Theorem 4sqexercise1 12916

Description: Exercise which may help in understanding the proof of 4sqlemsdc 12918. (Contributed by Jim Kingdon, 25-May-2025.)

Hypothesis

Ref	Expression
4sqexercise1.s	|- S = { n | E. x e. ZZ n = ( x ^ 2 ) }

Assertion

Ref	Expression
4sqexercise1	|- ( A e. NN0 -> DECID A e. S )

Distinct variable group:   A, n, x

Allowed substitution hints:   S( x, n)

Proof of Theorem 4sqexercise1

Step	Hyp	Ref	Expression
1		nn0negz 9476	. . . 4 |- ( A e. NN0 -> -u A e. ZZ )
2		nn0z 9462	. . . 4 |- ( A e. NN0 -> A e. ZZ )
3		elfzelz 10217	. . . . . . 7 |- ( x e. ( -u A ... A ) -> x e. ZZ )
4	3	adantl 277	. . . . . 6 |- ( ( A e. NN0 /\ x e. ( -u A ... A ) ) -> x e. ZZ )
5		zsqcl 10827	. . . . . 6 |- ( x e. ZZ -> ( x ^ 2 ) e. ZZ )
6	4, 5	syl 14	. . . . 5 |- ( ( A e. NN0 /\ x e. ( -u A ... A ) ) -> ( x ^ 2 ) e. ZZ )
7		zdceq 9518	. . . . 5 |- ( ( A e. ZZ /\ ( x ^ 2 ) e. ZZ ) -> DECID A = ( x ^ 2 ) )
8	2, 6, 7	syl2an2r 597	. . . 4 |- ( ( A e. NN0 /\ x e. ( -u A ... A ) ) -> DECID A = ( x ^ 2 ) )
9	1, 2, 8	exfzdc 10441	. . 3 |- ( A e. NN0 -> DECID E. x e. ( -u A ... A ) A = ( x ^ 2 ) )
10		simpr 110	. . . . . . . . . . 11 |- ( ( x e. ZZ /\ A = ( x ^ 2 ) ) -> A = ( x ^ 2 ) )
11		zsqcl2 10834	. . . . . . . . . . . 12 |- ( x e. ZZ -> ( x ^ 2 ) e. NN0 )
12	11	adantr 276	. . . . . . . . . . 11 |- ( ( x e. ZZ /\ A = ( x ^ 2 ) ) -> ( x ^ 2 ) e. NN0 )
13	10, 12	eqeltrd 2306	. . . . . . . . . 10 |- ( ( x e. ZZ /\ A = ( x ^ 2 ) ) -> A e. NN0 )
14	13	nn0zd 9563	. . . . . . . . 9 |- ( ( x e. ZZ /\ A = ( x ^ 2 ) ) -> A e. ZZ )
15	14	znegcld 9567	. . . . . . . 8 |- ( ( x e. ZZ /\ A = ( x ^ 2 ) ) -> -u A e. ZZ )
16		simpl 109	. . . . . . . 8 |- ( ( x e. ZZ /\ A = ( x ^ 2 ) ) -> x e. ZZ )
17		zre 9446	. . . . . . . . . 10 |- ( x e. ZZ -> x e. RR )
18	17	adantr 276	. . . . . . . . 9 |- ( ( x e. ZZ /\ A = ( x ^ 2 ) ) -> x e. RR )
19	13	nn0red 9419	. . . . . . . . 9 |- ( ( x e. ZZ /\ A = ( x ^ 2 ) ) -> A e. RR )
20		znegcl 9473	. . . . . . . . . . . . 13 |- ( x e. ZZ -> -u x e. ZZ )
21		zzlesq 10925	. . . . . . . . . . . . 13 |- ( -u x e. ZZ -> -u x <_ ( -u x ^ 2 ) )
22	20, 21	syl 14	. . . . . . . . . . . 12 |- ( x e. ZZ -> -u x <_ ( -u x ^ 2 ) )
23	22	adantr 276	. . . . . . . . . . 11 |- ( ( x e. ZZ /\ A = ( x ^ 2 ) ) -> -u x <_ ( -u x ^ 2 ) )
24		zcn 9447	. . . . . . . . . . . . 13 |- ( x e. ZZ -> x e. CC )
25		sqneg 10815	. . . . . . . . . . . . 13 |- ( x e. CC -> ( -u x ^ 2 ) = ( x ^ 2 ) )
26	24, 25	syl 14	. . . . . . . . . . . 12 |- ( x e. ZZ -> ( -u x ^ 2 ) = ( x ^ 2 ) )
27	26	adantr 276	. . . . . . . . . . 11 |- ( ( x e. ZZ /\ A = ( x ^ 2 ) ) -> ( -u x ^ 2 ) = ( x ^ 2 ) )
28	23, 27	breqtrd 4108	. . . . . . . . . 10 |- ( ( x e. ZZ /\ A = ( x ^ 2 ) ) -> -u x <_ ( x ^ 2 ) )
