4sqlem10 - Intuitionistic Logic Explorer

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Theorem 4sqlem10 12926

Description: Lemma for 4sq 12949. (Contributed by Mario Carneiro, 16-Jul-2014.)

**Hypotheses**
Ref	Expression
4sqlem5.2	⊢ (𝜑 → 𝐴 ∈ ℤ)
4sqlem5.3	⊢ (𝜑 → 𝑀 ∈ ℕ)
4sqlem5.4	⊢ 𝐵 = (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))
4sqlem10.5	⊢ ((𝜑 ∧ 𝜓) → ((((𝑀↑2) / 2) / 2) − (𝐵↑2)) = 0)

**Assertion**
Ref	Expression
4sqlem10	⊢ ((𝜑 ∧ 𝜓) → (𝑀↑2) ∥ ((𝐴↑2) − (((𝑀↑2) / 2) / 2)))

**Proof of Theorem 4sqlem10**
Step	Hyp	Ref	Expression
1		4sqlem5.3	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℕ)
2	1	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ ℕ)
3	2	nnzd 9579	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ ℤ)
4		zsqcl 10844	. . . 4 ⊢ (𝑀 ∈ ℤ → (𝑀↑2) ∈ ℤ)
5	3, 4	syl 14	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝑀↑2) ∈ ℤ)
6		4sqlem5.2	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℤ)
7	6	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ∈ ℤ)
8	2	nnred 9134	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ ℝ)
9	8	rehalfcld 9369	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → (𝑀 / 2) ∈ ℝ)
10	9	recnd 8186	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → (𝑀 / 2) ∈ ℂ)
11	10	negnegd 8459	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → --(𝑀 / 2) = (𝑀 / 2))
12		4sqlem5.4	. . . . . . . . . . . . . . 15 ⊢ 𝐵 = (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))
13	6, 1, 12	4sqlem5 12921	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐵 ∈ ℤ ∧ ((𝐴 − 𝐵) / 𝑀) ∈ ℤ))
14	13	adantr 276	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → (𝐵 ∈ ℤ ∧ ((𝐴 − 𝐵) / 𝑀) ∈ ℤ))
15	14	simpld 112	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝜓) → 𝐵 ∈ ℤ)
16	15	zred 9580	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → 𝐵 ∈ ℝ)
17	6, 1, 12	4sqlem6 12922	. . . . . . . . . . . . 13 ⊢ (𝜑 → (-(𝑀 / 2) ≤ 𝐵 ∧ 𝐵 < (𝑀 / 2)))
18	17	adantr 276	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝜓) → (-(𝑀 / 2) ≤ 𝐵 ∧ 𝐵 < (𝑀 / 2)))
19	18	simprd 114	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → 𝐵 < (𝑀 / 2))
20	16, 19	ltned 8271	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝜓) → 𝐵 ≠ (𝑀 / 2))
21	20	neneqd 2421	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜓) → ¬ 𝐵 = (𝑀 / 2))
22		2cnd 9194	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝜓) → 2 ∈ ℂ)
23	22	sqvald 10904	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → (2↑2) = (2 · 2))
24	23	oveq2d 6023	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → ((𝑀↑2) / (2↑2)) = ((𝑀↑2) / (2 · 2)))
25	2	nncnd 9135	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ ℂ)
26		2ap0 9214	. . . . . . . . . . . . . . 15 ⊢ 2 # 0
27	26	a1i 9	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → 2 # 0)
28	25, 22, 27	sqdivapd 10920	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → ((𝑀 / 2)↑2) = ((𝑀↑2) / (2↑2)))
29	25	sqcld 10905	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → (𝑀↑2) ∈ ℂ)
30	29, 22, 22, 27, 27	divdivap1d 8980	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → (((𝑀↑2) / 2) / 2) = ((𝑀↑2) / (2 · 2)))
31	24, 28, 30	3eqtr4d 2272	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝜓) → ((𝑀 / 2)↑2) = (((𝑀↑2) / 2) / 2))
32	29	halfcld 9367	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → ((𝑀↑2) / 2) ∈ ℂ)
33	32	halfcld 9367	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → (((𝑀↑2) / 2) / 2) ∈ ℂ)
34	15	zcnd 9581	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → 𝐵 ∈ ℂ)
