Theorem 4sqlemsdc 13157

Description: Lemma for 4sq 13167. The property of being the sum of four squares is decidable.

The proof involves showing that (for a particular 𝐴) there are only a finite number of possible ways that it could be the sum of four squares, so checking each of those possibilities in turn decides whether the number is the sum of four squares. If this proof is hard to follow, especially because of its length, the simplified versions at 4sqexercise1 13155 and 4sqexercise2 13156 may help clarify, as they are using very much the same techniques on simplified versions of this lemma. (Contributed by Jim Kingdon, 25-May-2025.)

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
4sqlemsdc	⊢ (𝐴 ∈ ℕ0 → DECID 𝐴 ∈ 𝑆)

Distinct variable group:   𝐴,𝑛,𝑤,𝑥,𝑦,𝑧

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

Proof of Theorem 4sqlemsdc

Step	Hyp	Ref	Expression
1		nn0negz 9657	. . . 4 ⊢ (𝐴 ∈ ℕ0 → -𝐴 ∈ ℤ)
2		nn0z 9643	. . . 4 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℤ)
3	1	adantr 276	. . . . . 6 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) → -𝐴 ∈ ℤ)
4	2	adantr 276	. . . . . 6 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) → 𝐴 ∈ ℤ)
5	3	adantr 276	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) → -𝐴 ∈ ℤ)
6	4	adantr 276	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) → 𝐴 ∈ ℤ)
7	5	adantr 276	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ (-𝐴...𝐴)) → -𝐴 ∈ ℤ)
8	6	adantr 276	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ (-𝐴...𝐴)) → 𝐴 ∈ ℤ)
9	8	adantr 276	. . . . . . . . . . 11 ⊢ (((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ (-𝐴...𝐴)) ∧ 𝑤 ∈ (-𝐴...𝐴)) → 𝐴 ∈ ℤ)
10		elfzelz 10407	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (-𝐴...𝐴) → 𝑥 ∈ ℤ)
11	10	ad4antlr 499	. . . . . . . . . . . . . . 15 ⊢ (((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ (-𝐴...𝐴)) ∧ 𝑤 ∈ (-𝐴...𝐴)) → 𝑥 ∈ ℤ)
12		zsqcl2 11032	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℤ → (𝑥↑2) ∈ ℕ0)
13	11, 12	syl 14	. . . . . . . . . . . . . 14 ⊢ (((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ (-𝐴...𝐴)) ∧ 𝑤 ∈ (-𝐴...𝐴)) → (𝑥↑2) ∈ ℕ0)
14		elfzelz 10407	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (-𝐴...𝐴) → 𝑦 ∈ ℤ)
15	14	ad3antlr 497	. . . . . . . . . . . . . . 15 ⊢ (((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ (-𝐴...𝐴)) ∧ 𝑤 ∈ (-𝐴...𝐴)) → 𝑦 ∈ ℤ)
16		zsqcl2 11032	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℤ → (𝑦↑2) ∈ ℕ0)
17	15, 16	syl 14	. . . . . . . . . . . . . 14 ⊢ (((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ (-𝐴...𝐴)) ∧ 𝑤 ∈ (-𝐴...𝐴)) → (𝑦↑2) ∈ ℕ0)
18	13, 17	nn0addcld 9603	. . . . . . . . . . . . 13 ⊢ (((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ (-𝐴...𝐴)) ∧ 𝑤 ∈ (-𝐴...𝐴)) → ((𝑥↑2) + (𝑦↑2)) ∈ ℕ0)
19		elfzelz 10407	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ (-𝐴...𝐴) → 𝑧 ∈ ℤ)
20	19	ad2antlr 493	. . . . . . . . . . . . . . 15 ⊢ (((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ (-𝐴...𝐴)) ∧ 𝑤 ∈ (-𝐴...𝐴)) → 𝑧 ∈ ℤ)
21		zsqcl2 11032	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ ℤ → (𝑧↑2) ∈ ℕ0)
22	20, 21	syl 14	. . . . . . . . . . . . . 14 ⊢ (((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ (-𝐴...𝐴)) ∧ 𝑤 ∈ (-𝐴...𝐴)) → (𝑧↑2) ∈ ℕ0)
23		elfzelz 10407	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ (-𝐴...𝐴) → 𝑤 ∈ ℤ)
24	23	adantl 277	. . . . . . . . . . . . . . 15 ⊢ (((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ (-𝐴...𝐴)) ∧ 𝑤 ∈ (-𝐴...𝐴)) → 𝑤 ∈ ℤ)
25		zsqcl2 11032	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ ℤ → (𝑤↑2) ∈ ℕ0)
26	24, 25	syl 14	. . . . . . . . . . . . . 14 ⊢ (((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ (-𝐴...𝐴)) ∧ 𝑤 ∈ (-𝐴...𝐴)) → (𝑤↑2) ∈ ℕ0)
27	22, 26	nn0addcld 9603	. . . . . . . . . . . . 13 ⊢ (((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ (-𝐴...𝐴)) ∧ 𝑤 ∈ (-𝐴...𝐴)) → ((𝑧↑2) + (𝑤↑2)) ∈ ℕ0)
