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Theorem dfgcd3 12025
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 11975 . . 3  |-  ( 0  gcd  0 )  =  0
2 simprl 529 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  /\  N  =  0 ) )  ->  M  =  0 )
3 simprr 531 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  /\  N  =  0 ) )  ->  N  =  0 )
42, 3oveq12d 5906 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  /\  N  =  0 ) )  -> 
( M  gcd  N
)  =  ( 0  gcd  0 ) )
5 0nn0 9205 . . . . 5  |-  0  e.  NN0
65a1i 9 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  /\  N  =  0 ) )  -> 
0  e.  NN0 )
7 0dvds 11832 . . . . . . . . . . 11  |-  ( M  e.  ZZ  ->  (
0  ||  M  <->  M  = 
0 ) )
87ad2antrr 488 . . . . . . . . . 10  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  /\  N  =  0 ) )  -> 
( 0  ||  M  <->  M  =  0 ) )
92, 8mpbird 167 . . . . . . . . 9  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  /\  N  =  0 ) )  -> 
0  ||  M )
10 0dvds 11832 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  (
0  ||  N  <->  N  = 
0 ) )
1110ad2antlr 489 . . . . . . . . . 10  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  /\  N  =  0 ) )  -> 
( 0  ||  N  <->  N  =  0 ) )
123, 11mpbird 167 . . . . . . . . 9  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  /\  N  =  0 ) )  -> 
0  ||  N )
139, 12jca 306 . . . . . . . 8  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  /\  N  =  0 ) )  -> 
( 0  ||  M  /\  0  ||  N ) )
1413ad2antrr 488 . . . . . . 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 9278 . . . . . . . . 9  |-  0  e.  ZZ
16 breq1 4018 . . . . . . . . . . 11  |-  ( z  =  0  ->  (
z  ||  d  <->  0  ||  d ) )
17 breq1 4018 . . . . . . . . . . . 12  |-  ( z  =  0  ->  (
z  ||  M  <->  0  ||  M ) )
18 breq1 4018 . . . . . . . . . . . 12  |-  ( z  =  0  ->  (
z  ||  N  <->  0  ||  N ) )
1917, 18anbi12d 473 . . . . . . . . . . 11  |-  ( z  =  0  ->  (
( z  ||  M  /\  z  ||  N )  <-> 
( 0  ||  M  /\  0  ||  N ) ) )
2016, 19bibi12d 235 . . . . . . . . . 10  |-  ( z  =  0  ->  (
( z  ||  d  <->  ( z  ||  M  /\  z  ||  N ) )  <-> 
( 0  ||  d  <->  ( 0  ||  M  /\  0  ||  N ) ) ) )
2120rspcv 2849 . . . . . . . . 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 277 . . . . . . 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 167 . . . . . 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 528 . . . . . . . 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 9387 . . . . . . 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 11832 . . . . . . 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 147 . . . . 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 11827 . . . . . . . . 9  |-  ( z  e.  ZZ  ->  z  ||  0 )
3130adantl 277 . . . . . . . 8  |-  ( ( ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  /\  N  =  0 ) )  /\  d  e.  NN0 )  /\  d  =  0 )  /\  z  e.  ZZ )  ->  z  ||  0 )
32 breq2 4019 . . . . . . . . 9  |-  ( d  =  0  ->  (
z  ||  d  <->  z  ||  0 ) )
3332ad2antlr 489 . . . . . . . 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 167 . . . . . . 7  |-  ( ( ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  /\  N  =  0 ) )  /\  d  e.  NN0 )  /\  d  =  0 )  /\  z  e.  ZZ )  ->  z  ||  d )
352ad3antrrr 492 . . . . . . . . 9  |-  ( ( ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  /\  N  =  0 ) )  /\  d  e.  NN0 )  /\  d  =  0 )  /\  z  e.  ZZ )  ->  M  =  0 )
3631, 35breqtrrd 4043 . . . . . . . 8  |-  ( ( ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  /\  N  =  0 ) )  /\  d  e.  NN0 )  /\  d  =  0 )  /\  z  e.  ZZ )  ->  z  ||  M )
373ad3antrrr 492 . . . . . . . . 9  |-  ( ( ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  /\  N  =  0 ) )  /\  d  e.  NN0 )  /\  d  =  0 )  /\  z  e.  ZZ )  ->  N  =  0 )
3831, 37breqtrrd 4043 . . . . . . . 8  |-  ( ( ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  /\  N  =  0 ) )  /\  d  e.  NN0 )  /\  d  =  0 )  /\  z  e.  ZZ )  ->  z  ||  N )
3936, 38jca 306 . . . . . . 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 175 . . . . . 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 2560 . . . . 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 596 . . . 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 5869 . . 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 2246 . 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 12020 . . . . 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 109 . . . . . 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 2584 . . . . 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 276 . . 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 533 . . . . 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 534 . . . . 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 529 . . . . 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 4018 . . . . . . . . 9  |-  ( w  =  z  ->  (