35	34	sqcld 10905	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → (𝐵↑2) ∈ ℂ)
36		4sqlem10.5	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → ((((𝑀↑2) / 2) / 2) − (𝐵↑2)) = 0)
37	33, 35, 36	subeq0d 8476	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝜓) → (((𝑀↑2) / 2) / 2) = (𝐵↑2))
38	31, 37	eqtr2d 2263	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → (𝐵↑2) = ((𝑀 / 2)↑2))
39		zq 9833	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ ℤ → 𝐵 ∈ ℚ)
40	15, 39	syl 14	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝜓) → 𝐵 ∈ ℚ)
41		2nn 9283	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℕ
42	41	a1i 9	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → 2 ∈ ℕ)
43		znq 9831	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℤ ∧ 2 ∈ ℕ) → (𝑀 / 2) ∈ ℚ)
44	3, 42, 43	syl2anc 411	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝜓) → (𝑀 / 2) ∈ ℚ)
45		qsqeqor 10884	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ℚ ∧ (𝑀 / 2) ∈ ℚ) → ((𝐵↑2) = ((𝑀 / 2)↑2) ↔ (𝐵 = (𝑀 / 2) ∨ 𝐵 = -(𝑀 / 2))))
46	40, 44, 45	syl2anc 411	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → ((𝐵↑2) = ((𝑀 / 2)↑2) ↔ (𝐵 = (𝑀 / 2) ∨ 𝐵 = -(𝑀 / 2))))
47	38, 46	mpbid 147	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝜓) → (𝐵 = (𝑀 / 2) ∨ 𝐵 = -(𝑀 / 2)))
48	47	ord 729	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜓) → (¬ 𝐵 = (𝑀 / 2) → 𝐵 = -(𝑀 / 2)))
49	21, 48	mpd 13	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → 𝐵 = -(𝑀 / 2))
50	49, 15	eqeltrrd 2307	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → -(𝑀 / 2) ∈ ℤ)
51	50	znegcld 9582	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → --(𝑀 / 2) ∈ ℤ)
52	11, 51	eqeltrrd 2307	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝑀 / 2) ∈ ℤ)
53	7, 52	zaddcld 9584	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 + (𝑀 / 2)) ∈ ℤ)
54		zsqcl 10844	. . . 4 ⊢ ((𝐴 + (𝑀 / 2)) ∈ ℤ → ((𝐴 + (𝑀 / 2))↑2) ∈ ℤ)
55	53, 54	syl 14	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2))↑2) ∈ ℤ)
56	53, 3	zmulcld 9586	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) · 𝑀) ∈ ℤ)
57		zq 9833	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℚ)
58	7, 57	syl 14	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ∈ ℚ)
59		qaddcl 9842	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℚ ∧ (𝑀 / 2) ∈ ℚ) → (𝐴 + (𝑀 / 2)) ∈ ℚ)
60	58, 44, 59	syl2anc 411	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 + (𝑀 / 2)) ∈ ℚ)
61		nnq 9840	. . . . . . . . 9 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℚ)
62	2, 61	syl 14	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ ℚ)
63	2	nngt0d 9165	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → 0 < 𝑀)
64	60, 62, 63	modqcld 10562	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) mod 𝑀) ∈ ℚ)
65		qcn 9841	. . . . . . 7 ⊢ (((𝐴 + (𝑀 / 2)) mod 𝑀) ∈ ℚ → ((𝐴 + (𝑀 / 2)) mod 𝑀) ∈ ℂ)
66	64, 65	syl 14	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) mod 𝑀) ∈ ℂ)
67		0cnd 8150	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 0 ∈ ℂ)
68		df-neg 8331	. . . . . . 7 ⊢ -(𝑀 / 2) = (0 − (𝑀 / 2))
69	49, 12, 68	3eqtr3g 2285	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) = (0 − (𝑀 / 2)))
70	66, 67, 10, 69	subcan2d 8510	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) mod 𝑀) = 0)
71		dvdsval3 12318	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ (𝐴 + (𝑀 / 2)) ∈ ℤ) → (𝑀 ∥ (𝐴 + (𝑀 / 2)) ↔ ((𝐴 + (𝑀 / 2)) mod 𝑀) = 0))
72	2, 53, 71	syl2anc 411	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝑀 ∥ (𝐴 + (𝑀 / 2)) ↔ ((𝐴 + (𝑀 / 2)) mod 𝑀) = 0))
