Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ztprmneprm Structured version   Visualization version   Unicode version

Theorem ztprmneprm 40636
Description: A prime is not an integer multiple of another prime. (Contributed by AV, 23-May-2019.)
Assertion
Ref Expression
ztprmneprm  |-  ( ( Z  e.  ZZ  /\  A  e.  Prime  /\  B  e.  Prime )  ->  (
( Z  x.  A
)  =  B  ->  A  =  B )
)

Proof of Theorem ztprmneprm
StepHypRef Expression
1 elznn0nn 10975 . . 3  |-  ( Z  e.  ZZ  <->  ( Z  e.  NN0  \/  ( Z  e.  RR  /\  -u Z  e.  NN ) ) )
2 elnn0 10895 . . . . 5  |-  ( Z  e.  NN0  <->  ( Z  e.  NN  \/  Z  =  0 ) )
3 elnn1uz2 11258 . . . . . . 7  |-  ( Z  e.  NN  <->  ( Z  =  1  \/  Z  e.  ( ZZ>= `  2 )
) )
4 oveq1 6315 . . . . . . . . . . . 12  |-  ( Z  =  1  ->  ( Z  x.  A )  =  ( 1  x.  A ) )
54adantr 472 . . . . . . . . . . 11  |-  ( ( Z  =  1  /\  ( A  e.  Prime  /\  B  e.  Prime )
)  ->  ( Z  x.  A )  =  ( 1  x.  A ) )
65eqeq1d 2473 . . . . . . . . . 10  |-  ( ( Z  =  1  /\  ( A  e.  Prime  /\  B  e.  Prime )
)  ->  ( ( Z  x.  A )  =  B  <->  ( 1  x.  A )  =  B ) )
7 prmz 14705 . . . . . . . . . . . . . . . 16  |-  ( A  e.  Prime  ->  A  e.  ZZ )
87zcnd 11064 . . . . . . . . . . . . . . 15  |-  ( A  e.  Prime  ->  A  e.  CC )
98mulid2d 9679 . . . . . . . . . . . . . 14  |-  ( A  e.  Prime  ->  ( 1  x.  A )  =  A )
109adantr 472 . . . . . . . . . . . . 13  |-  ( ( A  e.  Prime  /\  B  e.  Prime )  ->  (
1  x.  A )  =  A )
1110eqeq1d 2473 . . . . . . . . . . . 12  |-  ( ( A  e.  Prime  /\  B  e.  Prime )  ->  (
( 1  x.  A
)  =  B  <->  A  =  B ) )
1211biimpd 212 . . . . . . . . . . 11  |-  ( ( A  e.  Prime  /\  B  e.  Prime )  ->  (
( 1  x.  A
)  =  B  ->  A  =  B )
)
1312adantl 473 . . . . . . . . . 10  |-  ( ( Z  =  1  /\  ( A  e.  Prime  /\  B  e.  Prime )
)  ->  ( (
1  x.  A )  =  B  ->  A  =  B ) )
146, 13sylbid 223 . . . . . . . . 9  |-  ( ( Z  =  1  /\  ( A  e.  Prime  /\  B  e.  Prime )
)  ->  ( ( Z  x.  A )  =  B  ->  A  =  B ) )
1514ex 441 . . . . . . . 8  |-  ( Z  =  1  ->  (
( A  e.  Prime  /\  B  e.  Prime )  ->  ( ( Z  x.  A )  =  B  ->  A  =  B ) ) )
