Theorem 4sqlemsdc 12793

Description: Lemma for 4sq 12803. 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 12791 and 4sqexercise2 12792 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 9421	. . . 4 ⊢ (𝐴 ∈ ℕ0 → -𝐴 ∈ ℤ)
2		nn0z 9407	. . . 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 10162	. . . . . . . . . . . . . . . 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 10779	. . . . . . . . . . . . . . 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 10162	. . . . . . . . . . . . . . . 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 10779	. . . . . . . . . . . . . . 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 9367	. . . . . . . . . . . . 13 ⊢ (((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ (-𝐴...𝐴)) ∧ 𝑤 ∈ (-𝐴...𝐴)) → ((𝑥↑2) + (𝑦↑2)) ∈ ℕ0)
19		elfzelz 10162	. . . . . . . . . . . . . . . 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 10779	. . . . . . . . . . . . . . 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 10162	. . . . . . . . . . . . . . . 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 10779	. . . . . . . . . . . . . . 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 9367	. . . . . . . . . . . . 13 ⊢ (((((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) ∧ 𝑧 ∈ (-𝐴...𝐴)) ∧ 𝑤 ∈ (-𝐴...𝐴)) → ((𝑧↑2) + (𝑤↑2)) ∈ ℕ0)
28	18, 27	nn0addcld 9367	. . . . . . . . . . . 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 9508	. . . . . . . . . . 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 9463	. . . . . . . . . . 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 10386	. . . . . . . . 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 9510	. . . . . . . . . . . . . . . . 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 9510	. . . . . . . . . . . . . . . . 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 8467	. . . . . . . . . . . . . . . . . 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 10861	. . . . . . . . . . . . . . . . . 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 9512	. . . . . . . . . . . . . . . . . . . 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 10870	. . . . . . . . . . . . . . . . . . . 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 9511	. . . . . . . . . . . . . . . . . . . 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 10760	. . . . . . . . . . . . . . . . . . . 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 4076	. . . . . . . . . . . . . . . . . 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 9367	. . . . . . . . . . . . . . . . . . . . 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 9364	. . . . . . . . . . . . . . . . . . . 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 9367	. . . . . . . . . . . . . . . . . . . . . 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 9367	. . . . . . . . . . . . . . . . . . . . 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 9364	. . . . . . . . . . . . . . . . . . . 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 9357	. . . . . . . . . . . . . . . . . . . . 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 9357	. . . . . . . . . . . . . . . . . . . . 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 8211	. . . . . . . . . . . . . . . . . . 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 528	. . . . . . . . . . . . . . . . . . 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 4078	. . . . . . . . . . . . . . . . . 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 8211	. . . . . . . . . . . . . . . . 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 8615	. . . . . . . . . . . . . . . 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 10870	. . . . . . . . . . . . . . . . . 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 8211	. . . . . . . . . . . . . . . 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 10153	. . . . . . . . . . . . . . 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 2510	. . . . . . . . . 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 840	. . . . . . . . 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 10386	. . . . . . 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 9510	. . . . . . . . . . . . . . 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 9510	. . . . . . . . . . . . . . 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 8467	. . . . . . . . . . . . . . . 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 10861	. . . . . . . . . . . . . . . 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 9512	. . . . . . . . . . . . . . . . . 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 10870	. . . . . . . . . . . . . . . . . 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 9511	. . . . . . . . . . . . . . . . . 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 10760	. . . . . . . . . . . . . . . . . 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 4076	. . . . . . . . . . . . . . . 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 9367	. . . . . . . . . . . . . . . . . . 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 9364	. . . . . . . . . . . . . . . . . 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 9367	. . . . . . . . . . . . . . . . . . . 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 9367	. . . . . . . . . . . . . . . . . . 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 9364	. . . . . . . . . . . . . . . . . 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 9356	. . . . . . . . . . . . . . . . . . 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 8211	. . . . . . . . . . . . . . . . 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 528	. . . . . . . . . . . . . . . . 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 4078	. . . . . . . . . . . . . . . 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 8211	. . . . . . . . . . . . . . 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 8615	. . . . . . . . . . . . . 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 10870	. . . . . . . . . . . . . . . 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 8211	. . . . . . . . . . . . . 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 10153	. . . . . . . . . . . . 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 2627	. . . . . . . . . 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 2510	. . . . . . . 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 840	. . . . . . 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 10386	. . . . 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 9510	. . . . . . . . . . . . . 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 9510	. . . . . . . . . . . . . 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 8467	. . . . . . . . . . . . . . 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 10861	. . . . . . . . . . . . . . 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 9512	. . . . . . . . . . . . . . . . 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 10870	. . . . . . . . . . . . . . . . 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 9511	. . . . . . . . . . . . . . . . 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 10760	. . . . . . . . . . . . . . . . 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 4076	. . . . . . . . . . . . . . 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 9367	. . . . . . . . . . . . . . . . . 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 9364	. . . . . . . . . . . . . . . . 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 9367	. . . . . . . . . . . . . . . . . . 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 9367	. . . . . . . . . . . . . . . . . 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 9364	. . . . . . . . . . . . . . . . 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 9357	. . . . . . . . . . . . . . . . . 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 9356	. . . . . . . . . . . . . . . . . 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 8211	. . . . . . . . . . . . . . . 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 528	. . . . . . . . . . . . . . . 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 4078	. . . . . . . . . . . . . . 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 8211	. . . . . . . . . . . . . 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 8615	. . . . . . . . . . . . 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 10870	. . . . . . . . . . . . . . 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 8211	. . . . . . . . . . . . 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 10153	. . . . . . . . . . . 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 2649	. . . . . . . . 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 2627	. . . . . . . 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 2510	. . . . . 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 840	. . . . 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 10386	. . 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 9510	. . . . . . . . . . . . 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 9510	. . . . . . . . . . . . 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 8467	. . . . . . . . . . . . . 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 10861	. . . . . . . . . . . . . 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 9512	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑥 ∈ ℤ) → -𝑥 ∈ ℤ)
179		zzlesq 10870	. . . . . . . . . . . . . . . 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 9511	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑥 ∈ ℤ) → 𝑥 ∈ ℂ)
182		sqneg 10760	. . . . . . . . . . . . . . . 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 4076	. . . . . . . . . . . . . 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 9367	. . . . . . . . . . . . . . . . 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 9364	. . . . . . . . . . . . . . . 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 9367	. . . . . . . . . . . . . . . . . 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 9367	. . . . . . . . . . . . . . . . 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 9364	. . . . . . . . . . . . . . . 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 9356	. . . . . . . . . . . . . . . . 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 8211	. . . . . . . . . . . . . . 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 528	. . . . . . . . . . . . . . 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 4078	. . . . . . . . . . . . . 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 8211	. . . . . . . . . . . . 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 8615	. . . . . . . . . . . 12 ⊢ ((((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ ℤ) ∧ 𝑤 ∈ ℤ) ∧ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))) ∧ 𝑥 ∈ ℤ) → -𝐴 ≤ 𝑥)
202		zzlesq 10870	. . . . . . . . . . . . . 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 8211	. . . . . . . . . . . 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 10153	. . . . . . . . . . 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 2649	. . . . . . . 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 2649	. . . . . . 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 2627	. . . . . 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 2510	. . . 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 840	. . 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 2213	. . . . . 6 ⊢ (𝑛 = 𝐴 → (𝑛 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))) ↔ 𝐴 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))))
216	215	2rexbidv 2532	. . . . 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 2532	. . . 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 2924	. . 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 840	. 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 836   = wceq 1373   ∈ wcel 2177  {cab 2192  ∃wrex 2486   class class class wbr 4050  (class class class)co 5956  ℂcc 7938  ℝcr 7939   + caddc 7943   ≤ cle 8123  -cneg 8259  2c2 9102  ℕ₀cn0 9310  ℤcz 9387  ...cfz 10145  ↑cexp 10700

