-cntz 19175 df-cmn 19644 df-abl 19645 df-mgp 19982 df-ur 19999 df-srg 20003 df-ring 20051 df-cring 20052 df-oppr 20142 df-dvdsr 20163 df-unit 20164 df-invr 20194 df-dvr 20207 df-rnghom 20243 df-nzr 20284 df-drng 20309 df-field 20310 df-subrg 20353 df-lmod 20465 df-lss 20535 df-lsp 20575 df-sra 20777 df-rgmod 20778 df-lidl 20779 df-rsp 20780 df-2idl 20849 df-rlreg 20891 df-domn 20892 df-idom 20893 df-cnfld 20937 df-zring 21010 df-zrh 21044 df-zn 21047 df-assa 21399 df-asp 21400 df-ascl 21401 df-psr 21453 df-mvr 21454 df-mpl 21455 df-opsr 21457 df-evls 21626 df-evl 21627 df-psr1 21695 df-vr1 21696 df-ply1 21697 df-evl1 21826

This theorem is referenced by: lgsqrlem2 26839 lgsqrlem3 26840

W3C validator