Theorem 4sqlem19 13188

Description: Lemma for 4sq 13189. 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 13187. 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 13173 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 9565	. . . 4 ⊢ (𝑘 ∈ ℕ0 ↔ (𝑘 ∈ ℕ ∨ 𝑘 = 0))
2		eleq1 2301	. . . . . 6 ⊢ (𝑗 = 1 → (𝑗 ∈ 𝑆 ↔ 1 ∈ 𝑆))
3		eleq1 2301	. . . . . 6 ⊢ (𝑗 = 𝑚 → (𝑗 ∈ 𝑆 ↔ 𝑚 ∈ 𝑆))
4		eleq1 2301	. . . . . 6 ⊢ (𝑗 = 𝑖 → (𝑗 ∈ 𝑆 ↔ 𝑖 ∈ 𝑆))
5		eleq1 2301	. . . . . 6 ⊢ (𝑗 = (𝑚 · 𝑖) → (𝑗 ∈ 𝑆 ↔ (𝑚 · 𝑖) ∈ 𝑆))
6		eleq1 2301	. . . . . 6 ⊢ (𝑗 = 𝑘 → (𝑗 ∈ 𝑆 ↔ 𝑘 ∈ 𝑆))
7		abs1 11838	. . . . . . . . . . 11 ⊢ (abs‘1) = 1
8	7	oveq1i 6095	. . . . . . . . . 10 ⊢ ((abs‘1)↑2) = (1↑2)
9		sq1 11070	. . . . . . . . . 10 ⊢ (1↑2) = 1
10	8, 9	eqtri 2259	. . . . . . . . 9 ⊢ ((abs‘1)↑2) = 1
11		abs0 11824	. . . . . . . . . . 11 ⊢ (abs‘0) = 0
12	11	oveq1i 6095	. . . . . . . . . 10 ⊢ ((abs‘0)↑2) = (0↑2)
13		sq0 11067	. . . . . . . . . 10 ⊢ (0↑2) = 0
14	12, 13	eqtri 2259	. . . . . . . . 9 ⊢ ((abs‘0)↑2) = 0
15	10, 14	oveq12i 6097	. . . . . . . 8 ⊢ (((abs‘1)↑2) + ((abs‘0)↑2)) = (1 + 0)
16		1p0e1 9420	. . . . . . . 8 ⊢ (1 + 0) = 1
17	15, 16	eqtri 2259	. . . . . . 7 ⊢ (((abs‘1)↑2) + ((abs‘0)↑2)) = 1
18		1z 9670	. . . . . . . . 9 ⊢ 1 ∈ ℤ
19		zgz 13152	. . . . . . . . 9 ⊢ (1 ∈ ℤ → 1 ∈ ℤ[i])
20	18, 19	ax-mp 5	. . . . . . . 8 ⊢ 1 ∈ ℤ[i]
21		0z 9655	. . . . . . . . 9 ⊢ 0 ∈ ℤ
22		zgz 13152	. . . . . . . . 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 13170	. . . . . . . 8 ⊢ ((1 ∈ ℤ[i] ∧ 0 ∈ ℤ[i]) → (((abs‘1)↑2) + ((abs‘0)↑2)) ∈ 𝑆)
26	20, 23, 25	mp2an 430	. . . . . . 7 ⊢ (((abs‘1)↑2) + ((abs‘0)↑2)) ∈ 𝑆
27	17, 26	eqeltrri 2312	. . . . . 6 ⊢ 1 ∈ 𝑆
28	10, 10	oveq12i 6097	. . . . . . . . . 10 ⊢ (((abs‘1)↑2) + ((abs‘1)↑2)) = (1 + 1)
29		df-2 9363	. . . . . . . . . 10 ⊢ 2 = (1 + 1)
30	28, 29	eqtr4i 2262	. . . . . . . . 9 ⊢ (((abs‘1)↑2) + ((abs‘1)↑2)) = 2
31	24	4sqlem4a 13170	. . . . . . . . . 10 ⊢ ((1 ∈ ℤ[i] ∧ 1 ∈ ℤ[i]) → (((abs‘1)↑2) + ((abs‘1)↑2)) ∈ 𝑆)
32	20, 20, 31	mp2an 430	. . . . . . . . 9 ⊢ (((abs‘1)↑2) + ((abs‘1)↑2)) ∈ 𝑆
33	30, 32	eqeltrri 2312	. . . . . . . 8 ⊢ 2 ∈ 𝑆
34		eleq1 2301	. . . . . . . . 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 3841	. . . . . . . . 9 ⊢ (𝑗 ∈ (ℙ ∖ {2}) ↔ (𝑗 ∈ ℙ ∧ 𝑗 ≠ 2))
38		oddprm 13038	. . . . . . . . . . 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 3351	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (ℙ ∖ {2}) → 𝑗 ∈ ℙ)
41	40	adantr 276	. . . . . . . . . . . . . . 15 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → 𝑗 ∈ ℙ)
42		prmnn 12888	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ ℙ → 𝑗 ∈ ℕ)
43		nncn 9312	. . . . . . . . . . . . . . 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 8272	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℂ
46		subcl 8525	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑗 − 1) ∈ ℂ)
47	44, 45, 46	sylancl 417	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → (𝑗 − 1) ∈ ℂ)
48		2cnd 9377	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → 2 ∈ ℂ)
49		2ap0 9397	. . . . . . . . . . . . . 14 ⊢ 2 # 0
50	49	a1i 9	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → 2 # 0)
51	47, 48, 50	divcanap2d 9122	. . . . . . . . . . . 12 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → (2 · ((𝑗 − 1) / 2)) = (𝑗 − 1))
52	51	oveq1d 6100	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → ((2 · ((𝑗 − 1) / 2)) + 1) = ((𝑗 − 1) + 1))
53		npcan 8535	. . . . . . . . . . . 12 ⊢ ((𝑗 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑗 − 1) + 1) = 𝑗)
54	44, 45, 53	sylancl 417	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → ((𝑗 − 1) + 1) = 𝑗)
55	52, 54	eqtr2d 2272	. . . . . . . . . 10 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → 𝑗 = ((2 · ((𝑗 − 1) / 2)) + 1))
