Theorem sqnprm 12679

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

Step	Hyp	Ref	Expression
1		zre 9466	. . . . . 6 |- ( A e. ZZ -> A e. RR )
2	1	adantr 276	. . . . 5 |- ( ( A e. ZZ /\ ( A ^ 2 ) e. Prime ) -> A e. RR )
3		absresq 11610	. . . . 5 |- ( A e. RR -> ( ( abs ` A ) ^ 2 ) = ( A ^ 2 ) )
4	2, 3	syl 14	. . . 4 |- ( ( A e. ZZ /\ ( A ^ 2 ) e. Prime ) -> ( ( abs ` A ) ^ 2 ) = ( A ^ 2 ) )
5	2	recnd 8191	. . . . . . 7 |- ( ( A e. ZZ /\ ( A ^ 2 ) e. Prime ) -> A e. CC )
6	5	abscld 11713	. . . . . 6 |- ( ( A e. ZZ /\ ( A ^ 2 ) e. Prime ) -> ( abs ` A ) e. RR )
7	6	recnd 8191	. . . . 5 |- ( ( A e. ZZ /\ ( A ^ 2 ) e. Prime ) -> ( abs ` A ) e. CC )
8	7	sqvald 10909	. . . 4 |- ( ( A e. ZZ /\ ( A ^ 2 ) e. Prime ) -> ( ( abs ` A ) ^ 2 ) = ( ( abs ` A ) x. ( abs ` A ) ) )
9	4, 8	eqtr3d 2264	. . 3 |- ( ( A e. ZZ /\ ( A ^ 2 ) e. Prime ) -> ( A ^ 2 ) = ( ( abs ` A ) x. ( abs ` A ) ) )
10		simpr 110	. . 3 |- ( ( A e. ZZ /\ ( A ^ 2 ) e. Prime ) -> ( A ^ 2 ) e. Prime )
11	9, 10	eqeltrrd 2307	. 2 |- ( ( A e. ZZ /\ ( A ^ 2 ) e. Prime ) -> ( ( abs ` A ) x. ( abs ` A ) ) e. Prime )
12		nn0abscl 11617	. . . . . 6 |- ( A e. ZZ -> ( abs ` A ) e. NN0 )
13	12	adantr 276	. . . . 5 |- ( ( A e. ZZ /\ ( A ^ 2 ) e. Prime ) -> ( abs ` A ) e. NN0 )
14	13	nn0zd 9583	. . . 4 |- ( ( A e. ZZ /\ ( A ^ 2 ) e. Prime ) -> ( abs ` A ) e. ZZ )
15		sq1 10872	. . . . . 6 |- ( 1 ^ 2 ) = 1
16		prmuz2 12674	. . . . . . . . 9 |- ( ( A ^ 2 ) e. Prime -> ( A ^ 2 ) e. ( ZZ>= ` 2 ) )
17	16	adantl 277	. . . . . . . 8 |- ( ( A e. ZZ /\ ( A ^ 2 ) e. Prime ) -> ( A ^ 2 ) e. ( ZZ>= ` 2 ) )
18		eluz2b1 9813	. . . . . . . . 9 |- ( ( A ^ 2 ) e. ( ZZ>= ` 2 ) <-> ( ( A ^ 2 ) e. ZZ /\ 1 < ( A ^ 2 ) ) )
19	18	simprbi 275	. . . . . . . 8 |- ( ( A ^ 2 ) e. ( ZZ>= ` 2 ) -> 1 < ( A ^ 2 ) )
20	17, 19	syl 14	. . . . . . 7 |- ( ( A e. ZZ /\ ( A ^ 2 ) e. Prime ) -> 1 < ( A ^ 2 ) )
21	20, 4	breqtrrd 4111	. . . . . 6 |- ( ( A e. ZZ /\ ( A ^ 2 ) e. Prime ) -> 1 < ( ( abs ` A ) ^ 2 ) )
22	15, 21	eqbrtrid 4118	. . . . 5 |- ( ( A e. ZZ /\ ( A ^ 2 ) e. Prime ) -> ( 1 ^ 2 ) < ( ( abs ` A ) ^ 2 ) )
23	5	absge0d 11716	. . . . . 6 |- ( ( A e. ZZ /\ ( A ^ 2 ) e. Prime ) -> 0 <_ ( abs ` A ) )
24		1re 8161	. . . . . . 7 |- 1 e. RR
25		0le1 8644	. . . . . . 7 |- 0 <_ 1
26		lt2sq 10852	. . . . . . 7 |- ( ( ( 1 e. RR /\ 0 <_ 1 ) /\ ( ( abs ` A ) e. RR /\ 0 <_ ( abs ` A ) ) ) -> ( 1 < ( abs ` A ) <-> ( 1 ^ 2 ) < ( ( abs ` A ) ^ 2 ) ) )
27	24, 25, 26	mpanl12 436	. . . . . 6 |- ( ( ( abs ` A ) e. RR /\ 0 <_ ( abs ` A ) ) -> ( 1 < ( abs ` A ) <-> ( 1 ^ 2 ) < ( ( abs ` A ) ^ 2 ) ) )
28	6, 23, 27	syl2anc 411	. . . . 5 |- ( ( A e. ZZ /\ ( A ^ 2 ) e. Prime ) -> ( 1 < ( abs ` A ) <-> ( 1 ^ 2 ) < ( ( abs ` A ) ^ 2 ) ) )
29	22, 28	mpbird 167	. . . 4 |- ( ( A e. ZZ /\ ( A ^ 2 ) e. Prime ) -> 1 < ( abs ` A ) )
30		eluz2b1 9813	. . . 4 |- ( ( abs ` A ) e. ( ZZ>= ` 2 ) <-> ( ( abs ` A ) e. ZZ /\ 1 < ( abs ` A ) ) )
31	14, 29, 30	sylanbrc 417	. . 3 |- ( ( A e. ZZ /\ ( A ^ 2 ) e. Prime ) -> ( abs ` A ) e. ( ZZ>= ` 2 ) )
32		nprm 12666	. . 3 |- ( ( ( abs ` A ) e. ( ZZ>= ` 2 ) /\ ( abs ` A ) e. ( ZZ>= ` 2 ) ) -> -. ( ( abs ` A ) x. ( abs ` A ) ) e. Prime )
33	31, 31, 32	syl2anc 411	. 2 |- ( ( A e. ZZ /\ ( A ^ 2 ) e. Prime ) -> -. ( ( abs ` A ) x. ( abs ` A ) ) e. Prime )
34	11, 33	pm2.65da 665	1 |- ( A e. ZZ -> -. ( A ^ 2 ) e. Prime )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   class class class wbr 4083   ` cfv 5321  (class class class)co 6010   RRcr 8014   0cc0 8015   1c1 8016   x. cmul 8020   < clt 8197   <_ cle 8198   2c2 9177   NN0cn0 9385   ZZcz 9462   ZZ>=cuz 9738   ^cexp 10777   abscabs 11529   Primecprime 12650

