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Theorem sqnprm 13088
Description: A square is never prime. (Contributed by Mario Carneiro, 20-Jun-2015.)
Assertion
Ref Expression
sqnprm  |-  ( A  e.  ZZ  ->  -.  ( A ^ 2 )  e.  Prime )

Proof of Theorem sqnprm
StepHypRef Expression
1 zre 10276 . . . . . 6  |-  ( A  e.  ZZ  ->  A  e.  RR )
21adantr 452 . . . . 5  |-  ( ( A  e.  ZZ  /\  ( A ^ 2 )  e.  Prime )  ->  A  e.  RR )
3 absresq 12097 . . . . 5  |-  ( A  e.  RR  ->  (
( abs `  A
) ^ 2 )  =  ( A ^
2 ) )
42, 3syl 16 . . . 4  |-  ( ( A  e.  ZZ  /\  ( A ^ 2 )  e.  Prime )  ->  (
( abs `  A
) ^ 2 )  =  ( A ^
2 ) )
52recnd 9104 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  ( A ^ 2 )  e.  Prime )  ->  A  e.  CC )
65abscld 12228 . . . . . 6  |-  ( ( A  e.  ZZ  /\  ( A ^ 2 )  e.  Prime )  ->  ( abs `  A )  e.  RR )
76recnd 9104 . . . . 5  |-  ( ( A  e.  ZZ  /\  ( A ^ 2 )  e.  Prime )  ->  ( abs `  A )  e.  CC )
87sqvald 11510 . . . 4  |-  ( ( A  e.  ZZ  /\  ( A ^ 2 )  e.  Prime )  ->  (
( abs `  A
) ^ 2 )  =  ( ( abs `  A )  x.  ( abs `  A ) ) )
94, 8eqtr3d 2469 . . 3  |-  ( ( A  e.  ZZ  /\  ( A ^ 2 )  e.  Prime )  ->  ( A ^ 2 )  =  ( ( abs `  A
)  x.  ( abs `  A ) ) )
10 simpr 448 . . 3  |-  ( ( A  e.  ZZ  /\  ( A ^ 2 )  e.  Prime )  ->  ( A ^ 2 )  e. 
Prime )
119, 10eqeltrrd 2510 . 2  |-  ( ( A  e.  ZZ  /\  ( A ^ 2 )  e.  Prime )  ->  (
( abs `  A
)  x.  ( abs `  A ) )  e. 
Prime )
12 nn0abscl 12107 . . . . . 6  |-  ( A  e.  ZZ  ->  ( abs `  A )  e. 
NN0 )
1312adantr 452 . . . . 5  |-  ( ( A  e.  ZZ  /\  ( A ^ 2 )  e.  Prime )  ->  ( abs `  A )  e. 
NN0 )
1413nn0zd 10363 . . . 4  |-  ( ( A  e.  ZZ  /\  ( A ^ 2 )  e.  Prime )  ->  ( abs `  A )  e.  ZZ )
15 sq1 11466 . . . . . 6  |-  ( 1 ^ 2 )  =  1
16 prmuz2 13087 . . . . . . . . 9  |-  ( ( A ^ 2 )  e.  Prime  ->  ( A ^ 2 )  e.  ( ZZ>= `  2 )
)
1716adantl 453 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  ( A ^ 2 )  e.  Prime )  ->  ( A ^ 2 )  e.  ( ZZ>= `  2 )
)
18 eluz2b1 10537 . . . . . . . . 9  |-  ( ( A ^ 2 )  e.  ( ZZ>= `  2
)  <->  ( ( A ^ 2 )  e.  ZZ  /\  1  < 
( A ^ 2 ) ) )
1918simprbi 451 . . . . . . . 8  |-  ( ( A ^ 2 )  e.  ( ZZ>= `  2
)  ->  1  <  ( A ^ 2 ) )
2017, 19syl 16 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  ( A ^ 2 )  e.  Prime )  ->  1  <  ( A ^ 2 ) )
