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Theorem dfgcd3 11605
Description: Alternate definition of the  gcd operator. (Contributed by Jim Kingdon, 31-Dec-2021.)
Assertion
Ref Expression
dfgcd3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N
)  =  ( iota_ d  e.  NN0  A. z  e.  ZZ  ( z  ||  d 
<->  ( z  ||  M  /\  z  ||  N ) ) ) )
Distinct variable groups:    M, d, z    N, d, z

Proof of Theorem dfgcd3
Dummy variables  a  b  r  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gcd0val 11556 . . 3  |-  ( 0  gcd  0 )  =  0
2 simprl 503 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  /\  N  =  0 ) )  ->  M  =  0 )
3 simprr 504 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  /\  N  =  0 ) )  ->  N  =  0 )
42, 3oveq12d 5758 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  /\  N  =  0 ) )  -> 
( M  gcd  N
)  =  ( 0  gcd  0 ) )
5 0nn0 8946 . . . . 5  |-  0  e.  NN0
65a1i 9 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  /\  N  =  0 ) )  -> 
0  e.  NN0 )
7 0dvds 11420 . . . . . . . . . . 11  |-  ( M  e.  ZZ  ->  (
0  ||  M  <->  M  = 
0 ) )
87ad2antrr 477 . . . . . . . . . 10  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  /\  N  =  0 ) )  -> 
( 0  ||  M  <->  M  =  0 ) )
92, 8mpbird 166 . . . . . . . . 9  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  /\  N  =  0 ) )  -> 
0  ||  M )
10 0dvds 11420 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  (
0  ||  N  <->  N  = 
0 ) )
1110ad2antlr 478 . . . . . . . . . 10  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  /\  N  =  0 ) )  -> 
( 0  ||  N  <->  N  =  0 ) )
123, 11mpbird 166 . . . . . . . . 9  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  /\  N  =  0 ) )  -> 
0  ||  N )
139, 12jca 302 . . . . . . . 8  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  /\  N  =  0 ) )  -> 
( 0  ||  M  /\  0  ||  N ) )
1413ad2antrr 477 . . . . . . 7  |-  ( ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  /\  N  =  0 ) )  /\  d  e. 
NN0 )  /\  A. z  e.  ZZ  (
z  ||  d  <->  ( z  ||  M  /\  z  ||  N ) ) )  ->  ( 0  ||  M  /\  0  ||  N
) )
15 0z 9019 . . . . . . . . 9  |-  0  e.  ZZ
16 breq1 3900 . . . . . . . . . . 11  |-  ( z  =  0  ->  (
z  ||  d  <->  0  ||  d ) )
17 breq1 3900 . . . . . . . . . . . 12  |-  ( z  =  0  ->  (
z  ||  M  <->  0  ||  M ) )
18 breq1 3900 . . . . . . . . . . . 12  |-  ( z  =  0  ->  (
z  ||  N  <->  0  ||  N ) )
1917, 18anbi12d 462 . . . . . . . . . . 11  |-  ( z  =  0  ->  (
( z  ||  M  /\  z  ||  N )  <-> 
( 0  ||  M  /\  0  ||  N ) ) )
2016, 19bibi12d 234 . . . . . . . . . 10  |-  ( z  =  0  ->  (
( z  ||  d  <->  ( z  ||  M  /\  z  ||  N ) )  <-> 
( 0  ||  d  <->  ( 0  ||  M  /\  0  ||  N ) ) ) )
2120rspcv 2757 . . . . . . . . 9  |-  ( 0  e.  ZZ  ->  ( A. z  e.  ZZ  ( z  ||  d  <->  ( z  ||  M  /\  z  ||  N ) )  ->  ( 0  ||  d 
<->  ( 0  ||  M  /\  0  ||  N ) ) ) )
2215, 21ax-mp 5 . . . . . . . 8  |-  ( A. z  e.  ZZ  (
z  ||  d  <->  ( z  ||  M  /\  z  ||  N ) )  -> 
( 0  ||  d  <->  ( 0  ||  M  /\  0  ||  N ) ) )
2322adantl 273 . . . . . . 7  |-  ( ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  /\  N  =  0 ) )  /\  d  e. 