w  ||  r  <->  z  ||  r ) )
54 breq1 4018 . . . . . . . . . 10  |-  ( w  =  z  ->  (
w  ||  M  <->  z  ||  M ) )
55 breq1 4018 . . . . . . . . . 10  |-  ( w  =  z  ->  (
w  ||  N  <->  z  ||  N ) )
5654, 55anbi12d 473 . . . . . . . . 9  |-  ( w  =  z  ->  (
( w  ||  M  /\  w  ||  N )  <-> 
( z  ||  M  /\  z  ||  N ) ) )
5753, 56bibi12d 235 . . . . . . . 8  |-  ( w  =  z  ->  (
( w  ||  r  <->  ( w  ||  M  /\  w  ||  N ) )  <-> 
( z  ||  r  <->  ( z  ||  M  /\  z  ||  N ) ) ) )
5857cbvralv 2715 . . . . . . 7  |-  ( A. w  e.  ZZ  (
w  ||  r  <->  ( w  ||  M  /\  w  ||  N ) )  <->  A. z  e.  ZZ  ( z  ||  r 
<->  ( z  ||  M  /\  z  ||  N ) ) )
5958biimpi 120 . . . . . 6  |-  ( A. w  e.  ZZ  (
w  ||  r  <->  ( w  ||  M  /\  w  ||  N ) )  ->  A. z  e.  ZZ  ( z  ||  r  <->  ( z  ||  M  /\  z  ||  N ) ) )
6059ad2antll 491 . . . . 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 528 . . . . 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 12024 . . . 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 4018 . . . . . . . . 9  |-  ( w  =  z  ->  (
w  ||  d  <->  z  ||  d ) )
6463, 56bibi12d 235 . . . . . . . 8  |-  ( w  =  z  ->  (
( w  ||  d  <->  ( w  ||  M  /\  w  ||  N ) )  <-> 
( z  ||  d  <->  ( z  ||  M  /\  z  ||  N ) ) ) )
6564cbvralv 2715 . . . . . . 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 5861 . . . . 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 531 . . . . . 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 12022 . . . . . . 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 4019 . . . . . . . . . 10  |-  ( d  =  r  ->  (
w  ||  d  <->  w  ||  r
) )
7170bibi1d 233 . . . . . . . . 9  |-  ( d  =  r  ->  (
( w  ||  d  <->  ( w  ||  M  /\  w  ||  N ) )  <-> 
( w  ||  r  <->  ( w  ||  M  /\  w  ||  N ) ) ) )
7271ralbidv 2487 . . . . . . . 8  |-  ( d  =  r  ->  ( A. w  e.  ZZ  ( w  ||  d  <->  ( w  ||  M  /\  w  ||  N ) )  <->  A. w  e.  ZZ  ( w  ||  r 
<->  ( w  ||  M  /\  w  ||  N ) ) ) )
7372riota2 5866 . . . . . . 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 411 . . . . . 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 147 . . . . 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, 75eqtr3id 2234 . . . 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 11976 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  ( M  gcd  N )  =  sup ( { z  e.  ZZ  |  ( z  ||  M  /\  z  ||  N
) } ,  RR ,  <  ) )
7877adantr 276 . . . 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 2231 . . 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 2609 . 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 11959 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  -> DECID  ( M  =  0  /\  N  =  0 ) )
82 exmiddc 837 . . 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 799 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 104    <-> wb 105    \/ wo 709  DECID wdc 835    = wceq 1363    e. wcel 2158   A.wral 2465   E.wrex 2466   E!wreu 2467   {crab 2469   class class class wbr 4015   iota_crio 5843  (class class class)co 5888   supcsup 6995   RRcr 7824   0cc0 7825    + caddc 7828    x. cmul 7830    < clt 8006   NN0cn0 9190   ZZcz 9267    || cdvds 11808    gcd cgcd 11957
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599  ax-cnex 7916  ax-resscn 7917  ax-1cn 7918  ax-1re 7919  ax-icn 7920  ax-addcl 7921  ax-addrcl 7922  ax-mulcl 7923  ax-mulrcl 7924  ax-addcom 7925  ax-mulcom 7926  ax-addass 7927  ax-mulass 7928  ax-distr 7929  ax-i2m1 7930  ax-0lt1 7931  ax-1rid 7932  ax-0id 7933  ax-rnegex 7934  ax-precex 7935  ax-cnre 7936  ax-pre-ltirr 7937  ax-pre-ltwlin 7938  ax-pre-lttrn 7939  ax-pre-apti 7940  ax-pre-ltadd 7941  ax-pre-mulgt0 7942  ax-pre-mulext 7943  ax-arch 7944  ax-caucvg 7945
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-if 3547  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-ilim 4381  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6155  df-2nd 6156  df-recs 6320  df-frec 6406  df-sup 6997  df-pnf 8008  df-mnf 8009  df-xr 8010  df-ltxr 8011  df-le 8012  df-sub 8144  df-neg 8145  df-reap 8546  df-ap 8553  df-div 8644  df-inn 8934  df-2 8992  df-3 8993  df-4 8994  df-n0 9191  df-z 9268  df-uz 9543  df-q 9634  df-rp 9668  df-fz 10023  df-fzo 10157  df-fl 10284  df-mod 10337  df-seqfrec 10460  df-exp 10534  df-cj 10865  df-re 10866  df-im 10867  df-rsqrt 11021  df-abs 11022  df-dvds 11809  df-gcd 11958
This theorem is referenced by:  bezout  12026
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