Theorem 4sqlem19 16929

Description: Lemma for 4sq 16930. 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 16928. 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 16920 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 12434	. . . 4 ⊢ (𝑘 ∈ ℕ0 ↔ (𝑘 ∈ ℕ ∨ 𝑘 = 0))
2		eleq1 2825	. . . . . 6 ⊢ (𝑗 = 1 → (𝑗 ∈ 𝑆 ↔ 1 ∈ 𝑆))
3		eleq1 2825	. . . . . 6 ⊢ (𝑗 = 𝑚 → (𝑗 ∈ 𝑆 ↔ 𝑚 ∈ 𝑆))
4		eleq1 2825	. . . . . 6 ⊢ (𝑗 = 𝑖 → (𝑗 ∈ 𝑆 ↔ 𝑖 ∈ 𝑆))
5		eleq1 2825	. . . . . 6 ⊢ (𝑗 = (𝑚 · 𝑖) → (𝑗 ∈ 𝑆 ↔ (𝑚 · 𝑖) ∈ 𝑆))
6		eleq1 2825	. . . . . 6 ⊢ (𝑗 = 𝑘 → (𝑗 ∈ 𝑆 ↔ 𝑘 ∈ 𝑆))
7		abs1 15254	. . . . . . . . . . 11 ⊢ (abs‘1) = 1
8	7	oveq1i 7372	. . . . . . . . . 10 ⊢ ((abs‘1)↑2) = (1↑2)
9		sq1 14152	. . . . . . . . . 10 ⊢ (1↑2) = 1
10	8, 9	eqtri 2760	. . . . . . . . 9 ⊢ ((abs‘1)↑2) = 1
11		abs0 15242	. . . . . . . . . . 11 ⊢ (abs‘0) = 0
12	11	oveq1i 7372	. . . . . . . . . 10 ⊢ ((abs‘0)↑2) = (0↑2)
13		sq0 14149	. . . . . . . . . 10 ⊢ (0↑2) = 0
14	12, 13	eqtri 2760	. . . . . . . . 9 ⊢ ((abs‘0)↑2) = 0
15	10, 14	oveq12i 7374	. . . . . . . 8 ⊢ (((abs‘1)↑2) + ((abs‘0)↑2)) = (1 + 0)
16		1p0e1 12295	. . . . . . . 8 ⊢ (1 + 0) = 1
17	15, 16	eqtri 2760	. . . . . . 7 ⊢ (((abs‘1)↑2) + ((abs‘0)↑2)) = 1
18		1z 12552	. . . . . . . . 9 ⊢ 1 ∈ ℤ
19		zgz 16899	. . . . . . . . 9 ⊢ (1 ∈ ℤ → 1 ∈ ℤ[i])
20	18, 19	ax-mp 5	. . . . . . . 8 ⊢ 1 ∈ ℤ[i]
21		0z 12530	. . . . . . . . 9 ⊢ 0 ∈ ℤ
22		zgz 16899	. . . . . . . . 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 16917	. . . . . . . 8 ⊢ ((1 ∈ ℤ[i] ∧ 0 ∈ ℤ[i]) → (((abs‘1)↑2) + ((abs‘0)↑2)) ∈ 𝑆)
26	20, 23, 25	mp2an 693	. . . . . . 7 ⊢ (((abs‘1)↑2) + ((abs‘0)↑2)) ∈ 𝑆
27	17, 26	eqeltrri 2834	. . . . . 6 ⊢ 1 ∈ 𝑆
28		eleq1 2825	. . . . . . 7 ⊢ (𝑗 = 2 → (𝑗 ∈ 𝑆 ↔ 2 ∈ 𝑆))
29		eldifsn 4730	. . . . . . . . 9 ⊢ (𝑗 ∈ (ℙ ∖ {2}) ↔ (𝑗 ∈ ℙ ∧ 𝑗 ≠ 2))
30		oddprm 16776	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (ℙ ∖ {2}) → ((𝑗 − 1) / 2) ∈ ℕ)
31	30	adantr 480	. . . . . . . . . 10 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → ((𝑗 − 1) / 2) ∈ ℕ)
32		eldifi 4072	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (ℙ ∖ {2}) → 𝑗 ∈ ℙ)
33	32	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → 𝑗 ∈ ℙ)
34		prmnn 16638	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ ℙ → 𝑗 ∈ ℕ)
35		nncn 12177	. . . . . . . . . . . . . . 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 11091	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℂ
38		subcl 11387	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑗 − 1) ∈ ℂ)
39	36, 37, 38	sylancl 587	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → (𝑗 − 1) ∈ ℂ)
40		2cnd 12254	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → 2 ∈ ℂ)
41		2ne0 12280	. . . . . . . . . . . . . 14 ⊢ 2 ≠ 0
42	41	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → 2 ≠ 0)
43	39, 40, 42	divcan2d 11928	. . . . . . . . . . . 12 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → (2 · ((𝑗 − 1) / 2)) = (𝑗 − 1))
44	43	oveq1d 7377	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → ((2 · ((𝑗 − 1) / 2)) + 1) = ((𝑗 − 1) + 1))
45		npcan 11397	. . . . . . . . . . . 12 ⊢ ((𝑗 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑗 − 1) + 1) = 𝑗)
46	36, 37, 45	sylancl 587	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → ((𝑗 − 1) + 1) = 𝑗)
47	44, 46	eqtr2d 2773	. . . . . . . . . 10 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → 𝑗 = ((2 · ((𝑗 − 1) / 2)) + 1))
48	43	oveq2d 7378	. . . . . . . . . . . 12 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → (0...(2 · ((𝑗 − 1) / 2))) = (0...(𝑗 − 1)))
49		nnm1nn0 12473	. . . . . . . . . . . . . . 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 12824	. . . . . . . . . . . . . 14 ⊢ ((𝑗 − 1) ∈ ℕ0 ↔ (𝑗 − 1) ∈ (ℤ≥‘0))
52	50, 51	sylib 218	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → (𝑗 − 1) ∈ (ℤ≥‘0))
53		eluzfz1 13480	. . . . . . . . . . . . 13 ⊢ ((𝑗 − 1) ∈ (ℤ≥‘0) → 0 ∈ (0...(𝑗 − 1)))
