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Theorem ztprmneprm 33123
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 10899 . . 3  |-  ( Z  e.  ZZ  <->  ( Z  e.  NN0  \/  ( Z  e.  RR  /\  -u Z  e.  NN ) ) )
2 elnn0 10818 . . . . 5  |-  ( Z  e.  NN0  <->  ( Z  e.  NN  \/  Z  =  0 ) )
3 elnn1uz2 11183 . . . . . . 7  |-  ( Z  e.  NN  <->  ( Z  =  1  \/  Z  e.  ( ZZ>= `  2 )
) )
4 oveq1 6303 . . . . . . . . . . . 12  |-  ( Z  =  1  ->  ( Z  x.  A )  =  ( 1  x.  A ) )
54adantr 465 . . . . . . . . . . 11  |-  ( ( Z  =  1  /\  ( A  e.  Prime  /\  B  e.  Prime )
)  ->  ( Z  x.  A )  =  ( 1  x.  A ) )
65eqeq1d 2459 . . . . . . . . . 10  |-  ( ( Z  =  1  /\  ( A  e.  Prime  /\  B  e.  Prime )
)  ->  ( ( Z  x.  A )  =  B  <->  ( 1  x.  A )  =  B ) )
7 prmz 14324 . . . . . . . . . . . . . . . 16  |-  ( A  e.  Prime  ->  A  e.  ZZ )
87zcnd 10991 . . . . . . . . . . . . . . 15  |-  ( A  e.  Prime  ->  A  e.  CC )
98mulid2d 9631 . . . . . . . . . . . . . 14  |-  ( A  e.  Prime  ->  ( 1  x.  A )  =  A )
109adantr 465 . . . . . . . . . . . . 13  |-  ( ( A  e.  Prime  /\  B  e.  Prime )  ->  (
1  x.  A )  =  A )
1110eqeq1d 2459 . . . . . . . . . . . 12  |-  ( ( A  e.  Prime  /\  B  e.  Prime )  ->  (
( 1  x.  A
)  =  B  <->  A  =  B ) )
1211biimpd 207 . . . . . . . . . . 11  |-  ( ( A  e.  Prime  /\  B  e.  Prime )  ->  (
( 1  x.  A
)  =  B  ->  A  =  B )
)
1312adantl 466 . . . . . . . . . 10  |-  ( ( Z  =  1  /\  ( A  e.  Prime  /\  B  e.  Prime )
)  ->  ( (
1  x.  A )  =  B  ->  A  =  B ) )
146, 13sylbid 215 . . . . . . . . 9  |-  ( ( Z  =  1  /\  ( A  e.  Prime  /\  B  e.  Prime )
)  ->  ( ( Z  x.  A )  =  B  ->  A  =  B ) )
1514ex 434 . . . . . . . 8  |-  ( Z  =  1  ->  (
( A  e.  Prime  /\  B  e.  Prime )  ->  ( ( Z  x.  A )  =  B  ->  A  =  B ) ) )
16 prmuz2 14338 . . . . . . . . . . . 12  |-  ( A  e.  Prime  ->  A  e.  ( ZZ>= `  2 )
)
1716adantr 465 . . . . . . . . . . 11  |-  ( ( A  e.  Prime  /\  B  e.  Prime )  ->  A  e.  ( ZZ>= `  2 )
)
18 nprm 14334 . . . . . . . . . . 11  |-  ( ( Z  e.  ( ZZ>= ` 
2 )  /\  A  e.  ( ZZ>= `  2 )
)  ->  -.  ( Z  x.  A )  e.  Prime )
1917, 18sylan2 474 . . . . . . . . . 10  |-  ( ( Z  e.  ( ZZ>= ` 
2 )  /\  ( A  e.  Prime  /\  B  e.  Prime ) )  ->  -.  ( Z  x.  A
)  e.  Prime )
20 eleq1 2529 . . . . . . . . . . . . 13  |-  ( ( Z  x.  A )  =  B  ->  (
( Z  x.  A
)  e.  Prime  <->  B  e.  Prime ) )
2120notbid 294 . . . . . . . . . . . 12  |-  ( ( Z  x.  A )  =  B  ->  ( -.  ( Z  x.  A
)  e.  Prime  <->  -.  B  e.  Prime ) )
22 pm2.24 109 . . . . . . . . . . . . . . 15  |-  ( B  e.  Prime  ->  ( -.  B  e.  Prime  ->  A  =  B ) )
