Theorem 4sqexercise2 12593

Description: Exercise which may help in understanding the proof of 4sqlemsdc 12594. (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 9377	. . . 4 ⊢ (𝐴 ∈ ℕ0 → -𝐴 ∈ ℤ)
2		nn0z 9363	. . . 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 10117	. . . . . . . . . 10 ⊢ (𝑥 ∈ (-𝐴...𝐴) → 𝑥 ∈ ℤ)
7	6	ad2antlr 489	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) → 𝑥 ∈ ℤ)
8		zsqcl 10719	. . . . . . . . 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 10117	. . . . . . . . . 10 ⊢ (𝑦 ∈ (-𝐴...𝐴) → 𝑦 ∈ ℤ)
11	10	adantl 277	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) → 𝑦 ∈ ℤ)
12		zsqcl 10719	. . . . . . . . 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 9469	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ 𝑦 ∈ (-𝐴...𝐴)) → ((𝑥↑2) + (𝑦↑2)) ∈ ℤ)
15		zdceq 9418	. . . . . . 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 10333	. . . . 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 529	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ (𝑦 ∈ ℤ ∧ 𝐴 = ((𝑥↑2) + (𝑦↑2)))) → 𝑦 ∈ ℤ)
21	20	zred 9465	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ (𝑦 ∈ ℤ ∧ 𝐴 = ((𝑥↑2) + (𝑦↑2)))) → 𝑦 ∈ ℝ)
22	19	zred 9465	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ (𝑦 ∈ ℤ ∧ 𝐴 = ((𝑥↑2) + (𝑦↑2)))) → 𝐴 ∈ ℝ)
23	21	renegcld 8423	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ (𝑦 ∈ ℤ ∧ 𝐴 = ((𝑥↑2) + (𝑦↑2)))) → -𝑦 ∈ ℝ)
24		zsqcl2 10726	. . . . . . . . . . . . . . 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 9320	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ (𝑦 ∈ ℤ ∧ 𝐴 = ((𝑥↑2) + (𝑦↑2)))) → (𝑦↑2) ∈ ℝ)
27		znegcl 9374	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℤ → -𝑦 ∈ ℤ)
28		zzlesq 10817	. . . . . . . . . . . . . . . 16 ⊢ (-𝑦 ∈ ℤ → -𝑦 ≤ (-𝑦↑2))
29	27, 28	syl 14	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℤ → -𝑦 ≤ (-𝑦↑2))
30		zcn 9348	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℤ → 𝑦 ∈ ℂ)
31		sqneg 10707	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℂ → (-𝑦↑2) = (𝑦↑2))
32	30, 31	syl 14	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℤ → (-𝑦↑2) = (𝑦↑2))
33	29, 32	breqtrd 4060	. . . . . . . . . . . . . 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 10726	. . . . . . . . . . . . . . . 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 9313	. . . . . . . . . . . . . . 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 531	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ (𝑦 ∈ ℤ ∧ 𝐴 = ((𝑥↑2) + (𝑦↑2)))) → 𝐴 = ((𝑥↑2) + (𝑦↑2)))
41	39, 40	breqtrrd 4062	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ (𝑦 ∈ ℤ ∧ 𝐴 = ((𝑥↑2) + (𝑦↑2)))) → (𝑦↑2) ≤ 𝐴)
42	23, 26, 22, 34, 41	letrd 8167	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ (𝑦 ∈ ℤ ∧ 𝐴 = ((𝑥↑2) + (𝑦↑2)))) → -𝑦 ≤ 𝐴)
43	21, 22, 42	lenegcon1d 8571	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ (𝑦 ∈ ℤ ∧ 𝐴 = ((𝑥↑2) + (𝑦↑2)))) → -𝐴 ≤ 𝑦)
44		zzlesq 10817	. . . . . . . . . . . . 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 8167	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) ∧ (𝑦 ∈ ℤ ∧ 𝐴 = ((𝑥↑2) + (𝑦↑2)))) → 𝑦 ≤ 𝐴)
