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Theorem sqnprm 14646
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 10941 . . . . . 6  |-  ( A  e.  ZZ  ->  A  e.  RR )
21adantr 467 . . . . 5  |-  ( ( A  e.  ZZ  /\  ( A ^ 2 )  e.  Prime )  ->  A  e.  RR )
3 absresq 13365 . . . . 5  |-  ( A  e.  RR  ->  (
( abs `  A
) ^ 2 )  =  ( A ^
2 ) )
42, 3syl 17 . . . 4  |-  ( ( A  e.  ZZ  /\  ( A ^ 2 )  e.  Prime )  ->  (
( abs `  A
) ^ 2 )  =  ( A ^
2 ) )
52recnd 9669 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  ( A ^ 2 )  e.  Prime )  ->  A  e.  CC )
65abscld 13498 . . . . . 6  |-  ( ( A  e.  ZZ  /\  ( A ^ 2 )  e.  Prime )  ->  ( abs `  A )  e.  RR )
76recnd 9669 . . . . 5  |-  ( ( A  e.  ZZ  /\  ( A ^ 2 )  e.  Prime )  ->  ( abs `  A )  e.  CC )
87sqvald 12413 . . . 4  |-  ( ( A  e.  ZZ  /\  ( A ^ 2 )  e.  Prime )  ->  (
( abs `  A
) ^ 2 )  =  ( ( abs `  A )  x.  ( abs `  A ) ) )
94, 8eqtr3d 2487 . . 3  |-  ( ( A  e.  ZZ  /\  ( A ^ 2 )  e.  Prime )  ->  ( A ^ 2 )  =  ( ( abs `  A
)  x.  ( abs `  A ) ) )
10 simpr 463 . . 3  |-  ( ( A  e.  ZZ  /\  ( A ^ 2 )  e.  Prime )  ->  ( A ^ 2 )  e. 
Prime )
119, 10eqeltrrd 2530 . 2  |-  ( ( A  e.  ZZ  /\  ( A ^ 2 )  e.  Prime )  ->  (
( abs `  A
)  x.  ( abs `  A ) )  e. 
Prime )
12 nn0abscl 13375 . . . . . 6  |-  ( A  e.  ZZ  ->  ( abs `  A )  e. 
NN0 )
1312adantr 467 . . . . 5  |-  ( ( A  e.  ZZ  /\  ( A ^ 2 )  e.  Prime )  ->  ( abs `  A )  e. 
NN0 )
1413nn0zd 11038 . . . 4  |-  ( ( A  e.  ZZ  /\  ( A ^ 2 )  e.  Prime )  ->  ( abs `  A )  e.  ZZ )
15 sq1 12369 . . . . . 6  |-  ( 1 ^ 2 )  =  1
16 prmuz2 14642 . . . . . . . . 9  |-  ( ( A ^ 2 )  e.  Prime  ->  ( A ^ 2 )  e.  ( ZZ>= `  2 )
)
1716adantl 468 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  ( A ^ 2 )  e.  Prime )  ->  ( A ^ 2 )  e.  ( ZZ>= `  2 )
)
18 eluz2b1 11230 . . . . . . . . 9  |-  ( ( A ^ 2 )  e.  ( ZZ>= `  2
)  <->  ( ( A ^ 2 )  e.  ZZ  /\  1  < 
( A ^ 2 ) ) )
1918simprbi 466 . . . . . . . 8  |-  ( ( A ^ 2 )  e.  ( ZZ>= `  2
)  ->  1  <  ( A ^ 2 ) )
2017, 19syl 17 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  ( A ^ 2 )  e.  Prime )  ->  1  <  ( A ^ 2 ) )
2120, 4breqtrrd 4429 . . . . . 6  |-  ( ( A  e.  ZZ  /\  ( A ^ 2 )  e.  Prime )  ->  1  <  ( ( abs `  A
) ^ 2 ) )
2215, 21syl5eqbr 4436 . . . . 5  |-  ( ( A  e.  ZZ  /\  ( A ^ 2 )  e.  Prime )  ->  (
1 ^ 2 )  <  ( ( abs `  A ) ^ 2 ) )
235absge0d 13506 . . . . . 6  |-  ( ( A  e.  ZZ  /\  ( A ^ 2 )  e.  Prime )  ->  0  <_  ( abs `  A
) )
24 1re 9642 . . . . . . 7  |-  1  e.  RR
25 0le1 10137 . . . . . . 7  |-  0  <_  1
26 lt2sq 12348 . . . . . . 7  |-  ( ( ( 1  e.  RR  /\  0  <_  1 )  /\  ( ( abs `  A )  e.  RR  /\  0  <_  ( abs `  A ) ) )  ->  ( 1  < 
( abs `  A
)  <->  ( 1 ^ 2 )  <  (
( abs `  A
) ^ 2 ) ) )
2724, 25, 26mpanl12 688 . . . . . 6  |-  ( ( ( abs `  A
)  e.  RR  /\  0  <_  ( abs `  A
) )  ->  (
1  <  ( abs `  A )  <->  ( 1 ^ 2 )  < 
( ( abs `  A
) ^ 2 ) ) )
286, 23, 27syl2anc 667 . . . . 5  |-  ( ( A  e.  ZZ  /\  ( A ^ 2 )  e.  Prime )  ->  (
1  <  ( abs `  A )  <->  ( 1 ^ 2 )  < 
( ( abs `  A
) ^ 2 ) ) )
2922, 28mpbird 236 . . . 4  |-  ( ( A  e.  ZZ  /\  ( A ^ 2 )  e.  Prime )  ->  1  <  ( abs `  A
) )
30 eluz2b1 11230 . . . 4  |-  ( ( abs `  A )  e.  ( ZZ>= `  2
)  <->  ( ( abs `  A )  e.  ZZ  /\  1  <  ( abs `  A ) ) )
3114, 29, 30sylanbrc 670 . . 3  |-  ( ( A  e.  ZZ  /\  ( A ^ 2 )  e.  Prime )  ->  ( abs `  A )  e.  ( ZZ>= `  2 )
)
32 nprm 14638 . . 3  |-  ( ( ( abs `  A
)  e.  ( ZZ>= ` 
2 )  /\  ( abs `  A )  e.  ( ZZ>= `  2 )
)  ->  -.  (
( abs `  A
)  x.  ( abs `  A ) )  e. 
Prime )
3331, 31, 32syl2anc 667 . 2  |-  ( ( A  e.  ZZ  /\  ( A ^ 2 )  e.  Prime )  ->  -.  ( ( abs `  A
)  x.  ( abs `  A ) )  e. 
Prime )
3411, 33pm2.65da 580 1  |-  ( A  e.  ZZ  ->  -.  ( A ^ 2 )  e.  Prime )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1444    e. wcel 1887   class class class wbr 4402   ` cfv 5582  (class class class)co 6290   RRcr 9538   0cc0 9539   1c1 9540    x. cmul 9544    < clt 9675    <_ cle 9676   2c2 10659   NN0cn0 10869   ZZcz 10937   ZZ>=cuz 11159   ^cexp 12272   abscabs 13297   Primecprime 14622
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-2o 7183  df-oadd 7186  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-sup 7956  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-seq 12214  df-exp 12273  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-dvds 14306  df-prm 14623
This theorem is referenced by:  2sqblem  24305  2sqn0  28407  2sqcoprm  28408
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