28	18, 27	nn0addcld 9603	. . . . . . . . . . . 12 ⊢ (((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ (-𝐴...𝐴)) ∧ 𝑤 ∈ (-𝐴...𝐴)) → (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))) ∈ ℕ0)
29	28	nn0zd 9745	. . . . . . . . . . 11 ⊢ (((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ (-𝐴...𝐴)) ∧ 𝑤 ∈ (-𝐴...𝐴)) → (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))) ∈ ℤ)
30		zdceq 9699	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))) ∈ ℤ) → DECID 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))))
31	9, 29, 30	syl2anc 415	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ (-𝐴...𝐴)) ∧ 𝑤 ∈ (-𝐴...𝐴)) → DECID 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))))
32	7, 8, 31	exfzdc 10637	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ (-𝐴...𝐴)) → DECID ∃𝑤 ∈ (-𝐴...𝐴)𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))))
33	1	ad5antr 500	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ (-𝐴...𝐴)) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑤 ∈ ℤ) → -𝐴 ∈ ℤ)
34	2	ad5antr 500	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ (-𝐴...𝐴)) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑤 ∈ ℤ) → 𝐴 ∈ ℤ)
35		simpr 110	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ (-𝐴...𝐴)) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑤 ∈ ℤ) → 𝑤 ∈ ℤ)
36	35	zred 9747	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ (-𝐴...𝐴)) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑤 ∈ ℤ) → 𝑤 ∈ ℝ)
37	34	zred 9747	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ (-𝐴...𝐴)) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑤 ∈ ℤ) → 𝐴 ∈ ℝ)
38	36	renegcld 8697	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ (-𝐴...𝐴)) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑤 ∈ ℤ) → -𝑤 ∈ ℝ)
39	36	resqcld 11115	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ (-𝐴...𝐴)) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑤 ∈ ℤ) → (𝑤↑2) ∈ ℝ)
40	35	znegcld 9749	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ (-𝐴...𝐴)) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑤 ∈ ℤ) → -𝑤 ∈ ℤ)
41		zzlesq 11124	. . . . . . . . . . . . . . . . . . . 20 ⊢ (-𝑤 ∈ ℤ → -𝑤 ≤ (-𝑤↑2))
42	40, 41	syl 14	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ (-𝐴...𝐴)) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑤 ∈ ℤ) → -𝑤 ≤ (-𝑤↑2))
43	35	zcnd 9748	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ (-𝐴...𝐴)) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑤 ∈ ℤ) → 𝑤 ∈ ℂ)
44		sqneg 11013	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 ∈ ℂ → (-𝑤↑2) = (𝑤↑2))
45	43, 44	syl 14	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ (-𝐴...𝐴)) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑤 ∈ ℤ) → (-𝑤↑2) = (𝑤↑2))
46	42, 45	breqtrd 4151	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ (-𝐴...𝐴)) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑤 ∈ ℤ) → -𝑤 ≤ (𝑤↑2))
47	19	ad3antlr 497	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ (-𝐴...𝐴)) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑤 ∈ ℤ) → 𝑧 ∈ ℤ)
48	47, 21	syl 14	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ (-𝐴...𝐴)) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑤 ∈ ℤ) → (𝑧↑2) ∈ ℕ0)
49	25	adantl 277	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ (-𝐴...𝐴)) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑤 ∈ ℤ) → (𝑤↑2) ∈ ℕ0)
50	48, 49	nn0addcld 9603	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ (-𝐴...𝐴)) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑤 ∈ ℤ) → ((𝑧↑2) + (𝑤↑2)) ∈ ℕ0)
51	50	nn0red 9600	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ (-𝐴...𝐴)) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑤 ∈ ℤ) → ((𝑧↑2) + (𝑤↑2)) ∈ ℝ)
52	10	ad5antlr 501	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ (-𝐴...𝐴)) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑤 ∈ ℤ) → 𝑥 ∈ ℤ)