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4259  ax-pr 4294  ax-un 4525  ax-setind 4630  ax-iinf 4681  ax-cnex 8106  ax-resscn 8107  ax-1cn 8108  ax-1re 8109  ax-icn 8110  ax-addcl 8111  ax-addrcl 8112  ax-mulcl 8113  ax-mulrcl 8114  ax-addcom 8115  ax-mulcom 8116  ax-addass 8117  ax-mulass 8118  ax-distr 8119  ax-i2m1 8120  ax-0lt1 8121  ax-1rid 8122  ax-0id 8123  ax-rnegex 8124  ax-precex 8125  ax-cnre 8126  ax-pre-ltirr 8127  ax-pre-ltwlin 8128  ax-pre-lttrn 8129  ax-pre-apti 8130  ax-pre-ltadd 8131  ax-pre-mulgt0 8132  ax-pre-mulext 8133  ax-arch 8134  ax-caucvg 8135

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4385  df-po 4388  df-iso 4389  df-iord 4458  df-on 4460  df-ilim 4461  df-suc 4463  df-iom 4684  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-res 4732  df-ima 4733  df-iota 5281  df-fun 5323  df-fn 5324  df-f 5325  df-f1 5326  df-fo 5327  df-f1o 5328  df-fv 5329  df-riota 5963  df-ov 6013  df-oprab 6014  df-mpo 6015  df-1st 6295  df-2nd 6296  df-recs 6462  df-frec 6548  df-1o 6573  df-2o 6574  df-er 6693  df-en 6901  df-pnf 8199  df-mnf 8200  df-xr 8201  df-ltxr 8202  df-le 8203  df-sub 8335  df-neg 8336  df-reap 8738  df-ap 8745  df-div 8836  df-inn 9127  df-2 9185  df-3 9186  df-4 9187  df-n0 9386  df-z 9463  df-uz 9739  df-q 9832  df-rp 9867  df-seqfrec 10687  df-exp 10778  df-cj 11374  df-re 11375  df-im 11376  df-rsqrt 11530  df-abs 11531  df-dvds 12320  df-prm 12651

This theorem is referenced by: (None)