Theorem sqnprm 12458

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 9376	. . . . . 6 |- ( A e. ZZ -> A e. RR )
2	1	adantr 276	. . . . 5 |- ( ( A e. ZZ /\ ( A ^ 2 ) e. Prime ) -> A e. RR )
3		absresq 11389	. . . . 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 8101	. . . . . . 7 |- ( ( A e. ZZ /\ ( A ^ 2 ) e. Prime ) -> A e. CC )
6	5	abscld 11492	. . . . . 6 |- ( ( A e. ZZ /\ ( A ^ 2 ) e. Prime ) -> ( abs ` A ) e. RR )
7	6	recnd 8101	. . . . 5 |- ( ( A e. ZZ /\ ( A ^ 2 ) e. Prime ) -> ( abs ` A ) e. CC )
8	7	sqvald 10815	. . . 4 |- ( ( A e. ZZ /\ ( A ^ 2 ) e. Prime ) -> ( ( abs ` A ) ^ 2 ) = ( ( abs ` A ) x. ( abs ` A ) ) )
9	4, 8	eqtr3d 2240	. . 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 2283	. 2 |- ( ( A e. ZZ /\ ( A ^ 2 ) e. Prime ) -> ( ( abs ` A ) x. ( abs ` A ) ) e. Prime )
12		nn0abscl 11396	. . . . . 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 9493	. . . 4 |- ( ( A e. ZZ /\ ( A ^ 2 ) e. Prime ) -> ( abs ` A ) e. ZZ )
15		sq1 10778	. . . . . 6 |- ( 1 ^ 2 ) = 1
16		prmuz2 12453	. . . . . . . . 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 9722	. . . . . . . . 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 4072	. . . . . 6 |- ( ( A e. ZZ /\ ( A ^ 2 ) e. Prime ) -> 1 < ( ( abs ` A ) ^ 2 ) )
22	15, 21	eqbrtrid 4079	. . . . 5 |- ( ( A e. ZZ /\ ( A ^ 2 ) e. Prime ) -> ( 1 ^ 2 ) < ( ( abs ` A ) ^ 2 ) )
23	5	absge0d 11495	. . . . . 6 |- ( ( A e. ZZ /\ ( A ^ 2 ) e. Prime ) -> 0 <_ ( abs ` A ) )
24		1re 8071	. . . . . . 7 |- 1 e. RR
25		0le1 8554	. . . . . . 7 |- 0 <_ 1
26		lt2sq 10758	. . . . . . 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 9722	. . . 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 12445	. . 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 663	1 |- ( A e. ZZ -> -. ( A ^ 2 ) e. Prime )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2176   class class class wbr 4044   ` cfv 5271  (class class class)co 5944   RRcr 7924   0cc0 7925   1c1 7926   x. cmul 7930   < clt 8107   <_ cle 8108   2c2 9087   NN0cn0 9295   ZZcz 9372   ZZ>=cuz 9648   ^cexp 10683   abscabs 11308   Primecprime 12429

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-iinf 4636  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-mulrcl 8024  ax-addcom 8025  ax-mulcom 8026  ax-addass 8027  ax-mulass 8028  ax-distr 8029  ax-i2m1 8030  ax-0lt1 8031  ax-1rid 8032  ax-0id 8033  ax-rnegex 8034  ax-precex 8035  ax-cnre 8036  ax-pre-ltirr 8037  ax-pre-ltwlin 8038  ax-pre-lttrn 8039  ax-pre-apti 8040  ax-pre-ltadd 8041  ax-pre-mulgt0 8042  ax-pre-mulext 8043  ax-arch 8044  ax-caucvg 8045

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-tr 4143  df-id 4340  df-po 4343  df-iso 4344  df-iord 4413  df-on 4415  df-ilim 4416  df-suc 4418  df-iom 4639  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-recs 6391  df-frec 6477  df-1o 6502  df-2o 6503  df-er 6620  df-en 6828  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112  df-le 8113  df-sub 8245  df-neg 8246  df-reap 8648  df-ap 8655  df-div 8746  df-inn 9037  df-2 9095  df-3 9096  df-4 9097  df-n0 9296  df-z 9373  df-uz 9649  df-q 9741  df-rp 9776  df-seqfrec 10593  df-exp 10684  df-cj 11153  df-re 11154  df-im 11155  df-rsqrt 11309  df-abs 11310  df-dvds 12099  df-prm 12430

This theorem is referenced by: (None)