53	52, 12	syl 14	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ (-𝐴...𝐴)) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑤 ∈ ℤ) → (𝑥↑2) ∈ ℕ0)
54	14	ad4antlr 499	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ (-𝐴...𝐴)) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑤 ∈ ℤ) → 𝑦 ∈ ℤ)
55	54, 16	syl 14	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ (-𝐴...𝐴)) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑤 ∈ ℤ) → (𝑦↑2) ∈ ℕ0)
56	53, 55	nn0addcld 9603	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ (-𝐴...𝐴)) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑤 ∈ ℤ) → ((𝑥↑2) + (𝑦↑2)) ∈ ℕ0)
57	56, 50	nn0addcld 9603	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ (-𝐴...𝐴)) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑤 ∈ ℤ) → (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))) ∈ ℕ0)
58	57	nn0red 9600	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ (-𝐴...𝐴)) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑤 ∈ ℤ) → (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))) ∈ ℝ)
59		nn0addge2 9589	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑤↑2) ∈ ℝ ∧ (𝑧↑2) ∈ ℕ0) → (𝑤↑2) ≤ ((𝑧↑2) + (𝑤↑2)))
60	39, 48, 59	syl2anc 415	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ (-𝐴...𝐴)) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑤 ∈ ℤ) → (𝑤↑2) ≤ ((𝑧↑2) + (𝑤↑2)))
61		nn0addge2 9589	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑧↑2) + (𝑤↑2)) ∈ ℝ ∧ ((𝑥↑2) + (𝑦↑2)) ∈ ℕ0) → ((𝑧↑2) + (𝑤↑2)) ≤ (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))))
62	51, 56, 61	syl2anc 415	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ (-𝐴...𝐴)) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑤 ∈ ℤ) → ((𝑧↑2) + (𝑤↑2)) ≤ (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))))
63	39, 51, 58, 60, 62	letrd 8440	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ (-𝐴...𝐴)) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑤 ∈ ℤ) → (𝑤↑2) ≤ (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))))
64		simplr 533	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ (-𝐴...𝐴)) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑤 ∈ ℤ) → 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))))
65	63, 64	breqtrrd 4153	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ (-𝐴...𝐴)) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑤 ∈ ℤ) → (𝑤↑2) ≤ 𝐴)
66	38, 39, 37, 46, 65	letrd 8440	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ (-𝐴...𝐴)) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑤 ∈ ℤ) → -𝑤 ≤ 𝐴)
67	36, 37, 66	lenegcon1d 8845	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ (-𝐴...𝐴)) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑤 ∈ ℤ) → -𝐴 ≤ 𝑤)
68		zzlesq 11124	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ ℤ → 𝑤 ≤ (𝑤↑2))
69	68	adantl 277	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ (-𝐴...𝐴)) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑤 ∈ ℤ) → 𝑤 ≤ (𝑤↑2))
70	36, 39, 37, 69, 65	letrd 8440	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ (-𝐴...𝐴)) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑤 ∈ ℤ) → 𝑤 ≤ 𝐴)
71	33, 34, 35, 67, 70	elfzd 10398	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ (-𝐴...𝐴)) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑤 ∈ ℤ) → 𝑤 ∈ (-𝐴...𝐴))
72	71	ex 115	. . . . . . . . . . . . . 14 ⊢ (((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ (-𝐴...𝐴)) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) → (𝑤 ∈ ℤ → 𝑤 ∈ (-𝐴...𝐴)))
73	72, 23	impbid1 142	. . . . . . . . . . . . 13 ⊢ (((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ (-𝐴...𝐴)) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) → (𝑤 ∈ ℤ ↔ 𝑤 ∈ (-𝐴...𝐴)))
74	73	ex 115	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ (-𝐴...𝐴)) → (𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))) → (𝑤 ∈ ℤ ↔ 𝑤 ∈ (-𝐴...𝐴))))
