Theorem 4sqlem19 15667

Description: Lemma for 4sq 15668. 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 15666. 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 15658 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 11294	. . . 4 ⊢ (𝑘 ∈ ℕ0 ↔ (𝑘 ∈ ℕ ∨ 𝑘 = 0))
2		eleq1 2689	. . . . . 6 ⊢ (𝑗 = 1 → (𝑗 ∈ 𝑆 ↔ 1 ∈ 𝑆))
3		eleq1 2689	. . . . . 6 ⊢ (𝑗 = 𝑚 → (𝑗 ∈ 𝑆 ↔ 𝑚 ∈ 𝑆))
4		eleq1 2689	. . . . . 6 ⊢ (𝑗 = 𝑖 → (𝑗 ∈ 𝑆 ↔ 𝑖 ∈ 𝑆))
5		eleq1 2689	. . . . . 6 ⊢ (𝑗 = (𝑚 · 𝑖) → (𝑗 ∈ 𝑆 ↔ (𝑚 · 𝑖) ∈ 𝑆))
6		eleq1 2689	. . . . . 6 ⊢ (𝑗 = 𝑘 → (𝑗 ∈ 𝑆 ↔ 𝑘 ∈ 𝑆))
7		abs1 14037	. . . . . . . . . . 11 ⊢ (abs‘1) = 1
8	7	oveq1i 6660	. . . . . . . . . 10 ⊢ ((abs‘1)↑2) = (1↑2)
9		sq1 12958	. . . . . . . . . 10 ⊢ (1↑2) = 1
10	8, 9	eqtri 2644	. . . . . . . . 9 ⊢ ((abs‘1)↑2) = 1
11		abs0 14025	. . . . . . . . . . 11 ⊢ (abs‘0) = 0
12	11	oveq1i 6660	. . . . . . . . . 10 ⊢ ((abs‘0)↑2) = (0↑2)
13		sq0 12955	. . . . . . . . . 10 ⊢ (0↑2) = 0
14	12, 13	eqtri 2644	. . . . . . . . 9 ⊢ ((abs‘0)↑2) = 0
15	10, 14	oveq12i 6662	. . . . . . . 8 ⊢ (((abs‘1)↑2) + ((abs‘0)↑2)) = (1 + 0)
16		1p0e1 11133	. . . . . . . 8 ⊢ (1 + 0) = 1
17	15, 16	eqtri 2644	. . . . . . 7 ⊢ (((abs‘1)↑2) + ((abs‘0)↑2)) = 1
18		1z 11407	. . . . . . . . 9 ⊢ 1 ∈ ℤ
19		zgz 15637	. . . . . . . . 9 ⊢ (1 ∈ ℤ → 1 ∈ ℤ[i])
20	18, 19	ax-mp 5	. . . . . . . 8 ⊢ 1 ∈ ℤ[i]
21		0z 11388	. . . . . . . . 9 ⊢ 0 ∈ ℤ
22		zgz 15637	. . . . . . . . 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 15655	. . . . . . . 8 ⊢ ((1 ∈ ℤ[i] ∧ 0 ∈ ℤ[i]) → (((abs‘1)↑2) + ((abs‘0)↑2)) ∈ 𝑆)
26	20, 23, 25	mp2an 708	. . . . . . 7 ⊢ (((abs‘1)↑2) + ((abs‘0)↑2)) ∈ 𝑆
27	17, 26	eqeltrri 2698	. . . . . 6 ⊢ 1 ∈ 𝑆
28		eleq1 2689	. . . . . . 7 ⊢ (𝑗 = 2 → (𝑗 ∈ 𝑆 ↔ 2 ∈ 𝑆))
29		eldifsn 4317	. . . . . . . . 9 ⊢ (𝑗 ∈ (ℙ ∖ {2}) ↔ (𝑗 ∈ ℙ ∧ 𝑗 ≠ 2))
30		oddprm 15515	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (ℙ ∖ {2}) → ((𝑗 − 1) / 2) ∈ ℕ)
31	30	adantr 481	. . . . . . . . . 10 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → ((𝑗 − 1) / 2) ∈ ℕ)
32		eldifi 3732	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (ℙ ∖ {2}) → 𝑗 ∈ ℙ)
33	32	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → 𝑗 ∈ ℙ)
34		prmnn 15388	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ ℙ → 𝑗 ∈ ℕ)
35		nncn 11028	. . . . . . . . . . . . . . 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 9994	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℂ
38		subcl 10280	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑗 − 1) ∈ ℂ)
39	36, 37, 38	sylancl 694	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → (𝑗 − 1) ∈ ℂ)
40		2cnd 11093	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → 2 ∈ ℂ)
41		2ne0 11113	. . . . . . . . . . . . . 14 ⊢ 2 ≠ 0
42	41	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → 2 ≠ 0)
