Theorem 4sqexercise1 13155

Description: Exercise which may help in understanding the proof of 4sqlemsdc 13157. (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 9657	. . . 4 |- ( A e. NN0 -> -u A e. ZZ )
2		nn0z 9643	. . . 4 |- ( A e. NN0 -> A e. ZZ )
3		elfzelz 10407	. . . . . . 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 11025	. . . . . 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 9699	. . . . 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 10637	. . 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 11032	. . . . . . . . . . . 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 9745	. . . . . . . . 9 |- ( ( x e. ZZ /\ A = ( x ^ 2 ) ) -> A e. ZZ )
15	14	znegcld 9749	. . . . . . . 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 9627	. . . . . . . . . 10 |- ( x e. ZZ -> x e. RR )
18	17	adantr 276	. . . . . . . . 9 |- ( ( x e. ZZ /\ A = ( x ^ 2 ) ) -> x e. RR )
19	13	nn0red 9600	. . . . . . . . 9 |- ( ( x e. ZZ /\ A = ( x ^ 2 ) ) -> A e. RR )
20		znegcl 9654	. . . . . . . . . . . . 13 |- ( x e. ZZ -> -u x e. ZZ )
21		zzlesq 11124	. . . . . . . . . . . . 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 9628	. . . . . . . . . . . . 13 |- ( x e. ZZ -> x e. CC )
25		sqneg 11013	. . . . . . . . . . . . 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 4151	. . . . . . . . . 10 |- ( ( x e. ZZ /\ A = ( x ^ 2 ) ) -> -u x <_ ( x ^ 2 ) )
29	28, 10	breqtrrd 4153	. . . . . . . . 9 |- ( ( x e. ZZ /\ A = ( x ^ 2 ) ) -> -u x <_ A )
30	18, 19, 29	lenegcon1d 8845	. . . . . . . 8 |- ( ( x e. ZZ /\ A = ( x ^ 2 ) ) -> -u A <_ x )
31		zzlesq 11124	. . . . . . . . . 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 4153	. . . . . . . 8 |- ( ( x e. ZZ /\ A = ( x ^ 2 ) ) -> x <_ A )
34	15, 14, 16, 30, 33	elfzd 10398	. . . . . . 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 )

Syntax hints:   -> wi 4   /\ wa 104  DECID wdc 846   = wceq 1402   e. wcel 2209   {cab 2224   E.wrex 2529   class class class wbr 4125  (class class class)co 6075   CCcc 8167   RRcr 8168   <_ cle 8351   -ucneg 8488   2c2 9334   NN0cn0 9542   ZZcz 9623   ...cfz 10390   ^cexp 10953

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 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 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287

This theorem 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 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993  df-inn 9284  df-2 9342  df-n0 9543  df-z 9624  df-uz 9901  df-fz 10391  df-fzo 10528  df-seqfrec 10863  df-exp 10954

This theorem is referenced by: (None)