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Theorem sqnprm 13897
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 10756 . . . . . 6  |-  ( A  e.  ZZ  ->  A  e.  RR )
21adantr 465 . . . . 5  |-  ( ( A  e.  ZZ  /\  ( A ^ 2 )  e.  Prime )  ->  A  e.  RR )
3 absresq 12904 . . . . 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 9518 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  ( A ^ 2 )  e.  Prime )  ->  A  e.  CC )
65abscld 13035 . . . . . 6  |-  ( ( A  e.  ZZ  /\  ( A ^ 2 )  e.  Prime )  ->  ( abs `  A )  e.  RR )
76recnd 9518 . . . . 5  |-  ( ( A  e.  ZZ  /\  ( A ^ 2 )  e.  Prime )  ->  ( abs `  A )  e.  CC )
87sqvald 12117 . . . 4  |-  ( ( A  e.  ZZ  /\  ( A ^ 2 )  e.  Prime )  ->  (
( abs `  A
) ^ 2 )  =  ( ( abs `  A )  x.  ( abs `  A ) ) )
94, 8eqtr3d 2495 . . 3  |-  ( ( A  e.  ZZ  /\  ( A ^ 2 )  e.  Prime )  ->  ( A ^ 2 )  =  ( ( abs `  A
)  x.  ( abs `  A ) ) )
10 simpr 461 . . 3  |-  ( ( A  e.  ZZ  /\  ( A ^ 2 )  e.  Prime )  ->  ( A ^ 2 )  e. 
Prime )
119, 10eqeltrrd 2541 . 2  |-  ( ( A  e.  ZZ  /\  ( A ^ 2 )  e.  Prime )  ->  (
( abs `  A
)  x.  ( abs `  A ) )  e. 
Prime )
12 nn0abscl 12914 . . . . . 6  |-  ( A  e.  ZZ  ->  ( abs `  A )  e. 
NN0 )
1312adantr 465 . . . . 5  |-  ( ( A  e.  ZZ  /\  ( A ^ 2 )  e.  Prime )  ->  ( abs `  A )  e. 
NN0 )
1413nn0zd 10851 . . . 4  |-  ( ( A  e.  ZZ  /\  ( A ^ 2 )  e.  Prime )  ->  ( abs `  A )  e.  ZZ )
15 sq1 12072 . . . . . 6  |-  ( 1 ^ 2 )  =  1
16 prmuz2 13894 . . . . . . . . 9  |-  ( ( A ^ 2 )  e.  Prime  ->  ( A ^ 2 )  e.  ( ZZ>= `  2 )
)
1716adantl 466 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  ( A ^ 2 )  e.  Prime )  ->  ( A ^ 2 )  e.  ( ZZ>= `  2 )
)
18 eluz2b1 11032 . . . . . . . . 9  |-  ( ( A ^ 2 )  e.  ( ZZ>= `  2
)  <->  ( ( A ^ 2 )  e.  ZZ  /\  1  < 
( A ^ 2 ) ) )
1918simprbi 464 . . . . . . . 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 4421 . . . . . 6  |-  ( ( A  e.  ZZ  /\  ( A ^ 2 )  e.  Prime )  ->  1  <  ( ( abs `  A
) ^ 2 ) )
2215, 21syl5eqbr 4428 . . . . 5  |-  ( ( A  e.  ZZ  /\  ( A ^ 2 )  e.  Prime )  ->  (
1 ^ 2 )  <  ( ( abs `  A ) ^ 2 ) )
235absge0d 13043 . . . . . 6  |-  ( ( A  e.  ZZ  /\  ( A ^ 2 )  e.  Prime )  ->  0  <_  ( abs `  A
) )
24 1re 9491 . . . . . . 7  |-  1  e.  RR
25 0le1 9969 . . . . . . 7  |-  0  <_  1
26 lt2sq 12051 . . . . . . 7  |-  ( ( ( 1  e.  RR  /\  0  <_  1 )  /\  ( ( abs `  A )  e.  RR  /\  0  <_  ( abs `  A ) ) )  ->  ( 1  < 
( abs `  A
)  <->  ( 1 ^ 2 )  <  (
( abs `  A
) ^ 2 ) ) )
2724, 25, 26mpanl12 682 . . . . . 6  |-  ( ( ( abs `  A
)  e.  RR  /\  0  <_  ( abs `  A
) )  ->  (
1  <  ( abs `  A )  <->  ( 1 ^ 2 )  < 
( ( abs `  A
) ^ 2 ) ) )
286, 23, 27syl2anc 661 . . . . 5  |-  ( ( A  e.  ZZ  /\  ( A ^ 2 )  e.  Prime )  ->  (
1  <  ( abs `  A )  <->  ( 1 ^ 2 )  < 
( ( abs `  A
) ^ 2 ) ) )
2922, 28mpbird 232 . . . 4  |-  ( ( A  e.  ZZ  /\  ( A ^ 2 )  e.  Prime )  ->  1  <  ( abs `  A
) )
30 eluz2b1 11032 . . . 4  |-  ( ( abs `  A )  e.  ( ZZ>= `  2
)  <->  ( ( abs `  A )  e.  ZZ  /\  1  <  ( abs `  A ) ) )
3114, 29, 30sylanbrc 664 . . 3  |-  ( ( A  e.  ZZ  /\  ( A ^ 2 )  e.  Prime )  ->  ( abs `  A )  e.  ( ZZ>= `  2 )
)
32 nprm 13890 . . 3  |-  ( ( ( abs `  A
)  e.  ( ZZ>= ` 
2 )  /\  ( abs `  A )  e.  ( ZZ>= `  2 )
)  ->  -.  (
( abs `  A
)  x.  ( abs `  A ) )  e. 
Prime )
3331, 31, 32syl2anc 661 . 2  |-  ( ( A  e.  ZZ  /\  ( A ^ 2 )  e.  Prime )  ->  -.  ( ( abs `  A
)  x.  ( abs `  A ) )  e. 
Prime )
3411, 33pm2.65da 576 1  |-  ( A  e.  ZZ  ->  -.  ( A ^ 2 )  e.  Prime )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   class class class wbr 4395   ` cfv 5521  (class class class)co 6195   RRcr 9387   0cc0 9388   1c1 9389    x. cmul 9393    < clt 9524    <_ cle 9525   2c2 10477   NN0cn0 10685   ZZcz 10752   ZZ>=cuz 10967   ^cexp 11977   abscabs 12836   Primecprime 13876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465  ax-pre-sup 9466
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-int 4232  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-2nd 6683  df-recs 6937  df-rdg 6971  df-1o 7025  df-2o 7026  df-oadd 7029  df-er 7206  df-en 7416  df-dom 7417  df-sdom 7418  df-fin 7419  df-sup 7797  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-div 10100  df-nn 10429  df-2 10486  df-3 10487  df-n0 10686  df-z 10753  df-uz 10968  df-rp 11098  df-seq 11919  df-exp 11978  df-cj 12701  df-re 12702  df-im 12703  df-sqr 12837  df-abs 12838  df-dvds 13649  df-prm 13877
This theorem is referenced by:  2sqblem  22844
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