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Metamath Proof Explorer

Theorem shjcom

Description: Commutative law for Hilbert lattice join of subspaces. (Contributed by NM, 22-Jun-2004) (New usage is discouraged.)

Ref Expression
Assertion shjcom 
|- ( ( A e. SH /\ B e. SH ) -> ( A vH B ) = ( B vH A ) )

Proof

Step Hyp Ref Expression
1 shjval 
 |-  ( ( A e. SH /\ B e. SH ) -> ( A vH B ) = ( _|_ ` ( _|_ ` ( A u. B ) ) ) )
2 shjval 
 |-  ( ( B e. SH /\ A e. SH ) -> ( B vH A ) = ( _|_ ` ( _|_ ` ( B u. A ) ) ) )
3 2 ancoms 
 |-  ( ( A e. SH /\ B e. SH ) -> ( B vH A ) = ( _|_ ` ( _|_ ` ( B u. A ) ) ) )
4 uncom 
 |-  ( B u. A ) = ( A u. B )
5 4 fveq2i 
 |-  ( _|_ ` ( B u. A ) ) = ( _|_ ` ( A u. B ) )
6 5 fveq2i 
 |-  ( _|_ ` ( _|_ ` ( B u. A ) ) ) = ( _|_ ` ( _|_ ` ( A u. B ) ) )
7 3 6 eqtrdi 
 |-  ( ( A e. SH /\ B e. SH ) -> ( B vH A ) = ( _|_ ` ( _|_ ` ( A u. B ) ) ) )
8 1 7 eqtr4d 
 |-  ( ( A e. SH /\ B e. SH ) -> ( A vH B ) = ( B vH A ) )