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Theorem ztprmneprm 40181
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 10951 . . 3  |-  ( Z  e.  ZZ  <->  ( Z  e.  NN0  \/  ( Z  e.  RR  /\  -u Z  e.  NN ) ) )
2 elnn0 10871 . . . . 5  |-  ( Z  e.  NN0  <->  ( Z  e.  NN  \/  Z  =  0 ) )
3 elnn1uz2 11235 . . . . . . 7  |-  ( Z  e.  NN  <->  ( Z  =  1  \/  Z  e.  ( ZZ>= `  2 )
) )
4 oveq1 6297 . . . . . . . . . . . 12  |-  ( Z  =  1  ->  ( Z  x.  A )  =  ( 1  x.  A ) )
54adantr 467 . . . . . . . . . . 11  |-  ( ( Z  =  1  /\  ( A  e.  Prime  /\  B  e.  Prime )
)  ->  ( Z  x.  A )  =  ( 1  x.  A ) )
65eqeq1d 2453 . . . . . . . . . 10  |-  ( ( Z  =  1  /\  ( A  e.  Prime  /\  B  e.  Prime )
)  ->  ( ( Z  x.  A )  =  B  <->  ( 1  x.  A )  =  B ) )
7 prmz 14626 . . . . . . . . . . . . . . . 16  |-  ( A  e.  Prime  ->  A  e.  ZZ )
87zcnd 11041 . . . . . . . . . . . . . . 15  |-  ( A  e.  Prime  ->  A  e.  CC )
98mulid2d 9661 . . . . . . . . . . . . . 14  |-  ( A  e.  Prime  ->  ( 1  x.  A )  =  A )
109adantr 467 . . . . . . . . . . . . 13  |-  ( ( A  e.  Prime  /\  B  e.  Prime )  ->  (
1  x.  A )  =  A )
1110eqeq1d 2453 . . . . . . . . . . . 12  |-  ( ( A  e.  Prime  /\  B  e.  Prime )  ->  (
( 1  x.  A
)  =  B  <->  A  =  B ) )
1211biimpd 211 . . . . . . . . . . 11  |-  ( ( A  e.  Prime  /\  B  e.  Prime )  ->  (
( 1  x.  A
)  =  B  ->  A  =  B )
)
1312adantl 468 . . . . . . . . . 10  |-  ( ( Z  =  1  /\  ( A  e.  Prime  /\  B  e.  Prime )
)  ->  ( (
1  x.  A )  =  B  ->  A  =  B ) )
146, 13sylbid 219 . . . . . . . . 9  |-  ( ( Z  =  1  /\  ( A  e.  Prime  /\  B  e.  Prime )
)  ->  ( ( Z  x.  A )  =  B  ->  A  =  B ) )
1514ex 436 . . . . . . . 8  |-  ( Z  =  1  ->  (
( A  e.  Prime  /\  B  e.  Prime )  ->  ( ( Z  x.  A )  =  B  ->  A  =  B ) ) )
16 prmuz2 14642 . . . . . . . . . . . 12  |-  ( A  e.  Prime  ->  A  e.  ( ZZ>= `  2 )
)
1716adantr 467 . . . . . . . . . . 11  |-  ( ( A  e.  Prime  /\  B  e.  Prime )  ->  A  e.  ( ZZ>= `  2 )
)
18 nprm 14638 . . . . . . . . . . 11  |-  ( ( Z  e.  ( ZZ>= ` 
2 )  /\  A  e.  ( ZZ>= `  2 )
)  ->  -.  ( Z  x.  A )  e.  Prime )
1917, 18sylan2 477 . . . . . . . . . 10  |-  ( ( Z  e.  ( ZZ>= ` 
2 )  /\  ( A  e.  Prime  /\  B  e.  Prime ) )  ->  -.  ( Z  x.  A
)  e.  Prime )
20 eleq1 2517 . . . . . . . . . . . . 13  |-  ( ( Z  x.  A )  =  B  ->  (
( Z  x.  A
)  e.  Prime  <->  B  e.  Prime ) )
2120notbid 296 . . . . . . . . . . . 12  |-  ( ( Z  x.  A )  =  B  ->  ( -.  ( Z  x.  A
)  e.  Prime  <->  -.  B  e.  Prime ) )
22 pm2.24 113 . . . . . . . . . . . . . . 15  |-  ( B  e.  Prime  ->  ( -.  B  e.  Prime  ->  A  =  B ) )
2322adantl 468 . . . . . . . . . . . . . 14  |-  ( ( A  e.  Prime  /\  B  e.  Prime )  ->  ( -.  B  e.  Prime  ->  A  =  B )
)
2423adantl 468 . . . . . . . . . . . . 13  |-  ( ( Z  e.  ( ZZ>= ` 
2 )  /\  ( A  e.  Prime  /\  B  e.  Prime ) )  -> 
( -.  B  e. 