2322adantl 466 . . . . . . . . . . . . . 14  |-  ( ( A  e.  Prime  /\  B  e.  Prime )  ->  ( -.  B  e.  Prime  ->  A  =  B )
)
2423adantl 466 . . . . . . . . . . . . 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 228 . . . . . . . . . . 11  |-  ( ( Z  x.  A )  =  B  ->  ( -.  ( Z  x.  A
)  e.  Prime  ->  ( ( Z  e.  (
ZZ>= `  2 )  /\  ( A  e.  Prime  /\  B  e.  Prime )
)  ->  A  =  B ) ) )
2726com3l 81 . . . . . . . . . 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 434 . . . . . . . 8  |-  ( Z  e.  ( ZZ>= `  2
)  ->  ( ( A  e.  Prime  /\  B  e.  Prime )  ->  (
( Z  x.  A
)  =  B  ->  A  =  B )
) )
3015, 29jaoi 379 . . . . . . 7  |-  ( ( Z  =  1  \/  Z  e.  ( ZZ>= ` 
2 ) )  -> 
( ( A  e. 
Prime  /\  B  e.  Prime )  ->  ( ( Z  x.  A )  =  B  ->  A  =  B ) ) )
313, 30sylbi 195 . . . . . 6  |-  ( Z  e.  NN  ->  (
( A  e.  Prime  /\  B  e.  Prime )  ->  ( ( Z  x.  A )  =  B  ->  A  =  B ) ) )
32 oveq1 6303 . . . . . . . . 9  |-  ( Z  =  0  ->  ( Z  x.  A )  =  ( 0  x.  A ) )
3332eqeq1d 2459 . . . . . . . 8  |-  ( Z  =  0  ->  (
( Z  x.  A
)  =  B  <->  ( 0  x.  A )  =  B ) )
34 prmnn 14323 . . . . . . . . . . . . . 14  |-  ( A  e.  Prime  ->  A  e.  NN )
3534nnred 10571 . . . . . . . . . . . . 13  |-  ( A  e.  Prime  ->  A  e.  RR )
36 mul02lem2 9774 . . . . . . . . . . . . 13  |-  ( A  e.  RR  ->  (
0  x.  A )  =  0 )
3735, 36syl 16 . . . . . . . . . . . 12  |-  ( A  e.  Prime  ->  ( 0  x.  A )  =  0 )
3837adantr 465 . . . . . . . . . . 11  |-  ( ( A  e.  Prime  /\  B  e.  Prime )  ->  (
0  x.  A )  =  0 )
3938eqeq1d 2459 . . . . . . . . . 10  |-  ( ( A  e.  Prime  /\  B  e.  Prime )  ->  (
( 0  x.  A
)  =  B  <->  0  =  B ) )
40 prmnn 14323 . . . . . . . . . . . 12  |-  ( B  e.  Prime  ->  B  e.  NN )
41 elnnne0 10830 . . . . . . . . . . . . 13  |-  ( B  e.  NN  <->  ( B  e.  NN0  /\  B  =/=  0 ) )
42 eqneqall 2664 . . . . . . . . . . . . . . . 16  |-  ( B  =  0  ->  ( B  =/=  0  ->  A  =  B ) )
4342eqcoms 2469 . . . . . . . . . . . . . . 15  |-  ( 0  =  B  ->  ( B  =/=  0  ->  A  =  B ) )
4443com12 31 . . . . . . . . . . . . . 14  |-  ( B  =/=  0  ->  (
0  =  B  ->  A  =  B )
)
4544adantl 466 . . . . . . . . . . . . 13  |-  ( ( B  e.  NN0  /\  B  =/=  0 )  -> 
( 0  =  B  ->  A  =  B ) )
4641, 45sylbi 195 . . . . . . . . . . . 12  |-  ( B  e.  NN  ->  (
0  =  B  ->  A  =  B )
)
4740, 46syl 16 . . . . . . . . . . 11  |-  ( B  e.  Prime  ->  ( 0  =  B  ->  A  =  B ) )
4847adantl 466 . . . . . . . . . 10  |-  ( ( A  e.  Prime  /\  B  e.  Prime )  ->  (
0  =  B  ->  A  =  B )
)
4939, 48sylbid 215 . . . . . . . . 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 228 . . . . . . 7  |-  ( Z  =  0  ->  (
( Z  x.  A
)  =  B  -> 
( ( A  e. 
