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Theorem gcdcllem2 12691
Description: Lemma for gcdn0cl 12693, gcddvds 12694 and dvdslegcd 12695. (Contributed by Paul Chapman, 21-Mar-2011.)
Hypotheses
Ref Expression
gcdcllem2.1  |-  S  =  { z  e.  ZZ  |  A. n  e.  { M ,  N }
z  ||  n }
gcdcllem2.2  |-  R  =  { z  e.  ZZ  |  ( z  ||  M  /\  z  ||  N
) }
Assertion
Ref Expression
gcdcllem2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  R  =  S )
Distinct variable groups:    z, n, M    n, N, z
Allowed substitution hints:    R( z, n)    S( z, n)

Proof of Theorem gcdcllem2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 breq1 4026 . . . . . 6  |-  ( z  =  x  ->  (
z  ||  n  <->  x  ||  n
) )
21ralbidv 2563 . . . . 5  |-  ( z  =  x  ->  ( A. n  e.  { M ,  N } z  ||  n 
<-> 
A. n  e.  { M ,  N }
x  ||  n )
)
3 gcdcllem2.1 . . . . 5  |-  S  =  { z  e.  ZZ  |  A. n  e.  { M ,  N }
z  ||  n }
42, 3elrab2 2925 . . . 4  |-  ( x  e.  S  <->  ( x  e.  ZZ  /\  A. n  e.  { M ,  N } x  ||  n ) )
5 breq2 4027 . . . . . 6  |-  ( n  =  M  ->  (
x  ||  n  <->  x  ||  M
) )
6 breq2 4027 . . . . . 6  |-  ( n  =  N  ->  (
x  ||  n  <->  x  ||  N
) )
75, 6ralprg 3682 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( A. n  e. 
{ M ,  N } x  ||  n  <->  ( x  ||  M  /\  x  ||  N ) ) )
87anbi2d 684 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( x  e.  ZZ  /\  A. n  e.  { M ,  N } x  ||  n )  <-> 
( x  e.  ZZ  /\  ( x  ||  M  /\  x  ||  N ) ) ) )
94, 8syl5bb 248 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( x  e.  S  <->  ( x  e.  ZZ  /\  ( x  ||  M  /\  x  ||  N ) ) ) )
10 breq1 4026 . . . . 5  |-  ( z  =  x  ->  (
z  ||  M  <->  x  ||  M
) )
11 breq1 4026 . . . . 5  |-  ( z  =  x  ->  (
z  ||  N  <->  x  ||  N
) )
1210, 11anbi12d 691 . . . 4  |-  ( z  =  x  ->  (
( z  ||  M  /\  z  ||  N )  <-> 
( x  ||  M  /\  x  ||  N ) ) )
13 gcdcllem2.2 . . . 4  |-  R  =  { z  e.  ZZ  |  ( z  ||  M  /\  z  ||  N
) }
1412, 13elrab2 2925 . . 3  |-  ( x  e.  R  <->  ( x  e.  ZZ  /\  ( x 
||  M  /\  x  ||  N ) ) )
159, 14syl6rbbr 255 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( x  e.  R  <->  x  e.  S ) )
1615eqrdv 2281 1  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  R  =  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547   {cpr 3641   class class class wbr 4023   ZZcz 10024    || cdivides 12531
This theorem is referenced by:  gcdcllem3  12692
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024
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