Theorem 4sqlem19 12807

Description: Lemma for 4sq 12808. 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 12806. 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 12792 𝑘 ∈ 𝑆. (Contributed by Mario Carneiro, 14-Jul-2014.) (Revised by Mario Carneiro, 20-Jun-2015.)

Hypothesis

Ref	Expression
4sqlem11.1	⊢ 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:   𝑆,𝑛   𝑤,𝑛,𝑥,𝑦,𝑧

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

Proof of Theorem 4sqlem19

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

Step	Hyp	Ref	Expression
1		elnn0 9317	. . . 4 ⊢ (𝑘 ∈ ℕ0 ↔ (𝑘 ∈ ℕ ∨ 𝑘 = 0))
2		eleq1 2269	. . . . . 6 ⊢ (𝑗 = 1 → (𝑗 ∈ 𝑆 ↔ 1 ∈ 𝑆))
3		eleq1 2269	. . . . . 6 ⊢ (𝑗 = 𝑚 → (𝑗 ∈ 𝑆 ↔ 𝑚 ∈ 𝑆))
4		eleq1 2269	. . . . . 6 ⊢ (𝑗 = 𝑖 → (𝑗 ∈ 𝑆 ↔ 𝑖 ∈ 𝑆))
5		eleq1 2269	. . . . . 6 ⊢ (𝑗 = (𝑚 · 𝑖) → (𝑗 ∈ 𝑆 ↔ (𝑚 · 𝑖) ∈ 𝑆))
6		eleq1 2269	. . . . . 6 ⊢ (𝑗 = 𝑘 → (𝑗 ∈ 𝑆 ↔ 𝑘 ∈ 𝑆))
7		abs1 11458	. . . . . . . . . . 11 ⊢ (abs‘1) = 1
8	7	oveq1i 5967	. . . . . . . . . 10 ⊢ ((abs‘1)↑2) = (1↑2)
9		sq1 10800	. . . . . . . . . 10 ⊢ (1↑2) = 1
10	8, 9	eqtri 2227	. . . . . . . . 9 ⊢ ((abs‘1)↑2) = 1
11		abs0 11444	. . . . . . . . . . 11 ⊢ (abs‘0) = 0
12	11	oveq1i 5967	. . . . . . . . . 10 ⊢ ((abs‘0)↑2) = (0↑2)
13		sq0 10797	. . . . . . . . . 10 ⊢ (0↑2) = 0
14	12, 13	eqtri 2227	. . . . . . . . 9 ⊢ ((abs‘0)↑2) = 0
15	10, 14	oveq12i 5969	. . . . . . . 8 ⊢ (((abs‘1)↑2) + ((abs‘0)↑2)) = (1 + 0)
16		1p0e1 9172	. . . . . . . 8 ⊢ (1 + 0) = 1
17	15, 16	eqtri 2227	. . . . . . 7 ⊢ (((abs‘1)↑2) + ((abs‘0)↑2)) = 1
18		1z 9418	. . . . . . . . 9 ⊢ 1 ∈ ℤ
19		zgz 12771	. . . . . . . . 9 ⊢ (1 ∈ ℤ → 1 ∈ ℤ[i])
20	18, 19	ax-mp 5	. . . . . . . 8 ⊢ 1 ∈ ℤ[i]
21		0z 9403	. . . . . . . . 9 ⊢ 0 ∈ ℤ
22		zgz 12771	. . . . . . . . 9 ⊢ (0 ∈ ℤ → 0 ∈ ℤ[i])
23	21, 22	ax-mp 5	. . . . . . . 8 ⊢ 0 ∈ ℤ[i]
24		4sqlem11.1	. . . . . . . . 9 ⊢ 𝑆 = {𝑛 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))}
25	24	4sqlem4a 12789	. . . . . . . 8 ⊢ ((1 ∈ ℤ[i] ∧ 0 ∈ ℤ[i]) → (((abs‘1)↑2) + ((abs‘0)↑2)) ∈ 𝑆)
26	20, 23, 25	mp2an 426	. . . . . . 7 ⊢ (((abs‘1)↑2) + ((abs‘0)↑2)) ∈ 𝑆
27	17, 26	eqeltrri 2280	. . . . . 6 ⊢ 1 ∈ 𝑆
28	10, 10	oveq12i 5969	. . . . . . . . . 10 ⊢ (((abs‘1)↑2) + ((abs‘1)↑2)) = (1 + 1)
29		df-2 9115	. . . . . . . . . 10 ⊢ 2 = (1 + 1)
30	28, 29	eqtr4i 2230	. . . . . . . . 9 ⊢ (((abs‘1)↑2) + ((abs‘1)↑2)) = 2