16 prmuz2 14721 . . . . . . . . . . . 12  |-  ( A  e.  Prime  ->  A  e.  ( ZZ>= `  2 )
)
1716adantr 472 . . . . . . . . . . 11  |-  ( ( A  e.  Prime  /\  B  e.  Prime )  ->  A  e.  ( ZZ>= `  2 )
)
18 nprm 14717 . . . . . . . . . . 11  |-  ( ( Z  e.  ( ZZ>= ` 
2 )  /\  A  e.  ( ZZ>= `  2 )
)  ->  -.  ( Z  x.  A )  e.  Prime )
1917, 18sylan2 482 . . . . . . . . . 10  |-  ( ( Z  e.  ( ZZ>= ` 
2 )  /\  ( A  e.  Prime  /\  B  e.  Prime ) )  ->  -.  ( Z  x.  A
)  e.  Prime )
20 eleq1 2537 . . . . . . . . . . . . 13  |-  ( ( Z  x.  A )  =  B  ->  (
( Z  x.  A
)  e.  Prime  <->  B  e.  Prime ) )
2120notbid 301 . . . . . . . . . . . 12  |-  ( ( Z  x.  A )  =  B  ->  ( -.  ( Z  x.  A
)  e.  Prime  <->  -.  B  e.  Prime ) )
22 pm2.24 112 . . . . . . . . . . . . . . 15  |-  ( B  e.  Prime  ->  ( -.  B  e.  Prime  ->  A  =  B ) )
2322adantl 473 . . . . . . . . . . . . . 14  |-  ( ( A  e.  Prime  /\  B  e.  Prime )  ->  ( -.  B  e.  Prime  ->  A  =  B )
)
2423adantl 473 . . . . . . . . . . . . 13  |-  ( ( Z  e.  ( ZZ>= ` 
2 )  /\  ( A  e.  Prime  /\  B  e.  Prime ) )  -> 
( -.  B  e. 
Prime  ->  A  =  B ) )
2524com12 31 . . . . . . . . . . . 12  |-  ( -.  B  e.  Prime  ->  ( ( Z  e.  (
ZZ>= `  2 )  /\  ( A  e.  Prime  /\  B  e.  Prime )
)  ->  A  =  B ) )
2621, 25syl6bi 236 . . . . . . . . . . 11  |-  ( ( Z  x.  A )  =  B  ->  ( -.  ( Z  x.  A
)  e.  Prime  ->  ( ( Z  e.  (
ZZ>= `  2 )  /\  ( A  e.  Prime  /\  B  e.  Prime )
)  ->  A  =  B ) ) )
2726com3l 83 . . . . . . . . . 10  |-  ( -.  ( Z  x.  A
)  e.  Prime  ->  ( ( Z  e.  (
ZZ>= `  2 )  /\  ( A  e.  Prime  /\  B  e.  Prime )
)  ->  ( ( Z  x.  A )  =  B  ->  A  =  B ) ) )
2819, 27mpcom 36 . . . . . . . . 9  |-  ( ( Z  e.  ( ZZ>= ` 
2 )  /\  ( A  e.  Prime  /\  B  e.  Prime ) )  -> 
( ( Z  x.  A )  =  B  ->  A  =  B ) )
2928ex 441 . . . . . . . 8  |-  ( Z  e.  ( ZZ>= `  2
)  ->  ( ( A  e.  Prime  /\  B  e.  Prime )  ->  (
( Z  x.  A
)  =  B  ->  A  =  B )
) )
3015, 29jaoi 386 . . . . . . 7  |-  ( ( Z  =  1  \/  Z  e.  ( ZZ>= ` 
2 ) )  -> 
( ( A  e. 
