Theorem 4sqexercise1 12692

Description: Exercise which may help in understanding the proof of 4sqlemsdc 12694. (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 9405	. . . 4 |- ( A e. NN0 -> -u A e. ZZ )
2		nn0z 9391	. . . 4 |- ( A e. NN0 -> A e. ZZ )
3		elfzelz 10146	. . . . . . 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 10753	. . . . . 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 9447	. . . . 5 |- ( ( A e. ZZ /\ ( x ^ 2 ) e. ZZ ) -> DECID A = ( x ^ 2 ) )
8	2, 6, 7	syl2an2r 595	. . . 4 |- ( ( A e. NN0 /\ x e. ( -u A ... A ) ) -> DECID A = ( x ^ 2 ) )
9	1, 2, 8	exfzdc 10367	. . 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 10760	. . . . . . . . . . . 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 2281	. . . . . . . . . 10 |- ( ( x e. ZZ /\ A = ( x ^ 2 ) ) -> A e. NN0 )
14	13	nn0zd 9492	. . . . . . . . 9 |- ( ( x e. ZZ /\ A = ( x ^ 2 ) ) -> A e. ZZ )
15	14	znegcld 9496	. . . . . . . 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 9375	. . . . . . . . . 10 |- ( x e. ZZ -> x e. RR )
18	17	adantr 276	. . . . . . . . 9 |- ( ( x e. ZZ /\ A = ( x ^ 2 ) ) -> x e. RR )
19	13	nn0red 9348	. . . . . . . . 9 |- ( ( x e. ZZ /\ A = ( x ^ 2 ) ) -> A e. RR )
20		znegcl 9402	. . . . . . . . . . . . 13 |- ( x e. ZZ -> -u x e. ZZ )
21		zzlesq 10851	. . . . . . . . . . . . 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 9376	. . . . . . . . . . . . 13 |- ( x e. ZZ -> x e. CC )
25		sqneg 10741	. . . . . . . . . . . . 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 4069	. . . . . . . . . 10 |- ( ( x e. ZZ /\ A = ( x ^ 2 ) ) -> -u x <_ ( x ^ 2 ) )
29	28, 10	breqtrrd 4071	. . . . . . . . 9 |- ( ( x e. ZZ /\ A = ( x ^ 2 ) ) -> -u x <_ A )
30	18, 19, 29	lenegcon1d 8599	. . . . . . . 8 |- ( ( x e. ZZ /\ A = ( x ^ 2 ) ) -> -u A <_ x )
31		zzlesq 10851	. . . . . . . . . 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 4071	. . . . . . . 8 |- ( ( x e. ZZ /\ A = ( x ^ 2 ) ) -> x <_ A )
34	15, 14, 16, 30, 33	elfzd 10137	. . . . . . 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 2516	. . . 4 |- ( E. x e. ZZ A = ( x ^ 2 ) <-> E. x e. ( -u A ... A ) A = ( x ^ 2 ) )
39	38	dcbii 841	. . 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 2211	. . . . 5 |- ( n = A -> ( n = ( x ^ 2 ) <-> A = ( x ^ 2 ) ) )
42	41	rexbidv 2506	. . . 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 2919	. . 3 |- ( A e. NN0 -> ( A e. S <-> E. x e. ZZ A = ( x ^ 2 ) ) )
45	44	dcbid 839	. 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 835   = wceq 1372   e. wcel 2175   {cab 2190   E.wrex 2484   class class class wbr 4043  (class class class)co 5943   CCcc 7922   RRcr 7923   <_ cle 8107   -ucneg 8243   2c2 9086   NN0cn0 9294   ZZcz 9371   ...cfz 10129   ^cexp 10681

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-iinf 4635  ax-cnex 8015  ax-resscn 8016  ax-1cn 8017  ax-1re 8018  ax-icn 8019  ax-addcl 8020  ax-addrcl 8021  ax-mulcl 8022  ax-mulrcl 8023  ax-addcom 8024  ax-mulcom 8025  ax-addass 8026  ax-mulass 8027  ax-distr 8028  ax-i2m1 8029  ax-0lt1 8030  ax-1rid 8031  ax-0id 8032  ax-rnegex 8033  ax-precex 8034  ax-cnre 8035  ax-pre-ltirr 8036  ax-pre-ltwlin 8037  ax-pre-lttrn 8038  ax-pre-apti 8039  ax-pre-ltadd 8040  ax-pre-mulgt0 8041  ax-pre-mulext 8042

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-if 3571  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-id 4339  df-po 4342  df-iso 4343  df-iord 4412  df-on 4414  df-ilim 4415  df-suc 4417  df-iom 4638  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-1st 6225  df-2nd 6226  df-recs 6390  df-frec 6476  df-pnf 8108  df-mnf 8109  df-xr 8110  df-ltxr 8111  df-le 8112  df-sub 8244  df-neg 8245  df-reap 8647  df-ap 8654  df-div 8745  df-inn 9036  df-2 9094  df-n0 9295  df-z 9372  df-uz 9648  df-fz 10130  df-fzo 10264  df-seqfrec 10591  df-exp 10682

This theorem is referenced by: (None)