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Theorem dvdsprime 13020
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 13015 . . 3  |-  ( P  e.  Prime  <->  ( P  e.  ( ZZ>= `  2 )  /\  A. m  e.  NN  ( m  ||  P  -> 
( m  =  1  \/  m  =  P ) ) ) )
2 breq1 4157 . . . . . 6  |-  ( m  =  M  ->  (
m  ||  P  <->  M  ||  P
) )
3 eqeq1 2394 . . . . . . . 8  |-  ( m  =  M  ->  (
m  =  1  <->  M  =  1 ) )
4 eqeq1 2394 . . . . . . . 8  |-  ( m  =  M  ->  (
m  =  P  <->  M  =  P ) )
53, 4orbi12d 691 . . . . . . 7  |-  ( m  =  M  ->  (
( m  =  1  \/  m  =  P )  <->  ( M  =  1  \/  M  =  P ) ) )
6 orcom 377 . . . . . . 7  |-  ( ( M  =  1  \/  M  =  P )  <-> 
( M  =  P  \/  M  =  1 ) )
75, 6syl6bb 253 . . . . . 6  |-  ( m  =  M  ->  (
( m  =  1  \/  m  =  P )  <->  ( M  =  P  \/  M  =  1 ) ) )
82, 7imbi12d 312 . . . . 5  |-  ( m  =  M  ->  (
( m  ||  P  ->  ( m  =  1  \/  m  =  P ) )  <->  ( M  ||  P  ->  ( M  =  P  \/  M  =  1 ) ) ) )
98rspccva 2995 . . . 4  |-  ( ( A. m  e.  NN  ( m  ||  P  -> 
( m  =  1  \/  m  =  P ) )  /\  M  e.  NN )  ->  ( M  ||  P  ->  ( M  =  P  \/  M  =  1 ) ) )
109adantll 695 . . 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 459 . 2  |-  ( ( P  e.  Prime  /\  M  e.  NN )  ->  ( M  ||  P  ->  ( M  =  P  \/  M  =  1 ) ) )
12 prmz 13011 . . . . . 6  |-  ( P  e.  Prime  ->  P  e.  ZZ )
13 iddvds 12791 . . . . . 6  |-  ( P  e.  ZZ  ->  P  ||  P )
1412, 13syl 16 . . . . 5  |-  ( P  e.  Prime  ->  P  ||  P )
1514adantr 452 . . . 4  |-  ( ( P  e.  Prime  /\  M  e.  NN )  ->  P  ||  P )
16 breq1 4157 . . . 4  |-  ( M  =  P  ->  ( M  ||  P  <->  P  ||  P
) )
1715, 16syl5ibrcom 214 . . 3  |-  ( ( P  e.  Prime  /\  M  e.  NN )  ->  ( M  =  P  ->  M 
||  P ) )
18 1dvds 12792 . . . . . 6  |-  ( P  e.  ZZ  ->  1  ||  P )
1912, 18syl 16 . . . . 5  |-  ( P  e.  Prime  ->  1  ||  P )
2019adantr 452 . . . 4  |-  ( ( P  e.  Prime  /\  M  e.  NN )  ->  1  ||  P )
21 breq1 4157 . . . 4  |-  ( M  =  1  ->  ( M  ||  P  <->  1  ||  P ) )
2220, 21syl5ibrcom 214 . . 3  |-  ( ( P  e.  Prime  /\  M  e.  NN )  ->  ( M  =  1  ->  M 
||  P ) )
2317, 22jaod 370 . 2  |-  ( ( P  e.  Prime  /\  M  e.  NN )  ->  (
( M  =  P  \/  M  =  1 )  ->  M  ||  P
) )
2411, 23impbid 184 1  |-  ( ( P  e.  Prime  /\  M  e.  NN )  ->  ( M  ||  P  <->  ( M  =  P  \/  M  =  1 ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2650   class class class wbr 4154   ` cfv 5395   1c1 8925   NNcn 9933   2c2 9982   ZZcz 10215   ZZ>=cuz 10421    || cdivides 12780   Primecprime 13007
This theorem is referenced by:  pythagtriplem4  13121  odcau  15166  prmcyg  15431
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-riota 6486  df-recs 6570  df-rdg 6605  df-1o 6661  df-2o 6662  df-oadd 6665  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-nn 9934  df-2 9991  df-n0 10155  df-z 10216  df-uz 10422  df-dvds 12781  df-prm 13008
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