Theorem 4sqlem19 16354

Description: Lemma for 4sq 16355. The proof is by strong induction - we show that if all the integers less than 𝑘 are in 𝑆, then 𝑘 is as well. In this part of the proof we do the induction argument and dispense with all the cases except the odd prime case, which is sent to 4sqlem18 16353. If 𝑘 is 0, 1, 2, we show 𝑘 ∈ 𝑆 directly; otherwise if 𝑘 is composite, 𝑘 is the product of two numbers less than it (and hence in 𝑆 by assumption), so by mul4sq 16345 𝑘 ∈ 𝑆. (Contributed by Mario Carneiro, 14-Jul-2014.) (Revised by Mario Carneiro, 20-Jun-2015.)

Hypothesis

Ref	Expression
4sq.1	⊢ 𝑆 = {𝑛 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))}

Assertion

Ref	Expression
4sqlem19	⊢ ℕ0 = 𝑆

Distinct variable groups:   𝑤,𝑛,𝑥,𝑦,𝑧   𝑆,𝑛

Allowed substitution hints:   𝑆(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem 4sqlem19

Dummy variables 𝑗 𝑘 𝑖 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elnn0 11936	. . . 4 ⊢ (𝑘 ∈ ℕ0 ↔ (𝑘 ∈ ℕ ∨ 𝑘 = 0))
2		eleq1 2839	. . . . . 6 ⊢ (𝑗 = 1 → (𝑗 ∈ 𝑆 ↔ 1 ∈ 𝑆))
3		eleq1 2839	. . . . . 6 ⊢ (𝑗 = 𝑚 → (𝑗 ∈ 𝑆 ↔ 𝑚 ∈ 𝑆))
4		eleq1 2839	. . . . . 6 ⊢ (𝑗 = 𝑖 → (𝑗 ∈ 𝑆 ↔ 𝑖 ∈ 𝑆))
5		eleq1 2839	. . . . . 6 ⊢ (𝑗 = (𝑚 · 𝑖) → (𝑗 ∈ 𝑆 ↔ (𝑚 · 𝑖) ∈ 𝑆))
6		eleq1 2839	. . . . . 6 ⊢ (𝑗 = 𝑘 → (𝑗 ∈ 𝑆 ↔ 𝑘 ∈ 𝑆))
7		abs1 14705	. . . . . . . . . . 11 ⊢ (abs‘1) = 1
8	7	oveq1i 7160	. . . . . . . . . 10 ⊢ ((abs‘1)↑2) = (1↑2)
9		sq1 13608	. . . . . . . . . 10 ⊢ (1↑2) = 1
10	8, 9	eqtri 2781	. . . . . . . . 9 ⊢ ((abs‘1)↑2) = 1
11		abs0 14693	. . . . . . . . . . 11 ⊢ (abs‘0) = 0
12	11	oveq1i 7160	. . . . . . . . . 10 ⊢ ((abs‘0)↑2) = (0↑2)
13		sq0 13605	. . . . . . . . . 10 ⊢ (0↑2) = 0
14	12, 13	eqtri 2781	. . . . . . . . 9 ⊢ ((abs‘0)↑2) = 0
15	10, 14	oveq12i 7162	. . . . . . . 8 ⊢ (((abs‘1)↑2) + ((abs‘0)↑2)) = (1 + 0)
16		1p0e1 11798	. . . . . . . 8 ⊢ (1 + 0) = 1
17	15, 16	eqtri 2781	. . . . . . 7 ⊢ (((abs‘1)↑2) + ((abs‘0)↑2)) = 1
18		1z 12051	. . . . . . . . 9 ⊢ 1 ∈ ℤ
19		zgz 16324	. . . . . . . . 9 ⊢ (1 ∈ ℤ → 1 ∈ ℤ[i])
20	18, 19	ax-mp 5	. . . . . . . 8 ⊢ 1 ∈ ℤ[i]
21		0z 12031	. . . . . . . . 9 ⊢ 0 ∈ ℤ
22		zgz 16324	. . . . . . . . 9 ⊢ (0 ∈ ℤ → 0 ∈ ℤ[i])
23	21, 22	ax-mp 5	. . . . . . . 8 ⊢ 0 ∈ ℤ[i]
24		4sq.1	. . . . . . . . 9 ⊢ 𝑆 = {𝑛 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))}
25	24	4sqlem4a 16342	. . . . . . . 8 ⊢ ((1 ∈ ℤ[i] ∧ 0 ∈ ℤ[i]) → (((abs‘1)↑2) + ((abs‘0)↑2)) ∈ 𝑆)
26	20, 23, 25	mp2an 691	. . . . . . 7 ⊢ (((abs‘1)↑2) + ((abs‘0)↑2)) ∈ 𝑆
27	17, 26	eqeltrri 2849	. . . . . 6 ⊢ 1 ∈ 𝑆
28		eleq1 2839	. . . . . . 7 ⊢ (𝑗 = 2 → (𝑗 ∈ 𝑆 ↔ 2 ∈ 𝑆))
29		eldifsn 4677	. . . . . . . . 9 ⊢ (𝑗 ∈ (ℙ ∖ {2}) ↔ (𝑗 ∈ ℙ ∧ 𝑗 ≠ 2))
30		oddprm 16202	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (ℙ ∖ {2}) → ((𝑗 − 1) / 2) ∈ ℕ)
31	30	adantr 484	. . . . . . . . . 10 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → ((𝑗 − 1) / 2) ∈ ℕ)
32		eldifi 4032	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (ℙ ∖ {2}) → 𝑗 ∈ ℙ)
33	32	adantr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → 𝑗 ∈ ℙ)