Prime  ->  A  =  B ) )
2524com12 32 . . . . . . . . . . . 12  |-  ( -.  B  e.  Prime  ->  ( ( Z  e.  (
ZZ>= `  2 )  /\  ( A  e.  Prime  /\  B  e.  Prime )
)  ->  A  =  B ) )
2621, 25syl6bi 232 . . . . . . . . . . 11  |-  ( ( Z  x.  A )  =  B  ->  ( -.  ( Z  x.  A
)  e.  Prime  ->  ( ( Z  e.  (
ZZ>= `  2 )  /\  ( A  e.  Prime  /\  B  e.  Prime )
)  ->  A  =  B ) ) )
2726com3l 84 . . . . . . . . . 10  |-  ( -.  ( Z  x.  A
)  e.  Prime  ->  ( ( Z  e.  (
ZZ>= `  2 )  /\  ( A  e.  Prime  /\  B  e.  Prime )
)  ->  ( ( Z  x.  A )  =  B  ->  A  =  B ) ) )
2819, 27mpcom 37 . . . . . . . . 9  |-  ( ( Z  e.  ( ZZ>= ` 
2 )  /\  ( A  e.  Prime  /\  B  e.  Prime ) )  -> 
( ( Z  x.  A )  =  B  ->  A  =  B ) )
2928ex 436 . . . . . . . 8  |-  ( Z  e.  ( ZZ>= `  2
)  ->  ( ( A  e.  Prime  /\  B  e.  Prime )  ->  (
( Z  x.  A
)  =  B  ->  A  =  B )
) )
3015, 29jaoi 381 . . . . . . 7  |-  ( ( Z  =  1  \/  Z  e.  ( ZZ>= ` 
2 ) )  -> 
( ( A  e. 
Prime  /\  B  e.  Prime )  ->  ( ( Z  x.  A )  =  B  ->  A  =  B ) ) )
313, 30sylbi 199 . . . . . 6  |-  ( Z  e.  NN  ->  (
( A  e.  Prime  /\  B  e.  Prime )  ->  ( ( Z  x.  A )  =  B  ->  A  =  B ) ) )
32 oveq1 6297 . . . . . . . . 9  |-  ( Z  =  0  ->  ( Z  x.  A )  =  ( 0  x.  A ) )
3332eqeq1d 2453 . . . . . . . 8  |-  ( Z  =  0  ->  (
( Z  x.  A
)  =  B  <->  ( 0  x.  A )  =  B ) )
34 prmnn 14625 . . . . . . . . . . . . . 14  |-  ( A  e.  Prime  ->  A  e.  NN )
3534nnred 10624 . . . . . . . . . . . . 13  |-  ( A  e.  Prime  ->  A  e.  RR )
36 mul02lem2 9810 . . . . . . . . . . . . 13  |-  ( A  e.  RR  ->  (
0  x.  A )  =  0 )
3735, 36syl 17 . . . . . . . . . . . 12  |-  ( A  e.  Prime  ->  ( 0  x.  A )  =  0 )
3837adantr 467 . . . . . . . . . . 11  |-  ( ( A  e.  Prime  /\  B  e.  Prime )  ->  (
0  x.  A )  =  0 )
3938eqeq1d 2453 . . . . . . . . . 10  |-  ( ( A  e.  Prime  /\  B  e.  Prime )  ->  (
( 0  x.  A
)  =  B  <->  0  =  B ) )
40 prmnn 14625 . . . . . . . . . . . 12  |-  ( B  e.  Prime  ->  B  e.  NN )
41 elnnne0 10883 . . . . . . . . . . . . 13  |-  ( B  e.  NN  <->  ( B  e.  NN0  /\  B  =/=  0 ) )
42 eqneqall 2634 . . . . . . . . . . . . . . . 16  |-  ( B  =  0  ->  ( B  =/=  0  ->  A  =  B ) )
4342eqcoms 2459 . . . . . . . . . . . . . . 15  |-  ( 0  =  B  ->  ( B  =/=  0  ->  A  =  B ) )
4443com12 32 . . . . . . . . . . . . . 14  |-  ( B  =/=  0  ->  (
0  =  B  ->  A  =  B )
)
4544adantl 468 . . . . . . . . . . . . 13  |-  ( ( B  e.  NN0  /\  B  =/=  0 )  -> 
( 0  =  B  ->  A  =  B ) )
4641, 45sylbi 199 . . . . . . . . . . . 12  |-  ( B  e.  NN  ->  (
0  =  B  ->  A  =  B )
)
4740, 46syl 17 . . . . . . . . . . 11  |-  ( B  e.  Prime  ->  ( 0  =  B  ->  A  =  B ) )
4847adantl 468 . . . . . . . . . 10  |-  ( ( A  e.  Prime  /\  B  e.  Prime )  ->  (
0  =  B  ->  A  =  B )
)