43	39, 40, 42	divcan2d 10803	. . . . . . . . . . . 12 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → (2 · ((𝑗 − 1) / 2)) = (𝑗 − 1))
44	43	oveq1d 6665	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → ((2 · ((𝑗 − 1) / 2)) + 1) = ((𝑗 − 1) + 1))
45		npcan 10290	. . . . . . . . . . . 12 ⊢ ((𝑗 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑗 − 1) + 1) = 𝑗)
46	36, 37, 45	sylancl 694	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → ((𝑗 − 1) + 1) = 𝑗)
47	44, 46	eqtr2d 2657	. . . . . . . . . 10 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → 𝑗 = ((2 · ((𝑗 − 1) / 2)) + 1))
48	43	oveq2d 6666	. . . . . . . . . . . 12 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → (0...(2 · ((𝑗 − 1) / 2))) = (0...(𝑗 − 1)))
49		nnm1nn0 11334	. . . . . . . . . . . . . . 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 11725	. . . . . . . . . . . . . 14 ⊢ ((𝑗 − 1) ∈ ℕ0 ↔ (𝑗 − 1) ∈ (ℤ≥‘0))
52	50, 51	sylib 208	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → (𝑗 − 1) ∈ (ℤ≥‘0))
53		eluzfz1 12348	. . . . . . . . . . . . 13 ⊢ ((𝑗 − 1) ∈ (ℤ≥‘0) → 0 ∈ (0...(𝑗 − 1)))
54		fzsplit 12367	. . . . . . . . . . . . 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 2656	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → (0...(2 · ((𝑗 − 1) / 2))) = ((0...0) ∪ ((0 + 1)...(𝑗 − 1))))
57		fzsn 12383	. . . . . . . . . . . . . . 15 ⊢ (0 ∈ ℤ → (0...0) = {0})
58	21, 57	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (0...0) = {0}
59	14, 14	oveq12i 6662	. . . . . . . . . . . . . . . . 17 ⊢ (((abs‘0)↑2) + ((abs‘0)↑2)) = (0 + 0)
60		00id 10211	. . . . . . . . . . . . . . . . 17 ⊢ (0 + 0) = 0
61	59, 60	eqtri 2644	. . . . . . . . . . . . . . . 16 ⊢ (((abs‘0)↑2) + ((abs‘0)↑2)) = 0
62	24	4sqlem4a 15655	. . . . . . . . . . . . . . . . 17 ⊢ ((0 ∈ ℤ[i] ∧ 0 ∈ ℤ[i]) → (((abs‘0)↑2) + ((abs‘0)↑2)) ∈ 𝑆)
63	23, 23, 62	mp2an 708	. . . . . . . . . . . . . . . 16 ⊢ (((abs‘0)↑2) + ((abs‘0)↑2)) ∈ 𝑆
64	61, 63	eqeltrri 2698	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ 𝑆
65		snssi 4339	. . . . . . . . . . . . . . 15 ⊢ (0 ∈ 𝑆 → {0} ⊆ 𝑆)
66	64, 65	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ {0} ⊆ 𝑆
67	58, 66	eqsstri 3635	. . . . . . . . . . . . 13 ⊢ (0...0) ⊆ 𝑆
68	67	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → (0...0) ⊆ 𝑆)
69		0p1e1 11132	. . . . . . . . . . . . . 14 ⊢ (0 + 1) = 1
70	69	oveq1i 6660	. . . . . . . . . . . . 13 ⊢ ((0 + 1)...(𝑗 − 1)) = (1...(𝑗 − 1))
71		simpr 477	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆)
72		dfss3 3592	. . . . . . . . . . . . . 14 ⊢ ((1...(𝑗 − 1)) ⊆ 𝑆 ↔ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆)
73	71, 72	sylibr 224	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → (1...(𝑗 − 1)) ⊆ 𝑆)
74	70, 73	syl5eqss 3649	. . . . . . . . . . . 12 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → ((0 + 1)...(𝑗 − 1)) ⊆ 𝑆)
