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Metamath Proof Explorer

Theorem hoadd4

Description: Rearrangement of 4 terms in a sum of Hilbert space operators. (Contributed by NM, 24-Aug-2006) (New usage is discouraged.)

Ref Expression
Assertion hoadd4 
|- ( ( ( R : ~H --> ~H /\ S : ~H --> ~H ) /\ ( T : ~H --> ~H /\ U : ~H --> ~H ) ) -> ( ( R +op S ) +op ( T +op U ) ) = ( ( R +op T ) +op ( S +op U ) ) )

Proof

Step Hyp Ref Expression
1 hoadd32 
 |-  ( ( R : ~H --> ~H /\ S : ~H --> ~H /\ T : ~H --> ~H ) -> ( ( R +op S ) +op T ) = ( ( R +op T ) +op S ) )
2 1 oveq1d 
 |-  ( ( R : ~H --> ~H /\ S : ~H --> ~H /\ T : ~H --> ~H ) -> ( ( ( R +op S ) +op T ) +op U ) = ( ( ( R +op T ) +op S ) +op U ) )
3 2 3expa 
 |-  ( ( ( R : ~H --> ~H /\ S : ~H --> ~H ) /\ T : ~H --> ~H ) -> ( ( ( R +op S ) +op T ) +op U ) = ( ( ( R +op T ) +op S ) +op U ) )
4 3 adantrr 
 |-  ( ( ( R : ~H --> ~H /\ S : ~H --> ~H ) /\ ( T : ~H --> ~H /\ U : ~H --> ~H ) ) -> ( ( ( R +op S ) +op T ) +op U ) = ( ( ( R +op T ) +op S ) +op U ) )
5 hoaddcl 
 |-  ( ( R : ~H --> ~H /\ S : ~H --> ~H ) -> ( R +op S ) : ~H --> ~H )
6 hoaddass 
 |-  ( ( ( R +op S ) : ~H --> ~H /\ T : ~H --> ~H /\ U : ~H --> ~H ) -> ( ( ( R +op S ) +op T ) +op U ) = ( ( R +op S ) +op ( T +op U ) ) )
7 6 3expb 
 |-  ( ( ( R +op S ) : ~H --> ~H /\ ( T : ~H --> ~H /\ U : ~H --> ~H ) ) -> ( ( ( R +op S ) +op T ) +op U ) = ( ( R +op S ) +op ( T +op U ) ) )
8 5 7 sylan 
 |-  ( ( ( R : ~H --> ~H /\ S : ~H --> ~H ) /\ ( T : ~H --> ~H /\ U : ~H --> ~H ) ) -> ( ( ( R +op S ) +op T ) +op U ) = ( ( R +op S ) +op ( T +op U ) ) )
9 hoaddcl 
 |-  ( ( R : ~H --> ~H /\ T : ~H --> ~H ) -> ( R +op T ) : ~H --> ~H )
10 hoaddass 
 |-  ( ( ( R +op T ) : ~H --> ~H /\ S : ~H --> ~H /\ U : ~H --> ~H ) -> ( ( ( R +op T ) +op S ) +op U ) = ( ( R +op T ) +op ( S +op U ) ) )
11 10 3expb 
 |-  ( ( ( R +op T ) : ~H --> ~H /\ ( S : ~H --> ~H /\ U : ~H --> ~H ) ) -> ( ( ( R +op T ) +op S ) +op U ) = ( ( R +op T ) +op ( S +op U ) ) )
12 9 11 sylan 
 |-  ( ( ( R : ~H --> ~H /\ T : ~H --> ~H ) /\ ( S : ~H --> ~H /\ U : ~H --> ~H ) ) -> ( ( ( R +op T ) +op S ) +op U ) = ( ( R +op T ) +op ( S +op U ) ) )
13 12 an4s 
 |-  ( ( ( R : ~H --> ~H /\ S : ~H --> ~H ) /\ ( T : ~H --> ~H /\ U : ~H --> ~H ) ) -> ( ( ( R +op T ) +op S ) +op U ) = ( ( R +op T ) +op ( S +op U ) ) )
14 4 8 13 3eqtr3d 
 |-  ( ( ( R : ~H --> ~H /\ S : ~H --> ~H ) /\ ( T : ~H --> ~H /\ U : ~H --> ~H ) ) -> ( ( R +op S ) +op ( T +op U ) ) = ( ( R +op T ) +op ( S +op U ) ) )