Metamath Proof Explorer

Theorem homco1

Description: Associative law for scalar product and composition of operators. (Contributed by NM, 13-Aug-2006) (New usage is discouraged.)

Ref Expression
Assertion homco1 
|- ( ( A e. CC /\ T : ~H --> ~H /\ U : ~H --> ~H ) -> ( ( A .op T ) o. U ) = ( A .op ( T o. U ) ) )

Proof

Step Hyp Ref Expression
1 fvco3 
 |-  ( ( U : ~H --> ~H /\ x e. ~H ) -> ( ( ( A .op T ) o. U ) ` x ) = ( ( A .op T ) ` ( U ` x ) ) )
2 1 3ad2antl3 
 |-  ( ( ( A e. CC /\ T : ~H --> ~H /\ U : ~H --> ~H ) /\ x e. ~H ) -> ( ( ( A .op T ) o. U ) ` x ) = ( ( A .op T ) ` ( U ` x ) ) )
3 fvco3 
 |-  ( ( U : ~H --> ~H /\ x e. ~H ) -> ( ( T o. U ) ` x ) = ( T ` ( U ` x ) ) )
4 3 3ad2antl3 
 |-  ( ( ( A e. CC /\ T : ~H --> ~H /\ U : ~H --> ~H ) /\ x e. ~H ) -> ( ( T o. U ) ` x ) = ( T ` ( U ` x ) ) )
5 4 oveq2d 
 |-  ( ( ( A e. CC /\ T : ~H --> ~H /\ U : ~H --> ~H ) /\ x e. ~H ) -> ( A .h ( ( T o. U ) ` x ) ) = ( A .h ( T ` ( U ` x ) ) ) )
6 ffvelrn 
 |-  ( ( U : ~H --> ~H /\ x e. ~H ) -> ( U ` x ) e. ~H )
7 homval 
 |-  ( ( A e. CC /\ T : ~H --> ~H /\ ( U ` x ) e. ~H ) -> ( ( A .op T ) ` ( U ` x ) ) = ( A .h ( T ` ( U ` x ) ) ) )
8 6 7 syl3an3 
 |-  ( ( A e. CC /\ T : ~H --> ~H /\ ( U : ~H --> ~H /\ x e. ~H ) ) -> ( ( A .op T ) ` ( U ` x ) ) = ( A .h ( T ` ( U ` x ) ) ) )
9 8 3expa 
 |-  ( ( ( A e. CC /\ T : ~H --> ~H ) /\ ( U : ~H --> ~H /\ x e. ~H ) ) -> ( ( A .op T ) ` ( U ` x ) ) = ( A .h ( T ` ( U ` x ) ) ) )
10 9 exp43 
 |-  ( A e. CC -> ( T : ~H --> ~H -> ( U : ~H --> ~H -> ( x e. ~H -> ( ( A .op T ) ` ( U ` x ) ) = ( A .h ( T ` ( U ` x ) ) ) ) ) ) )
11 10 3imp1 
 |-  ( ( ( A e. CC /\ T : ~H --> ~H /\ U : ~H --> ~H ) /\ x e. ~H ) -> ( ( A .op T ) ` ( U ` x ) ) = ( A .h ( T ` ( U ` x ) ) ) )
12 5 11 eqtr4d 
 |-  ( ( ( A e. CC /\ T : ~H --> ~H /\ U : ~H --> ~H ) /\ x e. ~H ) -> ( A .h ( ( T o. U ) ` x ) ) = ( ( A .op T ) ` ( U ` x ) ) )
13 2 12 eqtr4d 
 |-  ( ( ( A e. CC /\ T : ~H --> ~H /\ U : ~H --> ~H ) /\ x e. ~H ) -> ( ( ( A .op T ) o. U ) ` x ) = ( A .h ( ( T o. U ) ` x ) ) )
14 fco 
 |-  ( ( T : ~H --> ~H /\ U : ~H --> ~H ) -> ( T o. U ) : ~H --> ~H )
15 homval 
