Theorem 4sqexercise1 12441

Description: Exercise which may help in understanding the proof of 4sqlemsdc 12443. (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 9322	. . . 4 ⊢ (𝐴 ∈ ℕ0 → -𝐴 ∈ ℤ)
2		nn0z 9308	. . . 4 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℤ)
3		elfzelz 10061	. . . . . . 7 ⊢ (𝑥 ∈ (-𝐴...𝐴) → 𝑥 ∈ ℤ)
4	3	adantl 277	. . . . . 6 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) → 𝑥 ∈ ℤ)
5		zsqcl 10631	. . . . . 6 ⊢ (𝑥 ∈ ℤ → (𝑥↑2) ∈ ℤ)
6	4, 5	syl 14	. . . . 5 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) → (𝑥↑2) ∈ ℤ)
7		zdceq 9363	. . . . 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 10276	. . 3 ⊢ (𝐴 ∈ ℕ0 → DECID ∃𝑥 ∈ (-𝐴...𝐴)𝐴 = (𝑥↑2))
10		simpr 110	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℤ ∧ 𝐴 = (𝑥↑2)) → 𝐴 = (𝑥↑2))
11		zsqcl2 10638	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℤ → (𝑥↑2) ∈ ℕ0)
12	11	adantr 276	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℤ ∧ 𝐴 = (𝑥↑2)) → (𝑥↑2) ∈ ℕ0)
13	10, 12	eqeltrd 2266	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℤ ∧ 𝐴 = (𝑥↑2)) → 𝐴 ∈ ℕ0)
14	13	nn0zd 9408	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℤ ∧ 𝐴 = (𝑥↑2)) → 𝐴 ∈ ℤ)
15	14	znegcld 9412	. . . . . . . 8 ⊢ ((𝑥 ∈ ℤ ∧ 𝐴 = (𝑥↑2)) → -𝐴 ∈ ℤ)
16		simpl 109	. . . . . . . 8 ⊢ ((𝑥 ∈ ℤ ∧ 𝐴 = (𝑥↑2)) → 𝑥 ∈ ℤ)
17		zre 9292	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℝ)
18	17	adantr 276	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℤ ∧ 𝐴 = (𝑥↑2)) → 𝑥 ∈ ℝ)
19	13	nn0red 9265	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℤ ∧ 𝐴 = (𝑥↑2)) → 𝐴 ∈ ℝ)
20		znegcl 9319	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℤ → -𝑥 ∈ ℤ)
21		zzlesq 10729	. . . . . . . . . . . . 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 9293	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℂ)
25		sqneg 10619	. . . . . . . . . . . . 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 4047	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℤ ∧ 𝐴 = (𝑥↑2)) → -𝑥 ≤ (𝑥↑2))
29	28, 10	breqtrrd 4049	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℤ ∧ 𝐴 = (𝑥↑2)) → -𝑥 ≤ 𝐴)
30	18, 19, 29	lenegcon1d 8519	. . . . . . . 8 ⊢ ((𝑥 ∈ ℤ ∧ 𝐴 = (𝑥↑2)) → -𝐴 ≤ 𝑥)
31		zzlesq 10729	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℤ → 𝑥 ≤ (𝑥↑2))
32	31	adantr 276	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℤ ∧ 𝐴 = (𝑥↑2)) → 𝑥 ≤ (𝑥↑2))
33	32, 10	breqtrrd 4049	. . . . . . . 8 ⊢ ((𝑥 ∈ ℤ ∧ 𝐴 = (𝑥↑2)) → 𝑥 ≤ 𝐴)
34	15, 14, 16, 30, 33	elfzd 10052	. . . . . . 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 2501	. . . 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 2196	. . . . 5 ⊢ (𝑛 = 𝐴 → (𝑛 = (𝑥↑2) ↔ 𝐴 = (𝑥↑2)))
42	41	rexbidv 2491	. . . 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 2899	. . 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 2160  {cab 2175  ∃wrex 2469   class class class wbr 4021  (class class class)co 5900  ℂcc 7844  ℝcr 7845   ≤ cle 8028  -cneg 8164  2c2 9005  ℕ₀cn0 9211  ℤcz 9288  ...cfz 10044  ↑cexp 10559

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 2162  ax-14 2163  ax-ext 2171  ax-coll 4136  ax-sep 4139  ax-nul 4147  ax-pow 4195  ax-pr 4230  ax-un 4454  ax-setind 4557  ax-iinf 4608  ax-cnex 7937  ax-resscn 7938  ax-1cn 7939  ax-1re 7940  ax-icn 7941  ax-addcl 7942  ax-addrcl 7943  ax-mulcl 7944  ax-mulrcl 7945  ax-addcom 7946  ax-mulcom 7947  ax-addass 7948  ax-mulass 7949  ax-distr 7950  ax-i2m1 7951  ax-0lt1 7952  ax-1rid 7953  ax-0id 7954  ax-rnegex 7955  ax-precex 7956  ax-cnre 7957  ax-pre-ltirr 7958  ax-pre-ltwlin 7959  ax-pre-lttrn 7960  ax-pre-apti 7961  ax-pre-ltadd 7962  ax-pre-mulgt0 7963  ax-pre-mulext 7964

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 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-int 3863  df-iun 3906  df-br 4022  df-opab 4083  df-mpt 4084  df-tr 4120  df-id 4314  df-po 4317  df-iso 4318  df-iord 4387  df-on 4389  df-ilim 4390  df-suc 4392  df-iom 4611  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-rn 4658  df-res 4659  df-ima 4660  df-iota 5199  df-fun 5240  df-fn 5241  df-f 5242  df-f1 5243  df-fo 5244  df-f1o 5245  df-fv 5246  df-riota 5855  df-ov 5903  df-oprab 5904  df-mpo 5905  df-1st 6169  df-2nd 6170  df-recs 6334  df-frec 6420  df-pnf 8029  df-mnf 8030  df-xr 8031  df-ltxr 8032  df-le 8033  df-sub 8165  df-neg 8166  df-reap 8567  df-ap 8574  df-div 8665  df-inn 8955  df-2 9013  df-n0 9212  df-z 9289  df-uz 9564  df-fz 10045  df-fzo 10179  df-seqfrec 10485  df-exp 10560

This theorem is referenced by: (None)