Theorem 4sqexercise1 13206

Description: Exercise which may help in understanding the proof of 4sqlemsdc 13208. (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 9689	. . . 4 |- ( A e. NN0 -> -u A e. ZZ )
2		nn0z 9675	. . . 4 |- ( A e. NN0 -> A e. ZZ )
3		elfzelz 10445	. . . . . . 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 11068	. . . . . 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 9731	. . . . 5 |- ( ( A e. ZZ /\ ( x ^ 2 ) e. ZZ ) -> DECID A = ( x ^ 2 ) )
8	2, 6, 7	syl2an2r 603	. . . 4 |- ( ( A e. NN0 /\ x e. ( -u A ... A ) ) -> DECID A = ( x ^ 2 ) )
9	1, 2, 8	exfzdc 10676	. . 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 11075	. . . . . . . . . . . 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 2315	. . . . . . . . . 10 |- ( ( x e. ZZ /\ A = ( x ^ 2 ) ) -> A e. NN0 )
14	13	nn0zd 9777	. . . . . . . . 9 |- ( ( x e. ZZ /\ A = ( x ^ 2 ) ) -> A e. ZZ )
15	14	znegcld 9781	. . . . . . . 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 9659	. . . . . . . . . 10 |- ( x e. ZZ -> x e. RR )
18	17	adantr 276	. . . . . . . . 9 |- ( ( x e. ZZ /\ A = ( x ^ 2 ) ) -> x e. RR )
19	13	nn0red 9632	. . . . . . . . 9 |- ( ( x e. ZZ /\ A = ( x ^ 2 ) ) -> A e. RR )
20		znegcl 9686	. . . . . . . . . . . . 13 |- ( x e. ZZ -> -u x e. ZZ )
21		zzlesq 11167	. . . . . . . . . . . . 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 9660	. . . . . . . . . . . . 13 |- ( x e. ZZ -> x e. CC )
25		sqneg 11056	. . . . . . . . . . . . 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 4157	. . . . . . . . . 10 |- ( ( x e. ZZ /\ A = ( x ^ 2 ) ) -> -u x <_ ( x ^ 2 ) )
29	28, 10	breqtrrd 4159	. . . . . . . . 9 |- ( ( x e. ZZ /\ A = ( x ^ 2 ) ) -> -u x <_ A )
30	18, 19, 29	lenegcon1d 8862	. . . . . . . 8 |- ( ( x e. ZZ /\ A = ( x ^ 2 ) ) -> -u A <_ x )
31		zzlesq 11167	. . . . . . . . . 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 4159	. . . . . . . 8 |- ( ( x e. ZZ /\ A = ( x ^ 2 ) ) -> x <_ A )
34	15, 14, 16, 30, 33	elfzd 10436	. . . . . . 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 2561	. . . 4 |- ( E. x e. ZZ A = ( x ^ 2 ) <-> E. x e. ( -u A ... A ) A = ( x ^ 2 ) )
39	38	dcbii 852	. . 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 2245	. . . . 5 |- ( n = A -> ( n = ( x ^ 2 ) <-> A = ( x ^ 2 ) ) )
42	41	rexbidv 2551	. . . 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 2973	. . 3 |- ( A e. NN0 -> ( A e. S <-> E. x e. ZZ A = ( x ^ 2 ) ) )
45	44	dcbid 850	. 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 )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104  DECID wdc 846   = wceq 1402   e. wcel 2209   {cab 2224   E.wrex 2529   class class class wbr 4131  (class class class)co 6088   CCcc 8183   RRcr 8184   <_ cle 8367   -ucneg 8505   2c2 9364   NN0cn0 9574   ZZcz 9655   ...cfz 10428   ^cexp 10996

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302  ax-pre-mulext 8303

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-frec 6666  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-reap 8911  df-ap 8918  df-div 9011  df-inn 9314  df-2 9372  df-n0 9575  df-z 9656  df-uz 9938  df-fz 10429  df-fzo 10567  df-seqfrec 10906  df-exp 10997

This theorem is used by: (None)