31	24	4sqlem4a 12789	. . . . . . . . . 10 ⊢ ((1 ∈ ℤ[i] ∧ 1 ∈ ℤ[i]) → (((abs‘1)↑2) + ((abs‘1)↑2)) ∈ 𝑆)
32	20, 20, 31	mp2an 426	. . . . . . . . 9 ⊢ (((abs‘1)↑2) + ((abs‘1)↑2)) ∈ 𝑆
33	30, 32	eqeltrri 2280	. . . . . . . 8 ⊢ 2 ∈ 𝑆
34		eleq1 2269	. . . . . . . . 9 ⊢ (𝑗 = 2 → (𝑗 ∈ 𝑆 ↔ 2 ∈ 𝑆))
35	34	adantl 277	. . . . . . . 8 ⊢ (((𝑗 ∈ ℙ ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) ∧ 𝑗 = 2) → (𝑗 ∈ 𝑆 ↔ 2 ∈ 𝑆))
36	33, 35	mpbiri 168	. . . . . . 7 ⊢ (((𝑗 ∈ ℙ ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) ∧ 𝑗 = 2) → 𝑗 ∈ 𝑆)
37		eldifsn 3766	. . . . . . . . 9 ⊢ (𝑗 ∈ (ℙ ∖ {2}) ↔ (𝑗 ∈ ℙ ∧ 𝑗 ≠ 2))
38		oddprm 12657	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (ℙ ∖ {2}) → ((𝑗 − 1) / 2) ∈ ℕ)
39	38	adantr 276	. . . . . . . . . 10 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → ((𝑗 − 1) / 2) ∈ ℕ)
40		eldifi 3299	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (ℙ ∖ {2}) → 𝑗 ∈ ℙ)
41	40	adantr 276	. . . . . . . . . . . . . . 15 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → 𝑗 ∈ ℙ)
42		prmnn 12507	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ ℙ → 𝑗 ∈ ℕ)
43		nncn 9064	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℂ)
44	41, 42, 43	3syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → 𝑗 ∈ ℂ)
45		ax-1cn 8038	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℂ
46		subcl 8291	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑗 − 1) ∈ ℂ)
47	44, 45, 46	sylancl 413	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → (𝑗 − 1) ∈ ℂ)
48		2cnd 9129	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → 2 ∈ ℂ)
49		2ap0 9149	. . . . . . . . . . . . . 14 ⊢ 2 # 0
50	49	a1i 9	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → 2 # 0)
51	47, 48, 50	divcanap2d 8885	. . . . . . . . . . . 12 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → (2 · ((𝑗 − 1) / 2)) = (𝑗 − 1))
52	51	oveq1d 5972	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → ((2 · ((𝑗 − 1) / 2)) + 1) = ((𝑗 − 1) + 1))
53		npcan 8301	. . . . . . . . . . . 12 ⊢ ((𝑗 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑗 − 1) + 1) = 𝑗)
54	44, 45, 53	sylancl 413	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → ((𝑗 − 1) + 1) = 𝑗)
55	52, 54	eqtr2d 2240	. . . . . . . . . 10 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → 𝑗 = ((2 · ((𝑗 − 1) / 2)) + 1))