73	70, 72	mpbird 167	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∥ (𝐴 + (𝑀 / 2)))
74		dvdssq 12568	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ (𝐴 + (𝑀 / 2)) ∈ ℤ) → (𝑀 ∥ (𝐴 + (𝑀 / 2)) ↔ (𝑀↑2) ∥ ((𝐴 + (𝑀 / 2))↑2)))
75	3, 53, 74	syl2anc 411	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝑀 ∥ (𝐴 + (𝑀 / 2)) ↔ (𝑀↑2) ∥ ((𝐴 + (𝑀 / 2))↑2)))
76	73, 75	mpbid 147	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝑀↑2) ∥ ((𝐴 + (𝑀 / 2))↑2))
77	25	sqvald 10904	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝑀↑2) = (𝑀 · 𝑀))
78	2	nnne0d 9166	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ≠ 0)
79		dvdsmulcr 12348	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ (𝐴 + (𝑀 / 2)) ∈ ℤ ∧ (𝑀 ∈ ℤ ∧ 𝑀 ≠ 0)) → ((𝑀 · 𝑀) ∥ ((𝐴 + (𝑀 / 2)) · 𝑀) ↔ 𝑀 ∥ (𝐴 + (𝑀 / 2))))
80	3, 53, 3, 78, 79	syl112anc 1275	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → ((𝑀 · 𝑀) ∥ ((𝐴 + (𝑀 / 2)) · 𝑀) ↔ 𝑀 ∥ (𝐴 + (𝑀 / 2))))
81	73, 80	mpbird 167	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝑀 · 𝑀) ∥ ((𝐴 + (𝑀 / 2)) · 𝑀))
82	77, 81	eqbrtrd 4105	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝑀↑2) ∥ ((𝐴 + (𝑀 / 2)) · 𝑀))
83	5, 55, 56, 76, 82	dvds2subd 12354	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝑀↑2) ∥ (((𝐴 + (𝑀 / 2))↑2) − ((𝐴 + (𝑀 / 2)) · 𝑀)))
84	53	zcnd 9581	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 + (𝑀 / 2)) ∈ ℂ)
85	84	sqvald 10904	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2))↑2) = ((𝐴 + (𝑀 / 2)) · (𝐴 + (𝑀 / 2))))
86	85	oveq1d 6022	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (((𝐴 + (𝑀 / 2))↑2) − ((𝐴 + (𝑀 / 2)) · 𝑀)) = (((𝐴 + (𝑀 / 2)) · (𝐴 + (𝑀 / 2))) − ((𝐴 + (𝑀 / 2)) · 𝑀)))
87	84, 84, 25	subdid 8571	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) · ((𝐴 + (𝑀 / 2)) − 𝑀)) = (((𝐴 + (𝑀 / 2)) · (𝐴 + (𝑀 / 2))) − ((𝐴 + (𝑀 / 2)) · 𝑀)))
88	25	2halvesd 9368	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → ((𝑀 / 2) + (𝑀 / 2)) = 𝑀)
89	88	oveq2d 6023	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) − ((𝑀 / 2) + (𝑀 / 2))) = ((𝐴 + (𝑀 / 2)) − 𝑀))
90	7	zcnd 9581	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ∈ ℂ)
91	90, 10, 10	pnpcan2d 8506	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) − ((𝑀 / 2) + (𝑀 / 2))) = (𝐴 − (𝑀 / 2)))
92	89, 91	eqtr3d 2264	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) − 𝑀) = (𝐴 − (𝑀 / 2)))
93	92	oveq2d 6023	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) · ((𝐴 + (𝑀 / 2)) − 𝑀)) = ((𝐴 + (𝑀 / 2)) · (𝐴 − (𝑀 / 2))))
94		subsq 10880	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (𝑀 / 2) ∈ ℂ) → ((𝐴↑2) − ((𝑀 / 2)↑2)) = ((𝐴 + (𝑀 / 2)) · (𝐴 − (𝑀 / 2))))
95	90, 10, 94	syl2anc 411	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴↑2) − ((𝑀 / 2)↑2)) = ((𝐴 + (𝑀 / 2)) · (𝐴 − (𝑀 / 2))))
96	31	oveq2d 6023	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴↑2) − ((𝑀 / 2)↑2)) = ((𝐴↑2) − (((𝑀↑2) / 2) / 2)))
97	93, 95, 96	3eqtr2d 2268	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) · ((𝐴 + (𝑀 / 2)) − 𝑀)) = ((𝐴↑2) − (((𝑀↑2) / 2) / 2)))
98	86, 87, 97	3eqtr2d 2268	. 2 ⊢ ((𝜑 ∧ 𝜓) → (((𝐴 + (𝑀 / 2))↑2) − ((𝐴 + (𝑀 / 2)) · 𝑀)) = ((𝐴↑2) − (((𝑀↑2) / 2) / 2)))
99	83, 98	breqtrd 4109	1 ⊢ ((𝜑 ∧ 𝜓) → (𝑀↑2) ∥ ((𝐴↑2) − (((𝑀↑2) / 2) / 2)))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 713 = wceq 1395 ∈ wcel 2200 ≠ wne 2400 class class class wbr 4083 (class class class)co 6007 ℂcc 8008 0cc0 8010 + caddc 8013 · cmul 8015 < clt 8192 ≤ cle 8193 − cmin 8328 -cneg 8329 # cap 8739 / cdiv 8830 ℕcn 9121 2c2 9172 ℤcz 9457 ℚcq 9826 mod cmo 10556 ↑cexp 10772 ∥ cdvds 12314

