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Theorem nprm 12445
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 9657 . . . . 5 |- ( A e. ( ZZ>= ` 2 ) -> A e. ZZ )
21adantr 276 . . . 4 |- ( ( A e. ( ZZ>= ` 2 ) /\ B e. ( ZZ>= ` 2 ) ) -> A e. ZZ )
32zred 9495 . . 3 |- ( ( A e. ( ZZ>= ` 2 ) /\ B e. ( ZZ>= ` 2 ) ) -> A e. RR )
4 eluz2b2 9724 . . . . . 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 9657 . . . . . . 7 |- ( B e. ( ZZ>= ` 2 ) -> B e. ZZ )
87adantl 277 . . . . . 6 |- ( ( A e. ( ZZ>= ` 2 ) /\ B e. ( ZZ>= ` 2 ) ) -> B e. ZZ )
98zred 9495 . . . . 5 |- ( ( A e. ( ZZ>= ` 2 ) /\ B e. ( ZZ>= ` 2 ) ) -> B e. RR )
10 eluz2nn 9687 . . . . . . 7 |- ( A e. ( ZZ>= ` 2 ) -> A e. NN )
1110adantr 276 . . . . . 6 |- ( ( A e. ( ZZ>= ` 2 ) /\ B e. ( ZZ>= ` 2 ) ) -> A e. NN )
1211nngt0d 9080 . . . . 5 |- ( ( A e. ( ZZ>= ` 2 ) /\ B e. ( ZZ>= ` 2 ) ) -> 0 < A )
13 ltmulgt11 8937 . . . . 5 |- ( ( A e. RR /\ B e. RR /\ 0 < A ) -> ( 1 < B <-> A < ( A x. B ) ) )
143, 9, 12, 13syl3anc 1250 . . . 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 8186 . 2 |- ( ( A e. ( ZZ>= ` 2 ) /\ B e. ( ZZ>= ` 2 ) ) -> A =/= ( A x. B ) )
17 dvdsmul1 12124 . . . . 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 12441 . . . . . . 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 4047 . . . . . . . 8 |- ( x = A -> ( x || ( A x. B ) <-> A || ( A x. B ) ) )
22 eqeq1 2212 . . . . . . . 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 2873 . . . . . 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 2418 . 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 1373   e. wcel 2176   =/= wne 2376   A.wral 2484   class class class wbr 4044   ` cfv 5271  (class class class)co 5944   RRcr 7924   0cc0 7925   1c1 7926   x. cmul 7930   < clt 8107   NNcn 9036   2c2 9087   ZZcz 9372   ZZ>=cuz 9648   || cdvds 12098   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:  nprmi  12446  dvdsnprmd  12447  sqnprm  12458  mersenne  15469
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