Metamath Proof Explorer

Theorem ehl2eudisval

Description: The value of the Euclidean distance function in a real Euclidean space of dimension 2. (Contributed by AV, 16-Jan-2023)

Ref Expression
Hypotheses ehl2eudis.e 
|- E = ( EEhil ` 2 )
ehl2eudis.x 
|- X = ( RR ^m { 1 , 2 } )
ehl2eudis.d 
|- D = ( dist ` E )
Assertion ehl2eudisval 
|- ( ( F e. X /\ G e. X ) -> ( F D G ) = ( sqrt ` ( ( ( ( F ` 1 ) - ( G ` 1 ) ) ^ 2 ) + ( ( ( F ` 2 ) - ( G ` 2 ) ) ^ 2 ) ) ) )

Proof

Step Hyp Ref Expression
1 ehl2eudis.e 
 |-  E = ( EEhil ` 2 )
2 ehl2eudis.x 
 |-  X = ( RR ^m { 1 , 2 } )
3 ehl2eudis.d 
 |-  D = ( dist ` E )
4 1 2 3 ehl2eudis 
 |-  D = ( f e. X , g e. X |-> ( sqrt ` ( ( ( ( f ` 1 ) - ( g ` 1 ) ) ^ 2 ) + ( ( ( f ` 2 ) - ( g ` 2 ) ) ^ 2 ) ) ) )
5 4 oveqi 
 |-  ( F D G ) = ( F ( f e. X , g e. X |-> ( sqrt ` ( ( ( ( f ` 1 ) - ( g ` 1 ) ) ^ 2 ) + ( ( ( f ` 2 ) - ( g ` 2 ) ) ^ 2 ) ) ) ) G )
6 eqidd 
 |-  ( ( F e. X /\ G e. X ) -> ( f e. X , g e. X |-> ( sqrt ` ( ( ( ( f ` 1 ) - ( g ` 1 ) ) ^ 2 ) + ( ( ( f ` 2 ) - ( g ` 2 ) ) ^ 2 ) ) ) ) = ( f e. X , g e. X |-> ( sqrt ` ( ( ( ( f ` 1 ) - ( g ` 1 ) ) ^ 2 ) + ( ( ( f ` 2 ) - ( g ` 2 ) ) ^ 2 ) ) ) ) )
7 fveq1 
 |-  ( f = F -> ( f ` 1 ) = ( F ` 1 ) )
8 fveq1 
 |-  ( g = G -> ( g ` 1 ) = ( G ` 1 ) )
9 7 8 oveqan12d 
 |-  ( ( f = F /\ g = G ) -> ( ( f ` 1 ) - ( g ` 1 ) ) = ( ( F ` 1 ) - ( G ` 1 ) ) )
10 9 oveq1d 
 |-  ( ( f = F /\ g = G ) -> ( ( ( f ` 1 ) - ( g ` 1 ) ) ^ 2 ) = ( ( ( F ` 1 ) - ( G ` 1 ) ) ^ 2 ) )
11 fveq1 
 |-  ( f = F -> ( f ` 2 ) = ( F ` 2 ) )
12 fveq1 
 |-  ( g = G -> ( g ` 2 ) = ( G ` 2 ) )
13 11 12 oveqan12d 
 |-  ( ( f = F /\ g = G ) -> ( ( f ` 2 ) - ( g ` 2 ) ) = ( ( F ` 2 ) - ( G ` 2 ) ) )
14 13 oveq1d 
 |-  ( ( f = F /\ g = G ) -> ( ( ( f ` 2 ) - ( g ` 2 ) ) ^ 2 ) = ( ( ( F ` 2 ) - ( G ` 2 ) ) ^ 2 ) )
15 10 14 oveq12d 
 |-  ( ( f = F /\ g = G ) -> ( ( ( ( f ` 1 ) - ( g ` 1 ) ) ^ 2 ) + ( ( ( f ` 2 ) - ( g ` 2 ) ) ^ 2 ) ) = ( ( ( ( F ` 1 ) - ( G ` 1 ) ) ^ 2 ) + ( ( ( F ` 2 ) - ( G ` 2 ) ) ^ 2 ) ) )
16 15 fveq2d 
 |-  ( ( f = F /\ g = G ) -> ( sqrt ` ( ( ( ( f ` 1 ) - ( g ` 1 ) ) ^ 2 ) + ( ( ( f ` 2 ) - ( g ` 2 ) ) ^ 2 ) ) ) = ( sqrt ` ( ( ( ( F ` 1 ) - ( G ` 1 ) ) ^ 2 ) + ( ( ( F ` 2 ) - ( G ` 2 ) ) ^ 2 ) ) ) )
17 16 adantl 
 |-  ( ( ( F e. X /\ G e. X ) /\ ( f = F /\ g = G ) ) -> ( sqrt ` ( ( ( ( f ` 1 ) - ( g ` 1 ) ) ^ 2 ) + ( ( ( f ` 2 ) - ( g ` 2 ) ) ^ 2 ) ) ) = ( sqrt ` ( ( ( ( F ` 1 ) - ( G ` 1 ) ) ^ 2 ) + ( ( ( F ` 2 ) - ( G ` 2 ) ) ^ 2 ) ) ) )
18 simpl 
 |-  ( ( F e. X /\ G e. X ) -> F e. X )
19 simpr 
 |-  ( ( F e. X /\ G e. X ) -> G e. X )
20 fvexd 
 |-  ( ( F e. X /\ G e. X ) -> ( sqrt ` ( ( ( ( F ` 1 ) - ( G ` 1 ) ) ^ 2 ) + ( ( ( F ` 2 ) - ( G ` 2 ) ) ^ 2 ) ) ) e. _V )
21 6 17 18 19 20 ovmpod 
 |-  ( ( F e. X /\ G e. X ) -> ( F ( f e. X , g e. X |-> ( sqrt ` ( ( ( ( f ` 1 ) - ( g ` 1 ) ) ^ 2 ) + ( ( ( f ` 2 ) - ( g ` 2 ) ) ^ 2 ) ) ) ) G ) = ( sqrt ` ( ( ( ( F ` 1 ) - ( G ` 1 ) ) ^ 2 ) + ( ( ( F ` 2 ) - ( G ` 2 ) ) ^ 2 ) ) ) )
22 5 21 eqtrid 
 |-  ( ( F e. X /\ G e. X ) -> ( F D G ) = ( sqrt ` ( ( ( ( F ` 1 ) - ( G ` 1 ) ) ^ 2 ) + ( ( ( F ` 2 ) - ( G ` 2 ) ) ^ 2 ) ) ) )