Theorem 4sqexercise2 13125

Description: Exercise which may help in understanding the proof of 4sqlemsdc 13126. (Contributed by Jim Kingdon, 30-May-2025.)

Hypothesis

Ref	Expression
4sqexercise2.s	⊢ 𝑆 = {𝑛 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑛 = ((𝑥↑2) + (𝑦↑2))}

Assertion

Ref	Expression
4sqexercise2	⊢ (𝐴 ∈ ℕ0 → DECID 𝐴 ∈ 𝑆)

Distinct variable group:   𝐴,𝑛,𝑥,𝑦

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

Proof of Theorem 4sqexercise2

Step	Hyp	Ref	Expression
1		nn0negz 9631	. . . 4 ⊢ (𝐴 ∈ ℕ0 → -𝐴 ∈ ℤ)
2		nn0z 9617	. . . 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	4	adantr 276	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) → 𝐴 ∈ ℤ)
6		elfzelz 10381	. . . . . . . . . 10 ⊢ (𝑥 ∈ (-𝐴...𝐴) → 𝑥 ∈ ℤ)
7	6	ad2antlr 489	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) → 𝑥 ∈ ℤ)
8		zsqcl 10999	. . . . . . . . 9 ⊢ (𝑥 ∈ ℤ → (𝑥↑2) ∈ ℤ)
9	7, 8	syl 14	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) → (𝑥↑2) ∈ ℤ)
10		elfzelz 10381	. . . . . . . . . 10 ⊢ (𝑦 ∈ (-𝐴...𝐴) → 𝑦 ∈ ℤ)
11	10	adantl 277	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) → 𝑦 ∈ ℤ)
12		zsqcl 10999	. . . . . . . . 9 ⊢ (𝑦 ∈ ℤ → (𝑦↑2) ∈ ℤ)
13	11, 12	syl 14	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) → (𝑦↑2) ∈ ℤ)
14	9, 13	zaddcld 9725	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) → ((𝑥↑2) + (𝑦↑2)) ∈ ℤ)
15		zdceq 9673	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ ((𝑥↑2) + (𝑦↑2)) ∈ ℤ) → DECID 𝐴 = ((𝑥↑2) + (𝑦↑2)))
16	5, 14, 15	syl2anc 411	. . . . . 6 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) → DECID 𝐴 = ((𝑥↑2) + (𝑦↑2)))
17	3, 4, 16	exfzdc 10611	. . . . 5 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) → DECID ∃𝑦 ∈ (-𝐴...𝐴)𝐴 = ((𝑥↑2) + (𝑦↑2)))
18	3	adantr 276	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ (𝑦 ∈ ℤ ∧ 𝐴 = ((𝑥↑2) + (𝑦↑2)))) → -𝐴 ∈ ℤ)
19	4	adantr 276	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ (𝑦 ∈ ℤ ∧ 𝐴 = ((𝑥↑2) + (𝑦↑2)))) → 𝐴 ∈ ℤ)
20		simprl 531	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ (𝑦 ∈ ℤ ∧ 𝐴 = ((𝑥↑2) + (𝑦↑2)))) → 𝑦 ∈ ℤ)
21	20	zred 9721	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ (𝑦 ∈ ℤ ∧ 𝐴 = ((𝑥↑2) + (𝑦↑2)))) → 𝑦 ∈ ℝ)
22	19	zred 9721	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ (𝑦 ∈ ℤ ∧ 𝐴 = ((𝑥↑2) + (𝑦↑2)))) → 𝐴 ∈ ℝ)
23	21	renegcld 8671	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ (𝑦 ∈ ℤ ∧ 𝐴 = ((𝑥↑2) + (𝑦↑2)))) → -𝑦 ∈ ℝ)
24		zsqcl2 11006	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℤ → (𝑦↑2) ∈ ℕ0)
25	20, 24	syl 14	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ (𝑦 ∈ ℤ ∧ 𝐴 = ((𝑥↑2) + (𝑦↑2)))) → (𝑦↑2) ∈ ℕ0)
26	25	nn0red 9574	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ (𝑦 ∈ ℤ ∧ 𝐴 = ((𝑥↑2) + (𝑦↑2)))) → (𝑦↑2) ∈ ℝ)