34		prmnn 16070	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ ℙ → 𝑗 ∈ ℕ)
35		nncn 11682	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℂ)
36	33, 34, 35	3syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → 𝑗 ∈ ℂ)
37		ax-1cn 10633	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℂ
38		subcl 10923	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑗 − 1) ∈ ℂ)
39	36, 37, 38	sylancl 589	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → (𝑗 − 1) ∈ ℂ)
40		2cnd 11752	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → 2 ∈ ℂ)
41		2ne0 11778	. . . . . . . . . . . . . 14 ⊢ 2 ≠ 0
42	41	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → 2 ≠ 0)
43	39, 40, 42	divcan2d 11456	. . . . . . . . . . . 12 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → (2 · ((𝑗 − 1) / 2)) = (𝑗 − 1))
44	43	oveq1d 7165	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → ((2 · ((𝑗 − 1) / 2)) + 1) = ((𝑗 − 1) + 1))
45		npcan 10933	. . . . . . . . . . . 12 ⊢ ((𝑗 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑗 − 1) + 1) = 𝑗)
46	36, 37, 45	sylancl 589	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → ((𝑗 − 1) + 1) = 𝑗)
47	44, 46	eqtr2d 2794	. . . . . . . . . 10 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → 𝑗 = ((2 · ((𝑗 − 1) / 2)) + 1))
48	43	oveq2d 7166	. . . . . . . . . . . 12 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → (0...(2 · ((𝑗 − 1) / 2))) = (0...(𝑗 − 1)))
49		nnm1nn0 11975	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ ℕ → (𝑗 − 1) ∈ ℕ0)
50	33, 34, 49	3syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → (𝑗 − 1) ∈ ℕ0)
51		elnn0uz 12323	. . . . . . . . . . . . . 14 ⊢ ((𝑗 − 1) ∈ ℕ0 ↔ (𝑗 − 1) ∈ (ℤ≥‘0))
52	50, 51	sylib 221	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → (𝑗 − 1) ∈ (ℤ≥‘0))
53		eluzfz1 12963	. . . . . . . . . . . . 13 ⊢ ((𝑗 − 1) ∈ (ℤ≥‘0) → 0 ∈ (0...(𝑗 − 1)))
54		fzsplit 12982	. . . . . . . . . . . . 13 ⊢ (0 ∈ (0...(𝑗 − 1)) → (0...(𝑗 − 1)) = ((0...0) ∪ ((0 + 1)...(𝑗 − 1))))
55	52, 53, 54	3syl 18	. . . . . . . . . . . 12 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → (0...(𝑗 − 1)) = ((0...0) ∪ ((0 + 1)...(𝑗 − 1))))
56	48, 55	eqtrd 2793	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → (0...(2 · ((𝑗 − 1) / 2))) = ((0...0) ∪ ((0 + 1)...(𝑗 − 1))))
57		fz0sn 13056	. . . . . . . . . . . . . 14 ⊢ (0...0) = {0}
58	14, 14	oveq12i 7162	. . . . . . . . . . . . . . . . 17 ⊢ (((abs‘0)↑2) + ((abs‘0)↑2)) = (0 + 0)
59		00id 10853	. . . . . . . . . . . . . . . . 17 ⊢ (0 + 0) = 0
60	58, 59	eqtri 2781	. . . . . . . . . . . . . . . 16 ⊢ (((abs‘0)↑2) + ((abs‘0)↑2)) = 0
61	24	4sqlem4a 16342	. . . . . . . . . . . . . . . . 17 ⊢ ((0 ∈ ℤ[i] ∧ 0 ∈ ℤ[i]) → (((abs‘0)↑2) + ((abs‘0)↑2)) ∈ 𝑆)
62	23, 23, 61	mp2an 691	. . . . . . . . . . . . . . . 16 ⊢ (((abs‘0)↑2) + ((abs‘0)↑2)) ∈ 𝑆
63	60, 62	eqeltrri 2849	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ 𝑆
64		snssi 4698	. . . . . . . . . . . . . . 15 ⊢ (0 ∈ 𝑆 → {0} ⊆ 𝑆)
