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Theorem dvdsprime 14102
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 14097 . . 3  |-  ( P  e.  Prime  <->  ( P  e.  ( ZZ>= `  2 )  /\  A. m  e.  NN  ( m  ||  P  -> 
( m  =  1  \/  m  =  P ) ) ) )
2 breq1 4436 . . . . . 6  |-  ( m  =  M  ->  (
m  ||  P  <->  M  ||  P
) )
3 eqeq1 2445 . . . . . . . 8  |-  ( m  =  M  ->  (
m  =  1  <->  M  =  1 ) )
4 eqeq1 2445 . . . . . . . 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 3193 . . . 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 14093 . . . . . 6  |-  ( P  e.  Prime  ->  P  e.  ZZ )
13 iddvds 13869 . . . . . 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 4436 . . . 4  |-  ( M  =  P  ->  ( M  ||  P  <->  P  ||  P
) )
1715, 16syl5ibrcom 222 . . 3  |-  ( ( P  e.  Prime  /\  M  e.  NN )  ->  ( M  =  P  ->  M 
||  P ) )
18 1dvds 13870 . . . . . 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 4436 . . . 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 1381    e. wcel 1802   A.wral 2791   class class class wbr 4433   ` cfv 5574   1c1 9491   NNcn 10537   2c2 10586   ZZcz 10865   ZZ>=cuz 11085    || cdvds 13858   Primecprime 14089
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-recs 7040  df-rdg 7074  df-1o 7128  df-2o 7129  df-oadd 7132  df-er 7309  df-en 7515  df-dom 7516  df-sdom 7517  df-fin 7518  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-nn 10538  df-2 10595  df-n0 10797  df-z 10866  df-uz 11086  df-dvds 13859  df-prm 14090
This theorem is referenced by:  pythagtriplem4  14215  odcau  16493  prmcyg  16765
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