Theorem nn0sqdcq 13013

Description: A nonnegative integer is a perfect square or not. This is similar to nn0sqdc 11168 but expresses the idea of being a perfect square as having a rational number which, when squared, gives the original number. (Contributed by Jim Kingdon, 25-Aug-2026.)

Assertion

Ref	Expression
nn0sqdcq	|- ( N e. NN0 -> DECID E. q e. QQ N = ( q ^ 2 ) )

Distinct variable group:   N, q

Proof of Theorem nn0sqdcq

Dummy variable r is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nn0sqdc 11168	. . . 4 |- ( N e. NN0 -> DECID E. q e. NN0 N = ( q ^ 2 ) )
2		exmiddc 848	. . . 4 |- (DECID E. q e. NN0 N = ( q ^ 2 ) -> ( E. q e. NN0 N = ( q ^ 2 ) \/ -. E. q e. NN0 N = ( q ^ 2 ) ) )
3	1, 2	syl 14	. . 3 |- ( N e. NN0 -> ( E. q e. NN0 N = ( q ^ 2 ) \/ -. E. q e. NN0 N = ( q ^ 2 ) ) )
4		nn0ssq 10044	. . . . 5 |- NN0 C_ QQ
5		ssrexv 3313	. . . . 5 |- ( NN0 C_ QQ -> ( E. q e. NN0 N = ( q ^ 2 ) -> E. q e. QQ N = ( q ^ 2 ) ) )
6	4, 5	mp1i 10	. . . 4 |- ( N e. NN0 -> ( E. q e. NN0 N = ( q ^ 2 ) -> E. q e. QQ N = ( q ^ 2 ) ) )
7		oveq1 6095	. . . . . . . 8 |- ( q = r -> ( q ^ 2 ) = ( r ^ 2 ) )
8	7	eqeq2d 2250	. . . . . . 7 |- ( q = r -> ( N = ( q ^ 2 ) <-> N = ( r ^ 2 ) ) )
9	8	cbvrexv 2787	. . . . . 6 |- ( E. q e. QQ N = ( q ^ 2 ) <-> E. r e. QQ N = ( r ^ 2 ) )
10		oveq1 6095	. . . . . . . . 9 |- ( q = ( abs ` r ) -> ( q ^ 2 ) = ( ( abs ` r ) ^ 2 ) )
11	10	eqeq2d 2250	. . . . . . . 8 |- ( q = ( abs ` r ) -> ( N = ( q ^ 2 ) <-> N = ( ( abs ` r ) ^ 2 ) ) )
12		simprr 537	. . . . . . . . . . . 12 |- ( ( N e. NN0 /\ ( r e. QQ /\ N = ( r ^ 2 ) ) ) -> N = ( r ^ 2 ) )
13	12	fveq2d 5702	. . . . . . . . . . 11 |- ( ( N e. NN0 /\ ( r e. QQ /\ N = ( r ^ 2 ) ) ) -> ( sqr ` N ) = ( sqr ` ( r ^ 2 ) ) )
14		qre 10041	. . . . . . . . . . . . 13 |- ( r e. QQ -> r e. RR )
15	14	ad2antrl 494	. . . . . . . . . . . 12 |- ( ( N e. NN0 /\ ( r e. QQ /\ N = ( r ^ 2 ) ) ) -> r e. RR )
16	15	absred 11951	. . . . . . . . . . 11 |- ( ( N e. NN0 /\ ( r e. QQ /\ N = ( r ^ 2 ) ) ) -> ( abs ` r ) = ( sqr ` ( r ^ 2 ) ) )
17	13, 16	eqtr4d 2274	. . . . . . . . . 10 |- ( ( N e. NN0 /\ ( r e. QQ /\ N = ( r ^ 2 ) ) ) -> ( sqr ` N ) = ( abs ` r ) )
18		qabscl 11865	. . . . . . . . . . . . 13 |- ( r e. QQ -> ( abs ` r ) e. QQ )
19	18	ad2antrl 494	. . . . . . . . . . . 12 |- ( ( N e. NN0 /\ ( r e. QQ /\ N = ( r ^ 2 ) ) ) -> ( abs ` r ) e. QQ )
20	17, 19	eqeltrd 2315	. . . . . . . . . . 11 |- ( ( N e. NN0 /\ ( r e. QQ /\ N = ( r ^ 2 ) ) ) -> ( sqr ` N ) e. QQ )
21		nn0sqrtelqelz 13011	. . . . . . . . . . 11 |- ( ( N e. NN0 /\ ( sqr ` N ) e. QQ ) -> ( sqr ` N ) e. ZZ )
22	20, 21	syldan 282	. . . . . . . . . 10 |- ( ( N e. NN0 /\ ( r e. QQ /\ N = ( r ^ 2 ) ) ) -> ( sqr ` N ) e. ZZ )
