Metamath Proof Explorer

Theorem norm3lemt

Description: Lemma involving norm of differences in Hilbert space. (Contributed by NM, 18-Aug-1999) (New usage is discouraged.)

Ref Expression
Assertion norm3lemt 
|- ( ( ( A e. ~H /\ B e. ~H ) /\ ( C e. ~H /\ D e. RR ) ) -> ( ( ( normh ` ( A -h C ) ) < ( D / 2 ) /\ ( normh ` ( C -h B ) ) < ( D / 2 ) ) -> ( normh ` ( A -h B ) ) < D ) )

Proof

Step Hyp Ref Expression
1 fvoveq1 
 |-  ( A = if ( A e. ~H , A , 0h ) -> ( normh ` ( A -h C ) ) = ( normh ` ( if ( A e. ~H , A , 0h ) -h C ) ) )
2 1 breq1d 
 |-  ( A = if ( A e. ~H , A , 0h ) -> ( ( normh ` ( A -h C ) ) < ( D / 2 ) <-> ( normh ` ( if ( A e. ~H , A , 0h ) -h C ) ) < ( D / 2 ) ) )
3 2 anbi1d 
 |-  ( A = if ( A e. ~H , A , 0h ) -> ( ( ( normh ` ( A -h C ) ) < ( D / 2 ) /\ ( normh ` ( C -h B ) ) < ( D / 2 ) ) <-> ( ( normh ` ( if ( A e. ~H , A , 0h ) -h C ) ) < ( D / 2 ) /\ ( normh ` ( C -h B ) ) < ( D / 2 ) ) ) )
4 fvoveq1 
 |-  ( A = if ( A e. ~H , A , 0h ) -> ( normh ` ( A -h B ) ) = ( normh ` ( if ( A e. ~H , A , 0h ) -h B ) ) )
5 4 breq1d 
 |-  ( A = if ( A e. ~H , A , 0h ) -> ( ( normh ` ( A -h B ) ) < D <-> ( normh ` ( if ( A e. ~H , A , 0h ) -h B ) ) < D ) )
6 3 5 imbi12d 
 |-  ( A = if ( A e. ~H , A , 0h ) -> ( ( ( ( normh ` ( A -h C ) ) < ( D / 2 ) /\ ( normh ` ( C -h B ) ) < ( D / 2 ) ) -> ( normh ` ( A -h B ) ) < D ) <-> ( ( ( normh ` ( if ( A e. ~H , A , 0h ) -h C ) ) < ( D / 2 ) /\ ( normh ` ( C -h B ) ) < ( D / 2 ) ) -> ( normh ` ( if ( A e. ~H , A , 0h ) -h B ) ) < D ) ) )
7 oveq2 
 |-  ( B = if ( B e. ~H , B , 0h ) -> ( C -h B ) = ( C -h if ( B e. ~H , B , 0h ) ) )
8 7 fveq2d 
 |-  ( B = if ( B e. ~H , B , 0h ) -> ( normh ` ( C -h B ) ) = ( normh ` ( C -h if ( B e. ~H , B , 0h ) ) ) )
9 8 breq1d 
 |-  ( B = if ( B e. ~H , B , 0h ) -> ( ( normh ` ( C -h B ) ) < ( D / 2 ) <-> ( normh ` ( C -h if ( B e. ~H , B , 0h ) ) ) < ( D / 2 ) ) )
10 9 anbi2d 
 |-  ( B = if ( B e. ~H , B , 0h ) -> ( ( ( normh ` ( if ( A e. ~H , A , 0h ) -h C ) ) < ( D / 2 ) /\ ( normh ` ( C -h B ) ) < ( D / 2 ) ) <-> ( ( normh ` ( if ( A e. ~H , A , 0h ) -h C ) ) < ( D / 2 ) /\ ( normh ` ( C -h if ( B e. ~H , B , 0h ) ) ) < ( D / 2 ) ) ) )
11 oveq2 
 |-  ( B = if ( B e. ~H , B , 0h ) -> ( if ( A e. ~H , A , 0h ) -h B ) = ( if ( A e. ~H , A , 0h ) -h if ( B e. ~H , B , 0h ) ) )
