Theorem 4sqlemsdc 13123

Description: Lemma for 4sq 13133. 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 13121 and 4sqexercise2 13122 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 9628	. . . 4 ⊢ (𝐴 ∈ ℕ0 → -𝐴 ∈ ℤ)
2		nn0z 9614	. . . 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 10378	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (-𝐴...𝐴) → 𝑥 ∈ ℤ)
11	10	ad4antlr 495	. . . . . . . . . . . . . . 15 ⊢ (((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ (-𝐴...𝐴)) ∧ 𝑤 ∈ (-𝐴...𝐴)) → 𝑥 ∈ ℤ)
12		zsqcl2 11003	. . . . . . . . . . . . . . 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 10378	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (-𝐴...𝐴) → 𝑦 ∈ ℤ)
15	14	ad3antlr 493	. . . . . . . . . . . . . . 15 ⊢ (((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ (-𝐴...𝐴)) ∧ 𝑤 ∈ (-𝐴...𝐴)) → 𝑦 ∈ ℤ)
16		zsqcl2 11003	. . . . . . . . . . . . . . 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 9574	. . . . . . . . . . . . 13 ⊢ (((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ (-𝐴...𝐴)) ∧ 𝑤 ∈ (-𝐴...𝐴)) → ((𝑥↑2) + (𝑦↑2)) ∈ ℕ0)
19		elfzelz 10378	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ (-𝐴...𝐴) → 𝑧 ∈ ℤ)
20	19	ad2antlr 489	. . . . . . . . . . . . . . 15 ⊢ (((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ (-𝐴...𝐴)) ∧ 𝑤 ∈ (-𝐴...𝐴)) → 𝑧 ∈ ℤ)
21		zsqcl2 11003	. . . . . . . . . . . . . . 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 10378	. . . . . . . . . . . . . . . 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 11003	. . . . . . . . . . . . . . 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 9574	. . . . . . . . . . . . 13 ⊢ (((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ (-𝐴...𝐴)) ∧ 𝑤 ∈ (-𝐴...𝐴)) → ((𝑧↑2) + (𝑤↑2)) ∈ ℕ0)
28	18, 27	nn0addcld 9574	. . . . . . . . . . . 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 9716	. . . . . . . . . . 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 9670	. . . . . . . . . . 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 411	. . . . . . . . . 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 10608	. . . . . . . . 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 496	. . . . . . . . . . . . . . . 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 496	. . . . . . . . . . . . . . . 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 9718	. . . . . . . . . . . . . . . . 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 9718	. . . . . . . . . . . . . . . . 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 8670	. . . . . . . . . . . . . . . . . 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 11086	. . . . . . . . . . . . . . . . . 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 9720	. . . . . . . . . . . . . . . . . . . 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 11095	. . . . . . . . . . . . . . . . . . . 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 9719	. . . . . . . . . . . . . . . . . . . 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 10984	. . . . . . . . . . . . . . . . . . . 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 4140	. . . . . . . . . . . . . . . . . 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 493	. . . . . . . . . . . . . . . . . . . . . . 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 9574	. . . . . . . . . . . . . . . . . . . . 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 9571	. . . . . . . . . . . . . . . . . . . 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 497	. . . . . . . . . . . . . . . . . . . . . . . 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 495	. . . . . . . . . . . . . . . . . . . . . . . 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 9574	. . . . . . . . . . . . . . . . . . . . . 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 9574	. . . . . . . . . . . . . . . . . . . . 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 9571	. . . . . . . . . . . . . . . . . . . 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 9560	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑤↑2) ∈ ℝ ∧ (𝑧↑2) ∈ ℕ0) → (𝑤↑2) ≤ ((𝑧↑2) + (𝑤↑2)))
60	39, 48, 59	syl2anc 411	. . . . . . . . . . . . . . . . . . . 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 9560	. . . . . . . . . . . . . . . . . . . . 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 411	. . . . . . . . . . . . . . . . . . . 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 8413	. . . . . . . . . . . . . . . . . . 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 529	. . . . . . . . . . . . . . . . . . 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 4142	. . . . . . . . . . . . . . . . . 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 8413	. . . . . . . . . . . . . . . . 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 8818	. . . . . . . . . . . . . . . 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 11095	. . . . . . . . . . . . . . . . . 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 8413	. . . . . . . . . . . . . . . 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 10369	. . . . . . . . . . . . . . 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 451	. . . . . . . . . . 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 2547	. . . . . . . . . 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 846	. . . . . . . . 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 10608	. . . . . . 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 496	. . . . . . . . . . . . . 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 496	. . . . . . . . . . . . . 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 9718	. . . . . . . . . . . . . . 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 9718	. . . . . . . . . . . . . . 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 8670	. . . . . . . . . . . . . . . 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 11086	. . . . . . . . . . . . . . . 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 9720	. . . . . . . . . . . . . . . . . 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 11095	. . . . . . . . . . . . . . . . . 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 9719	. . . . . . . . . . . . . . . . . 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 10984	. . . . . . . . . . . . . . . . . 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 4140	. . . . . . . . . . . . . . . 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 493	. . . . . . . . . . . . . . . . . . . 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 9574	. . . . . . . . . . . . . . . . . . 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 9571	. . . . . . . . . . . . . . . . . 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 497	. . . . . . . . . . . . . . . . . . . . . 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 495	. . . . . . . . . . . . . . . . . . . . . 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 9574	. . . . . . . . . . . . . . . . . . . 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 9574	. . . . . . . . . . . . . . . . . . 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 9571	. . . . . . . . . . . . . . . . . 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 9559	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑧↑2) ∈ ℝ ∧ (𝑤↑2) ∈ ℕ0) → (𝑧↑2) ≤ ((𝑧↑2) + (𝑤↑2)))