54		fzsplit 13499	. . . . . . . . . . . . 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 2772	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → (0...(2 · ((𝑗 − 1) / 2))) = ((0...0) ∪ ((0 + 1)...(𝑗 − 1))))
57		fz0sn 13576	. . . . . . . . . . . . . 14 ⊢ (0...0) = {0}
58	14, 14	oveq12i 7374	. . . . . . . . . . . . . . . . 17 ⊢ (((abs‘0)↑2) + ((abs‘0)↑2)) = (0 + 0)
59		00id 11316	. . . . . . . . . . . . . . . . 17 ⊢ (0 + 0) = 0
60	58, 59	eqtri 2760	. . . . . . . . . . . . . . . 16 ⊢ (((abs‘0)↑2) + ((abs‘0)↑2)) = 0
61	24	4sqlem4a 16917	. . . . . . . . . . . . . . . . 17 ⊢ ((0 ∈ ℤ[i] ∧ 0 ∈ ℤ[i]) → (((abs‘0)↑2) + ((abs‘0)↑2)) ∈ 𝑆)
62	23, 23, 61	mp2an 693	. . . . . . . . . . . . . . . 16 ⊢ (((abs‘0)↑2) + ((abs‘0)↑2)) ∈ 𝑆
63	60, 62	eqeltrri 2834	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ 𝑆
64		snssi 4752	. . . . . . . . . . . . . . 15 ⊢ (0 ∈ 𝑆 → {0} ⊆ 𝑆)
65	63, 64	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ {0} ⊆ 𝑆
66	57, 65	eqsstri 3969	. . . . . . . . . . . . 13 ⊢ (0...0) ⊆ 𝑆
67	66	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → (0...0) ⊆ 𝑆)
68		0p1e1 12293	. . . . . . . . . . . . . 14 ⊢ (0 + 1) = 1
69	68	oveq1i 7372	. . . . . . . . . . . . 13 ⊢ ((0 + 1)...(𝑗 − 1)) = (1...(𝑗 − 1))
70		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆)
71		dfss3 3911	. . . . . . . . . . . . . 14 ⊢ ((1...(𝑗 − 1)) ⊆ 𝑆 ↔ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆)
72	70, 71	sylibr 234	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → (1...(𝑗 − 1)) ⊆ 𝑆)
73	69, 72	eqsstrid 3961	. . . . . . . . . . . 12 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → ((0 + 1)...(𝑗 − 1)) ⊆ 𝑆)
74	67, 73	unssd 4133	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → ((0...0) ∪ ((0 + 1)...(𝑗 − 1))) ⊆ 𝑆)
75	56, 74	eqsstrd 3957	. . . . . . . . . 10 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → (0...(2 · ((𝑗 − 1) / 2))) ⊆ 𝑆)
76		oveq1 7369	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑖 → (𝑘 · 𝑗) = (𝑖 · 𝑗))
77	76	eleq1d 2822	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑖 → ((𝑘 · 𝑗) ∈ 𝑆 ↔ (𝑖 · 𝑗) ∈ 𝑆))
78	77	cbvrabv 3400	. . . . . . . . . 10 ⊢ {𝑘 ∈ ℕ ∣ (𝑘 · 𝑗) ∈ 𝑆} = {𝑖 ∈ ℕ ∣ (𝑖 · 𝑗) ∈ 𝑆}
79		eqid 2737	. . . . . . . . . 10 ⊢ inf({𝑘 ∈ ℕ ∣ (𝑘 · 𝑗) ∈ 𝑆}, ℝ, < ) = inf({𝑘 ∈ ℕ ∣ (𝑘 · 𝑗) ∈ 𝑆}, ℝ, < )
80	24, 31, 47, 33, 75, 78, 79	4sqlem18 16928	. . . . . . . . 9 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → 𝑗 ∈ 𝑆)
81	29, 80	sylanbr 583	. . . . . . . 8 ⊢ (((𝑗 ∈ ℙ ∧ 𝑗 ≠ 2) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → 𝑗 ∈ 𝑆)
82	81	an32s 653	. . . . . . 7 ⊢ (((𝑗 ∈ ℙ ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) ∧ 𝑗 ≠ 2) → 𝑗 ∈ 𝑆)
83	10, 10	oveq12i 7374	. . . . . . . . . 10 ⊢ (((abs‘1)↑2) + ((abs‘1)↑2)) = (1 + 1)
84		df-2 12239	. . . . . . . . . 10 ⊢ 2 = (1 + 1)
85	83, 84	eqtr4i 2763	. . . . . . . . 9 ⊢ (((abs‘1)↑2) + ((abs‘1)↑2)) = 2
86	24	4sqlem4a 16917	. . . . . . . . . 10 ⊢ ((1 ∈ ℤ[i] ∧ 1 ∈ ℤ[i]) → (((abs‘1)↑2) + ((abs‘1)↑2)) ∈ 𝑆)
87	20, 20, 86	mp2an 693	. . . . . . . . 9 ⊢ (((abs‘1)↑2) + ((abs‘1)↑2)) ∈ 𝑆
88	85, 87	eqeltrri 2834	. . . . . . . 8 ⊢ 2 ∈ 𝑆
89	88	a1i 11	. . . . . . 7 ⊢ ((𝑗 ∈ ℙ ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → 2 ∈ 𝑆)
90	28, 82, 89	pm2.61ne 3018	. . . . . 6 ⊢ ((𝑗 ∈ ℙ ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → 𝑗 ∈ 𝑆)
91	24	mul4sq 16920	. . . . . . 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 16649	. . . . 5 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ 𝑆)
94		id 22	. . . . . 6 ⊢ (𝑘 = 0 → 𝑘 = 0)
95	94, 63	eqeltrdi 2845	. . . . 5 ⊢ (𝑘 = 0 → 𝑘 ∈ 𝑆)
96	93, 95	jaoi 858	. . . 4 ⊢ ((𝑘 ∈ ℕ ∨ 𝑘 = 0) → 𝑘 ∈ 𝑆)
97	1, 96	sylbi 217	. . 3 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ 𝑆)
98	97	ssriv 3926	. 2 ⊢ ℕ0 ⊆ 𝑆
99	24	4sqlem1 16914	. 2 ⊢ 𝑆 ⊆ ℕ0
100	98, 99	eqssi 3939	1 ⊢ ℕ0 = 𝑆