56	51	oveq2d 5973	. . . . . . . . . . . 12 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → (0...(2 · ((𝑗 − 1) / 2))) = (0...(𝑗 − 1)))
57		nnm1nn0 9356	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ ℕ → (𝑗 − 1) ∈ ℕ0)
58	41, 42, 57	3syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → (𝑗 − 1) ∈ ℕ0)
59		elnn0uz 9706	. . . . . . . . . . . . . 14 ⊢ ((𝑗 − 1) ∈ ℕ0 ↔ (𝑗 − 1) ∈ (ℤ≥‘0))
60	58, 59	sylib 122	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → (𝑗 − 1) ∈ (ℤ≥‘0))
61		eluzfz1 10173	. . . . . . . . . . . . 13 ⊢ ((𝑗 − 1) ∈ (ℤ≥‘0) → 0 ∈ (0...(𝑗 − 1)))
62		fzsplit 10193	. . . . . . . . . . . . 13 ⊢ (0 ∈ (0...(𝑗 − 1)) → (0...(𝑗 − 1)) = ((0...0) ∪ ((0 + 1)...(𝑗 − 1))))
63	60, 61, 62	3syl 17	. . . . . . . . . . . 12 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → (0...(𝑗 − 1)) = ((0...0) ∪ ((0 + 1)...(𝑗 − 1))))
64	56, 63	eqtrd 2239	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → (0...(2 · ((𝑗 − 1) / 2))) = ((0...0) ∪ ((0 + 1)...(𝑗 − 1))))
65		fz0sn 10263	. . . . . . . . . . . . . 14 ⊢ (0...0) = {0}
66	14, 14	oveq12i 5969	. . . . . . . . . . . . . . . . 17 ⊢ (((abs‘0)↑2) + ((abs‘0)↑2)) = (0 + 0)
67		00id 8233	. . . . . . . . . . . . . . . . 17 ⊢ (0 + 0) = 0
68	66, 67	eqtri 2227	. . . . . . . . . . . . . . . 16 ⊢ (((abs‘0)↑2) + ((abs‘0)↑2)) = 0
69	24	4sqlem4a 12789	. . . . . . . . . . . . . . . . 17 ⊢ ((0 ∈ ℤ[i] ∧ 0 ∈ ℤ[i]) → (((abs‘0)↑2) + ((abs‘0)↑2)) ∈ 𝑆)
70	23, 23, 69	mp2an 426	. . . . . . . . . . . . . . . 16 ⊢ (((abs‘0)↑2) + ((abs‘0)↑2)) ∈ 𝑆
71	68, 70	eqeltrri 2280	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ 𝑆
72		snssi 3783	. . . . . . . . . . . . . . 15 ⊢ (0 ∈ 𝑆 → {0} ⊆ 𝑆)
73	71, 72	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ {0} ⊆ 𝑆
74	65, 73	eqsstri 3229	. . . . . . . . . . . . 13 ⊢ (0...0) ⊆ 𝑆
75	74	a1i 9	. . . . . . . . . . . 12 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → (0...0) ⊆ 𝑆)
76		0p1e1 9170	. . . . . . . . . . . . . 14 ⊢ (0 + 1) = 1
77	76	oveq1i 5967	. . . . . . . . . . . . 13 ⊢ ((0 + 1)...(𝑗 − 1)) = (1...(𝑗 − 1))
78		simpr 110	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆)
79		dfss3 3186	. . . . . . . . . . . . . 14 ⊢ ((1...(𝑗 − 1)) ⊆ 𝑆 ↔ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆)
80	78, 79	sylibr 134	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → (1...(𝑗 − 1)) ⊆ 𝑆)
81	77, 80	eqsstrid 3243	. . . . . . . . . . . 12 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → ((0 + 1)...(𝑗 − 1)) ⊆ 𝑆)
