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Theorem lgsqrlem1 27297

Description: Lemma for lgsqr 27302. (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 6895	. . . 4 ⊢ (𝑂‘𝑇) = (𝑂‘((((𝑃 − 1) / 2) ↑ 𝑋) − 1 ))
3	2	fveq1i 6893	. . 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 2725	. . . . 5 ⊢ (Base‘𝑌) = (Base‘𝑌)
7		lgsqr.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑆)
8		lgsqr.1	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))
9	8	eldifad 3951	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ ℙ)
10		lgsqr.y	. . . . . . . . 9 ⊢ 𝑌 = (ℤ/nℤ‘𝑃)
11	10	znfld 21498	. . . . . . . 8 ⊢ (𝑃 ∈ ℙ → 𝑌 ∈ Field)
12	9, 11	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ Field)
13		fldidom 21262	. . . . . . 7 ⊢ (𝑌 ∈ Field → 𝑌 ∈ IDomn)
14	12, 13	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ IDomn)
15		isidom 21258	. . . . . . 7 ⊢ (𝑌 ∈ IDomn ↔ (𝑌 ∈ CRing ∧ 𝑌 ∈ Domn))
16	15	simplbi 496	. . . . . 6 ⊢ (𝑌 ∈ IDomn → 𝑌 ∈ CRing)
17	14, 16	syl 17	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ CRing)
18		crngring 20189	. . . . . . . . 9 ⊢ (𝑌 ∈ CRing → 𝑌 ∈ Ring)
19	17, 18	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ Ring)
20		lgsqr.l	. . . . . . . . 9 ⊢ 𝐿 = (ℤRHom‘𝑌)
21	20	zrhrhm 21441	. . . . . . . 8 ⊢ (𝑌 ∈ Ring → 𝐿 ∈ (ℤ_ring RingHom 𝑌))
22	19, 21	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ (ℤ_ring RingHom 𝑌))
23		zringbas 21383	. . . . . . . 8 ⊢ ℤ = (Base‘ℤ_ring)
24	23, 6	rhmf 20428	. . . . . . 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 7090	. . . . 5 ⊢ (𝜑 → (𝐿‘𝐴) ∈ (Base‘𝑌))
28		lgsqr.x	. . . . . . . 8 ⊢ 𝑋 = (var₁‘𝑌)
29	4, 28, 6, 5, 7, 17, 27	evl1vard 22265	. . . . . . 7 ⊢ (𝜑 → (𝑋 ∈ 𝐵 ∧ ((𝑂‘𝑋)‘(𝐿‘𝐴)) = (𝐿‘𝐴)))
30		lgsqr.e	. . . . . . 7 ⊢ ↑ = (._g‘(mulGrp‘𝑆))
31		eqid 2725	. . . . . . 7 ⊢ (._g‘(mulGrp‘𝑌)) = (._g‘(mulGrp‘𝑌))
32		oddprm 16778	. . . . . . . . 9 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((𝑃 − 1) / 2) ∈ ℕ)
33	8, 32	syl 17	. . . . . . . 8 ⊢ (𝜑 → ((𝑃 − 1) / 2) ∈ ℕ)
34	33	nnnn0d 12562	. . . . . . 7 ⊢ (𝜑 → ((𝑃 − 1) / 2) ∈ ℕ₀)
35	4, 5, 6, 7, 17, 27, 29, 30, 31, 34	evl1expd 22273	. . . . . 6 ⊢ (𝜑 → ((((𝑃 − 1) / 2) ↑ 𝑋) ∈ 𝐵 ∧ ((𝑂‘(((𝑃 − 1) / 2) ↑ 𝑋))‘(𝐿‘𝐴)) = (((𝑃 − 1) / 2)(._g‘(mulGrp‘𝑌))(𝐿‘𝐴))))
36		zringmpg 21401	. . . . . . . . . . . 12 ⊢ ((mulGrp‘ℂ_fld) ↾_s ℤ) = (mulGrp‘ℤ_ring)
37		eqid 2725	. . . . . . . . . . . 12 ⊢ (mulGrp‘𝑌) = (mulGrp‘𝑌)
38	36, 37	rhmmhm 20422	. . . . . . . . . . 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 20084	. . . . . . . . . . 11 ⊢ ℤ = (Base‘((mulGrp‘ℂ_fld) ↾_s ℤ))
41		eqid 2725	. . . . . . . . . . 11 ⊢ (._g‘((mulGrp‘ℂ_fld) ↾_s ℤ)) = (._g‘((mulGrp‘ℂ_fld) ↾_s ℤ))
42	40, 41, 31	mhmmulg 19074	. . . . . . . . . 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 21357	. . . . . . . . . . . . . 14 ⊢ ℤ ∈ (SubRing‘ℂ_fld)
45		eqid 2725	. . . . . . . . . . . . . . 15 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
46	45	subrgsubm 20528	. . . . . . . . . . . . . 14 ⊢ (ℤ ∈ (SubRing‘ℂ_fld) → ℤ ∈ (SubMnd‘(mulGrp‘ℂ_fld)))
47	44, 46	mp1i 13	. . . . . . . . . . . . 13 ⊢ (𝜑 → ℤ ∈ (SubMnd‘(mulGrp‘ℂ_fld)))
48		eqid 2725	. . . . . . . . . . . . . 14 ⊢ (._g‘(mulGrp‘ℂ_fld)) = (._g‘(mulGrp‘ℂ_fld))
49		eqid 2725	. . . . . . . . . . . . . 14 ⊢ ((mulGrp‘ℂ_fld) ↾_s ℤ) = ((mulGrp‘ℂ_fld) ↾_s ℤ)
50	48, 49, 41	submmulg 19077	. . . . . . . . . . . . 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 12697	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ∈ ℂ)
53		cnfldexp 21336	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ ((𝑃 − 1) / 2) ∈ ℕ₀) → (((𝑃 − 1) / 2)(._g‘(mulGrp‘ℂ_fld))𝐴) = (𝐴↑((𝑃 − 1) / 2)))
