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Metamath Proof Explorer

Theorem hvpncan

Description: Addition/subtraction cancellation law for vectors in Hilbert space. (Contributed by NM, 7-Jun-2004) (New usage is discouraged.)

Ref Expression
Assertion hvpncan 
|- ( ( A e. ~H /\ B e. ~H ) -> ( ( A +h B ) -h B ) = A )

Proof

Step Hyp Ref Expression
1 hvaddcl 
 |-  ( ( A e. ~H /\ B e. ~H ) -> ( A +h B ) e. ~H )
2 hvsubval 
 |-  ( ( ( A +h B ) e. ~H /\ B e. ~H ) -> ( ( A +h B ) -h B ) = ( ( A +h B ) +h ( -u 1 .h B ) ) )
3 1 2 sylancom 
 |-  ( ( A e. ~H /\ B e. ~H ) -> ( ( A +h B ) -h B ) = ( ( A +h B ) +h ( -u 1 .h B ) ) )
4 neg1cn 
 |-  -u 1 e. CC
5 hvmulcl 
 |-  ( ( -u 1 e. CC /\ B e. ~H ) -> ( -u 1 .h B ) e. ~H )
6 4 5 mpan 
 |-  ( B e. ~H -> ( -u 1 .h B ) e. ~H )
7 6 ancli 
 |-  ( B e. ~H -> ( B e. ~H /\ ( -u 1 .h B ) e. ~H ) )
8 ax-hvass 
 |-  ( ( A e. ~H /\ B e. ~H /\ ( -u 1 .h B ) e. ~H ) -> ( ( A +h B ) +h ( -u 1 .h B ) ) = ( A +h ( B +h ( -u 1 .h B ) ) ) )
9 8 3expb 
 |-  ( ( A e. ~H /\ ( B e. ~H /\ ( -u 1 .h B ) e. ~H ) ) -> ( ( A +h B ) +h ( -u 1 .h B ) ) = ( A +h ( B +h ( -u 1 .h B ) ) ) )
10 7 9 sylan2 
 |-  ( ( A e. ~H /\ B e. ~H ) -> ( ( A +h B ) +h ( -u 1 .h B ) ) = ( A +h ( B +h ( -u 1 .h B ) ) ) )
11 hvnegid 
 |-  ( B e. ~H -> ( B +h ( -u 1 .h B ) ) = 0h )
12 11 oveq2d 
 |-  ( B e. ~H -> ( A +h ( B +h ( -u 1 .h B ) ) ) = ( A +h 0h ) )
13 ax-hvaddid 
 |-  ( A e. ~H -> ( A +h 0h ) = A )
14 12 13 sylan9eqr 
 |-  ( ( A e. ~H /\ B e. ~H ) -> ( A +h ( B +h ( -u 1 .h B ) ) ) = A )
15 3 10 14 3eqtrd 
 |-  ( ( A e. ~H /\ B e. ~H ) -> ( ( A +h B ) -h B ) = A )