56	51	oveq2d 6101	. . . . . . . . . . . 12 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → (0...(2 · ((𝑗 − 1) / 2))) = (0...(𝑗 − 1)))
57		nnm1nn0 9604	. . . . . . . . . . . . . . 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 9960	. . . . . . . . . . . . . 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 10435	. . . . . . . . . . . . 13 ⊢ ((𝑗 − 1) ∈ (ℤ≥‘0) → 0 ∈ (0...(𝑗 − 1)))
62		fzsplit 10456	. . . . . . . . . . . . 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 2271	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → (0...(2 · ((𝑗 − 1) / 2))) = ((0...0) ∪ ((0 + 1)...(𝑗 − 1))))
65		fz0sn 10528	. . . . . . . . . . . . . 14 ⊢ (0...0) = {0}
66	14, 14	oveq12i 6097	. . . . . . . . . . . . . . . . 17 ⊢ (((abs‘0)↑2) + ((abs‘0)↑2)) = (0 + 0)
67		00id 8467	. . . . . . . . . . . . . . . . 17 ⊢ (0 + 0) = 0
68	66, 67	eqtri 2259	. . . . . . . . . . . . . . . 16 ⊢ (((abs‘0)↑2) + ((abs‘0)↑2)) = 0
69	24	4sqlem4a 13170	. . . . . . . . . . . . . . . . 17 ⊢ ((0 ∈ ℤ[i] ∧ 0 ∈ ℤ[i]) → (((abs‘0)↑2) + ((abs‘0)↑2)) ∈ 𝑆)
70	23, 23, 69	mp2an 430	. . . . . . . . . . . . . . . 16 ⊢ (((abs‘0)↑2) + ((abs‘0)↑2)) ∈ 𝑆
71	68, 70	eqeltrri 2312	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ 𝑆
72		snssi 3859	. . . . . . . . . . . . . . 15 ⊢ (0 ∈ 𝑆 → {0} ⊆ 𝑆)
73	71, 72	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ {0} ⊆ 𝑆
74	65, 73	eqsstri 3280	. . . . . . . . . . . . 13 ⊢ (0...0) ⊆ 𝑆
75	74	a1i 9	. . . . . . . . . . . 12 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → (0...0) ⊆ 𝑆)
76		0p1e1 9418	. . . . . . . . . . . . . 14 ⊢ (0 + 1) = 1
77	76	oveq1i 6095	. . . . . . . . . . . . 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 3236	. . . . . . . . . . . . . 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 3294	. . . . . . . . . . . 12 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → ((0 + 1)...(𝑗 − 1)) ⊆ 𝑆)
82	75, 81	unssd 3405	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → ((0...0) ∪ ((0 + 1)...(𝑗 − 1))) ⊆ 𝑆)
83	64, 82	eqsstrd 3284	. . . . . . . . . 10 ⊢ ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → (0...(2 · ((𝑗 − 1) / 2))) ⊆ 𝑆)
84		oveq1 6092	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑖 → (𝑘 · 𝑗) = (𝑖 · 𝑗))
85	84	eleq1d 2307	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑖 → ((𝑘 · 𝑗) ∈ 𝑆 ↔ (𝑖 · 𝑗) ∈ 𝑆))
86	85	cbvrabv 2820	. . . . . . . . . 10 ⊢ {𝑘 ∈ ℕ ∣ (𝑘 · 𝑗) ∈ 𝑆} = {𝑖 ∈ ℕ ∣ (𝑖 · 𝑗) ∈ 𝑆}
87		eqid 2238	. . . . . . . . . 10 ⊢ inf({𝑘 ∈ ℕ ∣ (𝑘 · 𝑗) ∈ 𝑆}, ℝ, < ) = inf({𝑘 ∈ ℕ ∣ (𝑘 · 𝑗) ∈ 𝑆}, ℝ, < )
88	24, 39, 55, 41, 83, 86, 87	4sqlem18 13187	. . . . . . . . 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 574	. . . . . . 7 ⊢ (((𝑗 ∈ ℙ ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) ∧ 𝑗 ≠ 2) → 𝑗 ∈ 𝑆)
91		prmz 12889	. . . . . . . . . 10 ⊢ (𝑗 ∈ ℙ → 𝑗 ∈ ℤ)
92	91	adantr 276	. . . . . . . . 9 ⊢ ((𝑗 ∈ ℙ ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → 𝑗 ∈ ℤ)
93		2z 9672	. . . . . . . . 9 ⊢ 2 ∈ ℤ
94		zdceq 9720	. . . . . . . . 9 ⊢ ((𝑗 ∈ ℤ ∧ 2 ∈ ℤ) → DECID 𝑗 = 2)
95	92, 93, 94	sylancl 417	. . . . . . . 8 ⊢ ((𝑗 ∈ ℙ ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → DECID 𝑗 = 2)
96		dcne 2431	. . . . . . . 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 810	. . . . . 6 ⊢ ((𝑗 ∈ ℙ ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚 ∈ 𝑆) → 𝑗 ∈ 𝑆)
99	24	mul4sq 13173	. . . . . . 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 12898	. . . . 5 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ 𝑆)
102		id 19	. . . . . 6 ⊢ (𝑘 = 0 → 𝑘 = 0)
103	102, 71	eqeltrdi 2329	. . . . 5 ⊢ (𝑘 = 0 → 𝑘 ∈ 𝑆)
104	101, 103	jaoi 728	. . . 4 ⊢ ((𝑘 ∈ ℕ ∨ 𝑘 = 0) → 𝑘 ∈ 𝑆)
105	1, 104	sylbi 121	. . 3 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ 𝑆)
106	105	ssriv 3252	. 2 ⊢ ℕ0 ⊆ 𝑆
107	24	4sqlem1 13167	. 2 ⊢ 𝑆 ⊆ ℕ0
108	106, 107	eqssi 3264	1 ⊢ ℕ0 = 𝑆