54	52, 34, 53	syl2anc 582	. . . . . . . . . . . 12 ⊢ (𝜑 → (((𝑃 − 1) / 2)(._g‘(mulGrp‘ℂ_fld))𝐴) = (𝐴↑((𝑃 − 1) / 2)))
55	51, 54	eqtr3d 2767	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝑃 − 1) / 2)(._g‘((mulGrp‘ℂ_fld) ↾_s ℤ))𝐴) = (𝐴↑((𝑃 − 1) / 2)))
56	55	fveq2d 6896	. . . . . . . . . 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 16644	. . . . . . . . . . . . . 14 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
59	9, 58	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑃 ∈ ℕ)
60		zexpcl 14073	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℤ ∧ ((𝑃 − 1) / 2) ∈ ℕ₀) → (𝐴↑((𝑃 − 1) / 2)) ∈ ℤ)
61	26, 34, 60	syl2anc 582	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴↑((𝑃 − 1) / 2)) ∈ ℤ)
62		1zzd 12623	. . . . . . . . . . . . 13 ⊢ (𝜑 → 1 ∈ ℤ)
63		moddvds 16241	. . . . . . . . . . . . 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 12562	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑃 ∈ ℕ₀)
67	10, 20	zndvds 21487	. . . . . . . . . . . 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 256	. . . . . . . . . 10 ⊢ (𝜑 → (𝐿‘(𝐴↑((𝑃 − 1) / 2))) = (𝐿‘1))
70		zring1 21389	. . . . . . . . . . . 12 ⊢ 1 = (1_r‘ℤ_ring)
71		eqid 2725	. . . . . . . . . . . 12 ⊢ (1_r‘𝑌) = (1_r‘𝑌)
72	70, 71	rhm1 20432	. . . . . . . . . . 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 2769	. . . . . . . . 9 ⊢ (𝜑 → (𝐿‘(((𝑃 − 1) / 2)(._g‘((mulGrp‘ℂ_fld) ↾_s ℤ))𝐴)) = (1_r‘𝑌))
75	43, 74	eqtr3d 2767	. . . . . . . 8 ⊢ (𝜑 → (((𝑃 − 1) / 2)(._g‘(mulGrp‘𝑌))(𝐿‘𝐴)) = (1_r‘𝑌))
76	75	eqeq2d 2736	. . . . . . 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 2725	. . . . . . 7 ⊢ (algSc‘𝑆) = (algSc‘𝑆)
80	6, 71	ringidcl 20206	. . . . . . . 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 22263	. . . . . 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 22221	. . . . . . . . 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 2810	. . . . . . 7 ⊢ (𝜑 → (((algSc‘𝑆)‘(1_r‘𝑌)) ∈ 𝐵 ↔ 1 ∈ 𝐵))
87	85	fveq2d 6896	. . . . . . . . 9 ⊢ (𝜑 → (𝑂‘((algSc‘𝑆)‘(1_r‘𝑌))) = (𝑂‘ 1 ))
88	87	fveq1d 6894	. . . . . . . 8 ⊢ (𝜑 → ((𝑂‘((algSc‘𝑆)‘(1_r‘𝑌)))‘(𝐿‘𝐴)) = ((𝑂‘ 1 )‘(𝐿‘𝐴)))
89	88	eqeq1d 2727	. . . . . . 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 2725	. . . . 5 ⊢ (-_g‘𝑌) = (-_g‘𝑌)
94	4, 5, 6, 7, 17, 27, 78, 91, 92, 93	evl1subd 22270	. . . 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 494	. . 3 ⊢ (𝜑 → ((𝑂‘((((𝑃 − 1) / 2) ↑ 𝑋) − 1 ))‘(𝐿‘𝐴)) = ((1_r‘𝑌)(-_g‘𝑌)(1_r‘𝑌)))
96	3, 95	eqtrid 2777	. 2 ⊢ (𝜑 → ((𝑂‘𝑇)‘(𝐿‘𝐴)) = ((1_r‘𝑌)(-_g‘𝑌)(1_r‘𝑌)))
97		ringgrp 20182	. . . 4 ⊢ (𝑌 ∈ Ring → 𝑌 ∈ Grp)
98	19, 97	syl 17	. . 3 ⊢ (𝜑 → 𝑌 ∈ Grp)
99		eqid 2725	. . . 4 ⊢ (0_g‘𝑌) = (0_g‘𝑌)
100	6, 99, 93	grpsubid 18984	. . 3 ⊢ ((𝑌 ∈ Grp ∧ (1_r‘𝑌) ∈ (Base‘𝑌)) → ((1_r‘𝑌)(-_g‘𝑌)(1_r‘𝑌)) = (0_g‘𝑌))
101	98, 81, 100	syl2anc 582	. 2 ⊢ (𝜑 → ((1_r‘𝑌)(-_g‘𝑌)(1_r‘𝑌)) = (0_g‘𝑌))
102	96, 101	eqtrd 2765	1 ⊢ (𝜑 → ((𝑂‘𝑇)‘(𝐿‘𝐴)) = (0_g‘𝑌))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∖ cdif 3936 {csn 4624 class class class wbr 5143 ⟶wf 6539 ‘cfv 6543 (class class class)co 7416 ℂcc 11136 1c1 11139 − cmin 11474 / cdiv 11901 ℕcn 12242 2c2 12297 ℕ₀cn0 12502 ℤcz 12588 mod cmo 13866 ↑cexp 14058 ∥ cdvds 16230 ℙcprime 16641 Basecbs 17179 ↾_s cress 17208 0_gc0g 17420 MndHom cmhm 18737 SubMndcsubmnd 18738 Grpcgrp 18894 -_gcsg 18896 ._gcmg 19027 mulGrpcmgp 20078 1_rcur 20125 Ringcrg 20177 CRingccrg 20178 RingHom crh 20412 SubRingcsubrg 20510 Fieldcfield 20629 Domncdomn 21231 IDomncidom 21232 ℂ_fldccnfld 21283 ℤ_ringczring 21376 ℤRHomczrh 21429 ℤ/nℤczn 21432 algSccascl 21790 var₁cv1 22103 Poly₁cpl1 22104 eval₁ce1 22242 deg₁ cdg1 26005

