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Theorem dvdsprime 13775
Description: If  M divides a prime, then  M is either the prime or one. (Contributed by Scott Fenton, 8-Apr-2014.)
Assertion
Ref Expression
dvdsprime  |-  ( ( P  e.  Prime  /\  M  e.  NN )  ->  ( M  ||  P  <->  ( M  =  P  \/  M  =  1 ) ) )

Proof of Theorem dvdsprime
Dummy variable  m is distinct from all other variables.
StepHypRef Expression
1 isprm2 13770 . . 3  |-  ( P  e.  Prime  <->  ( P  e.  ( ZZ>= `  2 )  /\  A. m  e.  NN  ( m  ||  P  -> 
( m  =  1  \/  m  =  P ) ) ) )
2 breq1 4294 . . . . . 6  |-  ( m  =  M  ->  (
m  ||  P  <->  M  ||  P
) )
3 eqeq1 2448 . . . . . . . 8  |-  ( m  =  M  ->  (
m  =  1  <->  M  =  1 ) )
4 eqeq1 2448 . . . . . . . 8  |-  ( m  =  M  ->  (
m  =  P  <->  M  =  P ) )
53, 4orbi12d 709 . . . . . . 7  |-  ( m  =  M  ->  (
( m  =  1  \/  m  =  P )  <->  ( M  =  1  \/  M  =  P ) ) )
6 orcom 387 . . . . . . 7  |-  ( ( M  =  1  \/  M  =  P )  <-> 
( M  =  P  \/  M  =  1 ) )
75, 6syl6bb 261 . . . . . 6  |-  ( m  =  M  ->  (
( m  =  1  \/  m  =  P )  <->  ( M  =  P  \/  M  =  1 ) ) )
82, 7imbi12d 320 . . . . 5  |-  ( m  =  M  ->  (
( m  ||  P  ->  ( m  =  1  \/  m  =  P ) )  <->  ( M  ||  P  ->  ( M  =  P  \/  M  =  1 ) ) ) )
98rspccva 3071 . . . 4  |-  ( ( A. m  e.  NN  ( m  ||  P  -> 
( m  =  1  \/  m  =  P ) )  /\  M  e.  NN )  ->  ( M  ||  P  ->  ( M  =  P  \/  M  =  1 ) ) )
109adantll 713 . . 3  |-  ( ( ( P  e.  (
ZZ>= `  2 )  /\  A. m  e.  NN  (
m  ||  P  ->  ( m  =  1  \/  m  =  P ) ) )  /\  M  e.  NN )  ->  ( M  ||  P  ->  ( M  =  P  \/  M  =  1 ) ) )
111, 10sylanb 472 . 2  |-  ( ( P  e.  Prime  /\  M  e.  NN )  ->  ( M  ||  P  ->  ( M  =  P  \/  M  =  1 ) ) )
12 prmz 13766 . . . . . 6  |-  ( P  e.  Prime  ->  P  e.  ZZ )
13 iddvds 13545 . . . . . 6  |-  ( P  e.  ZZ  ->  P  ||  P )
1412, 13syl 16 . . . . 5  |-  ( P  e.  Prime  ->  P  ||  P )
1514adantr 465 . . . 4  |-  ( ( P  e.  Prime  /\  M  e.  NN )  ->  P  ||  P )
16 breq1 4294 . . . 4  |-  ( M  =  P  ->  ( M  ||  P  <->  P  ||  P
) )
1715, 16syl5ibrcom 222 . . 3  |-  ( ( P  e.  Prime  /\  M  e.  NN )  ->  ( M  =  P  ->  M 
||  P ) )
18 1dvds 13546 . . . . . 6  |-  ( P  e.  ZZ  ->  1  ||  P )
1912, 18syl 16 . . . . 5  |-  ( P  e.  Prime  ->  1  ||  P )
2019adantr 465 . . . 4  |-  ( ( P  e.  Prime  /\  M  e.  NN )  ->  1  ||  P )
21 breq1 4294 . . . 4  |-  ( M  =  1  ->  ( M  ||  P  <->  1  ||  P ) )
2220, 21syl5ibrcom 222 . . 3  |-  ( ( P  e.  Prime  /\  M  e.  NN )  ->  ( M  =  1  ->  M 
||  P ) )
2317, 22jaod 380 . 2  |-  ( ( P  e.  Prime  /\  M  e.  NN )  ->  (
( M  =  P  \/  M  =  1 )  ->  M  ||  P
) )
2411, 23impbid 191 1  |-  ( ( P  e.  Prime  /\  M  e.  NN )  ->  ( M  ||  P  <->  ( M  =  P  \/  M  =  1 ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2714   class class class wbr 4291   ` cfv 5417   1c1 9282   NNcn 10321   2c2 10370   ZZcz 10645   ZZ>=cuz 10860    || cdivides 13534   Primecprime 13762
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-int 4128  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  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-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-recs 6831  df-rdg 6865  df-1o 6919  df-2o 6920  df-oadd 6923  df-er 7100  df-en 7310  df-dom 7311  df-sdom 7312  df-fin 7313  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-nn 10322  df-2 10379  df-n0 10579  df-z 10646  df-uz 10861  df-dvds 13535  df-prm 13763
This theorem is referenced by:  pythagtriplem4  13885  odcau  16102  prmcyg  16369
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