75	74	pm5.32rd 455	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ (-𝐴...𝐴)) → ((𝑤 ∈ ℤ ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ↔ (𝑤 ∈ (-𝐴...𝐴) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))))))
76	75	rexbidv2 2553	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ (-𝐴...𝐴)) → (∃𝑤 ∈ ℤ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))) ↔ ∃𝑤 ∈ (-𝐴...𝐴)𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))))
77	76	dcbid 850	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ (-𝐴...𝐴)) → (DECID ∃𝑤 ∈ ℤ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))) ↔ DECID ∃𝑤 ∈ (-𝐴...𝐴)𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))))
78	32, 77	mpbird 167	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ (-𝐴...𝐴)) → DECID ∃𝑤 ∈ ℤ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))))
79	5, 6, 78	exfzdc 10637	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) → DECID ∃𝑧 ∈ (-𝐴...𝐴)∃𝑤 ∈ ℤ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))))
80	1	ad5antr 500	. . . . . . . . . . . . . 14 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑧 ∈ ℤ) → -𝐴 ∈ ℤ)
81	2	ad5antr 500	. . . . . . . . . . . . . 14 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑧 ∈ ℤ) → 𝐴 ∈ ℤ)
82		simpr 110	. . . . . . . . . . . . . 14 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑧 ∈ ℤ) → 𝑧 ∈ ℤ)
83	82	zred 9747	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑧 ∈ ℤ) → 𝑧 ∈ ℝ)
84	81	zred 9747	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑧 ∈ ℤ) → 𝐴 ∈ ℝ)
85	83	renegcld 8697	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑧 ∈ ℤ) → -𝑧 ∈ ℝ)
86	83	resqcld 11115	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑧 ∈ ℤ) → (𝑧↑2) ∈ ℝ)
87	82	znegcld 9749	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑧 ∈ ℤ) → -𝑧 ∈ ℤ)
88		zzlesq 11124	. . . . . . . . . . . . . . . . . 18 ⊢ (-𝑧 ∈ ℤ → -𝑧 ≤ (-𝑧↑2))
89	87, 88	syl 14	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑧 ∈ ℤ) → -𝑧 ≤ (-𝑧↑2))
90	82	zcnd 9748	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑧 ∈ ℤ) → 𝑧 ∈ ℂ)
91		sqneg 11013	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ ℂ → (-𝑧↑2) = (𝑧↑2))
92	90, 91	syl 14	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑧 ∈ ℤ) → (-𝑧↑2) = (𝑧↑2))
93	89, 92	breqtrd 4151	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑧 ∈ ℤ) → -𝑧 ≤ (𝑧↑2))
94	21	adantl 277	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑧 ∈ ℤ) → (𝑧↑2) ∈ ℕ0)
95	25	ad3antlr 497	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑧 ∈ ℤ) → (𝑤↑2) ∈ ℕ0)
96	94, 95	nn0addcld 9603	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑧 ∈ ℤ) → ((𝑧↑2) + (𝑤↑2)) ∈ ℕ0)
97	96	nn0red 9600	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑧 ∈ ℤ) → ((𝑧↑2) + (𝑤↑2)) ∈ ℝ)
98	10	ad5antlr 501	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑧 ∈ ℤ) → 𝑥 ∈ ℤ)
99	98, 12	syl 14	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑧 ∈ ℤ) → (𝑥↑2) ∈ ℕ0)
100	14	ad4antlr 499	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑧 ∈ ℤ) → 𝑦 ∈ ℤ)
101	100, 16	syl 14	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑧 ∈ ℤ) → (𝑦↑2) ∈ ℕ0)
102	99, 101	nn0addcld 9603	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑧 ∈ ℤ) → ((𝑥↑2) + (𝑦↑2)) ∈ ℕ0)
103	102, 96	nn0addcld 9603	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑧 ∈ ℤ) → (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))) ∈ ℕ0)
104	103	nn0red 9600	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑧 ∈ ℤ) → (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))) ∈ ℝ)
105		nn0addge1 9588	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑧↑2) ∈ ℝ ∧ (𝑤↑2) ∈ ℕ0) → (𝑧↑2) ≤ ((𝑧↑2) + (𝑤↑2)))
106	86, 95, 105	syl2anc 415	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑧 ∈ ℤ) → (𝑧↑2) ≤ ((𝑧↑2) + (𝑤↑2)))
107	97, 102, 61	syl2anc 415	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑧 ∈ ℤ) → ((𝑧↑2) + (𝑤↑2)) ≤ (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))))