27		znegcl 9628	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℤ → -𝑦 ∈ ℤ)
28		zzlesq 11098	. . . . . . . . . . . . . . . 16 ⊢ (-𝑦 ∈ ℤ → -𝑦 ≤ (-𝑦↑2))
29	27, 28	syl 14	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℤ → -𝑦 ≤ (-𝑦↑2))
30		zcn 9602	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℤ → 𝑦 ∈ ℂ)
31		sqneg 10987	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℂ → (-𝑦↑2) = (𝑦↑2))
32	30, 31	syl 14	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℤ → (-𝑦↑2) = (𝑦↑2))
33	29, 32	breqtrd 4140	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℤ → -𝑦 ≤ (𝑦↑2))
34	20, 33	syl 14	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ (𝑦 ∈ ℤ ∧ 𝐴 = ((𝑥↑2) + (𝑦↑2)))) → -𝑦 ≤ (𝑦↑2))
35	6	ad2antlr 489	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ (𝑦 ∈ ℤ ∧ 𝐴 = ((𝑥↑2) + (𝑦↑2)))) → 𝑥 ∈ ℤ)
36		zsqcl2 11006	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℤ → (𝑥↑2) ∈ ℕ0)
37	35, 36	syl 14	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ (𝑦 ∈ ℤ ∧ 𝐴 = ((𝑥↑2) + (𝑦↑2)))) → (𝑥↑2) ∈ ℕ0)
38		nn0addge2 9563	. . . . . . . . . . . . . . 15 ⊢ (((𝑦↑2) ∈ ℝ ∧ (𝑥↑2) ∈ ℕ0) → (𝑦↑2) ≤ ((𝑥↑2) + (𝑦↑2)))
39	26, 37, 38	syl2anc 411	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ (𝑦 ∈ ℤ ∧ 𝐴 = ((𝑥↑2) + (𝑦↑2)))) → (𝑦↑2) ≤ ((𝑥↑2) + (𝑦↑2)))
40		simprr 533	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ (𝑦 ∈ ℤ ∧ 𝐴 = ((𝑥↑2) + (𝑦↑2)))) → 𝐴 = ((𝑥↑2) + (𝑦↑2)))
41	39, 40	breqtrrd 4142	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ (𝑦 ∈ ℤ ∧ 𝐴 = ((𝑥↑2) + (𝑦↑2)))) → (𝑦↑2) ≤ 𝐴)
42	23, 26, 22, 34, 41	letrd 8414	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ (𝑦 ∈ ℤ ∧ 𝐴 = ((𝑥↑2) + (𝑦↑2)))) → -𝑦 ≤ 𝐴)
43	21, 22, 42	lenegcon1d 8819	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ (𝑦 ∈ ℤ ∧ 𝐴 = ((𝑥↑2) + (𝑦↑2)))) → -𝐴 ≤ 𝑦)
44		zzlesq 11098	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℤ → 𝑦 ≤ (𝑦↑2))
45	20, 44	syl 14	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ (𝑦 ∈ ℤ ∧ 𝐴 = ((𝑥↑2) + (𝑦↑2)))) → 𝑦 ≤ (𝑦↑2))
46	21, 26, 22, 45, 41	letrd 8414	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ (𝑦 ∈ ℤ ∧ 𝐴 = ((𝑥↑2) + (𝑦↑2)))) → 𝑦 ≤ 𝐴)
47	18, 19, 20, 43, 46	elfzd 10372	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ (𝑦 ∈ ℤ ∧ 𝐴 = ((𝑥↑2) + (𝑦↑2)))) → 𝑦 ∈ (-𝐴...𝐴))
48	47, 40	jca 306	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ (𝑦 ∈ ℤ ∧ 𝐴 = ((𝑥↑2) + (𝑦↑2)))) → (𝑦 ∈ (-𝐴...𝐴) ∧ 𝐴 = ((𝑥↑2) + (𝑦↑2))))
49	48	ex 115	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) → ((𝑦 ∈ ℤ ∧ 𝐴 = ((𝑥↑2) + (𝑦↑2))) → (𝑦 ∈ (-𝐴...𝐴) ∧ 𝐴 = ((𝑥↑2) + (𝑦↑2)))))
50	10	anim1i 340	. . . . . . . 8 ⊢ ((𝑦 ∈ (-𝐴...𝐴) ∧ 𝐴 = ((𝑥↑2) + (𝑦↑2))) → (𝑦 ∈ ℤ ∧ 𝐴 = ((𝑥↑2) + (𝑦↑2))))
