Theorem 4sqexercise1 12536

Description: Exercise which may help in understanding the proof of 4sqlemsdc 12538. (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 9351	. . . 4 ⊢ (𝐴 ∈ ℕ0 → -𝐴 ∈ ℤ)
2		nn0z 9337	. . . 4 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℤ)
3		elfzelz 10091	. . . . . . 7 ⊢ (𝑥 ∈ (-𝐴...𝐴) → 𝑥 ∈ ℤ)
4	3	adantl 277	. . . . . 6 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) → 𝑥 ∈ ℤ)
5		zsqcl 10681	. . . . . 6 ⊢ (𝑥 ∈ ℤ → (𝑥↑2) ∈ ℤ)
6	4, 5	syl 14	. . . . 5 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) → (𝑥↑2) ∈ ℤ)
7		zdceq 9392	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ (𝑥↑2) ∈ ℤ) → DECID 𝐴 = (𝑥↑2))
8	2, 6, 7	syl2an2r 595	. . . 4 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) → DECID 𝐴 = (𝑥↑2))
9	1, 2, 8	exfzdc 10307	. . 3 ⊢ (𝐴 ∈ ℕ0 → DECID ∃𝑥 ∈ (-𝐴...𝐴)𝐴 = (𝑥↑2))
10		simpr 110	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℤ ∧ 𝐴 = (𝑥↑2)) → 𝐴 = (𝑥↑2))
11		zsqcl2 10688	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℤ → (𝑥↑2) ∈ ℕ0)
12	11	adantr 276	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℤ ∧ 𝐴 = (𝑥↑2)) → (𝑥↑2) ∈ ℕ0)
13	10, 12	eqeltrd 2270	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℤ ∧ 𝐴 = (𝑥↑2)) → 𝐴 ∈ ℕ0)
14	13	nn0zd 9437	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℤ ∧ 𝐴 = (𝑥↑2)) → 𝐴 ∈ ℤ)
15	14	znegcld 9441	. . . . . . . 8 ⊢ ((𝑥 ∈ ℤ ∧ 𝐴 = (𝑥↑2)) → -𝐴 ∈ ℤ)
16		simpl 109	. . . . . . . 8 ⊢ ((𝑥 ∈ ℤ ∧ 𝐴 = (𝑥↑2)) → 𝑥 ∈ ℤ)
17		zre 9321	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℝ)
18	17	adantr 276	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℤ ∧ 𝐴 = (𝑥↑2)) → 𝑥 ∈ ℝ)
19	13	nn0red 9294	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℤ ∧ 𝐴 = (𝑥↑2)) → 𝐴 ∈ ℝ)
20		znegcl 9348	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℤ → -𝑥 ∈ ℤ)
21		zzlesq 10779	. . . . . . . . . . . . 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 9322	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℂ)
25		sqneg 10669	. . . . . . . . . . . . 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 4055	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℤ ∧ 𝐴 = (𝑥↑2)) → -𝑥 ≤ (𝑥↑2))
29	28, 10	breqtrrd 4057	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℤ ∧ 𝐴 = (𝑥↑2)) → -𝑥 ≤ 𝐴)
30	18, 19, 29	lenegcon1d 8546	. . . . . . . 8 ⊢ ((𝑥 ∈ ℤ ∧ 𝐴 = (𝑥↑2)) → -𝐴 ≤ 𝑥)
31		zzlesq 10779	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℤ → 𝑥 ≤ (𝑥↑2))
32	31	adantr 276	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℤ ∧ 𝐴 = (𝑥↑2)) → 𝑥 ≤ (𝑥↑2))
33	32, 10	breqtrrd 4057	. . . . . . . 8 ⊢ ((𝑥 ∈ ℤ ∧ 𝐴 = (𝑥↑2)) → 𝑥 ≤ 𝐴)
34	15, 14, 16, 30, 33	elfzd 10082	. . . . . . 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 2505	. . . 4 ⊢ (∃𝑥 ∈ ℤ 𝐴 = (𝑥↑2) ↔ ∃𝑥 ∈ (-𝐴...𝐴)𝐴 = (𝑥↑2))
39	38	dcbii 841	. . 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 2200	. . . . 5 ⊢ (𝑛 = 𝐴 → (𝑛 = (𝑥↑2) ↔ 𝐴 = (𝑥↑2)))
42	41	rexbidv 2495	. . . 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 2907	. . 3 ⊢ (𝐴 ∈ ℕ0 → (𝐴 ∈ 𝑆 ↔ ∃𝑥 ∈ ℤ 𝐴 = (𝑥↑2)))
45	44	dcbid 839	. 2 ⊢ (𝐴 ∈ ℕ0 → (DECID 𝐴 ∈ 𝑆 ↔ DECID ∃𝑥 ∈ ℤ 𝐴 = (𝑥↑2)))
46	40, 45	mpbird 167	1 ⊢ (𝐴 ∈ ℕ0 → DECID 𝐴 ∈ 𝑆)

Syntax hints:   → wi 4   ∧ wa 104  DECID wdc 835   = wceq 1364   ∈ wcel 2164  {cab 2179  ∃wrex 2473   class class class wbr 4029  (class class class)co 5918  ℂcc 7870  ℝcr 7871   ≤ cle 8055  -cneg 8191  2c2 9033  ℕ₀cn0 9240  ℤcz 9317  ...cfz 10074  ↑cexp 10609

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990

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 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-frec 6444  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-2 9041  df-n0 9241  df-z 9318  df-uz 9593  df-fz 10075  df-fzo 10209  df-seqfrec 10519  df-exp 10610

This theorem is referenced by: (None)