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 720  DECID wdc 846   = wceq 1402   ∈ wcel 2209  {cab 2224   ≠ wne 2420  ∀wral 2528  ∃wrex 2529  {crab 2532   ∖ cdif 3217   ∪ cun 3218   ⊆ wss 3220  {csn 3709   class class class wbr 4130  ‘cfv 5377  (class class class)co 6085  infcinf 7323  ℂcc 8177  ℝcr 8178  0cc0 8179  1c1 8180   + caddc 8182   · cmul 8184   < clt 8360   − cmin 8497   # cap 8909   / cdiv 9002  ℕcn 9304  2c2 9355  ℕ₀cn0 9563  ℤcz 9644  ℤ_≥cuz 9921  ...cfz 10411  ↑cexp 10975  abscabs 11763  ℙcprime 12885  ℤ[i]cgz 13148

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299

This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-xor 1425  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-isom 5386  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-frec 6662  df-1o 6687  df-2o 6688  df-oadd 6691  df-er 6807  df-en 7023  df-dom 7024  df-fin 7025  df-sup 7324  df-inf 7325  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-n0 9564  df-z 9645  df-uz 9922  df-q 10020  df-rp 10055  df-fz 10412  df-fzo 10550  df-fl 10705  df-mod 10760  df-seqfrec 10885  df-exp 10976  df-ihash 11215  df-cj 11607  df-re 11608  df-im 11609  df-rsqrt 11764  df-abs 11765  df-dvds 12555  df-gcd 12731  df-prm 12886  df-gz 13149

This theorem is used by:  4sq  13189