23	17, 22	eqeltrrd 2316	. . . . . . . . 9 |- ( ( N e. NN0 /\ ( r e. QQ /\ N = ( r ^ 2 ) ) ) -> ( abs ` r ) e. ZZ )
24	15	recnd 8360	. . . . . . . . . 10 |- ( ( N e. NN0 /\ ( r e. QQ /\ N = ( r ^ 2 ) ) ) -> r e. CC )
25	24	absge0d 11973	. . . . . . . . 9 |- ( ( N e. NN0 /\ ( r e. QQ /\ N = ( r ^ 2 ) ) ) -> 0 <_ ( abs ` r ) )
26		elnn0z 9668	. . . . . . . . 9 |- ( ( abs ` r ) e. NN0 <-> ( ( abs ` r ) e. ZZ /\ 0 <_ ( abs ` r ) ) )
27	23, 25, 26	sylanbrc 421	. . . . . . . 8 |- ( ( N e. NN0 /\ ( r e. QQ /\ N = ( r ^ 2 ) ) ) -> ( abs ` r ) e. NN0 )
28		absresq 11867	. . . . . . . . . 10 |- ( r e. RR -> ( ( abs ` r ) ^ 2 ) = ( r ^ 2 ) )
29	15, 28	syl 14	. . . . . . . . 9 |- ( ( N e. NN0 /\ ( r e. QQ /\ N = ( r ^ 2 ) ) ) -> ( ( abs ` r ) ^ 2 ) = ( r ^ 2 ) )
30	12, 29	eqtr4d 2274	. . . . . . . 8 |- ( ( N e. NN0 /\ ( r e. QQ /\ N = ( r ^ 2 ) ) ) -> N = ( ( abs ` r ) ^ 2 ) )
31	11, 27, 30	rspcedvdw 2936	. . . . . . 7 |- ( ( N e. NN0 /\ ( r e. QQ /\ N = ( r ^ 2 ) ) ) -> E. q e. NN0 N = ( q ^ 2 ) )
32	31	rexlimdvaa 2669	. . . . . 6 |- ( N e. NN0 -> ( E. r e. QQ N = ( r ^ 2 ) -> E. q e. NN0 N = ( q ^ 2 ) ) )
33	9, 32	biimtrid 152	. . . . 5 |- ( N e. NN0 -> ( E. q e. QQ N = ( q ^ 2 ) -> E. q e. NN0 N = ( q ^ 2 ) ) )
34	33	con3d 640	. . . 4 |- ( N e. NN0 -> ( -. E. q e. NN0 N = ( q ^ 2 ) -> -. E. q e. QQ N = ( q ^ 2 ) ) )
35	6, 34	orim12d 798	. . 3 |- ( N e. NN0 -> ( ( E. q e. NN0 N = ( q ^ 2 ) \/ -. E. q e. NN0 N = ( q ^ 2 ) ) -> ( E. q e. QQ N = ( q ^ 2 ) \/ -. E. q e. QQ N = ( q ^ 2 ) ) ) )
36	3, 35	mpd 13	. 2 |- ( N e. NN0 -> ( E. q e. QQ N = ( q ^ 2 ) \/ -. E. q e. QQ N = ( q ^ 2 ) ) )
37		df-dc 847	. 2 |- (DECID E. q e. QQ N = ( q ^ 2 ) <-> ( E. q e. QQ N = ( q ^ 2 ) \/ -. E. q e. QQ N = ( q ^ 2 ) ) )
38	36, 37	sylibr 134	1 |- ( N e. NN0 -> DECID E. q e. QQ N = ( q ^ 2 ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 720  DECID wdc 846   = wceq 1402   e. wcel 2209   E.wrex 2529   C_ wss 3220   class class class wbr 4131   ` cfv 5380  (class class class)co 6088   RRcr 8184   0cc0 8185   <_ cle 8367   2c2 9364   NN0cn0 9574   ZZcz 9655   QQcq 10035   ^cexp 10996   sqrcsqrt 11784   abscabs 11785

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  ax-arch 8304  ax-caucvg 8305

This proof depends on definitions:  df-bi 117  df-stab 843  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-sup 7330  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-3 9373  df-4 9374  df-n0 9575  df-z 9656  df-uz 9938  df-q 10036  df-rp 10072  df-fz 10429  df-fzo 10567  df-fl 10722  df-mod 10781  df-seqfrec 10906  df-exp 10997  df-cj 11629  df-re 11630  df-im 11631  df-rsqrt 11786  df-abs 11787  df-dvds 12580  df-gcd 12756  df-numer 12988  df-denom 12989

This theorem is used by:  sqrtrirr  13014