Prime  /\  B  e.  Prime )  ->  A  =  B ) ) )
5251com23 78 . . . . . 6  |-  ( Z  =  0  ->  (
( A  e.  Prime  /\  B  e.  Prime )  ->  ( ( Z  x.  A )  =  B  ->  A  =  B ) ) )
5331, 52jaoi 379 . . . . 5  |-  ( ( Z  e.  NN  \/  Z  =  0 )  ->  ( ( A  e.  Prime  /\  B  e. 
Prime )  ->  ( ( Z  x.  A )  =  B  ->  A  =  B ) ) )
542, 53sylbi 195 . . . 4  |-  ( Z  e.  NN0  ->  ( ( A  e.  Prime  /\  B  e.  Prime )  ->  (
( Z  x.  A
)  =  B  ->  A  =  B )
) )
55 elnnz 10895 . . . . . 6  |-  ( -u Z  e.  NN  <->  ( -u Z  e.  ZZ  /\  0  <  -u Z ) )
56 lt0neg1 10079 . . . . . . . 8  |-  ( Z  e.  RR  ->  ( Z  <  0  <->  0  <  -u Z ) )
5734nngt0d 10600 . . . . . . . . . . . . . . . 16  |-  ( A  e.  Prime  ->  0  < 
A )
5857adantr 465 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  Prime  /\  B  e.  Prime )  ->  0  <  A )
59 simpr 461 . . . . . . . . . . . . . . 15  |-  ( ( Z  e.  RR  /\  Z  <  0 )  ->  Z  <  0 )
6058, 59anim12ci 567 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  Prime  /\  B  e.  Prime )  /\  ( Z  e.  RR  /\  Z  <  0 ) )  ->  ( Z  <  0  /\  0  < 
A ) )
6160orcd 392 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  Prime  /\  B  e.  Prime )  /\  ( Z  e.  RR  /\  Z  <  0 ) )  ->  ( ( Z  <  0  /\  0  <  A )  \/  (
0  <  Z  /\  A  <  0 ) ) )
62 simprl 756 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  Prime  /\  B  e.  Prime )  /\  ( Z  e.  RR  /\  Z  <  0 ) )  ->  Z  e.  RR )
6335adantr 465 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  Prime  /\  B  e.  Prime )  ->  A  e.  RR )
6463adantr 465 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  Prime  /\  B  e.  Prime )  /\  ( Z  e.  RR  /\  Z  <  0 ) )  ->  A  e.  RR )
6562, 64mul2lt0bi 27813 . . . . . . . . . . . . 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 232 . . . . . . . . . . . 12  |-  ( ( ( A  e.  Prime  /\  B  e.  Prime )  /\  ( Z  e.  RR  /\  Z  <  0 ) )  ->  ( Z  x.  A )  <  0
)
6766ex 434 . . . . . . . . . . 11  |-  ( ( A  e.  Prime  /\  B  e.  Prime )  ->  (
( Z  e.  RR  /\  Z  <  0 )  ->  ( Z  x.  A )  <  0
) )
68 breq1 4459 . . . . . . . . . . . . . . 15  |-  ( ( Z  x.  A )  =  B  ->  (
( Z  x.  A
)  <  0  <->  B  <  0 ) )
6968adantl 466 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  Prime  /\  B  e.  Prime )  /\  ( Z  x.  A
)  =  B )  ->  ( ( Z  x.  A )  <  0  <->  B  <  0
) )
70 nnnn0 10823 . . . . . . . . . . . . . . . . . 18  |-  ( B  e.  NN  ->  B  e.  NN0 )