NN0 )  /\  A. z  e.  ZZ  (
z  ||  d  <->  ( z  ||  M  /\  z  ||  N ) ) )  ->  ( 0  ||  d 
<->  ( 0  ||  M  /\  0  ||  N ) ) )
2414, 23mpbird 166 . . . . . 6  |-  ( ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  /\  N  =  0 ) )  /\  d  e. 
NN0 )  /\  A. z  e.  ZZ  (
z  ||  d  <->  ( z  ||  M  /\  z  ||  N ) ) )  ->  0  ||  d
)
25 simplr 502 . . . . . . . 8  |-  ( ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  /\  N  =  0 ) )  /\  d  e. 
NN0 )  /\  A. z  e.  ZZ  (
z  ||  d  <->  ( z  ||  M  /\  z  ||  N ) ) )  ->  d  e.  NN0 )
2625nn0zd 9125 . . . . . . 7  |-  ( ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  /\  N  =  0 ) )  /\  d  e. 
NN0 )  /\  A. z  e.  ZZ  (
z  ||  d  <->  ( z  ||  M  /\  z  ||  N ) ) )  ->  d  e.  ZZ )
27 0dvds 11420 . . . . . . 7  |-  ( d  e.  ZZ  ->  (
0  ||  d  <->  d  = 
0 ) )
2826, 27syl 14 . . . . . 6  |-  ( ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  /\  N  =  0 ) )  /\  d  e. 
NN0 )  /\  A. z  e.  ZZ  (
z  ||  d  <->  ( z  ||  M  /\  z  ||  N ) ) )  ->  ( 0  ||  d 
<->  d  =  0 ) )
2924, 28mpbid 146 . . . . 5  |-  ( ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  /\  N  =  0 ) )  /\  d  e. 
NN0 )  /\  A. z  e.  ZZ  (
z  ||  d  <->  ( z  ||  M  /\  z  ||  N ) ) )  ->  d  =  0 )
30 dvds0 11415 . . . . . . . . 9  |-  ( z  e.  ZZ  ->  z  ||  0 )
3130adantl 273 . . . . . . . 8  |-  ( ( ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  /\  N  =  0 ) )  /\  d  e.  NN0 )  /\  d  =  0 )  /\  z  e.  ZZ )  ->  z  ||  0 )
32 breq2 3901 . . . . . . . . 9  |-  ( d  =  0  ->  (
z  ||  d  <->  z  ||  0 ) )
3332ad2antlr 478 . . . . . . . 8  |-  ( ( ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  /\  N  =  0 ) )  /\  d  e.  NN0 )  /\  d  =  0 )  /\  z  e.  ZZ )  ->  ( z  ||  d  <->  z 
||  0 ) )
3431, 33mpbird 166 . . . . . . 7  |-  ( ( ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  /\  N  =  0 ) )  /\  d  e.  NN0 )  /\  d  =  0 )  /\  z  e.  ZZ )  ->  z  ||  d )
352ad3antrrr 481 . . . . . . . . 9  |-  ( ( ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  /\  N  =  0 ) )  /\  d  e.  NN0 )  /\  d  =  0 )  /\  z  e.  ZZ )  ->  M  =  0 )
3631, 35breqtrrd 3924 . . . . . . . 8  |-  ( ( ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  /\  N  =  0 ) )  /\  d  e.  NN0 )  /\  d  =  0 )  /\  z  e.  ZZ )  ->  z  ||  M )
373ad3antrrr 481 . . . . . . . . 9  |-  ( ( ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  /\  N  =  0 ) )  /\  d  e.  NN0 )  /\  d  =  0 )  /\  z  e.  ZZ )  ->  N  =  0 )
3831, 37breqtrrd 3924 . . . . . . . 8  |-  ( ( ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  /\  N  =  0 ) )  /\  d  e.  NN0 )  /\  d  =  0 )  /\  z  e.  ZZ )  ->  z  ||  N )
3936, 38jca 302 . . . . . . 7  |-  ( ( ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  /\  N  =  0 ) )  /\  d  e.  NN0 )  /\  d  =  0 )  /\  z  e.  ZZ )  ->  ( z  ||  M  /\  z  ||  N ) )
4034, 392thd 174 . . . . . 6  |-  ( ( ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  /\  N  =  0 ) )  /\  d  e.  NN0 )  /\  d  =  0 )  /\  z  e.  ZZ )  ->  ( z  ||  d  <->  ( z  ||  M  /\  z  ||  N ) ) )
4140ralrimiva 2480 . . . . 5  |-  ( ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  /\  N  =  0 ) )  /\  d  e. 