29	28, 10	breqtrrd 4110	. . . . . . . . 9 |- ( ( x e. ZZ /\ A = ( x ^ 2 ) ) -> -u x <_ A )
30	18, 19, 29	lenegcon1d 8670	. . . . . . . 8 |- ( ( x e. ZZ /\ A = ( x ^ 2 ) ) -> -u A <_ x )
31		zzlesq 10925	. . . . . . . . . 10 |- ( x e. ZZ -> x <_ ( x ^ 2 ) )
32	31	adantr 276	. . . . . . . . 9 |- ( ( x e. ZZ /\ A = ( x ^ 2 ) ) -> x <_ ( x ^ 2 ) )
33	32, 10	breqtrrd 4110	. . . . . . . 8 |- ( ( x e. ZZ /\ A = ( x ^ 2 ) ) -> x <_ A )
34	15, 14, 16, 30, 33	elfzd 10208	. . . . . . 7 |- ( ( x e. ZZ /\ A = ( x ^ 2 ) ) -> x e. ( -u A ... A ) )
35	34, 10	jca 306	. . . . . 6 |- ( ( x e. ZZ /\ A = ( x ^ 2 ) ) -> ( x e. ( -u A ... A ) /\ A = ( x ^ 2 ) ) )
36	3	anim1i 340	. . . . . 6 |- ( ( x e. ( -u A ... A ) /\ A = ( x ^ 2 ) ) -> ( x e. ZZ /\ A = ( x ^ 2 ) ) )
37	35, 36	impbii 126	. . . . 5 |- ( ( x e. ZZ /\ A = ( x ^ 2 ) ) <-> ( x e. ( -u A ... A ) /\ A = ( x ^ 2 ) ) )
38	37	rexbii2 2541	. . . 4 |- ( E. x e. ZZ A = ( x ^ 2 ) <-> E. x e. ( -u A ... A ) A = ( x ^ 2 ) )
39	38	dcbii 845	. . 3 |- (DECID E. x e. ZZ A = ( x ^ 2 ) <-> DECID E. x e. ( -u A ... A ) A = ( x ^ 2 ) )
40	9, 39	sylibr 134	. 2 |- ( A e. NN0 -> DECID E. x e. ZZ A = ( x ^ 2 ) )
41		eqeq1 2236	. . . . 5 |- ( n = A -> ( n = ( x ^ 2 ) <-> A = ( x ^ 2 ) ) )
42	41	rexbidv 2531	. . . 4 |- ( n = A -> ( E. x e. ZZ n = ( x ^ 2 ) <-> E. x e. ZZ A = ( x ^ 2 ) ) )
43		4sqexercise1.s	. . . 4 |- S = { n | E. x e. ZZ n = ( x ^ 2 ) }
44	42, 43	elab2g 2950	. . 3 |- ( A e. NN0 -> ( A e. S <-> E. x e. ZZ A = ( x ^ 2 ) ) )
45	44	dcbid 843	. 2 |- ( A e. NN0 -> (DECID A e. S <-> DECID E. x e. ZZ A = ( x ^ 2 ) ) )
46	40, 45	mpbird 167	1 |- ( A e. NN0 -> DECID A e. S )

Syntax hints:   -> wi 4   /\ wa 104  DECID wdc 839   = wceq 1395   e. wcel 2200   {cab 2215   E.wrex 2509   class class class wbr 4082  (class class class)co 6000   CCcc 7993   RRcr 7994   <_ cle 8178   -ucneg 8314   2c2 9157   NN0cn0 9365   ZZcz 9442   ...cfz 10200   ^cexp 10755

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-mulrcl 8094  ax-addcom 8095  ax-mulcom 8096  ax-addass 8097  ax-mulass 8098  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-1rid 8102  ax-0id 8103  ax-rnegex 8104  ax-precex 8105  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-apti 8110  ax-pre-ltadd 8111  ax-pre-mulgt0 8112  ax-pre-mulext 8113

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-po 4386  df-iso 4387  df-iord 4456  df-on 4458  df-ilim 4459  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-recs 6449  df-frec 6535  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-reap 8718  df-ap 8725  df-div 8816  df-inn 9107  df-2 9165  df-n0 9366  df-z 9443  df-uz 9719  df-fz 10201  df-fzo 10335  df-seqfrec 10665  df-exp 10756

This theorem is referenced by: (None)