75	68, 74	unssd 3789	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → ((0...0) ∪ ((0 + 1)...(𝑗 − 1))) ⊆ 𝑆)
76	56, 75	eqsstrd 3639	. . . . . . . . . 10 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → (0...(2 · ((𝑗 − 1) / 2))) ⊆ 𝑆)
77		oveq1 6657	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑖 → (𝑘 · 𝑗) = (𝑖 · 𝑗))
78	77	eleq1d 2686	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑖 → ((𝑘 · 𝑗) ∈ 𝑆 ↔ (𝑖 · 𝑗) ∈ 𝑆))
79	78	cbvrabv 3199	. . . . . . . . . 10 ⊢ {𝑘 ∈ ℕ ∣ (𝑘 · 𝑗) ∈ 𝑆} = {𝑖 ∈ ℕ ∣ (𝑖 · 𝑗) ∈ 𝑆}
80		eqid 2622	. . . . . . . . . 10 ⊢ inf({𝑘 ∈ ℕ ∣ (𝑘 · 𝑗) ∈ 𝑆}, ℝ, < ) = inf({𝑘 ∈ ℕ ∣ (𝑘 · 𝑗) ∈ 𝑆}, ℝ, < )
81	24, 31, 47, 33, 76, 79, 80	4sqlem18 15666	. . . . . . . . 9 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → 𝑗 ∈ 𝑆)
82	29, 81	sylanbr 490	. . . . . . . 8 ⊢ (((𝑗 ∈ ℙ ∧ 𝑗 ≠ 2) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → 𝑗 ∈ 𝑆)
83	82	an32s 846	. . . . . . 7 ⊢ (((𝑗 ∈ ℙ ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) ∧ 𝑗 ≠ 2) → 𝑗 ∈ 𝑆)
84	10, 10	oveq12i 6662	. . . . . . . . . 10 ⊢ (((abs‘1)↑2) + ((abs‘1)↑2)) = (1 + 1)
85		df-2 11079	. . . . . . . . . 10 ⊢ 2 = (1 + 1)
86	84, 85	eqtr4i 2647	. . . . . . . . 9 ⊢ (((abs‘1)↑2) + ((abs‘1)↑2)) = 2
87	24	4sqlem4a 15655	. . . . . . . . . 10 ⊢ ((1 ∈ ℤ[i] ∧ 1 ∈ ℤ[i]) → (((abs‘1)↑2) + ((abs‘1)↑2)) ∈ 𝑆)
88	20, 20, 87	mp2an 708	. . . . . . . . 9 ⊢ (((abs‘1)↑2) + ((abs‘1)↑2)) ∈ 𝑆
89	86, 88	eqeltrri 2698	. . . . . . . 8 ⊢ 2 ∈ 𝑆
90	89	a1i 11	. . . . . . 7 ⊢ ((𝑗 ∈ ℙ ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → 2 ∈ 𝑆)
91	28, 83, 90	pm2.61ne 2879	. . . . . 6 ⊢ ((𝑗 ∈ ℙ ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → 𝑗 ∈ 𝑆)
92	24	mul4sq 15658	. . . . . . 7 ⊢ ((𝑚 ∈ 𝑆 ∧ 𝑖 ∈ 𝑆) → (𝑚 · 𝑖) ∈ 𝑆)
93	92	a1i 11	. . . . . 6 ⊢ ((𝑚 ∈ (ℤ≥‘2) ∧ 𝑖 ∈ (ℤ≥‘2)) → ((𝑚 ∈ 𝑆 ∧ 𝑖 ∈ 𝑆) → (𝑚 · 𝑖) ∈ 𝑆))
94	2, 3, 4, 5, 6, 27, 91, 93	prmind2 15398	. . . . 5 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ 𝑆)
95		id 22	. . . . . 6 ⊢ (𝑘 = 0 → 𝑘 = 0)
96	95, 64	syl6eqel 2709	. . . . 5 ⊢ (𝑘 = 0 → 𝑘 ∈ 𝑆)
97	94, 96	jaoi 394	. . . 4 ⊢ ((𝑘 ∈ ℕ ∨ 𝑘 = 0) → 𝑘 ∈ 𝑆)
98	1, 97	sylbi 207	. . 3 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ 𝑆)
99	98	ssriv 3607	. 2 ⊢ ℕ0 ⊆ 𝑆
100	24	4sqlem1 15652	. 2 ⊢ 𝑆 ⊆ ℕ0
101	99, 100	eqssi 3619	1 ⊢ ℕ0 = 𝑆

Syntax hints:   → wi 4   ∨ wo 383   ∧ wa 384   = wceq 1483   ∈ wcel 1990  {cab 2608   ≠ wne 2794  ∀wral 2912  ∃wrex 2913  {crab 2916   ∖ cdif 3571   ∪ cun 3572   ⊆ wss 3574  {csn 4177  ‘cfv 5888  (class class class)co 6650  infcinf 8347  ℂcc 9934  ℝcr 9935  0cc0 9936  1c1 9937   + caddc 9939   · cmul 9941   < clt 10074   − cmin 10266   / cdiv 10684  ℕcn 11020  2c2 11070  ℕ₀cn0 11292  ℤcz 11377  ℤ_≥cuz 11687  ...cfz 12326  ↑cexp 12860  abscabs 13974  ℙcprime 15385  ℤ[i]cgz 15633

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-gcd 15217  df-prm 15386  df-gz 15634

This theorem is referenced by:  4sq  15668