2120, 4breqtrrd 4230 . . . . . 6  |-  ( ( A  e.  ZZ  /\  ( A ^ 2 )  e.  Prime )  ->  1  <  ( ( abs `  A
) ^ 2 ) )
2215, 21syl5eqbr 4237 . . . . 5  |-  ( ( A  e.  ZZ  /\  ( A ^ 2 )  e.  Prime )  ->  (
1 ^ 2 )  <  ( ( abs `  A ) ^ 2 ) )
235absge0d 12236 . . . . . 6  |-  ( ( A  e.  ZZ  /\  ( A ^ 2 )  e.  Prime )  ->  0  <_  ( abs `  A
) )
24 1re 9080 . . . . . . 7  |-  1  e.  RR
25 0le1 9541 . . . . . . 7  |-  0  <_  1
26 lt2sq 11445 . . . . . . 7  |-  ( ( ( 1  e.  RR  /\  0  <_  1 )  /\  ( ( abs `  A )  e.  RR  /\  0  <_  ( abs `  A ) ) )  ->  ( 1  < 
( abs `  A
)  <->  ( 1 ^ 2 )  <  (
( abs `  A
) ^ 2 ) ) )
2724, 25, 26mpanl12 664 . . . . . 6  |-  ( ( ( abs `  A
)  e.  RR  /\  0  <_  ( abs `  A
) )  ->  (
1  <  ( abs `  A )  <->  ( 1 ^ 2 )  < 
( ( abs `  A
) ^ 2 ) ) )
286, 23, 27syl2anc 643 . . . . 5  |-  ( ( A  e.  ZZ  /\  ( A ^ 2 )  e.  Prime )  ->  (
1  <  ( abs `  A )  <->  ( 1 ^ 2 )  < 
( ( abs `  A
) ^ 2 ) ) )
2922, 28mpbird 224 . . . 4  |-  ( ( A  e.  ZZ  /\  ( A ^ 2 )  e.  Prime )  ->  1  <  ( abs `  A
) )
30 eluz2b1 10537 . . . 4  |-  ( ( abs `  A )  e.  ( ZZ>= `  2
)  <->  ( ( abs `  A )  e.  ZZ  /\  1  <  ( abs `  A ) ) )
3114, 29, 30sylanbrc 646 . . 3  |-  ( ( A  e.  ZZ  /\  ( A ^ 2 )  e.  Prime )  ->  ( abs `  A )  e.  ( ZZ>= `  2 )
)
32 nprm 13083 . . 3  |-  ( ( ( abs `  A
)  e.  ( ZZ>= ` 
2 )  /\  ( abs `  A )  e.  ( ZZ>= `  2 )
)  ->  -.  (
( abs `  A
)  x.  ( abs `  A ) )  e. 
Prime )
3331, 31, 32syl2anc 643 . 2  |-  ( ( A  e.  ZZ  /\  ( A ^ 2 )  e.  Prime )  ->  -.  ( ( abs `  A
)  x.  ( abs `  A ) )  e. 
Prime )
3411, 33pm2.65da 560 1  |-  ( A  e.  ZZ  ->  -.  ( A ^ 2 )  e.  Prime )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   RRcr 8979   0cc0 8980   1c1 8981    x. cmul 8985    < clt 9110    <_ cle 9111   2c2 10039   NN0cn0 10211   ZZcz 10272   ZZ>=cuz 10478   ^cexp 11372   abscabs 12029   Primecprime 13069
This theorem is referenced by:  2sqblem  21151
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9036  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056  ax-pre-mulgt0 9057  ax-pre-sup 9058
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-div 9668  df-nn 9991  df-2 10048  df-3 10049  df-n0 10212  df-z 10273  df-uz 10479  df-rp 10603  df-seq 11314  df-exp 11373  df-cj 11894  df-re 11895  df-im 11896  df-sqr 12030  df-abs 12031  df-dvds 12843  df-prm 13070
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