Theorem 4sqexercise1 13179

Description: Exercise which may help in understanding the proof of 4sqlemsdc 13181. (Contributed by Jim Kingdon, 25-May-2025.)

Hypothesis

Ref	Expression
4sqexercise1.s	⊢ 𝑆 = {𝑛 ∣ ∃𝑥 ∈ ℤ 𝑛 = (𝑥↑2)}

Assertion

Ref	Expression
4sqexercise1	⊢ (𝐴 ∈ ℕ0 → DECID 𝐴 ∈ 𝑆)

Distinct variable group:   𝐴,𝑛,𝑥

Allowed substitution hints:   𝑆(𝑥, 𝑛)

Proof of Theorem 4sqexercise1

Step	Hyp	Ref	Expression
1		nn0negz 9680	. . . 4 ⊢ (𝐴 ∈ ℕ0 → -𝐴 ∈ ℤ)
2		nn0z 9666	. . . 4 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℤ)
3		elfzelz 10430	. . . . . . 7 ⊢ (𝑥 ∈ (-𝐴...𝐴) → 𝑥 ∈ ℤ)
4	3	adantl 277	. . . . . 6 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) → 𝑥 ∈ ℤ)
5		zsqcl 11049	. . . . . 6 ⊢ (𝑥 ∈ ℤ → (𝑥↑2) ∈ ℤ)
6	4, 5	syl 14	. . . . 5 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) → (𝑥↑2) ∈ ℤ)
7		zdceq 9722	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ (𝑥↑2) ∈ ℤ) → DECID 𝐴 = (𝑥↑2))
8	2, 6, 7	syl2an2r 603	. . . 4 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) → DECID 𝐴 = (𝑥↑2))
9	1, 2, 8	exfzdc 10661	. . 3 ⊢ (𝐴 ∈ ℕ0 → DECID ∃𝑥 ∈ (-𝐴...𝐴)𝐴 = (𝑥↑2))
10		simpr 110	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℤ ∧ 𝐴 = (𝑥↑2)) → 𝐴 = (𝑥↑2))
11		zsqcl2 11056	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℤ → (𝑥↑2) ∈ ℕ0)
12	11	adantr 276	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℤ ∧ 𝐴 = (𝑥↑2)) → (𝑥↑2) ∈ ℕ0)
13	10, 12	eqeltrd 2315	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℤ ∧ 𝐴 = (𝑥↑2)) → 𝐴 ∈ ℕ0)
14	13	nn0zd 9768	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℤ ∧ 𝐴 = (𝑥↑2)) → 𝐴 ∈ ℤ)
15	14	znegcld 9772	. . . . . . . 8 ⊢ ((𝑥 ∈ ℤ ∧ 𝐴 = (𝑥↑2)) → -𝐴 ∈ ℤ)
16		simpl 109	. . . . . . . 8 ⊢ ((𝑥 ∈ ℤ ∧ 𝐴 = (𝑥↑2)) → 𝑥 ∈ ℤ)
17		zre 9650	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℝ)
18	17	adantr 276	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℤ ∧ 𝐴 = (𝑥↑2)) → 𝑥 ∈ ℝ)
19	13	nn0red 9623	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℤ ∧ 𝐴 = (𝑥↑2)) → 𝐴 ∈ ℝ)
20		znegcl 9677	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℤ → -𝑥 ∈ ℤ)
21		zzlesq 11148	. . . . . . . . . . . . 13 ⊢ (-𝑥 ∈ ℤ → -𝑥 ≤ (-𝑥↑2))
22	20, 21	syl 14	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℤ → -𝑥 ≤ (-𝑥↑2))
23	22	adantr 276	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℤ ∧ 𝐴 = (𝑥↑2)) → -𝑥 ≤ (-𝑥↑2))