108	86, 97, 104, 106, 107	letrd 8440	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑧 ∈ ℤ) → (𝑧↑2) ≤ (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))))
109		simplr 533	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑧 ∈ ℤ) → 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))))
110	108, 109	breqtrrd 4153	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑧 ∈ ℤ) → (𝑧↑2) ≤ 𝐴)
111	85, 86, 84, 93, 110	letrd 8440	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑧 ∈ ℤ) → -𝑧 ≤ 𝐴)
112	83, 84, 111	lenegcon1d 8845	. . . . . . . . . . . . . 14 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑧 ∈ ℤ) → -𝐴 ≤ 𝑧)
113		zzlesq 11124	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ ℤ → 𝑧 ≤ (𝑧↑2))
114	113	adantl 277	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑧 ∈ ℤ) → 𝑧 ≤ (𝑧↑2))
115	83, 86, 84, 114, 110	letrd 8440	. . . . . . . . . . . . . 14 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑧 ∈ ℤ) → 𝑧 ≤ 𝐴)
116	80, 81, 82, 112, 115	elfzd 10398	. . . . . . . . . . . . 13 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑧 ∈ ℤ) → 𝑧 ∈ (-𝐴...𝐴))
117	116	ex 115	. . . . . . . . . . . 12 ⊢ (((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) → (𝑧 ∈ ℤ → 𝑧 ∈ (-𝐴...𝐴)))
118	117, 19	impbid1 142	. . . . . . . . . . 11 ⊢ (((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) → (𝑧 ∈ ℤ ↔ 𝑧 ∈ (-𝐴...𝐴)))
119	118	rexlimdva2 2671	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) → (∃𝑤 ∈ ℤ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))) → (𝑧 ∈ ℤ ↔ 𝑧 ∈ (-𝐴...𝐴))))
120	119	pm5.32rd 455	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) → ((𝑧 ∈ ℤ ∧ ∃𝑤 ∈ ℤ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ↔ (𝑧 ∈ (-𝐴...𝐴) ∧ ∃𝑤 ∈ ℤ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))))))
121	120	rexbidv2 2553	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) → (∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))) ↔ ∃𝑧 ∈ (-𝐴...𝐴)∃𝑤 ∈ ℤ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))))
122	121	dcbid 850	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) → (DECID ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))) ↔ DECID ∃𝑧 ∈ (-𝐴...𝐴)∃𝑤 ∈ ℤ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))))
123	79, 122	mpbird 167	. . . . . 6 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) → DECID ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))))
124	3, 4, 123	exfzdc 10637	. . . . 5 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) → DECID ∃𝑦 ∈ (-𝐴...𝐴)∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))))
125	1	ad5antr 500	. . . . . . . . . . . . 13 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑦 ∈ ℤ) → -𝐴 ∈ ℤ)
126	2	ad5antr 500	. . . . . . . . . . . . 13 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑦 ∈ ℤ) → 𝐴 ∈ ℤ)
127		simpr 110	. . . . . . . . . . . . 13 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑦 ∈ ℤ) → 𝑦 ∈ ℤ)
128	127	zred 9747	. . . . . . . . . . . . . 14 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑦 ∈ ℤ) → 𝑦 ∈ ℝ)
129	126	zred 9747	. . . . . . . . . . . . . 14 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑦 ∈ ℤ) → 𝐴 ∈ ℝ)
130	128	renegcld 8697	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑦 ∈ ℤ) → -𝑦 ∈ ℝ)
131	128	resqcld 11115	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑦 ∈ ℤ) → (𝑦↑2) ∈ ℝ)
132	127	znegcld 9749	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑦 ∈ ℤ) → -𝑦 ∈ ℤ)
133		zzlesq 11124	. . . . . . . . . . . . . . . . 17 ⊢ (-𝑦 ∈ ℤ → -𝑦 ≤ (-𝑦↑2))
134	132, 133	syl 14	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑦 ∈ ℤ) → -𝑦 ≤ (-𝑦↑2))
135	127	zcnd 9748	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑦 ∈ ℤ) → 𝑦 ∈ ℂ)
136		sqneg 11013	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ℂ → (-𝑦↑2) = (𝑦↑2))
137	135, 136	syl 14	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑦 ∈ ℤ) → (-𝑦↑2) = (𝑦↑2))