71 nn0nlt0 10843 . . . . . . . . . . . . . . . . . . 19  |-  ( B  e.  NN0  ->  -.  B  <  0 )
7271pm2.21d 106 . . . . . . . . . . . . . . . . . 18  |-  ( B  e.  NN0  ->  ( B  <  0  ->  A  =  B ) )
7370, 72syl 16 . . . . . . . . . . . . . . . . 17  |-  ( B  e.  NN  ->  ( B  <  0  ->  A  =  B ) )
7440, 73syl 16 . . . . . . . . . . . . . . . 16  |-  ( B  e.  Prime  ->  ( B  <  0  ->  A  =  B ) )
7574adantl 466 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  Prime  /\  B  e.  Prime )  ->  ( B  <  0  ->  A  =  B ) )
7675adantr 465 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  Prime  /\  B  e.  Prime )  /\  ( Z  x.  A
)  =  B )  ->  ( B  <  0  ->  A  =  B ) )
7769, 76sylbid 215 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  Prime  /\  B  e.  Prime )  /\  ( Z  x.  A
)  =  B )  ->  ( ( Z  x.  A )  <  0  ->  A  =  B ) )
7877ex 434 . . . . . . . . . . . 12  |-  ( ( A  e.  Prime  /\  B  e.  Prime )  ->  (
( Z  x.  A
)  =  B  -> 
( ( Z  x.  A )  <  0  ->  A  =  B ) ) )
7978com23 78 . . . . . . . . . . 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 434 . . . . . . . 8  |-  ( Z  e.  RR  ->  ( Z  <  0  ->  (
( A  e.  Prime  /\  B  e.  Prime )  ->  ( ( Z  x.  A )  =  B  ->  A  =  B ) ) ) )
8356, 82sylbird 235 . . . . . . 7  |-  ( Z  e.  RR  ->  (
0  <  -u Z  -> 
( ( A  e. 
Prime  /\  B  e.  Prime )  ->  ( ( Z  x.  A )  =  B  ->  A  =  B ) ) ) )
8483adantld 467 . . . . . 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 217 . . . . 5  |-  ( Z  e.  RR  ->  ( -u Z  e.  NN  ->  ( ( A  e.  Prime  /\  B  e.  Prime )  ->  ( ( Z  x.  A )  =  B  ->  A  =  B ) ) ) )
8685imp 429 . . . 4  |-  ( ( Z  e.  RR  /\  -u Z  e.  NN )  ->  ( ( A  e.  Prime  /\  B  e. 
Prime )  ->  ( ( Z  x.  A )  =  B  ->  A  =  B ) ) )
8754, 86jaoi 379 . . 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 195 . 2  |-  ( Z  e.  ZZ  ->  (
( A  e.  Prime  /\  B  e.  Prime )  ->  ( ( Z  x.  A )  =  B  ->  A  =  B ) ) )
89883impib 1194 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 184    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    =/= wne 2652   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   RRcr 9508   0cc0 9509   1c1 9510    x. cmul 9514    < clt 9645   -ucneg 9825   NNcn 10556   2c2 10606   NN0cn0 10816   ZZcz 10885   ZZ>=cuz 11106   Primecprime 14320
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-rp 11246  df-dvds 14090  df-prm 14321
This theorem is referenced by:  zlmodzxznm  33285
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