4939, 48sylbid 219 . . . . . . . . 9  |-  ( ( A  e.  Prime  /\  B  e.  Prime )  ->  (
( 0  x.  A
)  =  B  ->  A  =  B )
)
5049com12 32 . . . . . . . 8  |-  ( ( 0  x.  A )  =  B  ->  (
( A  e.  Prime  /\  B  e.  Prime )  ->  A  =  B ) )
5133, 50syl6bi 232 . . . . . . 7  |-  ( Z  =  0  ->  (
( Z  x.  A
)  =  B  -> 
( ( A  e. 
Prime  /\  B  e.  Prime )  ->  A  =  B ) ) )
5251com23 81 . . . . . 6  |-  ( Z  =  0  ->  (
( A  e.  Prime  /\  B  e.  Prime )  ->  ( ( Z  x.  A )  =  B  ->  A  =  B ) ) )
5331, 52jaoi 381 . . . . 5  |-  ( ( Z  e.  NN  \/  Z  =  0 )  ->  ( ( A  e.  Prime  /\  B  e. 
Prime )  ->  ( ( Z  x.  A )  =  B  ->  A  =  B ) ) )
542, 53sylbi 199 . . . 4  |-  ( Z  e.  NN0  ->  ( ( A  e.  Prime  /\  B  e.  Prime )  ->  (
( Z  x.  A
)  =  B  ->  A  =  B )
) )
55 elnnz 10947 . . . . . 6  |-  ( -u Z  e.  NN  <->  ( -u Z  e.  ZZ  /\  0  <  -u Z ) )
56 lt0neg1 10120 . . . . . . . 8  |-  ( Z  e.  RR  ->  ( Z  <  0  <->  0  <  -u Z ) )
5734nngt0d 10653 . . . . . . . . . . . . . . . 16  |-  ( A  e.  Prime  ->  0  < 
A )
5857adantr 467 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  Prime  /\  B  e.  Prime )  ->  0  <  A )
59 simpr 463 . . . . . . . . . . . . . . 15  |-  ( ( Z  e.  RR  /\  Z  <  0 )  ->  Z  <  0 )
6058, 59anim12ci 571 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  Prime  /\  B  e.  Prime )  /\  ( Z  e.  RR  /\  Z  <  0 ) )  ->  ( Z  <  0  /\  0  < 
A ) )
6160orcd 394 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  Prime  /\  B  e.  Prime )  /\  ( Z  e.  RR  /\  Z  <  0 ) )  ->  ( ( Z  <  0  /\  0  <  A )  \/  (
0  <  Z  /\  A  <  0 ) ) )
62 simprl 764 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  Prime  /\  B  e.  Prime )  /\  ( Z  e.  RR  /\  Z  <  0 ) )  ->  Z  e.  RR )
6335adantr 467 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  Prime  /\  B  e.  Prime )  ->  A  e.  RR )
6463adantr 467 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  Prime  /\  B  e.  Prime )  /\  ( Z  e.  RR  /\  Z  <  0 ) )  ->  A  e.  RR )
6562, 64mul2lt0bi 11402 . . . . . . . . . . . . 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 236 . . . . . . . . . . . 12  |-  ( ( ( A  e.  Prime  /\  B  e.  Prime )  /\  ( Z  e.  RR  /\  Z  <  0 ) )  ->  ( Z  x.  A )  <  0
)
6766ex 436 . . . . . . . . . . 11  |-  ( ( A  e.  Prime  /\  B  e.  Prime )  ->  (
( Z  e.  RR  /\  Z  <  0 )  ->  ( Z  x.  A )  <  0
) )
68 breq1 4405 . . . . . . . . . . . . . . 15  |-  ( ( Z  x.  A )  =  B  ->  (
( Z  x.  A
)  <  0  <->  B  <  0 ) )
6968adantl 468 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  Prime  /\  B  e.  Prime )  /\  ( Z  x.  A
)  =  B )  ->  ( ( Z  x.  A )  <  0  <->  B  <  0
) )
70 nnnn0 10876 . . . . . . . . . . . . . . . . . 18  |-  ( B  e.  NN  ->  B  e.  NN0 )