12 11 fveq2d 
 |-  ( B = if ( B e. ~H , B , 0h ) -> ( normh ` ( if ( A e. ~H , A , 0h ) -h B ) ) = ( normh ` ( if ( A e. ~H , A , 0h ) -h if ( B e. ~H , B , 0h ) ) ) )
13 12 breq1d 
 |-  ( B = if ( B e. ~H , B , 0h ) -> ( ( normh ` ( if ( A e. ~H , A , 0h ) -h B ) ) < D <-> ( normh ` ( if ( A e. ~H , A , 0h ) -h if ( B e. ~H , B , 0h ) ) ) < D ) )
14 10 13 imbi12d 
 |-  ( B = if ( B e. ~H , B , 0h ) -> ( ( ( ( normh ` ( if ( A e. ~H , A , 0h ) -h C ) ) < ( D / 2 ) /\ ( normh ` ( C -h B ) ) < ( D / 2 ) ) -> ( normh ` ( if ( A e. ~H , A , 0h ) -h B ) ) < D ) <-> ( ( ( normh ` ( if ( A e. ~H , A , 0h ) -h C ) ) < ( D / 2 ) /\ ( normh ` ( C -h if ( B e. ~H , B , 0h ) ) ) < ( D / 2 ) ) -> ( normh ` ( if ( A e. ~H , A , 0h ) -h if ( B e. ~H , B , 0h ) ) ) < D ) ) )
15 oveq2 
 |-  ( C = if ( C e. ~H , C , 0h ) -> ( if ( A e. ~H , A , 0h ) -h C ) = ( if ( A e. ~H , A , 0h ) -h if ( C e. ~H , C , 0h ) ) )
16 15 fveq2d 
 |-  ( C = if ( C e. ~H , C , 0h ) -> ( normh ` ( if ( A e. ~H , A , 0h ) -h C ) ) = ( normh ` ( if ( A e. ~H , A , 0h ) -h if ( C e. ~H , C , 0h ) ) ) )
17 16 breq1d 
 |-  ( C = if ( C e. ~H , C , 0h ) -> ( ( normh ` ( if ( A e. ~H , A , 0h ) -h C ) ) < ( D / 2 ) <-> ( normh ` ( if ( A e. ~H , A , 0h ) -h if ( C e. ~H , C , 0h ) ) ) < ( D / 2 ) ) )
18 fvoveq1 
 |-  ( C = if ( C e. ~H , C , 0h ) -> ( normh ` ( C -h if ( B e. ~H , B , 0h ) ) ) = ( normh ` ( if ( C e. ~H , C , 0h ) -h if ( B e. ~H , B , 0h ) ) ) )
19 18 breq1d 
 |-  ( C = if ( C e. ~H , C , 0h ) -> ( ( normh ` ( C -h if ( B e. ~H , B , 0h ) ) ) < ( D / 2 ) <-> ( normh ` ( if ( C e. ~H , C , 0h ) -h if ( B e. ~H , B , 0h ) ) ) < ( D / 2 ) ) )
20 17 19 anbi12d 
 |-  ( C = if ( C e. ~H , C , 0h ) -> ( ( ( normh ` ( if ( A e. ~H , A , 0h ) -h C ) ) < ( D / 2 ) /\ ( normh ` ( C -h if ( B e. ~H , B , 0h ) ) ) < ( D / 2 ) ) <-> ( ( normh ` ( if ( A e. ~H , A , 0h ) -h if ( C e. ~H , C , 0h ) ) ) < ( D / 2 ) /\ ( normh ` ( if ( C e. ~H , C , 0h ) -h if ( B e. ~H , B , 0h ) ) ) < ( D / 2 ) ) ) )
21 20 imbi1d 
 |-  ( C = if ( C e. ~H , C , 0h ) -> ( ( ( ( normh ` ( if ( A e. ~H , A , 0h ) -h C ) ) < ( D / 2 ) /\ ( normh ` ( C -h if ( B e. ~H , B , 0h ) ) ) < ( D / 2 ) ) -> ( normh ` ( if ( A e. ~H , A , 0h ) -h if ( B e. ~H , B , 0h ) ) ) < D ) <-> ( ( ( normh ` ( if ( A e. ~H , A , 0h ) -h if ( C e. ~H , C , 0h ) ) ) < ( D / 2 ) /\ ( normh ` ( if ( C e. ~H , C , 0h ) -h if ( B e. ~H , B , 0h ) ) ) < ( D / 2 ) ) -> ( normh ` ( if ( A e. ~H , A , 0h ) -h if ( B e. ~H , B , 0h ) ) ) < D ) ) )