NN0 )  /\  d  =  0 )  ->  A. z  e.  ZZ  ( z  ||  d  <->  ( z  ||  M  /\  z  ||  N ) ) )
4229, 41impbida 568 . . . 4  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  /\  N  =  0 ) )  /\  d  e. 
NN0 )  ->  ( A. z  e.  ZZ  ( z  ||  d  <->  ( z  ||  M  /\  z  ||  N ) )  <-> 
d  =  0 ) )
436, 42riota5 5721 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  /\  N  =  0 ) )  -> 
( iota_ d  e.  NN0  A. z  e.  ZZ  (
z  ||  d  <->  ( z  ||  M  /\  z  ||  N ) ) )  =  0 )
441, 4, 433eqtr4a 2174 . 2  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  /\  N  =  0 ) )  -> 
( M  gcd  N
)  =  ( iota_ d  e.  NN0  A. z  e.  ZZ  ( z  ||  d 
<->  ( z  ||  M  /\  z  ||  N ) ) ) )
45 bezoutlembi 11600 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  E. r  e.  NN0  ( A. w  e.  ZZ  ( w  ||  r  <->  ( w  ||  M  /\  w  ||  N ) )  /\  E. a  e.  ZZ  E. b  e.  ZZ  r  =  ( ( M  x.  a )  +  ( N  x.  b
) ) ) )
46 simpl 108 . . . . . 6  |-  ( ( A. w  e.  ZZ  ( w  ||  r  <->  ( w  ||  M  /\  w  ||  N ) )  /\  E. a  e.  ZZ  E. b  e.  ZZ  r  =  ( ( M  x.  a )  +  ( N  x.  b
) ) )  ->  A. w  e.  ZZ  ( w  ||  r  <->  ( w  ||  M  /\  w  ||  N ) ) )
4746reximi 2504 . . . . 5  |-  ( E. r  e.  NN0  ( A. w  e.  ZZ  ( w  ||  r  <->  ( w  ||  M  /\  w  ||  N ) )  /\  E. a  e.  ZZ  E. b  e.  ZZ  r  =  ( ( M  x.  a )  +  ( N  x.  b
) ) )  ->  E. r  e.  NN0  A. w  e.  ZZ  (
w  ||  r  <->  ( w  ||  M  /\  w  ||  N ) ) )
4845, 47syl 14 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  E. r  e.  NN0  A. w  e.  ZZ  (
w  ||  r  <->  ( w  ||  M  /\  w  ||  N ) ) )
4948adantr 272 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  E. r  e.  NN0  A. w  e.  ZZ  (
w  ||  r  <->  ( w  ||  M  /\  w  ||  N ) ) )
50 simplll 505 . . . . 5  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0
) )  /\  (
r  e.  NN0  /\  A. w  e.  ZZ  (
w  ||  r  <->  ( w  ||  M  /\  w  ||  N ) ) ) )  ->  M  e.  ZZ )
51 simpllr 506 . . . . 5  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0
) )  /\  (
r  e.  NN0  /\  A. w  e.  ZZ  (
w  ||  r  <->  ( w  ||  M  /\  w  ||  N ) ) ) )  ->  N  e.  ZZ )
52 simprl 503 . . . . 5  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0
) )  /\  (
r  e.  NN0  /\  A. w  e.  ZZ  (
w  ||  r  <->  ( w  ||  M  /\  w  ||  N ) ) ) )  ->  r  e.  NN0 )
53 breq1 3900 . . . . . . . . 9  |-  ( w  =  z  ->  (
w  ||  r  <->  z  ||  r ) )
54 breq1 3900 . . . . . . . . . 10  |-  ( w  =  z  ->  (
w  ||  M  <->  z  ||  M ) )