Prime  /\  B  e.  Prime )  ->  ( ( Z  x.  A )  =  B  ->  A  =  B ) ) )
313, 30sylbi 200 . . . . . 6  |-  ( Z  e.  NN  ->  (
( A  e.  Prime  /\  B  e.  Prime )  ->  ( ( Z  x.  A )  =  B  ->  A  =  B ) ) )
32 oveq1 6315 . . . . . . . . 9  |-  ( Z  =  0  ->  ( Z  x.  A )  =  ( 0  x.  A ) )
3332eqeq1d 2473 . . . . . . . 8  |-  ( Z  =  0  ->  (
( Z  x.  A
)  =  B  <->  ( 0  x.  A )  =  B ) )
34 prmnn 14704 . . . . . . . . . . . . . 14  |-  ( A  e.  Prime  ->  A  e.  NN )
3534nnred 10646 . . . . . . . . . . . . 13  |-  ( A  e.  Prime  ->  A  e.  RR )
36 mul02lem2 9828 . . . . . . . . . . . . 13  |-  ( A  e.  RR  ->  (
0  x.  A )  =  0 )
3735, 36syl 17 . . . . . . . . . . . 12  |-  ( A  e.  Prime  ->  ( 0  x.  A )  =  0 )
3837adantr 472 . . . . . . . . . . 11  |-  ( ( A  e.  Prime  /\  B  e.  Prime )  ->  (
0  x.  A )  =  0 )
3938eqeq1d 2473 . . . . . . . . . 10  |-  ( ( A  e.  Prime  /\  B  e.  Prime )  ->  (
( 0  x.  A
)  =  B  <->  0  =  B ) )
40 prmnn 14704 . . . . . . . . . . . 12  |-  ( B  e.  Prime  ->  B  e.  NN )
41 elnnne0 10907 . . . . . . . . . . . . 13  |-  ( B  e.  NN  <->  ( B  e.  NN0  /\  B  =/=  0 ) )
42 eqneqall 2654 . . . . . . . . . . . . . . . 16  |-  ( B  =  0  ->  ( B  =/=  0  ->  A  =  B ) )
4342eqcoms 2479 . . . . . . . . . . . . . . 15  |-  ( 0  =  B  ->  ( B  =/=  0  ->  A  =  B ) )
4443com12 31 . . . . . . . . . . . . . 14  |-  ( B  =/=  0  ->  (
0  =  B  ->  A  =  B )
)
4544adantl 473 . . . . . . . . . . . . 13  |-  ( ( B  e.  NN0  /\  B  =/=  0 )  -> 
( 0  =  B  ->  A  =  B ) )
4641, 45sylbi 200 . . . . . . . . . . . 12  |-  ( B  e.  NN  ->  (
0  =  B  ->  A  =  B )
)
4740, 46syl 17 . . . . . . . . . . 11  |-  ( B  e.  Prime  ->  ( 0  =  B  ->  A  =  B ) )
4847adantl 473 . . . . . . . . . 10  |-  ( ( A  e.  Prime  /\  B  e.  Prime )  ->  (
0  =  B  ->  A  =  B )
)
4939, 48sylbid 223 . . . . . . . . 9  |-  ( ( A  e.  Prime  /\  B  e.  Prime )  ->  (
( 0  x.  A
)  =  B  ->  A  =  B )
)
5049com12 31 . . . . . . . 8  |-  ( ( 0  x.  A )  =  B  ->  (
( A  e.  Prime  /\  B  e.  Prime )  ->  A  =  B ) )
5133, 50syl6bi 236 . . . . . . 7  |-  ( Z  =  0  ->  (
( Z  x.  A
)  =  B  -> 
( ( A  e. 
Prime  /\  B  e.  Prime )  ->  A  =  B ) ) )
5251com23 80 . . . . . 6  |-  ( Z  =  0  ->  (
( A  e.  Prime  /\  B  e.  Prime )  ->  ( ( Z  x.  A )  =  B  ->  A  =  B ) ) )
5331, 52jaoi 386 . . . . 5  |-  ( ( Z  e.  NN  \/  Z  =  0 )  ->  ( ( A  e.  Prime  /\  B  e. 