138	134, 137	breqtrd 4151	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑦 ∈ ℤ) → -𝑦 ≤ (𝑦↑2))
139	10	ad5antlr 501	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑦 ∈ ℤ) → 𝑥 ∈ ℤ)
140	139, 12	syl 14	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑦 ∈ ℤ) → (𝑥↑2) ∈ ℕ0)
141	16	adantl 277	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑦 ∈ ℤ) → (𝑦↑2) ∈ ℕ0)
142	140, 141	nn0addcld 9603	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑦 ∈ ℤ) → ((𝑥↑2) + (𝑦↑2)) ∈ ℕ0)
143	142	nn0red 9600	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑦 ∈ ℤ) → ((𝑥↑2) + (𝑦↑2)) ∈ ℝ)
144	21	ad4antlr 499	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑦 ∈ ℤ) → (𝑧↑2) ∈ ℕ0)
145	25	ad3antlr 497	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑦 ∈ ℤ) → (𝑤↑2) ∈ ℕ0)
146	144, 145	nn0addcld 9603	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑦 ∈ ℤ) → ((𝑧↑2) + (𝑤↑2)) ∈ ℕ0)
147	142, 146	nn0addcld 9603	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑦 ∈ ℤ) → (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))) ∈ ℕ0)
148	147	nn0red 9600	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑦 ∈ ℤ) → (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))) ∈ ℝ)
149		nn0addge2 9589	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦↑2) ∈ ℝ ∧ (𝑥↑2) ∈ ℕ0) → (𝑦↑2) ≤ ((𝑥↑2) + (𝑦↑2)))
150	131, 140, 149	syl2anc 415	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑦 ∈ ℤ) → (𝑦↑2) ≤ ((𝑥↑2) + (𝑦↑2)))
151		nn0addge1 9588	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑥↑2) + (𝑦↑2)) ∈ ℝ ∧ ((𝑧↑2) + (𝑤↑2)) ∈ ℕ0) → ((𝑥↑2) + (𝑦↑2)) ≤ (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))))
152	143, 146, 151	syl2anc 415	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑦 ∈ ℤ) → ((𝑥↑2) + (𝑦↑2)) ≤ (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))))
153	131, 143, 148, 150, 152	letrd 8440	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑦 ∈ ℤ) → (𝑦↑2) ≤ (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))))
154		simplr 533	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑦 ∈ ℤ) → 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))))
155	153, 154	breqtrrd 4153	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑦 ∈ ℤ) → (𝑦↑2) ≤ 𝐴)
156	130, 131, 129, 138, 155	letrd 8440	. . . . . . . . . . . . . 14 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑦 ∈ ℤ) → -𝑦 ≤ 𝐴)
157	128, 129, 156	lenegcon1d 8845	. . . . . . . . . . . . 13 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑦 ∈ ℤ) → -𝐴 ≤ 𝑦)
158		zzlesq 11124	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℤ → 𝑦 ≤ (𝑦↑2))
159	158	adantl 277	. . . . . . . . . . . . . 14 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑦 ∈ ℤ) → 𝑦 ≤ (𝑦↑2))
160	128, 131, 129, 159, 155	letrd 8440	. . . . . . . . . . . . 13 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑦 ∈ ℤ) → 𝑦 ≤ 𝐴)
161	125, 126, 127, 157, 160	elfzd 10398	. . . . . . . . . . . 12 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑦 ∈ ℤ) → 𝑦 ∈ (-𝐴...𝐴))
162	161	ex 115	. . . . . . . . . . 11 ⊢ (((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) → (𝑦 ∈ ℤ → 𝑦 ∈ (-𝐴...𝐴)))
163	162, 14	impbid1 142	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) → (𝑦 ∈ ℤ ↔ 𝑦 ∈ (-𝐴...𝐴)))
164	163	r19.29an 2693	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ ℤ) ∧ ∃𝑤 ∈ ℤ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) → (𝑦 ∈ ℤ ↔ 𝑦 ∈ (-𝐴...𝐴)))
165	164	rexlimdva2 2671	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) → (∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))) → (𝑦 ∈ ℤ ↔ 𝑦 ∈ (-𝐴...𝐴))))
166	165	pm5.32rd 455	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) → ((𝑦 ∈ ℤ ∧ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ↔ (𝑦 ∈ (-𝐴...𝐴) ∧ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))))))