106	86, 95, 105	syl2anc 411	. . . . . . . . . . . . . . . . . 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 411	. . . . . . . . . . . . . . . . . 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 8413	. . . . . . . . . . . . . . . . 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 529	. . . . . . . . . . . . . . . . 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 4142	. . . . . . . . . . . . . . . 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 8413	. . . . . . . . . . . . . . 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 8818	. . . . . . . . . . . . . 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 11095	. . . . . . . . . . . . . . . 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 8413	. . . . . . . . . . . . . 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 10369	. . . . . . . . . . . . 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 2665	. . . . . . . . . 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 451	. . . . . . . . 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 2547	. . . . . . . 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 846	. . . . . . 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 10608	. . . . 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 496	. . . . . . . . . . . . 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 496	. . . . . . . . . . . . 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 9718	. . . . . . . . . . . . . 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 9718	. . . . . . . . . . . . . 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 8670	. . . . . . . . . . . . . . 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 11086	. . . . . . . . . . . . . . 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 9720	. . . . . . . . . . . . . . . . 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 11095	. . . . . . . . . . . . . . . . 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 9719	. . . . . . . . . . . . . . . . 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 10984	. . . . . . . . . . . . . . . . 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 4140	. . . . . . . . . . . . . . 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 497	. . . . . . . . . . . . . . . . . . . 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 9574	. . . . . . . . . . . . . . . . . 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 9571	. . . . . . . . . . . . . . . . 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 495	. . . . . . . . . . . . . . . . . . . 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 493	. . . . . . . . . . . . . . . . . . . 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 9574	. . . . . . . . . . . . . . . . . . 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 9574	. . . . . . . . . . . . . . . . . 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 9571	. . . . . . . . . . . . . . . . 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 9560	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦↑2) ∈ ℝ ∧ (𝑥↑2) ∈ ℕ0) → (𝑦↑2) ≤ ((𝑥↑2) + (𝑦↑2)))
150	131, 140, 149	syl2anc 411	. . . . . . . . . . . . . . . . 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 9559	. . . . . . . . . . . . . . . . . 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 411	. . . . . . . . . . . . . . . . 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 8413	. . . . . . . . . . . . . . . 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 529	. . . . . . . . . . . . . . . 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 4142	. . . . . . . . . . . . . . 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 8413	. . . . . . . . . . . . . 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 8818	. . . . . . . . . . . . 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 11095	. . . . . . . . . . . . . . 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 8413	. . . . . . . . . . . . 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 10369	. . . . . . . . . . . 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 2687	. . . . . . . . 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 2665	. . . . . . . 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 451	. . . . . . 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 2547	. . . . . 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 846	. . . . 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 10608	. . 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 496	. . . . . . . . . . . 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 496	. . . . . . . . . . . 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 9718	. . . . . . . . . . . . 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 9718	. . . . . . . . . . . . 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 8670	. . . . . . . . . . . . . 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 11086	. . . . . . . . . . . . . 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 9720	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑥 ∈ ℤ) → -𝑥 ∈ ℤ)
179		zzlesq 11095	. . . . . . . . . . . . . . . 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 9719	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑥 ∈ ℤ) → 𝑥 ∈ ℂ)
182		sqneg 10984	. . . . . . . . . . . . . . . 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 4140	. . . . . . . . . . . . . 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 497	. . . . . . . . . . . . . . . . . 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 9574	. . . . . . . . . . . . . . . . 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 9571	. . . . . . . . . . . . . . . 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 495	. . . . . . . . . . . . . . . . . . 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 493	. . . . . . . . . . . . . . . . . . 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 9574	. . . . . . . . . . . . . . . . . 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 9574	. . . . . . . . . . . . . . . . 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 9571	. . . . . . . . . . . . . . . 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 9559	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥↑2) ∈ ℝ ∧ (𝑦↑2) ∈ ℕ0) → (𝑥↑2) ≤ ((𝑥↑2) + (𝑦↑2)))
195	177, 186, 194	syl2anc 411	. . . . . . . . . . . . . . . 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 411	. . . . . . . . . . . . . . . 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 8413	. . . . . . . . . . . . . . 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 529	. . . . . . . . . . . . . . 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 4142	. . . . . . . . . . . . . 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 8413	. . . . . . . . . . . . 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 8818	. . . . . . . . . . . 12 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑥 ∈ ℤ) → -𝐴 ≤ 𝑥)
202		zzlesq 11095	. . . . . . . . . . . . . 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 8413	. . . . . . . . . . . 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 10369	. . . . . . . . . . 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 2687	. . . . . . . 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 2687	. . . . . . 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 2665	. . . . . 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 451	. . . . 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 2547	. . . 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 846	. . 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 2241	. . . . . 6 ⊢ (𝑛 = 𝐴 → (𝑛 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))) ↔ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))))
216	215	2rexbidv 2569	. . . . 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 2569	. . . 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 2967	. . 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 846	. 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 842   = wceq 1398   ∈ wcel 2205  {cab 2220  ∃wrex 2523   class class class wbr 4114  (class class class)co 6058  ℂcc 8141  ℝcr 8142   + caddc 8146   ≤ cle 8325  -cneg 8461  2c2 9305  ℕ₀cn0 9513  ℤcz 9594  ...cfz 10361  ↑cexp 10924

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-n0 9514  df-z 9595  df-uz 9872  df-fz 10362  df-fzo 10499  df-seqfrec 10834  df-exp 10925

This theorem is referenced by:  4sqlem13m  13126  4sqlem14  13127  4sqlem17  13130  4sqlem18  13131