Prime )  ->  ( ( Z  x.  A )  =  B  ->  A  =  B ) ) )
542, 53sylbi 200 . . . 4  |-  ( Z  e.  NN0  ->  ( ( A  e.  Prime  /\  B  e.  Prime )  ->  (
( Z  x.  A
)  =  B  ->  A  =  B )
) )
55 elnnz 10971 . . . . . 6  |-  ( -u Z  e.  NN  <->  ( -u Z  e.  ZZ  /\  0  <  -u Z ) )
56 lt0neg1 10141 . . . . . . . 8  |-  ( Z  e.  RR  ->  ( Z  <  0  <->  0  <  -u Z ) )
5734nngt0d 10675 . . . . . . . . . . . . . . . 16  |-  ( A  e.  Prime  ->  0  < 
A )
5857adantr 472 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  Prime  /\  B  e.  Prime )  ->  0  <  A )
59 simpr 468 . . . . . . . . . . . . . . 15  |-  ( ( Z  e.  RR  /\  Z  <  0 )  ->  Z  <  0 )
6058, 59anim12ci 577 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  Prime  /\  B  e.  Prime )  /\  ( Z  e.  RR  /\  Z  <  0 ) )  ->  ( Z  <  0  /\  0  < 
A ) )
6160orcd 399 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  Prime  /\  B  e.  Prime )  /\  ( Z  e.  RR  /\  Z  <  0 ) )  ->  ( ( Z  <  0  /\  0  <  A )  \/  (
0  <  Z  /\  A  <  0 ) ) )
62 simprl 772 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  Prime  /\  B  e.  Prime )  /\  ( Z  e.  RR  /\  Z  <  0 ) )  ->  Z  e.  RR )
6335adantr 472 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  Prime  /\  B  e.  Prime )  ->  A  e.  RR )
6463adantr 472 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  Prime  /\  B  e.  Prime )  /\  ( Z  e.  RR  /\  Z  <  0 ) )  ->  A  e.  RR )
6562, 64mul2lt0bi 11425 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  Prime  /\  B  e.  Prime )  /\  ( Z  e.  RR  /\  Z  <  0 ) )  ->  ( ( Z  x.  A )  <  0  <->  ( ( Z  <  0  /\  0  <  A )  \/  (
0  <  Z  /\  A  <  0 ) ) ) )
6661, 65mpbird 240 . . . . . . . . . . . 12  |-  ( ( ( A  e.  Prime  /\  B  e.  Prime )  /\  ( Z  e.  RR  /\  Z  <  0 ) )  ->  ( Z  x.  A )  <  0
)
6766ex 441 . . . . . . . . . . 11  |-  ( ( A  e.  Prime  /\  B  e.  Prime )  ->  (
( Z  e.  RR  /\  Z  <  0 )  ->  ( Z  x.  A )  <  0
) )
68 breq1 4398 . . . . . . . . . . . . . . 15  |-  ( ( Z  x.  A )  =  B  ->  (
( Z  x.  A
)  <  0  <->  B  <  0 ) )
6968adantl 473 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  Prime  /\  B  e.  Prime )  /\  ( Z  x.  A
)  =  B )  ->  ( ( Z  x.  A )  <  0  <->  B  <  0
) )
70 nnnn0 10900 . . . . . . . . . . . . . . . . . 18  |-  ( B  e.  NN  ->  B  e.  NN0 )
71 nn0nlt0 10920 . . . . . . . . . . . . . . . . . . 19  |-  ( B  e.  NN0  ->  -.  B  <  0 )
7271pm2.21d 109 . . . . . . . . . . . . . . . . . 18  |-  ( B  e.  NN0  ->  ( B  <  0  ->  A  =  B ) )
7370, 72syl 17 . . . . . . . . . . . . . . . . 17  |-  ( B  e.  NN  ->  ( B  <  0  ->  A  =  B ) )
7440, 73syl 17 . . . . . . . . . . . . . . . 16  |-  ( B  e.  Prime  ->  ( B  <  0  ->  A  =  B ) )
7574adantl 473 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  Prime  /\  B  e.  Prime )  ->  ( B  <  0  ->  A  =  B ) )
7675adantr 472 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  Prime  /\  B  e.  Prime )  /\  ( Z  x.  A
)  =  B )  ->  ( B  <  0  ->  A  =  B ) )
7769, 76sylbid 223 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  Prime  /\  B  e.  Prime )  /\  ( Z  x.  A
)  =  B )  ->  ( ( Z  x.  A )  <  0  ->  A  =  B ) )
7877ex 441 . . . . . . . . . . . 12  |-  ( ( A  e.  Prime  /\  B  e.  Prime )  ->  (
( Z  x.  A
)  =  B  -> 
( ( Z  x.  A )  <  0  ->  A  =  B ) ) )
7978com23 80 . . . . . . . . . . 11  |-  ( ( A  e.  Prime  /\  B  e.  Prime )  ->  (
( Z  x.  A
)  <  0  ->  ( ( Z  x.  A
)  =  B  ->  A  =  B )
) )
8067, 79syld 44 . . . . . . . . . 10  |-  ( ( A  e.  Prime  /\  B  e.  Prime )  ->  (
( Z  e.  RR  /\  Z  <  0 )  ->  ( ( Z  x.  A )  =  B  ->  A  =  B ) ) )
8180com12 31 . . . . . . . . 9  |-  ( ( Z  e.  RR  /\  Z  <  0 )  -> 
( ( A  e. 
