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Metamath Proof Explorer

Theorem gcdval

Description: The value of the gcd operator. ( M gcd N ) is the greatest common divisor of M and N . If M and N are both 0 , the result is defined conventionally as 0 . (Contributed by Paul Chapman, 21-Mar-2011) (Revised by Mario Carneiro, 10-Nov-2013)

Ref Expression
Assertion gcdval 
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) = if ( ( M = 0 /\ N = 0 ) , 0 , sup ( { n e. ZZ | ( n || M /\ n || N ) } , RR , < ) ) )

Proof

Step Hyp Ref Expression
1 eqeq1 
 |-  ( x = M -> ( x = 0 <-> M = 0 ) )
2 1 anbi1d 
 |-  ( x = M -> ( ( x = 0 /\ y = 0 ) <-> ( M = 0 /\ y = 0 ) ) )
3 breq2 
 |-  ( x = M -> ( n || x <-> n || M ) )
4 3 anbi1d 
 |-  ( x = M -> ( ( n || x /\ n || y ) <-> ( n || M /\ n || y ) ) )
5 4 rabbidv 
 |-  ( x = M -> { n e. ZZ | ( n || x /\ n || y ) } = { n e. ZZ | ( n || M /\ n || y ) } )
6 5 supeq1d 
 |-  ( x = M -> sup ( { n e. ZZ | ( n || x /\ n || y ) } , RR , < ) = sup ( { n e. ZZ | ( n || M /\ n || y ) } , RR , < ) )
7 2 6 ifbieq2d 
 |-  ( x = M -> if ( ( x = 0 /\ y = 0 ) , 0 , sup ( { n e. ZZ | ( n || x /\ n || y ) } , RR , < ) ) = if ( ( M = 0 /\ y = 0 ) , 0 , sup ( { n e. ZZ | ( n || M /\ n || y ) } , RR , < ) ) )
8 eqeq1 
 |-  ( y = N -> ( y = 0 <-> N = 0 ) )
9 8 anbi2d 
 |-  ( y = N -> ( ( M = 0 /\ y = 0 ) <-> ( M = 0 /\ N = 0 ) ) )
10 breq2 
 |-  ( y = N -> ( n || y <-> n || N ) )
11 10 anbi2d 
 |-  ( y = N -> ( ( n || M /\ n || y ) <-> ( n || M /\ n || N ) ) )
12 11 rabbidv 
 |-  ( y = N -> { n e. ZZ | ( n || M /\ n || y ) } = { n e. ZZ | ( n || M /\ n || N ) } )
13 12 supeq1d 
 |-  ( y = N -> sup ( { n e. ZZ | ( n || M /\ n || y ) } , RR , < ) = sup ( { n e. ZZ | ( n || M /\ n || N ) } , RR , < ) )
14 9 13 ifbieq2d 
 |-  ( y = N -> if ( ( M = 0 /\ y = 0 ) , 0 , sup ( { n e. ZZ | ( n || M /\ n || y ) } , RR , < ) ) = if ( ( M = 0 /\ N = 0 ) , 0 , sup ( { n e. ZZ | ( n || M /\ n || N ) } , RR , < ) ) )
15 df-gcd 
 |-  gcd = ( x e. ZZ , y e. ZZ |-> if ( ( x = 0 /\ y = 0 ) , 0 , sup ( { n e. ZZ | ( n || x /\ n || y ) } , RR , < ) ) )
16 c0ex 
 |-  0 e. _V
17 ltso 
 |-  < Or RR
18 17 supex 
 |-  sup ( { n e. ZZ | ( n || M /\ n || N ) } , RR , < ) e. _V
19 16 18 ifex 
 |-  if ( ( M = 0 /\ N = 0 ) , 0 , sup ( { n e. ZZ | ( n || M /\ n || N ) } , RR , < ) ) e. _V
20 7 14 15 19 ovmpo 
 |-  ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) = if ( ( M = 0 /\ N = 0 ) , 0 , sup ( { n e. ZZ | ( n || M /\ n || N ) } , RR , < ) ) )