24		zcn 9651	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℂ)
25		sqneg 11037	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℂ → (-𝑥↑2) = (𝑥↑2))
26	24, 25	syl 14	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℤ → (-𝑥↑2) = (𝑥↑2))
27	26	adantr 276	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℤ ∧ 𝐴 = (𝑥↑2)) → (-𝑥↑2) = (𝑥↑2))
28	23, 27	breqtrd 4156	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℤ ∧ 𝐴 = (𝑥↑2)) → -𝑥 ≤ (𝑥↑2))
29	28, 10	breqtrrd 4158	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℤ ∧ 𝐴 = (𝑥↑2)) → -𝑥 ≤ 𝐴)
30	18, 19, 29	lenegcon1d 8855	. . . . . . . 8 ⊢ ((𝑥 ∈ ℤ ∧ 𝐴 = (𝑥↑2)) → -𝐴 ≤ 𝑥)
31		zzlesq 11148	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℤ → 𝑥 ≤ (𝑥↑2))
32	31	adantr 276	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℤ ∧ 𝐴 = (𝑥↑2)) → 𝑥 ≤ (𝑥↑2))
33	32, 10	breqtrrd 4158	. . . . . . . 8 ⊢ ((𝑥 ∈ ℤ ∧ 𝐴 = (𝑥↑2)) → 𝑥 ≤ 𝐴)
34	15, 14, 16, 30, 33	elfzd 10421	. . . . . . 7 ⊢ ((𝑥 ∈ ℤ ∧ 𝐴 = (𝑥↑2)) → 𝑥 ∈ (-𝐴...𝐴))
35	34, 10	jca 306	. . . . . 6 ⊢ ((𝑥 ∈ ℤ ∧ 𝐴 = (𝑥↑2)) → (𝑥 ∈ (-𝐴...𝐴) ∧ 𝐴 = (𝑥↑2)))
36	3	anim1i 340	. . . . . 6 ⊢ ((𝑥 ∈ (-𝐴...𝐴) ∧ 𝐴 = (𝑥↑2)) → (𝑥 ∈ ℤ ∧ 𝐴 = (𝑥↑2)))
37	35, 36	impbii 126	. . . . 5 ⊢ ((𝑥 ∈ ℤ ∧ 𝐴 = (𝑥↑2)) ↔ (𝑥 ∈ (-𝐴...𝐴) ∧ 𝐴 = (𝑥↑2)))
38	37	rexbii2 2561	. . . 4 ⊢ (∃𝑥 ∈ ℤ 𝐴 = (𝑥↑2) ↔ ∃𝑥 ∈ (-𝐴...𝐴)𝐴 = (𝑥↑2))
39	38	dcbii 852	. . 3 ⊢ (DECID ∃𝑥 ∈ ℤ 𝐴 = (𝑥↑2) ↔ DECID ∃𝑥 ∈ (-𝐴...𝐴)𝐴 = (𝑥↑2))
40	9, 39	sylibr 134	. 2 ⊢ (𝐴 ∈ ℕ0 → DECID ∃𝑥 ∈ ℤ 𝐴 = (𝑥↑2))
41		eqeq1 2245	. . . . 5 ⊢ (𝑛 = 𝐴 → (𝑛 = (𝑥↑2) ↔ 𝐴 = (𝑥↑2)))
42	41	rexbidv 2551	. . . 4 ⊢ (𝑛 = 𝐴 → (∃𝑥 ∈ ℤ 𝑛 = (𝑥↑2) ↔ ∃𝑥 ∈ ℤ 𝐴 = (𝑥↑2)))
43		4sqexercise1.s	. . . 4 ⊢ 𝑆 = {𝑛 ∣ ∃𝑥 ∈ ℤ 𝑛 = (𝑥↑2)}
44	42, 43	elab2g 2973	. . 3 ⊢ (𝐴 ∈ ℕ0 → (𝐴 ∈ 𝑆 ↔ ∃𝑥 ∈ ℤ 𝐴 = (𝑥↑2)))
45	44	dcbid 850	. 2 ⊢ (𝐴 ∈ ℕ0 → (DECID 𝐴 ∈ 𝑆 ↔ DECID ∃𝑥 ∈ ℤ 𝐴 = (𝑥↑2)))
46	40, 45	mpbird 167	1 ⊢ (𝐴 ∈ ℕ0 → DECID 𝐴 ∈ 𝑆)

This proof depends on syntax axioms:   → wi 4   ∧ wa 104  DECID wdc 846   = wceq 1402   ∈ wcel 2209  {cab 2224  ∃wrex 2529   class class class wbr 4130  (class class class)co 6085  ℂcc 8177  ℝcr 8178   ≤ cle 8361  -cneg 8498  2c2 9356  ℕ₀cn0 9565  ℤcz 9646  ...cfz 10413  ↑cexp 10977

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8904  df-ap 8911  df-div 9004  df-inn 9306  df-2 9364  df-n0 9566  df-z 9647  df-uz 9924  df-fz 10414  df-fzo 10552  df-seqfrec 10887  df-exp 10978

This theorem is used by: (None)