65	63, 64	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ {0} ⊆ 𝑆
66	57, 65	eqsstri 3926	. . . . . . . . . . . . 13 ⊢ (0...0) ⊆ 𝑆
67	66	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → (0...0) ⊆ 𝑆)
68		0p1e1 11796	. . . . . . . . . . . . . 14 ⊢ (0 + 1) = 1
69	68	oveq1i 7160	. . . . . . . . . . . . 13 ⊢ ((0 + 1)...(𝑗 − 1)) = (1...(𝑗 − 1))
70		simpr 488	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆)
71		dfss3 3880	. . . . . . . . . . . . . 14 ⊢ ((1...(𝑗 − 1)) ⊆ 𝑆 ↔ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆)
72	70, 71	sylibr 237	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → (1...(𝑗 − 1)) ⊆ 𝑆)
73	69, 72	eqsstrid 3940	. . . . . . . . . . . 12 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → ((0 + 1)...(𝑗 − 1)) ⊆ 𝑆)
74	67, 73	unssd 4091	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → ((0...0) ∪ ((0 + 1)...(𝑗 − 1))) ⊆ 𝑆)
75	56, 74	eqsstrd 3930	. . . . . . . . . 10 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → (0...(2 · ((𝑗 − 1) / 2))) ⊆ 𝑆)
76		oveq1 7157	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑖 → (𝑘 · 𝑗) = (𝑖 · 𝑗))
77	76	eleq1d 2836	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑖 → ((𝑘 · 𝑗) ∈ 𝑆 ↔ (𝑖 · 𝑗) ∈ 𝑆))
78	77	cbvrabv 3404	. . . . . . . . . 10 ⊢ {𝑘 ∈ ℕ ∣ (𝑘 · 𝑗) ∈ 𝑆} = {𝑖 ∈ ℕ ∣ (𝑖 · 𝑗) ∈ 𝑆}
79		eqid 2758	. . . . . . . . . 10 ⊢ inf({𝑘 ∈ ℕ ∣ (𝑘 · 𝑗) ∈ 𝑆}, ℝ, < ) = inf({𝑘 ∈ ℕ ∣ (𝑘 · 𝑗) ∈ 𝑆}, ℝ, < )
80	24, 31, 47, 33, 75, 78, 79	4sqlem18 16353	. . . . . . . . 9 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → 𝑗 ∈ 𝑆)
81	29, 80	sylanbr 585	. . . . . . . 8 ⊢ (((𝑗 ∈ ℙ ∧ 𝑗 ≠ 2) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → 𝑗 ∈ 𝑆)
82	81	an32s 651	. . . . . . 7 ⊢ (((𝑗 ∈ ℙ ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) ∧ 𝑗 ≠ 2) → 𝑗 ∈ 𝑆)
83	10, 10	oveq12i 7162	. . . . . . . . . 10 ⊢ (((abs‘1)↑2) + ((abs‘1)↑2)) = (1 + 1)
84		df-2 11737	. . . . . . . . . 10 ⊢ 2 = (1 + 1)
85	83, 84	eqtr4i 2784	. . . . . . . . 9 ⊢ (((abs‘1)↑2) + ((abs‘1)↑2)) = 2
86	24	4sqlem4a 16342	. . . . . . . . . 10 ⊢ ((1 ∈ ℤ[i] ∧ 1 ∈ ℤ[i]) → (((abs‘1)↑2) + ((abs‘1)↑2)) ∈ 𝑆)
87	20, 20, 86	mp2an 691	. . . . . . . . 9 ⊢ (((abs‘1)↑2) + ((abs‘1)↑2)) ∈ 𝑆
88	85, 87	eqeltrri 2849	. . . . . . . 8 ⊢ 2 ∈ 𝑆
89	88	a1i 11	. . . . . . 7 ⊢ ((𝑗 ∈ ℙ ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → 2 ∈ 𝑆)
90	28, 82, 89	pm2.61ne 3036	. . . . . 6 ⊢ ((𝑗 ∈ ℙ ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → 𝑗 ∈ 𝑆)
91	24	mul4sq 16345	. . . . . . 7 ⊢ ((𝑚 ∈ 𝑆 ∧ 𝑖 ∈ 𝑆) → (𝑚 · 𝑖) ∈ 𝑆)
92	91	a1i 11	. . . . . 6 ⊢ ((𝑚 ∈ (ℤ≥‘2) ∧ 𝑖 ∈ (ℤ≥‘2)) → ((𝑚 ∈ 𝑆 ∧ 𝑖 ∈ 𝑆) → (𝑚 · 𝑖) ∈ 𝑆))
93	2, 3, 4, 5, 6, 27, 90, 92	prmind2 16081	. . . . 5 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ 𝑆)
94		id 22	. . . . . 6 ⊢ (𝑘 = 0 → 𝑘 = 0)
95	94, 63	eqeltrdi 2860	. . . . 5 ⊢ (𝑘 = 0 → 𝑘 ∈ 𝑆)
96	93, 95	jaoi 854	. . . 4 ⊢ ((𝑘 ∈ ℕ ∨ 𝑘 = 0) → 𝑘 ∈ 𝑆)
97	1, 96	sylbi 220	. . 3 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ 𝑆)
98	97	ssriv 3896	. 2 ⊢ ℕ0 ⊆ 𝑆
99	24	4sqlem1 16339	. 2 ⊢ 𝑆 ⊆ ℕ0
100	98, 99	eqssi 3908	1 ⊢ ℕ0 = 𝑆