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 848   = wceq 1542   ∈ wcel 2114  {cab 2715   ≠ wne 2933  ∀wral 3052  ∃wrex 3062  {crab 3390   ∖ cdif 3887   ∪ cun 3888   ⊆ wss 3890  {csn 4568  ‘cfv 6494  (class class class)co 7362  infcinf 9349  ℂcc 11031  ℝcr 11032  0cc0 11033  1c1 11034   + caddc 11036   · cmul 11038   < clt 11174   − cmin 11372   / cdiv 11802  ℕcn 12169  2c2 12231  ℕ₀cn0 12432  ℤcz 12519  ℤ_≥cuz 12783  ...cfz 13456  ↑cexp 14018  abscabs 15191  ℙcprime 16635  ℤ[i]cgz 16895

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5304  ax-pr 5372  ax-un 7684  ax-cnex 11089  ax-resscn 11090  ax-1cn 11091  ax-icn 11092  ax-addcl 11093  ax-addrcl 11094  ax-mulcl 11095  ax-mulrcl 11096  ax-mulcom 11097  ax-addass 11098  ax-mulass 11099  ax-distr 11100  ax-i2m1 11101  ax-1ne0 11102  ax-1rid 11103  ax-rnegex 11104  ax-rrecex 11105  ax-cnre 11106  ax-pre-lttri 11107  ax-pre-lttrn 11108  ax-pre-ltadd 11109  ax-pre-mulgt0 11110  ax-pre-sup 11111

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5521  df-eprel 5526  df-po 5534  df-so 5535  df-fr 5579  df-we 5581  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-pred 6261  df-ord 6322  df-on 6323  df-lim 6324  df-suc 6325  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-riota 7319  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7813  df-1st 7937  df-2nd 7938  df-frecs 8226  df-wrecs 8257  df-recs 8306  df-rdg 8344  df-1o 8400  df-2o 8401  df-oadd 8404  df-er 8638  df-en 8889  df-dom 8890  df-sdom 8891  df-fin 8892  df-sup 9350  df-inf 9351  df-dju 9820  df-card 9858  df-pnf 11176  df-mnf 11177  df-xr 11178  df-ltxr 11179  df-le 11180  df-sub 11374  df-neg 11375  df-div 11803  df-nn 12170  df-2 12239  df-3 12240  df-4 12241  df-n0 12433  df-xnn0 12506  df-z 12520  df-uz 12784  df-rp 12938  df-fz 13457  df-fl 13746  df-mod 13824  df-seq 13959  df-exp 14019  df-hash 14288  df-cj 15056  df-re 15057  df-im 15058  df-sqrt 15192  df-abs 15193  df-dvds 16217  df-gcd 16459  df-prm 16636  df-gz 16896

This theorem is referenced by:  4sq  16930