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4166  ax-sep 4169  ax-nul 4177  ax-pow 4225  ax-pr 4260  ax-un 4487  ax-setind 4592  ax-iinf 4643  ax-cnex 8031  ax-resscn 8032  ax-1cn 8033  ax-1re 8034  ax-icn 8035  ax-addcl 8036  ax-addrcl 8037  ax-mulcl 8038  ax-mulrcl 8039  ax-addcom 8040  ax-mulcom 8041  ax-addass 8042  ax-mulass 8043  ax-distr 8044  ax-i2m1 8045  ax-0lt1 8046  ax-1rid 8047  ax-0id 8048  ax-rnegex 8049  ax-precex 8050  ax-cnre 8051  ax-pre-ltirr 8052  ax-pre-ltwlin 8053  ax-pre-lttrn 8054  ax-pre-apti 8055  ax-pre-ltadd 8056  ax-pre-mulgt0 8057  ax-pre-mulext 8058

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-if 3576  df-pw 3622  df-sn 3643  df-pr 3644  df-op 3646  df-uni 3856  df-int 3891  df-iun 3934  df-br 4051  df-opab 4113  df-mpt 4114  df-tr 4150  df-id 4347  df-po 4350  df-iso 4351  df-iord 4420  df-on 4422  df-ilim 4423  df-suc 4425  df-iom 4646  df-xp 4688  df-rel 4689  df-cnv 4690  df-co 4691  df-dm 4692  df-rn 4693  df-res 4694  df-ima 4695  df-iota 5240  df-fun 5281  df-fn 5282  df-f 5283  df-f1 5284  df-fo 5285  df-f1o 5286  df-fv 5287  df-riota 5911  df-ov 5959  df-oprab 5960  df-mpo 5961  df-1st 6238  df-2nd 6239  df-recs 6403  df-frec 6489  df-pnf 8124  df-mnf 8125  df-xr 8126  df-ltxr 8127  df-le 8128  df-sub 8260  df-neg 8261  df-reap 8663  df-ap 8670  df-div 8761  df-inn 9052  df-2 9110  df-n0 9311  df-z 9388  df-uz 9664  df-fz 10146  df-fzo 10280  df-seqfrec 10610  df-exp 10701

This theorem is referenced by:  4sqlem13m  12796  4sqlem14  12797  4sqlem17  12800  4sqlem18  12801