Theorem sqnprm 15414

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 11381	. . . . . 6 |- ( A e. ZZ -> A e. RR )
2	1	adantr 481	. . . . 5 |- ( ( A e. ZZ /\ ( A ^ 2 ) e. Prime ) -> A e. RR )
3		absresq 14042	. . . . 5 |- ( A e. RR -> ( ( abs ` A ) ^ 2 ) = ( A ^ 2 ) )
4	2, 3	syl 17	. . . 4 |- ( ( A e. ZZ /\ ( A ^ 2 ) e. Prime ) -> ( ( abs ` A ) ^ 2 ) = ( A ^ 2 ) )
5	2	recnd 10068	. . . . . . 7 |- ( ( A e. ZZ /\ ( A ^ 2 ) e. Prime ) -> A e. CC )
6	5	abscld 14175	. . . . . 6 |- ( ( A e. ZZ /\ ( A ^ 2 ) e. Prime ) -> ( abs ` A ) e. RR )
7	6	recnd 10068	. . . . 5 |- ( ( A e. ZZ /\ ( A ^ 2 ) e. Prime ) -> ( abs ` A ) e. CC )
8	7	sqvald 13005	. . . 4 |- ( ( A e. ZZ /\ ( A ^ 2 ) e. Prime ) -> ( ( abs ` A ) ^ 2 ) = ( ( abs ` A ) x. ( abs ` A ) ) )
9	4, 8	eqtr3d 2658	. . 3 |- ( ( A e. ZZ /\ ( A ^ 2 ) e. Prime ) -> ( A ^ 2 ) = ( ( abs ` A ) x. ( abs ` A ) ) )
10		simpr 477	. . 3 |- ( ( A e. ZZ /\ ( A ^ 2 ) e. Prime ) -> ( A ^ 2 ) e. Prime )
11	9, 10	eqeltrrd 2702	. 2 |- ( ( A e. ZZ /\ ( A ^ 2 ) e. Prime ) -> ( ( abs ` A ) x. ( abs ` A ) ) e. Prime )
12		nn0abscl 14052	. . . . . 6 |- ( A e. ZZ -> ( abs ` A ) e. NN0 )
13	12	adantr 481	. . . . 5 |- ( ( A e. ZZ /\ ( A ^ 2 ) e. Prime ) -> ( abs ` A ) e. NN0 )
14	13	nn0zd 11480	. . . 4 |- ( ( A e. ZZ /\ ( A ^ 2 ) e. Prime ) -> ( abs ` A ) e. ZZ )
15		sq1 12958	. . . . . 6 |- ( 1 ^ 2 ) = 1
16		prmuz2 15408	. . . . . . . . 9 |- ( ( A ^ 2 ) e. Prime -> ( A ^ 2 ) e. ( ZZ>= ` 2 ) )
17	16	adantl 482	. . . . . . . 8 |- ( ( A e. ZZ /\ ( A ^ 2 ) e. Prime ) -> ( A ^ 2 ) e. ( ZZ>= ` 2 ) )
18		eluz2b1 11759	. . . . . . . . 9 |- ( ( A ^ 2 ) e. ( ZZ>= ` 2 ) <-> ( ( A ^ 2 ) e. ZZ /\ 1 < ( A ^ 2 ) ) )
19	18	simprbi 480	. . . . . . . 8 |- ( ( A ^ 2 ) e. ( ZZ>= ` 2 ) -> 1 < ( A ^ 2 ) )
20	17, 19	syl 17	. . . . . . 7 |- ( ( A e. ZZ /\ ( A ^ 2 ) e. Prime ) -> 1 < ( A ^ 2 ) )
21	20, 4	breqtrrd 4681	. . . . . 6 |- ( ( A e. ZZ /\ ( A ^ 2 ) e. Prime ) -> 1 < ( ( abs ` A ) ^ 2 ) )
22	15, 21	syl5eqbr 4688	. . . . 5 |- ( ( A e. ZZ /\ ( A ^ 2 ) e. Prime ) -> ( 1 ^ 2 ) < ( ( abs ` A ) ^ 2 ) )
23	5	absge0d 14183	. . . . . 6 |- ( ( A e. ZZ /\ ( A ^ 2 ) e. Prime ) -> 0 <_ ( abs ` A ) )
24		1re 10039	. . . . . . 7 |- 1 e. RR
25		0le1 10551	. . . . . . 7 |- 0 <_ 1
26		lt2sq 12937	. . . . . . 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 718	. . . . . 6 |- ( ( ( abs ` A ) e. RR /\ 0 <_ ( abs ` A ) ) -> ( 1 < ( abs ` A ) <-> ( 1 ^ 2 ) < ( ( abs ` A ) ^ 2 ) ) )
28	6, 23, 27	syl2anc 693	. . . . 5 |- ( ( A e. ZZ /\ ( A ^ 2 ) e. Prime ) -> ( 1 < ( abs ` A ) <-> ( 1 ^ 2 ) < ( ( abs ` A ) ^ 2 ) ) )
29	22, 28	mpbird 247	. . . 4 |- ( ( A e. ZZ /\ ( A ^ 2 ) e. Prime ) -> 1 < ( abs ` A ) )
30		eluz2b1 11759	. . . 4 |- ( ( abs ` A ) e. ( ZZ>= ` 2 ) <-> ( ( abs ` A ) e. ZZ /\ 1 < ( abs ` A ) ) )
31	14, 29, 30	sylanbrc 698	. . 3 |- ( ( A e. ZZ /\ ( A ^ 2 ) e. Prime ) -> ( abs ` A ) e. ( ZZ>= ` 2 ) )
32		nprm 15401	. . 3 |- ( ( ( abs ` A ) e. ( ZZ>= ` 2 ) /\ ( abs ` A ) e. ( ZZ>= ` 2 ) ) -> -. ( ( abs ` A ) x. ( abs ` A ) ) e. Prime )
33	31, 31, 32	syl2anc 693	. 2 |- ( ( A e. ZZ /\ ( A ^ 2 ) e. Prime ) -> -. ( ( abs ` A ) x. ( abs ` A ) ) e. Prime )
34	11, 33	pm2.65da 600	1 |- ( A e. ZZ -> -. ( A ^ 2 ) e. Prime )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   1c1 9937   x. cmul 9941   < clt 10074   <_ cle 10075   2c2 11070   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   ^cexp 12860   abscabs 13974   Primecprime 15385

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-prm 15386

This theorem is referenced by:  2sqblem  25156  2sqn0  29646  2sqcoprm  29647