82	75, 81	unssd 3353	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → ((0...0) ∪ ((0 + 1)...(𝑗 − 1))) ⊆ 𝑆)
83	64, 82	eqsstrd 3233	. . . . . . . . . 10 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → (0...(2 · ((𝑗 − 1) / 2))) ⊆ 𝑆)
84		oveq1 5964	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑖 → (𝑘 · 𝑗) = (𝑖 · 𝑗))
85	84	eleq1d 2275	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑖 → ((𝑘 · 𝑗) ∈ 𝑆 ↔ (𝑖 · 𝑗) ∈ 𝑆))
86	85	cbvrabv 2772	. . . . . . . . . 10 ⊢ {𝑘 ∈ ℕ ∣ (𝑘 · 𝑗) ∈ 𝑆} = {𝑖 ∈ ℕ ∣ (𝑖 · 𝑗) ∈ 𝑆}
87		eqid 2206	. . . . . . . . . 10 ⊢ inf({𝑘 ∈ ℕ ∣ (𝑘 · 𝑗) ∈ 𝑆}, ℝ, < ) = inf({𝑘 ∈ ℕ ∣ (𝑘 · 𝑗) ∈ 𝑆}, ℝ, < )
88	24, 39, 55, 41, 83, 86, 87	4sqlem18 12806	. . . . . . . . 9 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → 𝑗 ∈ 𝑆)
89	37, 88	sylanbr 285	. . . . . . . 8 ⊢ (((𝑗 ∈ ℙ ∧ 𝑗 ≠ 2) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → 𝑗 ∈ 𝑆)
90	89	an32s 568	. . . . . . 7 ⊢ (((𝑗 ∈ ℙ ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) ∧ 𝑗 ≠ 2) → 𝑗 ∈ 𝑆)
91		prmz 12508	. . . . . . . . . 10 ⊢ (𝑗 ∈ ℙ → 𝑗 ∈ ℤ)
92	91	adantr 276	. . . . . . . . 9 ⊢ ((𝑗 ∈ ℙ ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → 𝑗 ∈ ℤ)
93		2z 9420	. . . . . . . . 9 ⊢ 2 ∈ ℤ
94		zdceq 9468	. . . . . . . . 9 ⊢ ((𝑗 ∈ ℤ ∧ 2 ∈ ℤ) → DECID 𝑗 = 2)
95	92, 93, 94	sylancl 413	. . . . . . . 8 ⊢ ((𝑗 ∈ ℙ ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → DECID 𝑗 = 2)
96		dcne 2388	. . . . . . . 8 ⊢ (DECID 𝑗 = 2 ↔ (𝑗 = 2 ∨ 𝑗 ≠ 2))
97	95, 96	sylib 122	. . . . . . 7 ⊢ ((𝑗 ∈ ℙ ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → (𝑗 = 2 ∨ 𝑗 ≠ 2))
98	36, 90, 97	mpjaodan 800	. . . . . 6 ⊢ ((𝑗 ∈ ℙ ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → 𝑗 ∈ 𝑆)
99	24	mul4sq 12792	. . . . . . 7 ⊢ ((𝑚 ∈ 𝑆 ∧ 𝑖 ∈ 𝑆) → (𝑚 · 𝑖) ∈ 𝑆)
100	99	a1i 9	. . . . . 6 ⊢ ((𝑚 ∈ (ℤ≥‘2) ∧ 𝑖 ∈ (ℤ≥‘2)) → ((𝑚 ∈ 𝑆 ∧ 𝑖 ∈ 𝑆) → (𝑚 · 𝑖) ∈ 𝑆))
101	2, 3, 4, 5, 6, 27, 98, 100	prmind2 12517	. . . . 5 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ 𝑆)
102		id 19	. . . . . 6 ⊢ (𝑘 = 0 → 𝑘 = 0)
103	102, 71	eqeltrdi 2297	. . . . 5 ⊢ (𝑘 = 0 → 𝑘 ∈ 𝑆)
104	101, 103	jaoi 718	. . . 4 ⊢ ((𝑘 ∈ ℕ ∨ 𝑘 = 0) → 𝑘 ∈ 𝑆)
105	1, 104	sylbi 121	. . 3 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ 𝑆)
106	105	ssriv 3201	. 2 ⊢ ℕ0 ⊆ 𝑆
107	24	4sqlem1 12786	. 2 ⊢ 𝑆 ⊆ ℕ0
108	106, 107	eqssi 3213	1 ⊢ ℕ0 = 𝑆

