Theorem 4sqlemsdc 13098

Description: Lemma for 4sq 13108. 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 13096 and 4sqexercise2 13097 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 9611	. . . 4 ⊢ (𝐴 ∈ ℕ0 → -𝐴 ∈ ℤ)
2		nn0z 9597	. . . 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 10359	. . . . . . . . . . . . . . . 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 10979	. . . . . . . . . . . . . . 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 10359	. . . . . . . . . . . . . . . 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 10979	. . . . . . . . . . . . . . 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 9557	. . . . . . . . . . . . 13 ⊢ (((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ (-𝐴...𝐴)) ∧ 𝑤 ∈ (-𝐴...𝐴)) → ((𝑥↑2) + (𝑦↑2)) ∈ ℕ0)
19		elfzelz 10359	. . . . . . . . . . . . . . . 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 10979	. . . . . . . . . . . . . . 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 10359	. . . . . . . . . . . . . . . 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 10979	. . . . . . . . . . . . . . 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 9557	. . . . . . . . . . . . 13 ⊢ (((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ (-𝐴...𝐴)) ∧ 𝑤 ∈ (-𝐴...𝐴)) → ((𝑧↑2) + (𝑤↑2)) ∈ ℕ0)
28	18, 27	nn0addcld 9557	. . . . . . . . . . . 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 9698	. . . . . . . . . . 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 9653	. . . . . . . . . . 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 10586	. . . . . . . . 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 9700	. . . . . . . . . . . . . . . . 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 9700	. . . . . . . . . . . . . . . . 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 8653	. . . . . . . . . . . . . . . . . 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 11061	. . . . . . . . . . . . . . . . . 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 9702	. . . . . . . . . . . . . . . . . . . 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 11070	. . . . . . . . . . . . . . . . . . . 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 9701	. . . . . . . . . . . . . . . . . . . 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 10960	. . . . . . . . . . . . . . . . . . . 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 4135	. . . . . . . . . . . . . . . . . 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 9557	. . . . . . . . . . . . . . . . . . . . 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 9554	. . . . . . . . . . . . . . . . . . . 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 9557	. . . . . . . . . . . . . . . . . . . . . 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 9557	. . . . . . . . . . . . . . . . . . . . 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 9554	. . . . . . . . . . . . . . . . . . . 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 9543	. . . . . . . . . . . . . . . . . . . . 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 9543	. . . . . . . . . . . . . . . . . . . . 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 8397	. . . . . . . . . . . . . . . . . . 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 4137	. . . . . . . . . . . . . . . . . 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 8397	. . . . . . . . . . . . . . . . 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 8801	. . . . . . . . . . . . . . . 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 11070	. . . . . . . . . . . . . . . . . 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 8397	. . . . . . . . . . . . . . . 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 10350	. . . . . . . . . . . . . . 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 2545	. . . . . . . . . 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 10586	. . . . . . 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 9700	. . . . . . . . . . . . . . 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 9700	. . . . . . . . . . . . . . 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 8653	. . . . . . . . . . . . . . . 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 11061	. . . . . . . . . . . . . . . 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 9702	. . . . . . . . . . . . . . . . . 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 11070	. . . . . . . . . . . . . . . . . 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 9701	. . . . . . . . . . . . . . . . . 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 10960	. . . . . . . . . . . . . . . . . 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 4135	. . . . . . . . . . . . . . . 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 9557	. . . . . . . . . . . . . . . . . . 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 9554	. . . . . . . . . . . . . . . . . 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 9557	. . . . . . . . . . . . . . . . . . . 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 9557	. . . . . . . . . . . . . . . . . . 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 9554	. . . . . . . . . . . . . . . . . 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 9542	. . . . . . . . . . . . . . . . . . 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 8397	. . . . . . . . . . . . . . . . 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 4137	. . . . . . . . . . . . . . . 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 8397	. . . . . . . . . . . . . . 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 8801	. . . . . . . . . . . . . 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 11070	. . . . . . . . . . . . . . . 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 8397	. . . . . . . . . . . . . 