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Theorem dvdsprime 14317
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 14312 . . 3  |-  ( P  e.  Prime  <->  ( P  e.  ( ZZ>= `  2 )  /\  A. m  e.  NN  ( m  ||  P  -> 
( m  =  1  \/  m  =  P ) ) ) )
2 breq1 4442 . . . . . 6  |-  ( m  =  M  ->  (
m  ||  P  <->  M  ||  P
) )
3 eqeq1 2458 . . . . . . . 8  |-  ( m  =  M  ->  (
m  =  1  <->  M  =  1 ) )
4 eqeq1 2458 . . . . . . . 8  |-  ( m  =  M  ->  (
m  =  P  <->  M  =  P ) )
53, 4orbi12d 707 . . . . . . 7  |-  ( m  =  M  ->  (
( m  =  1  \/  m  =  P )  <->  ( M  =  1  \/  M  =  P ) ) )
6 orcom 385 . . . . . . 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 318 . . . . 5  |-  ( m  =  M  ->  (
( m  ||  P  ->  ( m  =  1  \/  m  =  P ) )  <->  ( M  ||  P  ->  ( M  =  P  \/  M  =  1 ) ) ) )
98rspccva 3206 . . . 4  |-  ( ( A. m  e.  NN  ( m  ||  P  -> 
( m  =  1  \/  m  =  P ) )  /\  M  e.  NN )  ->  ( M  ||  P  ->  ( M  =  P  \/  M  =  1 ) ) )
109adantll 711 . . 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 470 . 2  |-  ( ( P  e.  Prime  /\  M  e.  NN )  ->  ( M  ||  P  ->  ( M  =  P  \/  M  =  1 ) ) )
12 prmz 14308 . . . . . 6  |-  ( P  e.  Prime  ->  P  e.  ZZ )
13 iddvds 14084 . . . . . 6  |-  ( P  e.  ZZ  ->  P  ||  P )
1412, 13syl 16 . . . . 5  |-  ( P  e.  Prime  ->  P  ||  P )
1514adantr 463 . . . 4  |-  ( ( P  e.  Prime  /\  M  e.  NN )  ->  P  ||  P )
16 breq1 4442 . . . 4  |-  ( M  =  P  ->  ( M  ||  P  <->  P  ||  P
) )
1715, 16syl5ibrcom 222 . . 3  |-  ( ( P  e.  Prime  /\  M  e.  NN )  ->  ( M  =  P  ->  M 
||  P ) )
18 1dvds 14085 . . . . . 6  |-  ( P  e.  ZZ  ->  1  ||  P )
1912, 18syl 16 . . . . 5  |-  ( P  e.  Prime  ->  1  ||  P )
2019adantr 463 . . . 4  |-  ( ( P  e.  Prime  /\  M  e.  NN )  ->  1  ||  P )
21 breq1 4442 . . . 4  |-  ( M  =  1  ->  ( M  ||  P  <->  1  ||  P ) )
2220, 21syl5ibrcom 222 . . 3  |-  ( ( P  e.  Prime  /\  M  e.  NN )  ->  ( M  =  1  ->  M 
||  P ) )
2317, 22jaod 378 . 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 366    /\ wa 367    = wceq 1398    e. wcel 1823   A.wral 2804   class class class wbr 4439   ` cfv 5570   1c1 9482   NNcn 10531   2c2 10581   ZZcz 10860   ZZ>=cuz 11082    || cdvds 14073   Primecprime 14304
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-dvds 14074  df-prm 14305
This theorem is referenced by:  pythagtriplem4  14430  odcau  16826  prmcyg  17098  oddprmALTV  32601
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