71 nn0nlt0 10896 . . . . . . . . . . . . . . . . . . 19  |-  ( B  e.  NN0  ->  -.  B  <  0 )
7271pm2.21d 110 . . . . . . . . . . . . . . . . . 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 468 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  Prime  /\  B  e.  Prime )  ->  ( B  <  0  ->  A  =  B ) )
7675adantr 467 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  Prime  /\  B  e.  Prime )  /\  ( Z  x.  A
)  =  B )  ->  ( B  <  0  ->  A  =  B ) )
7769, 76sylbid 219 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  Prime  /\  B  e.  Prime )  /\  ( Z  x.  A
)  =  B )  ->  ( ( Z  x.  A )  <  0  ->  A  =  B ) )
7877ex 436 . . . . . . . . . . . 12  |-  ( ( A  e.  Prime  /\  B  e.  Prime )  ->  (
( Z  x.  A
)  =  B  -> 
( ( Z  x.  A )  <  0  ->  A  =  B ) ) )
7978com23 81 . . . . . . . . . . 11  |-  ( ( A  e.  Prime  /\  B  e.  Prime )  ->  (
( Z  x.  A
)  <  0  ->  ( ( Z  x.  A
)  =  B  ->  A  =  B )
) )
8067, 79syld 45 . . . . . . . . . 10  |-  ( ( A  e.  Prime  /\  B  e.  Prime )  ->  (
( Z  e.  RR  /\  Z  <  0 )  ->  ( ( Z  x.  A )  =  B  ->  A  =  B ) ) )
8180com12 32 . . . . . . . . 9  |-  ( ( Z  e.  RR  /\  Z  <  0 )  -> 
( ( A  e. 
Prime  /\  B  e.  Prime )  ->  ( ( Z  x.  A )  =  B  ->  A  =  B ) ) )
8281ex 436 . . . . . . . 8  |-  ( Z  e.  RR  ->  ( Z  <  0  ->  (
( A  e.  Prime  /\  B  e.  Prime )  ->  ( ( Z  x.  A )  =  B  ->  A  =  B ) ) ) )
8356, 82sylbird 239 . . . . . . 7  |-  ( Z  e.  RR  ->  (
0  <  -u Z  -> 
( ( A  e. 
Prime  /\  B  e.  Prime )  ->  ( ( Z  x.  A )  =  B  ->  A  =  B ) ) ) )
8483adantld 469 . . . . . 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 221 . . . . 5  |-  ( Z  e.  RR  ->  ( -u Z  e.  NN  ->  ( ( A  e.  Prime  /\  B  e.  Prime )  ->  ( ( Z  x.  A )  =  B  ->  A  =  B ) ) ) )
8685imp 431 . . . 4  |-  ( ( Z  e.  RR  /\  -u Z  e.  NN )  ->  ( ( A  e.  Prime  /\  B  e. 
Prime )  ->  ( ( Z  x.  A )  =  B  ->  A  =  B ) ) )
8754, 86jaoi 381 . . 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 199 . 2  |-  ( Z  e.  ZZ  ->  (
( A  e.  Prime  /\  B  e.  Prime )  ->  ( ( Z  x.  A )  =  B  ->  A  =  B ) ) )
89883impib 1206 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 188    \/ wo 370    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887    =/= wne 2622   class class class wbr 4402   ` cfv 5582  (class class class)co 6290   RRcr 9538   0cc0 9539   1c1 9540    x. cmul 9544    < clt 9675   -ucneg 9861   NNcn 10609   2c2 10659   NN0cn0 10869   ZZcz 10937   ZZ>=cuz 11159   Primecprime 14622
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-2o 7183  df-oadd 7186  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-dvds 14306  df-prm 14623
This theorem is referenced by:  zlmodzxznm  40343
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