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 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 ax-addf 11217 ax-mulf 11218

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7682 df-ofr 7683 df-om 7869 df-1st 7991 df-2nd 7992 df-supp 8164 df-tpos 8230 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-oadd 8489 df-er 8723 df-ec 8725 df-qs 8729 df-map 8845 df-pm 8846 df-ixp 8915 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-fsupp 9386 df-sup 9465 df-inf 9466 df-oi 9533 df-dju 9924 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-xnn0 12575 df-z 12589 df-dec 12708 df-uz 12853 df-rp 13007 df-fz 13517 df-fzo 13660 df-fl 13789 df-mod 13867 df-seq 13999 df-exp 14059 df-hash 14322 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-abs 15215 df-dvds 16231 df-gcd 16469 df-prm 16642 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-starv 17247 df-sca 17248 df-vsca 17249 df-ip 17250 df-tset 17251 df-ple 17252 df-ds 17254 df-unif 17255 df-hom 17256 df-cco 17257 df-0g 17422 df-gsum 17423 df-prds 17428 df-pws 17430 df-imas 17489 df-qus 17490 df-mre 17565 df-mrc 17566 df-acs 17568 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-mhm 18739 df-submnd 18740 df-grp 18897 df-minusg 18898 df-sbg 18899 df-mulg 19028 df-subg 19082 df-nsg 19083 df-eqg 19084 df-ghm 19172 df-cntz 19272 df-cmn 19741 df-abl 19742 df-mgp 20079 df-rng 20097 df-ur 20126 df-srg 20131 df-ring 20179 df-cring 20180 df-oppr 20277 df-dvdsr 20300 df-unit 20301 df-invr 20331 df-dvr 20344 df-rhm 20415 df-nzr 20456 df-subrng 20487 df-subrg 20512 df-drng 20630 df-field 20631 df-lmod 20749 df-lss 20820 df-lsp 20860 df-sra 21062 df-rgmod 21063 df-lidl 21108 df-rsp 21109 df-2idl 21148 df-rlreg 21234 df-domn 21235 df-idom 21236 df-cnfld 21284 df-zring 21377 df-zrh 21433 df-zn 21436 df-assa 21791 df-asp 21792 df-ascl 21793 df-psr 21846 df-mvr 21847 df-mpl 21848 df-opsr 21850 df-evls 22025 df-evl 22026 df-psr1 22107 df-vr1 22108 df-ply1 22109 df-evl1 22244

This theorem is referenced by: lgsqrlem2 27298 lgsqrlem3 27299

W3C validator