Step |
Hyp |
Ref |
Expression |
1 |
|
hosubcl |
|- ( ( R : ~H --> ~H /\ S : ~H --> ~H ) -> ( R -op S ) : ~H --> ~H ) |
2 |
|
hosubsub2 |
|- ( ( ( R -op S ) : ~H --> ~H /\ T : ~H --> ~H /\ U : ~H --> ~H ) -> ( ( R -op S ) -op ( T -op U ) ) = ( ( R -op S ) +op ( U -op T ) ) ) |
3 |
2
|
3expb |
|- ( ( ( R -op S ) : ~H --> ~H /\ ( T : ~H --> ~H /\ U : ~H --> ~H ) ) -> ( ( R -op S ) -op ( T -op U ) ) = ( ( R -op S ) +op ( U -op T ) ) ) |
4 |
1 3
|
sylan |
|- ( ( ( R : ~H --> ~H /\ S : ~H --> ~H ) /\ ( T : ~H --> ~H /\ U : ~H --> ~H ) ) -> ( ( R -op S ) -op ( T -op U ) ) = ( ( R -op S ) +op ( U -op T ) ) ) |
5 |
|
hosub4 |
|- ( ( ( R : ~H --> ~H /\ U : ~H --> ~H ) /\ ( S : ~H --> ~H /\ T : ~H --> ~H ) ) -> ( ( R +op U ) -op ( S +op T ) ) = ( ( R -op S ) +op ( U -op T ) ) ) |
6 |
5
|
an42s |
|- ( ( ( R : ~H --> ~H /\ S : ~H --> ~H ) /\ ( T : ~H --> ~H /\ U : ~H --> ~H ) ) -> ( ( R +op U ) -op ( S +op T ) ) = ( ( R -op S ) +op ( U -op T ) ) ) |
7 |
4 6
|
eqtr4d |
|- ( ( ( R : ~H --> ~H /\ S : ~H --> ~H ) /\ ( T : ~H --> ~H /\ U : ~H --> ~H ) ) -> ( ( R -op S ) -op ( T -op U ) ) = ( ( R +op U ) -op ( S +op T ) ) ) |