Theorem sqnprm 12892

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 9627	. . . . . 6 |- ( A e. ZZ -> A e. RR )
2	1	adantr 276	. . . . 5 |- ( ( A e. ZZ /\ ( A ^ 2 ) e. Prime ) -> A e. RR )
3		absresq 11822	. . . . 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 8344	. . . . . . 7 |- ( ( A e. ZZ /\ ( A ^ 2 ) e. Prime ) -> A e. CC )
6	5	abscld 11925	. . . . . 6 |- ( ( A e. ZZ /\ ( A ^ 2 ) e. Prime ) -> ( abs ` A ) e. RR )
7	6	recnd 8344	. . . . 5 |- ( ( A e. ZZ /\ ( A ^ 2 ) e. Prime ) -> ( abs ` A ) e. CC )
8	7	sqvald 11086	. . . 4 |- ( ( A e. ZZ /\ ( A ^ 2 ) e. Prime ) -> ( ( abs ` A ) ^ 2 ) = ( ( abs ` A ) x. ( abs ` A ) ) )
9	4, 8	eqtr3d 2273	. . 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 2316	. 2 |- ( ( A e. ZZ /\ ( A ^ 2 ) e. Prime ) -> ( ( abs ` A ) x. ( abs ` A ) ) e. Prime )
12		nn0abscl 11829	. . . . . 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 9745	. . . 4 |- ( ( A e. ZZ /\ ( A ^ 2 ) e. Prime ) -> ( abs ` A ) e. ZZ )
15		sq1 11048	. . . . . 6 |- ( 1 ^ 2 ) = 1
16		prmuz2 12887	. . . . . . . . 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 9980	. . . . . . . . 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 4153	. . . . . 6 |- ( ( A e. ZZ /\ ( A ^ 2 ) e. Prime ) -> 1 < ( ( abs ` A ) ^ 2 ) )
22	15, 21	eqbrtrid 4160	. . . . 5 |- ( ( A e. ZZ /\ ( A ^ 2 ) e. Prime ) -> ( 1 ^ 2 ) < ( ( abs ` A ) ^ 2 ) )
23	5	absge0d 11928	. . . . . 6 |- ( ( A e. ZZ /\ ( A ^ 2 ) e. Prime ) -> 0 <_ ( abs ` A ) )
24		1re 8315	. . . . . . 7 |- 1 e. RR
25		0le1 8799	. . . . . . 7 |- 0 <_ 1
26		lt2sq 11028	. . . . . . 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 440	. . . . . 6 |- ( ( ( abs ` A ) e. RR /\ 0 <_ ( abs ` A ) ) -> ( 1 < ( abs ` A ) <-> ( 1 ^ 2 ) < ( ( abs ` A ) ^ 2 ) ) )
28	6, 23, 27	syl2anc 415	. . . . 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 9980	. . . 4 |- ( ( abs ` A ) e. ( ZZ>= ` 2 ) <-> ( ( abs ` A ) e. ZZ /\ 1 < ( abs ` A ) ) )
31	14, 29, 30	sylanbrc 421	. . 3 |- ( ( A e. ZZ /\ ( A ^ 2 ) e. Prime ) -> ( abs ` A ) e. ( ZZ>= ` 2 ) )
32		nprm 12879	. . 3 |- ( ( ( abs ` A ) e. ( ZZ>= ` 2 ) /\ ( abs ` A ) e. ( ZZ>= ` 2 ) ) -> -. ( ( abs ` A ) x. ( abs ` A ) ) e. Prime )
33	31, 31, 32	syl2anc 415	. 2 |- ( ( A e. ZZ /\ ( A ^ 2 ) e. Prime ) -> -. ( ( abs ` A ) x. ( abs ` A ) ) e. Prime )
34	11, 33	pm2.65da 671	1 |- ( A e. ZZ -> -. ( A ^ 2 ) e. Prime )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   class class class wbr 4125   ` cfv 5372  (class class class)co 6075   RRcr 8168   0cc0 8169   1c1 8170   x. cmul 8174   < clt 8350   <_ cle 8351   2c2 9334   NN0cn0 9542   ZZcz 9623   ZZ>=cuz 9900   ^cexp 10953   abscabs 11741   Primecprime 12863

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287  ax-arch 8288  ax-caucvg 8289

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-1o 6677  df-2o 6678  df-er 6797  df-en 7013  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-n0 9543  df-z 9624  df-uz 9901  df-q 9999  df-rp 10034  df-seqfrec 10863  df-exp 10954  df-cj 11585  df-re 11586  df-im 11587  df-rsqrt 11742  df-abs 11743  df-dvds 12533  df-prm 12864

This theorem is referenced by: (None)