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4199 ax-sep 4202 ax-nul 4210 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-setind 4629 ax-iinf 4680 ax-cnex 8101 ax-resscn 8102 ax-1cn 8103 ax-1re 8104 ax-icn 8105 ax-addcl 8106 ax-addrcl 8107 ax-mulcl 8108 ax-mulrcl 8109 ax-addcom 8110 ax-mulcom 8111 ax-addass 8112 ax-mulass 8113 ax-distr 8114 ax-i2m1 8115 ax-0lt1 8116 ax-1rid 8117 ax-0id 8118 ax-rnegex 8119 ax-precex 8120 ax-cnre 8121 ax-pre-ltirr 8122 ax-pre-ltwlin 8123 ax-pre-lttrn 8124 ax-pre-apti 8125 ax-pre-ltadd 8126 ax-pre-mulgt0 8127 ax-pre-mulext 8128 ax-arch 8129 ax-caucvg 8130

This theorem depends on definitions: df-bi 117 df-stab 836 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-if 3603 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-int 3924 df-iun 3967 df-br 4084 df-opab 4146 df-mpt 4147 df-tr 4183 df-id 4384 df-po 4387 df-iso 4388 df-iord 4457 df-on 4459 df-ilim 4460 df-suc 4462 df-iom 4683 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-rn 4730 df-res 4731 df-ima 4732 df-iota 5278 df-fun 5320 df-fn 5321 df-f 5322 df-f1 5323 df-fo 5324 df-f1o 5325 df-fv 5326 df-riota 5960 df-ov 6010 df-oprab 6011 df-mpo 6012 df-1st 6292 df-2nd 6293 df-recs 6457 df-frec 6543 df-sup 7162 df-pnf 8194 df-mnf 8195 df-xr 8196 df-ltxr 8197 df-le 8198 df-sub 8330 df-neg 8331 df-reap 8733 df-ap 8740 df-div 8831 df-inn 9122 df-2 9180 df-3 9181 df-4 9182 df-n0 9381 df-z 9458 df-uz 9734 df-q 9827 df-rp 9862 df-fz 10217 df-fzo 10351 df-fl 10502 df-mod 10557 df-seqfrec 10682 df-exp 10773 df-cj 11369 df-re 11370 df-im 11371 df-rsqrt 11525 df-abs 11526 df-dvds 12315 df-gcd 12491

This theorem is referenced by: 4sqlem16 12945

W3C validator