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Theorem nprm 12645
Description: A product of two integers greater than one is composite. (Contributed by Mario Carneiro, 20-Jun-2015.)
Assertion
Ref Expression
nprm |- ( ( A e. ( ZZ>= ` 2 ) /\ B e. ( ZZ>= ` 2 ) ) -> -. ( A x. B ) e. Prime )
Proof of Theorem nprm
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 eluzelz 9731 . . . . 5 |- ( A e. ( ZZ>= ` 2 ) -> A e. ZZ )
21adantr 276 . . . 4 |- ( ( A e. ( ZZ>= ` 2 ) /\ B e. ( ZZ>= ` 2 ) ) -> A e. ZZ )
32zred 9569 . . 3 |- ( ( A e. ( ZZ>= ` 2 ) /\ B e. ( ZZ>= ` 2 ) ) -> A e. RR )
4 eluz2b2 9798 . . . . . 6 |- ( B e. ( ZZ>= ` 2 ) <-> ( B e. NN /\ 1 < B ) )
54simprbi 275 . . . . 5 |- ( B e. ( ZZ>= ` 2 ) -> 1 < B )
65adantl 277 . . . 4 |- ( ( A e. ( ZZ>= ` 2 ) /\ B e. ( ZZ>= ` 2 ) ) -> 1 < B )
7 eluzelz 9731 . . . . . . 7 |- ( B e. ( ZZ>= ` 2 ) -> B e. ZZ )
87adantl 277 . . . . . 6 |- ( ( A e. ( ZZ>= ` 2 ) /\ B e. ( ZZ>= ` 2 ) ) -> B e. ZZ )
98zred 9569 . . . . 5 |- ( ( A e. ( ZZ>= ` 2 ) /\ B e. ( ZZ>= ` 2 ) ) -> B e. RR )
10 eluz2nn 9761 . . . . . . 7 |- ( A e. ( ZZ>= ` 2 ) -> A e. NN )
1110adantr 276 . . . . . 6 |- ( ( A e. ( ZZ>= ` 2 ) /\ B e. ( ZZ>= ` 2 ) ) -> A e. NN )
1211nngt0d 9154 . . . . 5 |- ( ( A e. ( ZZ>= ` 2 ) /\ B e. ( ZZ>= ` 2 ) ) -> 0 < A )
13 ltmulgt11 9011 . . . . 5 |- ( ( A e. RR /\ B e. RR /\ 0 < A ) -> ( 1 < B <-> A < ( A x. B ) ) )
143, 9, 12, 13syl3anc 1271 . . . 4 |- ( ( A e. ( ZZ>= ` 2 ) /\ B e. ( ZZ>= ` 2 ) ) -> ( 1 < B <-> A < ( A x. B ) ) )
156, 14mpbid 147 . . 3 |- ( ( A e. ( ZZ>= ` 2 ) /\ B e. ( ZZ>= ` 2 ) ) -> A < ( A x. B ) )
163, 15ltned 8260 . 2 |- ( ( A e. ( ZZ>= ` 2 ) /\ B e. ( ZZ>= ` 2 ) ) -> A =/= ( A x. B ) )
17 dvdsmul1 12324 . . . . 5 |- ( ( A e. ZZ /\ B e. ZZ ) -> A || ( A x. B ) )
181, 7, 17syl2an 289 . . . 4 |- ( ( A e. ( ZZ>= ` 2 ) /\ B e. ( ZZ>= ` 2 ) ) -> A || ( A x. B ) )
19 isprm4 12641 . . . . . . 7 |- ( ( A x. B ) e. Prime <-> ( ( A x. B ) e. ( ZZ>= ` 2 ) /\ A. x e. ( ZZ>= ` 2 ) ( x || ( A x. B ) -> x = ( A x. B ) ) ) )
2019simprbi 275 . . . . . 6 |- ( ( A x. B ) e. Prime -> A. x e. ( ZZ>= ` 2 ) ( x || ( A x. B ) -> x = ( A x. B ) ) )
21 breq1 4086 . . . . . . . 8 |- ( x = A -> ( x || ( A x. B ) <-> A || ( A x. B ) ) )
22 eqeq1 2236 . . . . . . . 8 |- ( x = A -> ( x = ( A x. B ) <-> A = ( A x. B ) ) )
2321, 22imbi12d 234 . . . . . . 7 |- ( x = A -> ( ( x || ( A x. B ) -> x = ( A x. B ) ) <-> ( A || ( A x. B ) -> A = ( A x. B ) ) ) )
2423rspcv 2903 . . . . . 6 |- ( A e. ( ZZ>= ` 2 ) -> ( A. x e. ( ZZ>= ` 2 ) ( x || ( A x. B ) -> x = ( A x. B ) ) -> ( A || ( A x. B ) -> A = ( A x. B ) ) ) )
2520, 24syl5 32 . . . . 5 |- ( A e. ( ZZ>= ` 2 ) -> ( ( A x. B ) e. Prime -> ( A || ( A x. B ) -> A = ( A x. B ) ) ) )
2625adantr 276 . . . 4 |- ( ( A e. ( ZZ>= ` 2 ) /\ B e. ( ZZ>= ` 2 ) ) -> ( ( A x. B ) e. Prime -> ( A || ( A x. B ) -> A = ( A x. B ) ) ) )
2718, 26mpid 42 . . 3 |- ( ( A e. ( ZZ>= ` 2 ) /\ B e. ( ZZ>= ` 2 ) ) -> ( ( A x. B ) e. Prime -> A = ( A x. B ) ) )
2827necon3ad 2442 . 2 |- ( ( A e. ( ZZ>= ` 2 ) /\ B e. ( ZZ>= ` 2 ) ) -> ( A =/= ( A x. B ) -> -. ( A x. B ) e. Prime ) )
2916, 28mpd 13 1 |- ( ( A e. ( ZZ>= ` 2 ) /\ B e. ( ZZ>= ` 2 ) ) -> -. ( A x. B ) e. Prime )
Colors of variables: wff set class
Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   =/= wne 2400   A.wral 2508   class class class wbr 4083   ` cfv 5318  (class class class)co 6001   RRcr 7998   0cc0 7999   1c1 8000   x. cmul 8004   < clt 8181   NNcn 9110   2c2 9161   ZZcz 9446   ZZ>=cuz 9722   || cdvds 12298   Primecprime 12629
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 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-mulrcl 8098  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-precex 8109  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115  ax-pre-mulgt0 8116  ax-pre-mulext 8117  ax-arch 8118  ax-caucvg 8119
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 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-frec 6537  df-1o 6562  df-2o 6563  df-er 6680  df-en 6888  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-reap 8722  df-ap 8729  df-div 8820  df-inn 9111  df-2 9169  df-3 9170  df-4 9171  df-n0 9370  df-z 9447  df-uz 9723  df-q 9815  df-rp 9850  df-seqfrec 10670  df-exp 10761  df-cj 11353  df-re 11354  df-im 11355  df-rsqrt 11509  df-abs 11510  df-dvds 12299  df-prm 12630
This theorem is referenced by:  nprmi  12646  dvdsnprmd  12647  sqnprm  12658  mersenne  15671
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