Step |
Hyp |
Ref |
Expression |
1 |
|
mapdh.q |
|- Q = ( 0g ` C ) |
2 |
|
mapdh.i |
|- I = ( x e. _V |-> if ( ( 2nd ` x ) = .0. , Q , ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` x ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` x ) ) .- ( 2nd ` x ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` x ) ) R h ) } ) ) ) ) ) |
3 |
|
mapdh2.x |
|- ( ph -> X e. A ) |
4 |
|
mapdh2.f |
|- ( ph -> F e. B ) |
5 |
|
mapdh2.y |
|- ( ph -> Y e. ( V \ { .0. } ) ) |
6 |
1 2 3 4 5
|
mapdhval |
|- ( ph -> ( I ` <. X , F , Y >. ) = if ( Y = .0. , Q , ( iota_ h e. D ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( F R h ) } ) ) ) ) ) |
7 |
|
eldifsni |
|- ( Y e. ( V \ { .0. } ) -> Y =/= .0. ) |
8 |
7
|
neneqd |
|- ( Y e. ( V \ { .0. } ) -> -. Y = .0. ) |
9 |
|
iffalse |
|- ( -. Y = .0. -> if ( Y = .0. , Q , ( iota_ h e. D ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( F R h ) } ) ) ) ) = ( iota_ h e. D ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( F R h ) } ) ) ) ) |
10 |
5 8 9
|
3syl |
|- ( ph -> if ( Y = .0. , Q , ( iota_ h e. D ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( F R h ) } ) ) ) ) = ( iota_ h e. D ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( F R h ) } ) ) ) ) |
11 |
6 10
|
eqtrd |
|- ( ph -> ( I ` <. X , F , Y >. ) = ( iota_ h e. D ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( F R h ) } ) ) ) ) |