51	49, 50	impbid1 142	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) → ((𝑦 ∈ ℤ ∧ 𝐴 = ((𝑥↑2) + (𝑦↑2))) ↔ (𝑦 ∈ (-𝐴...𝐴) ∧ 𝐴 = ((𝑥↑2) + (𝑦↑2)))))
52	51	rexbidv2 2547	. . . . . 6 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) → (∃𝑦 ∈ ℤ 𝐴 = ((𝑥↑2) + (𝑦↑2)) ↔ ∃𝑦 ∈ (-𝐴...𝐴)𝐴 = ((𝑥↑2) + (𝑦↑2))))
53	52	dcbid 846	. . . . 5 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) → (DECID ∃𝑦 ∈ ℤ 𝐴 = ((𝑥↑2) + (𝑦↑2)) ↔ DECID ∃𝑦 ∈ (-𝐴...𝐴)𝐴 = ((𝑥↑2) + (𝑦↑2))))
54	17, 53	mpbird 167	. . . 4 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) → DECID ∃𝑦 ∈ ℤ 𝐴 = ((𝑥↑2) + (𝑦↑2)))
55	1, 2, 54	exfzdc 10611	. . 3 ⊢ (𝐴 ∈ ℕ0 → DECID ∃𝑥 ∈ (-𝐴...𝐴)∃𝑦 ∈ ℤ 𝐴 = ((𝑥↑2) + (𝑦↑2)))
56	1	ad3antrrr 492	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝐴 = ((𝑥↑2) + (𝑦↑2))) ∧ 𝑥 ∈ ℤ) → -𝐴 ∈ ℤ)
57	2	ad3antrrr 492	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝐴 = ((𝑥↑2) + (𝑦↑2))) ∧ 𝑥 ∈ ℤ) → 𝐴 ∈ ℤ)
58		simpr 110	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝐴 = ((𝑥↑2) + (𝑦↑2))) ∧ 𝑥 ∈ ℤ) → 𝑥 ∈ ℤ)
59	58	zred 9721	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝐴 = ((𝑥↑2) + (𝑦↑2))) ∧ 𝑥 ∈ ℤ) → 𝑥 ∈ ℝ)
60	57	zred 9721	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝐴 = ((𝑥↑2) + (𝑦↑2))) ∧ 𝑥 ∈ ℤ) → 𝐴 ∈ ℝ)
61	59	renegcld 8671	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝐴 = ((𝑥↑2) + (𝑦↑2))) ∧ 𝑥 ∈ ℤ) → -𝑥 ∈ ℝ)
62	59	resqcld 11089	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝐴 = ((𝑥↑2) + (𝑦↑2))) ∧ 𝑥 ∈ ℤ) → (𝑥↑2) ∈ ℝ)
63	58	znegcld 9723	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝐴 = ((𝑥↑2) + (𝑦↑2))) ∧ 𝑥 ∈ ℤ) → -𝑥 ∈ ℤ)
64		zzlesq 11098	. . . . . . . . . . . . . 14 ⊢ (-𝑥 ∈ ℤ → -𝑥 ≤ (-𝑥↑2))
65	63, 64	syl 14	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝐴 = ((𝑥↑2) + (𝑦↑2))) ∧ 𝑥 ∈ ℤ) → -𝑥 ≤ (-𝑥↑2))
66	58	zcnd 9722	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝐴 = ((𝑥↑2) + (𝑦↑2))) ∧ 𝑥 ∈ ℤ) → 𝑥 ∈ ℂ)
67		sqneg 10987	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℂ → (-𝑥↑2) = (𝑥↑2))
68	66, 67	syl 14	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝐴 = ((𝑥↑2) + (𝑦↑2))) ∧ 𝑥 ∈ ℤ) → (-𝑥↑2) = (𝑥↑2))
69	65, 68	breqtrd 4140	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝐴 = ((𝑥↑2) + (𝑦↑2))) ∧ 𝑥 ∈ ℤ) → -𝑥 ≤ (𝑥↑2))
70	24	ad3antlr 493	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝐴 = ((𝑥↑2) + (𝑦↑2))) ∧ 𝑥 ∈ ℤ) → (𝑦↑2) ∈ ℕ0)
71		nn0addge1 9562	. . . . . . . . . . . . . 14 ⊢ (((𝑥↑2) ∈ ℝ ∧ (𝑦↑2) ∈ ℕ0) → (𝑥↑2) ≤ ((𝑥↑2) + (𝑦↑2)))
72	62, 70, 71	syl2anc 411	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝐴 = ((𝑥↑2) + (𝑦↑2))) ∧ 𝑥 ∈ ℤ) → (𝑥↑2) ≤ ((𝑥↑2) + (𝑦↑2)))
73		simplr 529	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝐴 = ((𝑥↑2) + (𝑦↑2))) ∧ 𝑥 ∈ ℤ) → 𝐴 = ((𝑥↑2) + (𝑦↑2)))
74	72, 73	breqtrrd 4142	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝐴 = ((𝑥↑2) + (𝑦↑2))) ∧ 𝑥 ∈ ℤ) → (𝑥↑2) ≤ 𝐴)