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 10350	. . . . . . . . . . . . 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 2663	. . . . . . . . . 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 2545	. . . . . . . 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 10586	. . . . 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 9700	. . . . . . . . . . . . . 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 9700	. . . . . . . . . . . . . 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 8653	. . . . . . . . . . . . . . 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 11061	. . . . . . . . . . . . . . 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 9702	. . . . . . . . . . . . . . . . 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 11070	. . . . . . . . . . . . . . . . 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 9701	. . . . . . . . . . . . . . . . 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 10960	. . . . . . . . . . . . . . . . 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 4135	. . . . . . . . . . . . . . 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 9557	. . . . . . . . . . . . . . . . . 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 9554	. . . . . . . . . . . . . . . . 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 9557	. . . . . . . . . . . . . . . . . . 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 9557	. . . . . . . . . . . . . . . . . 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 9554	. . . . . . . . . . . . . . . . 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 9543	. . . . . . . . . . . . . . . . . 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 9542	. . . . . . . . . . . . . . . . . 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 8397	. . . . . . . . . . . . . . . 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 4137	. . . . . . . . . . . . . . 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 8397	. . . . . . . . . . . . . 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 8801	. . . . . . . . . . . . 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 11070	. . . . . . . . . . . . . . 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 8397	. . . . . . . . . . . . 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 10350	. . . . . . . . . . . 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 2685	. . . . . . . . 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 2663	. . . . . . . 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 2545	. . . . . 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 10586	. . 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 9700	. . . . . . . . . . . . 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 9700	. . . . . . . . . . . . 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 8653	. . . . . . . . . . . . . 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 11061	. . . . . . . . . . . . . 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 9702	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑥 ∈ ℤ) → -𝑥 ∈ ℤ)
179		zzlesq 11070	. . . . . . . . . . . . . . . 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 9701	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑥 ∈ ℤ) → 𝑥 ∈ ℂ)
182		sqneg 10960	. . . . . . . . . . . . . . . 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 4135	. . . . . . . . . . . . . 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 9557	. . . . . . . . . . . . . . . . 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 9554	. . . . . . . . . . . . . . . 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 9557	. . . . . . . . . . . . . . . . . 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 9557	. . . . . . . . . . . . . . . . 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 9554	. . . . . . . . . . . . . . . 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 9542	. . . . . . . . . . . . . . . . 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 8397	. . . . . . . . . . . . . . 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 4137	. . . . . . . . . . . . . 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 8397	. . . . . . . . . . . . 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 8801	. . . . . . . . . . . 12 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑥 ∈ ℤ) → -𝐴 ≤ 𝑥)
202		zzlesq 11070	. . . . . . . . . . . . . 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 8397	. . . . . . . . . . . 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 10350	. . . . . . . . . . 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 2685	. . . . . . . 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 2685	. . . . . . 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 2663	. . . . . 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 2545	. . . 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 2239	. . . . . 6 ⊢ (𝑛 = 𝐴 → (𝑛 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))) ↔ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))))
216	215	2rexbidv 2567	. . . . 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 2567	. . . 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 2964	. . 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 2203  {cab 2218  ∃wrex 2521   class class class wbr 4109  (class class class)co 6050  ℂcc 8125  ℝcr 8126   + caddc 8130   ≤ cle 8309  -cneg 8445  2c2 9288  ℕ₀cn0 9496  ℤcz 9577  ...cfz 10342  ↑cexp 10900

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244  ax-pre-mulext 8245

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-frec 6622  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-reap 8849  df-ap 8856  df-div 8947  df-inn 9238  df-2 9296  df-n0 9497  df-z 9578  df-uz 9854  df-fz 10343  df-fzo 10477  df-seqfrec 10810  df-exp 10901

This theorem is referenced by:  4sqlem13m  13101  4sqlem14  13102  4sqlem17  13105  4sqlem18  13106