Prime  /\  B  e.  Prime )  ->  ( ( Z  x.  A )  =  B  ->  A  =  B ) ) )
8281ex 441 . . . . . . . 8  |-  ( Z  e.  RR  ->  ( Z  <  0  ->  (
( A  e.  Prime  /\  B  e.  Prime )  ->  ( ( Z  x.  A )  =  B  ->  A  =  B ) ) ) )
8356, 82sylbird 243 . . . . . . 7  |-  ( Z  e.  RR  ->  (
0  <  -u Z  -> 
( ( A  e. 
Prime  /\  B  e.  Prime )  ->  ( ( Z  x.  A )  =  B  ->  A  =  B ) ) ) )
8483adantld 474 . . . . . 6  |-  ( Z  e.  RR  ->  (
( -u Z  e.  ZZ  /\  0  <  -u Z
)  ->  ( ( A  e.  Prime  /\  B  e.  Prime )  ->  (
( Z  x.  A
)  =  B  ->  A  =  B )
) ) )
8555, 84syl5bi 225 . . . . 5  |-  ( Z  e.  RR  ->  ( -u Z  e.  NN  ->  ( ( A  e.  Prime  /\  B  e.  Prime )  ->  ( ( Z  x.  A )  =  B  ->  A  =  B ) ) ) )
8685imp 436 . . . 4  |-  ( ( Z  e.  RR  /\  -u Z  e.  NN )  ->  ( ( A  e.  Prime  /\  B  e. 
Prime )  ->  ( ( Z  x.  A )  =  B  ->  A  =  B ) ) )
8754, 86jaoi 386 . . 3  |-  ( ( Z  e.  NN0  \/  ( Z  e.  RR  /\  -u Z  e.  NN ) )  ->  (
( A  e.  Prime  /\  B  e.  Prime )  ->  ( ( Z  x.  A )  =  B  ->  A  =  B ) ) )
881, 87sylbi 200 . 2  |-  ( Z  e.  ZZ  ->  (
( A  e.  Prime  /\  B  e.  Prime )  ->  ( ( Z  x.  A )  =  B  ->  A  =  B ) ) )
89883impib 1229 1  |-  ( ( Z  e.  ZZ  /\  A  e.  Prime  /\  B  e.  Prime )  ->  (
( Z  x.  A
)  =  B  ->  A  =  B )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    \/ wo 375    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   class class class wbr 4395   ` cfv 5589  (class class class)co 6308   RRcr 9556   0cc0 9557   1c1 9558    x. cmul 9562    < clt 9693   -ucneg 9881   NNcn 10631   2c2 10681   NN0cn0 10893   ZZcz 10961   ZZ>=cuz 11182   Primecprime 14701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-n0 10894  df-z 10962  df-uz 11183  df-rp 11326  df-dvds 14383  df-prm 14702
This theorem is referenced by:  zlmodzxznm  40798
  Copyright terms: Public domain W3C validator