167	166	rexbidv2 2553	. . . . . 6 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) → (∃𝑦 ∈ ℤ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))) ↔ ∃𝑦 ∈ (-𝐴...𝐴)∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))))
168	167	dcbid 850	. . . . 5 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) → (DECID ∃𝑦 ∈ ℤ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))) ↔ DECID ∃𝑦 ∈ (-𝐴...𝐴)∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))))
169	124, 168	mpbird 167	. . . 4 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) → DECID ∃𝑦 ∈ ℤ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))))
170	1, 2, 169	exfzdc 10637	. . 3 ⊢ (𝐴 ∈ ℕ0 → DECID ∃𝑥 ∈ (-𝐴...𝐴)∃𝑦 ∈ ℤ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))))
171	1	ad5antr 500	. . . . . . . . . . . 12 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑥 ∈ ℤ) → -𝐴 ∈ ℤ)
172	2	ad5antr 500	. . . . . . . . . . . 12 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑥 ∈ ℤ) → 𝐴 ∈ ℤ)
173		simpr 110	. . . . . . . . . . . 12 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑥 ∈ ℤ) → 𝑥 ∈ ℤ)
174	173	zred 9747	. . . . . . . . . . . . 13 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑥 ∈ ℤ) → 𝑥 ∈ ℝ)
175	172	zred 9747	. . . . . . . . . . . . 13 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑥 ∈ ℤ) → 𝐴 ∈ ℝ)
176	174	renegcld 8697	. . . . . . . . . . . . . 14 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑥 ∈ ℤ) → -𝑥 ∈ ℝ)
177	174	resqcld 11115	. . . . . . . . . . . . . 14 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑥 ∈ ℤ) → (𝑥↑2) ∈ ℝ)
178	173	znegcld 9749	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑥 ∈ ℤ) → -𝑥 ∈ ℤ)
179		zzlesq 11124	. . . . . . . . . . . . . . . 16 ⊢ (-𝑥 ∈ ℤ → -𝑥 ≤ (-𝑥↑2))
180	178, 179	syl 14	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑥 ∈ ℤ) → -𝑥 ≤ (-𝑥↑2))
181	173	zcnd 9748	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑥 ∈ ℤ) → 𝑥 ∈ ℂ)
182		sqneg 11013	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℂ → (-𝑥↑2) = (𝑥↑2))
183	181, 182	syl 14	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑥 ∈ ℤ) → (-𝑥↑2) = (𝑥↑2))
184	180, 183	breqtrd 4151	. . . . . . . . . . . . . 14 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑥 ∈ ℤ) → -𝑥 ≤ (𝑥↑2))
185	12	adantl 277	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑥 ∈ ℤ) → (𝑥↑2) ∈ ℕ0)
186	16	ad5antlr 501	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑥 ∈ ℤ) → (𝑦↑2) ∈ ℕ0)
187	185, 186	nn0addcld 9603	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑥 ∈ ℤ) → ((𝑥↑2) + (𝑦↑2)) ∈ ℕ0)
188	187	nn0red 9600	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑥 ∈ ℤ) → ((𝑥↑2) + (𝑦↑2)) ∈ ℝ)
189	21	ad4antlr 499	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑥 ∈ ℤ) → (𝑧↑2) ∈ ℕ0)
190	25	ad3antlr 497	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑥 ∈ ℤ) → (𝑤↑2) ∈ ℕ0)
191	189, 190	nn0addcld 9603	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑥 ∈ ℤ) → ((𝑧↑2) + (𝑤↑2)) ∈ ℕ0)
192	187, 191	nn0addcld 9603	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑥 ∈ ℤ) → (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))) ∈ ℕ0)
193	192	nn0red 9600	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑥 ∈ ℤ) → (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))) ∈ ℝ)
194		nn0addge1 9588	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥↑2) ∈ ℝ ∧ (𝑦↑2) ∈ ℕ0) → (𝑥↑2) ≤ ((𝑥↑2) + (𝑦↑2)))
195	177, 186, 194	syl2anc 415	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑥 ∈ ℤ) → (𝑥↑2) ≤ ((𝑥↑2) + (𝑦↑2)))
196	188, 191, 151	syl2anc 415	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑥 ∈ ℤ) → ((𝑥↑2) + (𝑦↑2)) ≤ (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))))
197	177, 188, 193, 195, 196	letrd 8440	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑥 ∈ ℤ) → (𝑥↑2) ≤ (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))))
198		simplr 533	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑥 ∈ ℤ) → 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))))