Syntax hints:   → wi 4   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2111  {cab 2735   ≠ wne 2951  ∀wral 3070  ∃wrex 3071  {crab 3074   ∖ cdif 3855   ∪ cun 3856   ⊆ wss 3858  {csn 4522  ‘cfv 6335  (class class class)co 7150  infcinf 8938  ℂcc 10573  ℝcr 10574  0cc0 10575  1c1 10576   + caddc 10578   · cmul 10580   < clt 10713   − cmin 10908   / cdiv 11335  ℕcn 11674  2c2 11729  ℕ₀cn0 11934  ℤcz 12020  ℤ_≥cuz 12282  ...cfz 12939  ↑cexp 13479  abscabs 14641  ℙcprime 16067  ℤ[i]cgz 16320

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459  ax-cnex 10631  ax-resscn 10632  ax-1cn 10633  ax-icn 10634  ax-addcl 10635  ax-addrcl 10636  ax-mulcl 10637  ax-mulrcl 10638  ax-mulcom 10639  ax-addass 10640  ax-mulass 10641  ax-distr 10642  ax-i2m1 10643  ax-1ne0 10644  ax-1rid 10645  ax-rnegex 10646  ax-rrecex 10647  ax-cnre 10648  ax-pre-lttri 10649  ax-pre-lttrn 10650  ax-pre-ltadd 10651  ax-pre-mulgt0 10652  ax-pre-sup 10653

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-int 4839  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-tr 5139  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-we 5485  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7580  df-1st 7693  df-2nd 7694  df-wrecs 7957  df-recs 8018  df-rdg 8056  df-1o 8112  df-2o 8113  df-oadd 8116  df-er 8299  df-en 8528  df-dom 8529  df-sdom 8530  df-fin 8531  df-sup 8939  df-inf 8940  df-dju 9363  df-card 9401  df-pnf 10715  df-mnf 10716  df-xr 10717  df-ltxr 10718  df-le 10719  df-sub 10910  df-neg 10911  df-div 11336  df-nn 11675  df-2 11737  df-3 11738  df-4 11739  df-n0 11935  df-xnn0 12007  df-z 12021  df-uz 12283  df-rp 12431  df-fz 12940  df-fl 13211  df-mod 13287  df-seq 13419  df-exp 13480  df-hash 13741  df-cj 14506  df-re 14507  df-im 14508  df-sqrt 14642  df-abs 14643  df-dvds 15656  df-gcd 15894  df-prm 16068  df-gz 16321

This theorem is referenced by:  4sq  16355