55 breq1 3900 . . . . . . . . . 10  |-  ( w  =  z  ->  (
w  ||  N  <->  z  ||  N ) )
5654, 55anbi12d 462 . . . . . . . . 9  |-  ( w  =  z  ->  (
( w  ||  M  /\  w  ||  N )  <-> 
( z  ||  M  /\  z  ||  N ) ) )
5753, 56bibi12d 234 . . . . . . . 8  |-  ( w  =  z  ->  (
( w  ||  r  <->  ( w  ||  M  /\  w  ||  N ) )  <-> 
( z  ||  r  <->  ( z  ||  M  /\  z  ||  N ) ) ) )
5857cbvralv 2629 . . . . . . 7  |-  ( A. w  e.  ZZ  (
w  ||  r  <->  ( w  ||  M  /\  w  ||  N ) )  <->  A. z  e.  ZZ  ( z  ||  r 
<->  ( z  ||  M  /\  z  ||  N ) ) )
5958biimpi 119 . . . . . 6  |-  ( A. w  e.  ZZ  (
w  ||  r  <->  ( w  ||  M  /\  w  ||  N ) )  ->  A. z  e.  ZZ  ( z  ||  r  <->  ( z  ||  M  /\  z  ||  N ) ) )
6059ad2antll 480 . . . . 5  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0
) )  /\  (
r  e.  NN0  /\  A. w  e.  ZZ  (
w  ||  r  <->  ( w  ||  M  /\  w  ||  N ) ) ) )  ->  A. z  e.  ZZ  ( z  ||  r 
<->  ( z  ||  M  /\  z  ||  N ) ) )
61 simplr 502 . . . . 5  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0
) )  /\  (
r  e.  NN0  /\  A. w  e.  ZZ  (
w  ||  r  <->  ( w  ||  M  /\  w  ||  N ) ) ) )  ->  -.  ( M  =  0  /\  N  =  0 ) )
6250, 51, 52, 60, 61bezoutlemsup 11604 . . . 4  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0
) )  /\  (
r  e.  NN0  /\  A. w  e.  ZZ  (
w  ||  r  <->  ( w  ||  M  /\  w  ||  N ) ) ) )  ->  r  =  sup ( { z  e.  ZZ  |  ( z 
||  M  /\  z  ||  N ) } ,  RR ,  <  ) )
63 breq1 3900 . . . . . . . . 9  |-  ( w  =  z  ->  (
w  ||  d  <->  z  ||  d ) )
6463, 56bibi12d 234 . . . . . . . 8  |-  ( w  =  z  ->  (
( w  ||  d  <->  ( w  ||  M  /\  w  ||  N ) )  <-> 
( z  ||  d  <->  ( z  ||  M  /\  z  ||  N ) ) ) )
6564cbvralv 2629 . . . . . . 7  |-  ( A. w  e.  ZZ  (
w  ||  d  <->  ( w  ||  M  /\  w  ||  N ) )  <->  A. z  e.  ZZ  ( z  ||  d 
<->  ( z  ||  M  /\  z  ||  N ) ) )
6665a1i 9 . . . . . 6  |-  ( d  e.  NN0  ->  ( A. w  e.  ZZ  (
w  ||  d  <->  ( w  ||  M  /\  w  ||  N ) )  <->  A. z  e.  ZZ  ( z  ||  d 
<->  ( z  ||  M  /\  z  ||  N ) ) ) )
6766riotabiia 5713 . . . . 5  |-  ( iota_ d  e.  NN0  A. w  e.  ZZ  ( w  ||  d 
<->  ( w  ||  M  /\  w  ||  N ) ) )  =  (
iota_ d  e.  NN0  A. z  e.  ZZ  (
z  ||  d  <->  ( z  ||  M  /\  z  ||  N ) ) )
68 simprr 504 . . . . . 6  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0
) )  /\  (
r  e.  NN0  /\  A. w  e.  ZZ  (
w  ||  r  <->  ( w  ||  M  /\  w  ||  N ) ) ) )  ->  A. w  e.  ZZ  ( w  ||  r 
<->  ( w  ||  M  /\  w  ||  N ) ) )