47	18, 19, 20, 43, 46	elfzd 10108	. . . . . . . . . 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 2500	. . . . . 6 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) → (∃𝑦 ∈ ℤ 𝐴 = ((𝑥↑2) + (𝑦↑2)) ↔ ∃𝑦 ∈ (-𝐴...𝐴)𝐴 = ((𝑥↑2) + (𝑦↑2))))
53	52	dcbid 839	. . . . 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 10333	. . 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 9465	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝐴 = ((𝑥↑2) + (𝑦↑2))) ∧ 𝑥 ∈ ℤ) → 𝑥 ∈ ℝ)
60	57	zred 9465	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝐴 = ((𝑥↑2) + (𝑦↑2))) ∧ 𝑥 ∈ ℤ) → 𝐴 ∈ ℝ)
61	59	renegcld 8423	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝐴 = ((𝑥↑2) + (𝑦↑2))) ∧ 𝑥 ∈ ℤ) → -𝑥 ∈ ℝ)
62	59	resqcld 10808	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝐴 = ((𝑥↑2) + (𝑦↑2))) ∧ 𝑥 ∈ ℤ) → (𝑥↑2) ∈ ℝ)
63	58	znegcld 9467	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝐴 = ((𝑥↑2) + (𝑦↑2))) ∧ 𝑥 ∈ ℤ) → -𝑥 ∈ ℤ)
64		zzlesq 10817	. . . . . . . . . . . . . 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 9466	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝐴 = ((𝑥↑2) + (𝑦↑2))) ∧ 𝑥 ∈ ℤ) → 𝑥 ∈ ℂ)
67		sqneg 10707	. . . . . . . . . . . . . 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 4060	. . . . . . . . . . . 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 9312	. . . . . . . . . . . . . 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 528	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝐴 = ((𝑥↑2) + (𝑦↑2))) ∧ 𝑥 ∈ ℤ) → 𝐴 = ((𝑥↑2) + (𝑦↑2)))
74	72, 73	breqtrrd 4062	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝐴 = ((𝑥↑2) + (𝑦↑2))) ∧ 𝑥 ∈ ℤ) → (𝑥↑2) ≤ 𝐴)
75	61, 62, 60, 69, 74	letrd 8167	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝐴 = ((𝑥↑2) + (𝑦↑2))) ∧ 𝑥 ∈ ℤ) → -𝑥 ≤ 𝐴)
76	59, 60, 75	lenegcon1d 8571	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝐴 = ((𝑥↑2) + (𝑦↑2))) ∧ 𝑥 ∈ ℤ) → -𝐴 ≤ 𝑥)
77		zzlesq 10817	. . . . . . . . . . . 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 8167	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℕ0 ∧ 𝑦 ∈ ℤ) ∧ 𝐴 = ((𝑥↑2) + (𝑦↑2))) ∧ 𝑥 ∈ ℤ) → 𝑥 ≤ 𝐴)
80	56, 57, 58, 76, 79	elfzd 10108	. . . . . . . . 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 2617	. . . . . 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 2500	. . . 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 839	. . 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 2203	. . . . 5 ⊢ (𝑛 = 𝐴 → (𝑛 = ((𝑥↑2) + (𝑦↑2)) ↔ 𝐴 = ((𝑥↑2) + (𝑦↑2))))
89	88	2rexbidv 2522	. . . 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 2911	. . 3 ⊢ (𝐴 ∈ ℕ0 → (𝐴 ∈ 𝑆 ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝐴 = ((𝑥↑2) + (𝑦↑2))))
92	91	dcbid 839	. 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 835   = wceq 1364   ∈ wcel 2167  {cab 2182  ∃wrex 2476   class class class wbr 4034  (class class class)co 5925  ℂcc 7894  ℝcr 7895   + caddc 7899   ≤ cle 8079  -cneg 8215  2c2 9058  ℕ₀cn0 9266  ℤcz 9343  ...cfz 10100  ↑cexp 10647

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-frec 6458  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626  df-div 8717  df-inn 9008  df-2 9066  df-n0 9267  df-z 9344  df-uz 9619  df-fz 10101  df-fzo 10235  df-seqfrec 10557  df-exp 10648

This theorem is referenced by: (None)