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 710  DECID wdc 836   = wceq 1373   ∈ wcel 2177  {cab 2192   ≠ wne 2377  ∀wral 2485  ∃wrex 2486  {crab 2489   ∖ cdif 3167   ∪ cun 3168   ⊆ wss 3170  {csn 3638   class class class wbr 4051  ‘cfv 5280  (class class class)co 5957  infcinf 7100  ℂcc 7943  ℝcr 7944  0cc0 7945  1c1 7946   + caddc 7948   · cmul 7950   < clt 8127   − cmin 8263   # cap 8674   / cdiv 8765  ℕcn 9056  2c2 9107  ℕ₀cn0 9315  ℤcz 9392  ℤ_≥cuz 9668  ...cfz 10150  ↑cexp 10705  abscabs 11383  ℙcprime 12504  ℤ[i]cgz 12767

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4167  ax-sep 4170  ax-nul 4178  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593  ax-iinf 4644  ax-cnex 8036  ax-resscn 8037  ax-1cn 8038  ax-1re 8039  ax-icn 8040  ax-addcl 8041  ax-addrcl 8042  ax-mulcl 8043  ax-mulrcl 8044  ax-addcom 8045  ax-mulcom 8046  ax-addass 8047  ax-mulass 8048  ax-distr 8049  ax-i2m1 8050  ax-0lt1 8051  ax-1rid 8052  ax-0id 8053  ax-rnegex 8054  ax-precex 8055  ax-cnre 8056  ax-pre-ltirr 8057  ax-pre-ltwlin 8058  ax-pre-lttrn 8059  ax-pre-apti 8060  ax-pre-ltadd 8061  ax-pre-mulgt0 8062  ax-pre-mulext 8063  ax-arch 8064  ax-caucvg 8065

This theorem depends on definitions:  df-bi 117  df-stab 833  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-xor 1396  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-if 3576  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-iun 3935  df-br 4052  df-opab 4114  df-mpt 4115  df-tr 4151  df-id 4348  df-po 4351  df-iso 4352  df-iord 4421  df-on 4423  df-ilim 4424  df-suc 4426  df-iom 4647  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-isom 5289  df-riota 5912  df-ov 5960  df-oprab 5961  df-mpo 5962  df-1st 6239  df-2nd 6240  df-recs 6404  df-irdg 6469  df-frec 6490  df-1o 6515  df-2o 6516  df-oadd 6519  df-er 6633  df-en 6841  df-dom 6842  df-fin 6843  df-sup 7101  df-inf 7102  df-pnf 8129  df-mnf 8130  df-xr 8131  df-ltxr 8132  df-le 8133  df-sub 8265  df-neg 8266  df-reap 8668  df-ap 8675  df-div 8766  df-inn 9057  df-2 9115  df-3 9116  df-4 9117  df-n0 9316  df-z 9393  df-uz 9669  df-q 9761  df-rp 9796  df-fz 10151  df-fzo 10285  df-fl 10435  df-mod 10490  df-seqfrec 10615  df-exp 10706  df-ihash 10943  df-cj 11228  df-re 11229  df-im 11230  df-rsqrt 11384  df-abs 11385  df-dvds 12174  df-gcd 12350  df-prm 12505  df-gz 12768

This theorem is referenced by:  4sq  12808