6950, 51, 52, 68bezoutlemeu 11602 . . . . . . 7  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0
) )  /\  (
r  e.  NN0  /\  A. w  e.  ZZ  (
w  ||  r  <->  ( w  ||  M  /\  w  ||  N ) ) ) )  ->  E! d  e.  NN0  A. w  e.  ZZ  ( w  ||  d 
<->  ( w  ||  M  /\  w  ||  N ) ) )
70 breq2 3901 . . . . . . . . . 10  |-  ( d  =  r  ->  (
w  ||  d  <->  w  ||  r
) )
7170bibi1d 232 . . . . . . . . 9  |-  ( d  =  r  ->  (
( w  ||  d  <->  ( w  ||  M  /\  w  ||  N ) )  <-> 
( w  ||  r  <->  ( w  ||  M  /\  w  ||  N ) ) ) )
7271ralbidv 2412 . . . . . . . 8  |-  ( d  =  r  ->  ( A. w  e.  ZZ  ( w  ||  d  <->  ( w  ||  M  /\  w  ||  N ) )  <->  A. w  e.  ZZ  ( w  ||  r 
<->  ( w  ||  M  /\  w  ||  N ) ) ) )
7372riota2 5718 . . . . . . 7  |-  ( ( r  e.  NN0  /\  E! d  e.  NN0  A. w  e.  ZZ  (
w  ||  d  <->  ( w  ||  M  /\  w  ||  N ) ) )  ->  ( A. w  e.  ZZ  ( w  ||  r 
<->  ( w  ||  M  /\  w  ||  N ) )  <->  ( iota_ d  e. 
NN0  A. w  e.  ZZ  ( w  ||  d  <->  ( w  ||  M  /\  w  ||  N ) ) )  =  r ) )
7452, 69, 73syl2anc 406 . . . . . 6  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0
) )  /\  (
r  e.  NN0  /\  A. w  e.  ZZ  (
w  ||  r  <->  ( w  ||  M  /\  w  ||  N ) ) ) )  ->  ( A. w  e.  ZZ  (
w  ||  r  <->  ( w  ||  M  /\  w  ||  N ) )  <->  ( iota_ d  e.  NN0  A. w  e.  ZZ  ( w  ||  d 
<->  ( w  ||  M  /\  w  ||  N ) ) )  =  r ) )
7568, 74mpbid 146 . . . . 5  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0
) )  /\  (
r  e.  NN0  /\  A. w  e.  ZZ  (
w  ||  r  <->  ( w  ||  M  /\  w  ||  N ) ) ) )  ->  ( iota_ d  e.  NN0  A. w  e.  ZZ  ( w  ||  d 
<->  ( w  ||  M  /\  w  ||  N ) ) )  =  r )
7667, 75syl5eqr 2162 . . . 4  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0
) )  /\  (
r  e.  NN0  /\  A. w  e.  ZZ  (
w  ||  r  <->  ( w  ||  M  /\  w  ||  N ) ) ) )  ->  ( iota_ d  e.  NN0  A. z  e.  ZZ  ( z  ||  d 
<->  ( z  ||  M  /\  z  ||  N ) ) )  =  r )
77 gcdn0val 11557 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  ( M  gcd  N )  =  sup ( { z  e.  ZZ  |  ( z  ||  M  /\  z  ||  N
) } ,  RR ,  <  ) )
7877adantr 272 . . . 4  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0
) )  /\  (
r  e.  NN0  /\  A. w  e.  ZZ  (
w  ||  r  <->  ( w  ||  M  /\  w  ||  N ) ) ) )  ->  ( M  gcd  N )  =  sup ( { z  e.  ZZ  |  ( z  ||  M  /\  z  ||  N
) } ,  RR ,  <  ) )
7962, 76, 783eqtr4rd 2159 . . 3  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0
) )  /\  (
r  e.  NN0  /\  A. w  e.  ZZ  (