75	61, 62, 60, 69, 74	letrd 8414	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝐴 = ((𝑥↑2) + (𝑦↑2))) ∧ 𝑥 ∈ ℤ) → -𝑥 ≤ 𝐴)
76	59, 60, 75	lenegcon1d 8819	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝐴 = ((𝑥↑2) + (𝑦↑2))) ∧ 𝑥 ∈ ℤ) → -𝐴 ≤ 𝑥)
77		zzlesq 11098	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℤ → 𝑥 ≤ (𝑥↑2))
78	77	adantl 277	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝐴 = ((𝑥↑2) + (𝑦↑2))) ∧ 𝑥 ∈ ℤ) → 𝑥 ≤ (𝑥↑2))
79	59, 62, 60, 78, 74	letrd 8414	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝐴 = ((𝑥↑2) + (𝑦↑2))) ∧ 𝑥 ∈ ℤ) → 𝑥 ≤ 𝐴)
80	56, 57, 58, 76, 79	elfzd 10372	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝐴 = ((𝑥↑2) + (𝑦↑2))) ∧ 𝑥 ∈ ℤ) → 𝑥 ∈ (-𝐴...𝐴))
81	80	ex 115	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝐴 = ((𝑥↑2) + (𝑦↑2))) → (𝑥 ∈ ℤ → 𝑥 ∈ (-𝐴...𝐴)))
82	81, 6	impbid1 142	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝐴 = ((𝑥↑2) + (𝑦↑2))) → (𝑥 ∈ ℤ ↔ 𝑥 ∈ (-𝐴...𝐴)))
83	82	rexlimdva2 2665	. . . . . 6 ⊢ (𝐴 ∈ ℕ0 → (∃𝑦 ∈ ℤ 𝐴 = ((𝑥↑2) + (𝑦↑2)) → (𝑥 ∈ ℤ ↔ 𝑥 ∈ (-𝐴...𝐴))))
84	83	pm5.32rd 451	. . . . 5 ⊢ (𝐴 ∈ ℕ0 → ((𝑥 ∈ ℤ ∧ ∃𝑦 ∈ ℤ 𝐴 = ((𝑥↑2) + (𝑦↑2))) ↔ (𝑥 ∈ (-𝐴...𝐴) ∧ ∃𝑦 ∈ ℤ 𝐴 = ((𝑥↑2) + (𝑦↑2)))))
85	84	rexbidv2 2547	. . . 4 ⊢ (𝐴 ∈ ℕ0 → (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝐴 = ((𝑥↑2) + (𝑦↑2)) ↔ ∃𝑥 ∈ (-𝐴...𝐴)∃𝑦 ∈ ℤ 𝐴 = ((𝑥↑2) + (𝑦↑2))))
86	85	dcbid 846	. . 3 ⊢ (𝐴 ∈ ℕ0 → (DECID ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝐴 = ((𝑥↑2) + (𝑦↑2)) ↔ DECID ∃𝑥 ∈ (-𝐴...𝐴)∃𝑦 ∈ ℤ 𝐴 = ((𝑥↑2) + (𝑦↑2))))
87	55, 86	mpbird 167	. 2 ⊢ (𝐴 ∈ ℕ0 → DECID ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝐴 = ((𝑥↑2) + (𝑦↑2)))
88		eqeq1 2241	. . . . 5 ⊢ (𝑛 = 𝐴 → (𝑛 = ((𝑥↑2) + (𝑦↑2)) ↔ 𝐴 = ((𝑥↑2) + (𝑦↑2))))
89	88	2rexbidv 2569	. . . 4 ⊢ (𝑛 = 𝐴 → (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑛 = ((𝑥↑2) + (𝑦↑2)) ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝐴 = ((𝑥↑2) + (𝑦↑2))))
90		4sqexercise2.s	. . . 4 ⊢ 𝑆 = {𝑛 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑛 = ((𝑥↑2) + (𝑦↑2))}
91	89, 90	elab2g 2967	. . 3 ⊢ (𝐴 ∈ ℕ0 → (𝐴 ∈ 𝑆 ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝐴 = ((𝑥↑2) + (𝑦↑2))))
92	91	dcbid 846	. 2 ⊢ (𝐴 ∈ ℕ0 → (DECID 𝐴 ∈ 𝑆 ↔ DECID ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝐴 = ((𝑥↑2) + (𝑦↑2))))
93	87, 92	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 8462  2c2 9308  ℕ₀cn0 9516  ℤcz 9597  ...cfz 10364  ↑cexp 10927

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 8463  df-neg 8464  df-reap 8867  df-ap 8874  df-div 8967  df-inn 9258  df-2 9316  df-n0 9517  df-z 9598  df-uz 9875  df-fz 10365  df-fzo 10502  df-seqfrec 10837  df-exp 10928

This theorem is referenced by: (None)