 |-  ( ( A e. CC /\ ( T o. U ) : ~H --> ~H /\ x e. ~H ) -> ( ( A .op ( T o. U ) ) ` x ) = ( A .h ( ( T o. U ) ` x ) ) )
16 14 15 syl3an2 
 |-  ( ( A e. CC /\ ( T : ~H --> ~H /\ U : ~H --> ~H ) /\ x e. ~H ) -> ( ( A .op ( T o. U ) ) ` x ) = ( A .h ( ( T o. U ) ` x ) ) )
17 16 3expia 
 |-  ( ( A e. CC /\ ( T : ~H --> ~H /\ U : ~H --> ~H ) ) -> ( x e. ~H -> ( ( A .op ( T o. U ) ) ` x ) = ( A .h ( ( T o. U ) ` x ) ) ) )
18 17 3impb 
 |-  ( ( A e. CC /\ T : ~H --> ~H /\ U : ~H --> ~H ) -> ( x e. ~H -> ( ( A .op ( T o. U ) ) ` x ) = ( A .h ( ( T o. U ) ` x ) ) ) )
19 18 imp 
 |-  ( ( ( A e. CC /\ T : ~H --> ~H /\ U : ~H --> ~H ) /\ x e. ~H ) -> ( ( A .op ( T o. U ) ) ` x ) = ( A .h ( ( T o. U ) ` x ) ) )
20 13 19 eqtr4d 
 |-  ( ( ( A e. CC /\ T : ~H --> ~H /\ U : ~H --> ~H ) /\ x e. ~H ) -> ( ( ( A .op T ) o. U ) ` x ) = ( ( A .op ( T o. U ) ) ` x ) )
21 20 ralrimiva 
 |-  ( ( A e. CC /\ T : ~H --> ~H /\ U : ~H --> ~H ) -> A. x e. ~H ( ( ( A .op T ) o. U ) ` x ) = ( ( A .op ( T o. U ) ) ` x ) )
22 homulcl 
 |-  ( ( A e. CC /\ T : ~H --> ~H ) -> ( A .op T ) : ~H --> ~H )
23 fco 
 |-  ( ( ( A .op T ) : ~H --> ~H /\ U : ~H --> ~H ) -> ( ( A .op T ) o. U ) : ~H --> ~H )
24 22 23 stoic3 
 |-  ( ( A e. CC /\ T : ~H --> ~H /\ U : ~H --> ~H ) -> ( ( A .op T ) o. U ) : ~H --> ~H )
25 homulcl 
 |-  ( ( A e. CC /\ ( T o. U ) : ~H --> ~H ) -> ( A .op ( T o. U ) ) : ~H --> ~H )
26 14 25 sylan2 
 |-  ( ( A e. CC /\ ( T : ~H --> ~H /\ U : ~H --> ~H ) ) -> ( A .op ( T o. U ) ) : ~H --> ~H )
27 26 3impb 
 |-  ( ( A e. CC /\ T : ~H --> ~H /\ U : ~H --> ~H ) -> ( A .op ( T o. U ) ) : ~H --> ~H )
28 hoeq 
 |-  ( ( ( ( A .op T ) o. U ) : ~H --> ~H /\ ( A .op ( T o. U ) ) : ~H --> ~H ) -> ( A. x e. ~H ( ( ( A .op T ) o. U ) ` x ) = ( ( A .op ( T o. U ) ) ` x ) <-> ( ( A .op T ) o. U ) = ( A .op ( T o. U ) ) ) )
29 24 27 28 syl2anc 
 |-  ( ( A e. CC /\ T : ~H --> ~H /\ U : ~H --> ~H ) -> ( A. x e. ~H ( ( ( A .op T ) o. U ) ` x ) = ( ( A .op ( T o. U ) ) ` x ) <-> ( ( A .op T ) o. U ) = ( A .op ( T o. U ) ) ) )
30 21 29 mpbid 
 |-  ( ( A e. CC /\ T : ~H --> ~H /\ U : ~H --> ~H ) -> ( ( A .op T ) o. U ) = ( A .op ( T o. U ) ) )