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Theorem dfgcd3 12014
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 11964 . . 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 5896 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  /\  N  =  0 ) )  -> 
( M  gcd  N
)  =  ( 0  gcd  0 ) )
5 0nn0 9194 . . . . 5  |-  0  e.  NN0
65a1i 9 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  /\  N  =  0 ) )  -> 
0  e.  NN0 )
7 0dvds 11821 . . . . . . . . . . 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 11821 . . . . . . . . . . 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 9267 . . . . . . . . 9  |-  0  e.  ZZ
16 breq1 4008 . . . . . . . . . . 11  |-  ( z  =  0  ->  (
z  ||  d  <->  0  ||  d ) )
17 breq1 4008 . . . . . . . . . . . 12  |-  ( z  =  0  ->  (
z  ||  M  <->  0  ||  M ) )
18 breq1 4008 . . . . . . . . . . . 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 2839 . . . . . . . . 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 9376 . . . . . . 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 11821 . . . . . . 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 11816 . . . . . . . . 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 4009 . . . . . . . . 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 4033 . . . . . . . 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 4033 . . . . . . . 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 2550 . . . . 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 5859 . . 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 2236 . 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 12009 . . . . 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 2574 . . . . 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 4008 . . . . . . . . 9  |-  ( w  =  z  ->  (
w  ||  r  <->  z  ||  r ) )
54 breq1 4008 . . . . . . . . . 10  |-  ( w  =  z  ->  (
w  ||  M  <->  z  ||  M ) )
55 breq1 4008 . . . . . . . . . 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 2705 . . . . . . 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 12013 . . . 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 4008 . . . . . . . . 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 2705 . . . . . . 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 5851 . . . . 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 12011 . . . . . . 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 4009 . . . . . . . . . 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 2477 . . . . . . . 8  |-  ( d  =  r  ->  ( A. w  e.  ZZ  ( w  ||  d  <->  ( w  ||  M  /\  w  ||  N ) )  <->  A. w  e.  ZZ  ( w  ||  r 
<->  ( w  ||  M  /\  w  ||  N ) ) ) )
7372riota2 5856 . . . . . . 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 2224 . . . 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 11965 . . . . 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 2221 . . 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 2599 . 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 11948 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  -> DECID  ( M  =  0  /\  N  =  0 ) )
82 exmiddc 836 . . 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 798 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 708  DECID wdc 834    = wceq 1353    e. wcel 2148   A.wral 2455   E.wrex 2456   E!wreu 2457   {crab 2459   class class class wbr 4005   iota_crio 5833  (class class class)co 5878   supcsup 6984   RRcr 7813   0cc0 7814    + caddc 7817    x. cmul 7819    < clt 7995   NN0cn0 9179   ZZcz 9256    || cdvds 11797    gcd cgcd 11946
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7905  ax-resscn 7906  ax-1cn 7907  ax-1re 7908  ax-icn 7909  ax-addcl 7910  ax-addrcl 7911  ax-mulcl 7912  ax-mulrcl 7913  ax-addcom 7914  ax-mulcom 7915  ax-addass 7916  ax-mulass 7917  ax-distr 7918  ax-i2m1 7919  ax-0lt1 7920  ax-1rid 7921  ax-0id 7922  ax-rnegex 7923  ax-precex 7924  ax-cnre 7925  ax-pre-ltirr 7926  ax-pre-ltwlin 7927  ax-pre-lttrn 7928  ax-pre-apti 7929  ax-pre-ltadd 7930  ax-pre-mulgt0 7931  ax-pre-mulext 7932  ax-arch 7933  ax-caucvg 7934
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5834  df-ov 5881  df-oprab 5882  df-mpo 5883  df-1st 6144  df-2nd 6145  df-recs 6309  df-frec 6395  df-sup 6986  df-pnf 7997  df-mnf 7998  df-xr 7999  df-ltxr 8000  df-le 8001  df-sub 8133  df-neg 8134  df-reap 8535  df-ap 8542  df-div 8633  df-inn 8923  df-2 8981  df-3 8982  df-4 8983  df-n0 9180  df-z 9257  df-uz 9532  df-q 9623  df-rp 9657  df-fz 10012  df-fzo 10146  df-fl 10273  df-mod 10326  df-seqfrec 10449  df-exp 10523  df-cj 10854  df-re 10855  df-im 10856  df-rsqrt 11010  df-abs 11011  df-dvds 11798  df-gcd 11947
This theorem is referenced by:  bezout  12015
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