w  ||  r  <->  ( w  ||  M  /\  w  ||  N ) ) ) )  ->  ( M  gcd  N )  =  (
iota_ d  e.  NN0  A. z  e.  ZZ  (
z  ||  d  <->  ( z  ||  M  /\  z  ||  N ) ) ) )
8049, 79rexlimddv 2529 . 2  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  ( M  gcd  N )  =  ( iota_ d  e.  NN0  A. z  e.  ZZ  ( z  ||  d 
<->  ( z  ||  M  /\  z  ||  N ) ) ) )
81 gcdmndc 11544 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  -> DECID  ( M  =  0  /\  N  =  0 ) )
82 exmiddc 804 . . 3  |-  (DECID  ( M  =  0  /\  N  =  0 )  -> 
( ( M  =  0  /\  N  =  0 )  \/  -.  ( M  =  0  /\  N  =  0
) ) )
8381, 82syl 14 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M  =  0  /\  N  =  0 )  \/  -.  ( M  =  0  /\  N  =  0
) ) )
8444, 80, 83mpjaodan 770 1  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N
)  =  ( iota_ d  e.  NN0  A. z  e.  ZZ  ( z  ||  d 
<->  ( z  ||  M  /\  z  ||  N ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 680  DECID wdc 802    = wceq 1314    e. wcel 1463   A.wral 2391   E.wrex 2392   E!wreu 2393   {crab 2395   class class class wbr 3897   iota_crio 5695  (class class class)co 5740   supcsup 6835   RRcr 7583   0cc0 7584    + caddc 7587    x. cmul 7589    < clt 7764   NN0cn0 8931   ZZcz 9008    || cdvds 11400    gcd cgcd 11542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-mulrcl 7683  ax-addcom 7684  ax-mulcom 7685  ax-addass 7686  ax-mulass 7687  ax-distr 7688  ax-i2m1 7689  ax-0lt1 7690  ax-1rid 7691  ax-0id 7692  ax-rnegex 7693  ax-precex 7694  ax-cnre 7695  ax-pre-ltirr 7696  ax-pre-ltwlin 7697  ax-pre-lttrn 7698  ax-pre-apti 7699  ax-pre-ltadd 7700  ax-pre-mulgt0 7701  ax-pre-mulext 7702  ax-arch 7703  ax-caucvg 7704
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-reu 2398  df-rmo 2399  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-if 3443  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-ilim 4259  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-frec 6254  df-sup 6837  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769  df-le 7770  df-sub 7899  df-neg 7900  df-reap 8300  df-ap 8307  df-div 8396  df-inn 8681  df-2 8739  df-3 8740  df-4 8741  df-n0 8932  df-z 9009  df-uz 9279  df-q 9364  df-rp 9394  df-fz 9742  df-fzo 9871  df-fl 9994  df-mod 10047  df-seqfrec 10170  df-exp 10244  df-cj 10565  df-re 10566  df-im 10567  df-rsqrt 10721  df-abs 10722  df-dvds 11401  df-gcd 11543
This theorem is referenced by:  bezout  11606
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