Theorem 4sqexercise1 13089

Description: Exercise which may help in understanding the proof of 4sqlemsdc 13091. (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 9607	. . . 4 ⊢ (𝐴 ∈ ℕ0 → -𝐴 ∈ ℤ)
2		nn0z 9593	. . . 4 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℤ)
3		elfzelz 10355	. . . . . . 7 ⊢ (𝑥 ∈ (-𝐴...𝐴) → 𝑥 ∈ ℤ)
4	3	adantl 277	. . . . . 6 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) → 𝑥 ∈ ℤ)
5		zsqcl 10968	. . . . . 6 ⊢ (𝑥 ∈ ℤ → (𝑥↑2) ∈ ℤ)
6	4, 5	syl 14	. . . . 5 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) → (𝑥↑2) ∈ ℤ)
7		zdceq 9649	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ (𝑥↑2) ∈ ℤ) → DECID 𝐴 = (𝑥↑2))
8	2, 6, 7	syl2an2r 599	. . . 4 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑥 ∈ (-𝐴...𝐴)) → DECID 𝐴 = (𝑥↑2))
9	1, 2, 8	exfzdc 10582	. . 3 ⊢ (𝐴 ∈ ℕ0 → DECID ∃𝑥 ∈ (-𝐴...𝐴)𝐴 = (𝑥↑2))
10		simpr 110	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℤ ∧ 𝐴 = (𝑥↑2)) → 𝐴 = (𝑥↑2))
11		zsqcl2 10975	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℤ → (𝑥↑2) ∈ ℕ0)
12	11	adantr 276	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℤ ∧ 𝐴 = (𝑥↑2)) → (𝑥↑2) ∈ ℕ0)
13	10, 12	eqeltrd 2309	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℤ ∧ 𝐴 = (𝑥↑2)) → 𝐴 ∈ ℕ0)
14	13	nn0zd 9694	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℤ ∧ 𝐴 = (𝑥↑2)) → 𝐴 ∈ ℤ)
15	14	znegcld 9698	. . . . . . . 8 ⊢ ((𝑥 ∈ ℤ ∧ 𝐴 = (𝑥↑2)) → -𝐴 ∈ ℤ)
16		simpl 109	. . . . . . . 8 ⊢ ((𝑥 ∈ ℤ ∧ 𝐴 = (𝑥↑2)) → 𝑥 ∈ ℤ)
17		zre 9577	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℝ)
18	17	adantr 276	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℤ ∧ 𝐴 = (𝑥↑2)) → 𝑥 ∈ ℝ)
19	13	nn0red 9550	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℤ ∧ 𝐴 = (𝑥↑2)) → 𝐴 ∈ ℝ)
20		znegcl 9604	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℤ → -𝑥 ∈ ℤ)
21		zzlesq 11066	. . . . . . . . . . . . 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 9578	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℂ)
25		sqneg 10956	. . . . . . . . . . . . 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 4134	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℤ ∧ 𝐴 = (𝑥↑2)) → -𝑥 ≤ (𝑥↑2))
29	28, 10	breqtrrd 4136	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℤ ∧ 𝐴 = (𝑥↑2)) → -𝑥 ≤ 𝐴)
30	18, 19, 29	lenegcon1d 8797	. . . . . . . 8 ⊢ ((𝑥 ∈ ℤ ∧ 𝐴 = (𝑥↑2)) → -𝐴 ≤ 𝑥)
31		zzlesq 11066	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℤ → 𝑥 ≤ (𝑥↑2))
32	31	adantr 276	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℤ ∧ 𝐴 = (𝑥↑2)) → 𝑥 ≤ (𝑥↑2))
33	32, 10	breqtrrd 4136	. . . . . . . 8 ⊢ ((𝑥 ∈ ℤ ∧ 𝐴 = (𝑥↑2)) → 𝑥 ≤ 𝐴)
34	15, 14, 16, 30, 33	elfzd 10346	. . . . . . 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 2553	. . . 4 ⊢ (∃𝑥 ∈ ℤ 𝐴 = (𝑥↑2) ↔ ∃𝑥 ∈ (-𝐴...𝐴)𝐴 = (𝑥↑2))
39	38	dcbii 848	. . 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 2239	. . . . 5 ⊢ (𝑛 = 𝐴 → (𝑛 = (𝑥↑2) ↔ 𝐴 = (𝑥↑2)))
42	41	rexbidv 2543	. . . 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 2963	. . 3 ⊢ (𝐴 ∈ ℕ0 → (𝐴 ∈ 𝑆 ↔ ∃𝑥 ∈ ℤ 𝐴 = (𝑥↑2)))
45	44	dcbid 846	. 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 842   = wceq 1398   ∈ wcel 2203  {cab 2218  ∃wrex 2521   class class class wbr 4108  (class class class)co 6049  ℂcc 8121  ℝcr 8122   ≤ cle 8305  -cneg 8441  2c2 9284  ℕ₀cn0 9492  ℤcz 9573  ...cfz 10338  ↑cexp 10896

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709  ax-cnex 8214  ax-resscn 8215  ax-1cn 8216  ax-1re 8217  ax-icn 8218  ax-addcl 8219  ax-addrcl 8220  ax-mulcl 8221  ax-mulrcl 8222  ax-addcom 8223  ax-mulcom 8224  ax-addass 8225  ax-mulass 8226  ax-distr 8227  ax-i2m1 8228  ax-0lt1 8229  ax-1rid 8230  ax-0id 8231  ax-rnegex 8232  ax-precex 8233  ax-cnre 8234  ax-pre-ltirr 8235  ax-pre-ltwlin 8236  ax-pre-lttrn 8237  ax-pre-apti 8238  ax-pre-ltadd 8239  ax-pre-mulgt0 8240  ax-pre-mulext 8241

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-po 4416  df-iso 4417  df-iord 4486  df-on 4488  df-ilim 4489  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-recs 6535  df-frec 6621  df-pnf 8306  df-mnf 8307  df-xr 8308  df-ltxr 8309  df-le 8310  df-sub 8442  df-neg 8443  df-reap 8845  df-ap 8852  df-div 8943  df-inn 9234  df-2 9292  df-n0 9493  df-z 9574  df-uz 9850  df-fz 10339  df-fzo 10473  df-seqfrec 10806  df-exp 10897

This theorem is referenced by: (None)