199	197, 198	breqtrrd 4153	. . . . . . . . . . . . . 14 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑥 ∈ ℤ) → (𝑥↑2) ≤ 𝐴)
200	176, 177, 175, 184, 199	letrd 8440	. . . . . . . . . . . . 13 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑥 ∈ ℤ) → -𝑥 ≤ 𝐴)
201	174, 175, 200	lenegcon1d 8845	. . . . . . . . . . . 12 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑥 ∈ ℤ) → -𝐴 ≤ 𝑥)
202		zzlesq 11124	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℤ → 𝑥 ≤ (𝑥↑2))
203	202	adantl 277	. . . . . . . . . . . . 13 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑥 ∈ ℤ) → 𝑥 ≤ (𝑥↑2))
204	174, 177, 175, 203, 199	letrd 8440	. . . . . . . . . . . 12 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑥 ∈ ℤ) → 𝑥 ≤ 𝐴)
205	171, 172, 173, 201, 204	elfzd 10398	. . . . . . . . . . 11 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑥 ∈ ℤ) → 𝑥 ∈ (-𝐴...𝐴))
206	205	ex 115	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) → (𝑥 ∈ ℤ → 𝑥 ∈ (-𝐴...𝐴)))
207	206, 10	impbid1 142	. . . . . . . . 9 ⊢ (((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) → (𝑥 ∈ ℤ ↔ 𝑥 ∈ (-𝐴...𝐴)))
208	207	r19.29an 2693	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ ℤ) ∧ ∃𝑤 ∈ ℤ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) → (𝑥 ∈ ℤ ↔ 𝑥 ∈ (-𝐴...𝐴)))
209	208	r19.29an 2693	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) → (𝑥 ∈ ℤ ↔ 𝑥 ∈ (-𝐴...𝐴)))
210	209	rexlimdva2 2671	. . . . . 6 ⊢ (𝐴 ∈ ℕ0 → (∃𝑦 ∈ ℤ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))) → (𝑥 ∈ ℤ ↔ 𝑥 ∈ (-𝐴...𝐴))))
211	210	pm5.32rd 455	. . . . 5 ⊢ (𝐴 ∈ ℕ0 → ((𝑥 ∈ ℤ ∧ ∃𝑦 ∈ ℤ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ↔ (𝑥 ∈ (-𝐴...𝐴) ∧ ∃𝑦 ∈ ℤ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))))))
212	211	rexbidv2 2553	. . . 4 ⊢ (𝐴 ∈ ℕ0 → (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))) ↔ ∃𝑥 ∈ (-𝐴...𝐴)∃𝑦 ∈ ℤ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))))
213	212	dcbid 850	. . 3 ⊢ (𝐴 ∈ ℕ0 → (DECID ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))) ↔ DECID ∃𝑥 ∈ (-𝐴...𝐴)∃𝑦 ∈ ℤ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))))
214	170, 213	mpbird 167	. 2 ⊢ (𝐴 ∈ ℕ0 → DECID ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))))
215		eqeq1 2245	. . . . . 6 ⊢ (𝑛 = 𝐴 → (𝑛 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))) ↔ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))))
216	215	2rexbidv 2575	. . . . 5 ⊢ (𝑛 = 𝐴 → (∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))) ↔ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))))
217	216	2rexbidv 2575	. . . 4 ⊢ (𝑛 = 𝐴 → (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))) ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))))
218		4sqlem11.1	. . . 4 ⊢ 𝑆 = {𝑛 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))}
219	217, 218	elab2g 2973	. . 3 ⊢ (𝐴 ∈ ℕ0 → (𝐴 ∈ 𝑆 ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))))
220	219	dcbid 850	. 2 ⊢ (𝐴 ∈ ℕ0 → (DECID 𝐴 ∈ 𝑆 ↔ DECID ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))))
221	214, 220	mpbird 167	1 ⊢ (𝐴 ∈ ℕ0 → DECID 𝐴 ∈ 𝑆)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  DECID wdc 846   = wceq 1402   ∈ wcel 2209  {cab 2224  ∃wrex 2529   class class class wbr 4125  (class class class)co 6075  ℂcc 8167  ℝcr 8168   + caddc 8172   ≤ cle 8351  -cneg 8488  2c2 9334  ℕ₀cn0 9542  ℤcz 9623  ...cfz 10390  ↑cexp 10953

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 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 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  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 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993  df-inn 9284  df-2 9342  df-n0 9543  df-z 9624  df-uz 9901  df-fz 10391  df-fzo 10528  df-seqfrec 10863  df-exp 10954

This theorem is referenced by:  4sqlem13m  13160  4sqlem14  13161  4sqlem17  13164  4sqlem18  13165