22 oveq1 
 |-  ( D = if ( D e. RR , D , 2 ) -> ( D / 2 ) = ( if ( D e. RR , D , 2 ) / 2 ) )
23 22 breq2d 
 |-  ( D = if ( D e. RR , D , 2 ) -> ( ( normh ` ( if ( A e. ~H , A , 0h ) -h if ( C e. ~H , C , 0h ) ) ) < ( D / 2 ) <-> ( normh ` ( if ( A e. ~H , A , 0h ) -h if ( C e. ~H , C , 0h ) ) ) < ( if ( D e. RR , D , 2 ) / 2 ) ) )
24 22 breq2d 
 |-  ( D = if ( D e. RR , D , 2 ) -> ( ( normh ` ( if ( C e. ~H , C , 0h ) -h if ( B e. ~H , B , 0h ) ) ) < ( D / 2 ) <-> ( normh ` ( if ( C e. ~H , C , 0h ) -h if ( B e. ~H , B , 0h ) ) ) < ( if ( D e. RR , D , 2 ) / 2 ) ) )
25 23 24 anbi12d 
 |-  ( D = if ( D e. RR , D , 2 ) -> ( ( ( normh ` ( if ( A e. ~H , A , 0h ) -h if ( C e. ~H , C , 0h ) ) ) < ( D / 2 ) /\ ( normh ` ( if ( C e. ~H , C , 0h ) -h if ( B e. ~H , B , 0h ) ) ) < ( D / 2 ) ) <-> ( ( normh ` ( if ( A e. ~H , A , 0h ) -h if ( C e. ~H , C , 0h ) ) ) < ( if ( D e. RR , D , 2 ) / 2 ) /\ ( normh ` ( if ( C e. ~H , C , 0h ) -h if ( B e. ~H , B , 0h ) ) ) < ( if ( D e. RR , D , 2 ) / 2 ) ) ) )
26 breq2 
 |-  ( D = if ( D e. RR , D , 2 ) -> ( ( normh ` ( if ( A e. ~H , A , 0h ) -h if ( B e. ~H , B , 0h ) ) ) < D <-> ( normh ` ( if ( A e. ~H , A , 0h ) -h if ( B e. ~H , B , 0h ) ) ) < if ( D e. RR , D , 2 ) ) )
27 25 26 imbi12d 
 |-  ( D = if ( D e. RR , D , 2 ) -> ( ( ( ( normh ` ( if ( A e. ~H , A , 0h ) -h if ( C e. ~H , C , 0h ) ) ) < ( D / 2 ) /\ ( normh ` ( if ( C e. ~H , C , 0h ) -h if ( B e. ~H , B , 0h ) ) ) < ( D / 2 ) ) -> ( normh ` ( if ( A e. ~H , A , 0h ) -h if ( B e. ~H , B , 0h ) ) ) < D ) <-> ( ( ( normh ` ( if ( A e. ~H , A , 0h ) -h if ( C e. ~H , C , 0h ) ) ) < ( if ( D e. RR , D , 2 ) / 2 ) /\ ( normh ` ( if ( C e. ~H , C , 0h ) -h if ( B e. ~H , B , 0h ) ) ) < ( if ( D e. RR , D , 2 ) / 2 ) ) -> ( normh ` ( if ( A e. ~H , A , 0h ) -h if ( B e. ~H , B , 0h ) ) ) < if ( D e. RR , D , 2 ) ) ) )
28 ifhvhv0 
 |-  if ( A e. ~H , A , 0h ) e. ~H
29 ifhvhv0 
 |-  if ( B e. ~H , B , 0h ) e. ~H
30 ifhvhv0 
 |-  if ( C e. ~H , C , 0h ) e. ~H
31 2re 
 |-  2 e. RR
32 31 elimel 
 |-  if ( D e. RR , D , 2 ) e. RR
33 28 29 30 32 norm3lem 
 |-  ( ( ( normh ` ( if ( A e. ~H , A , 0h ) -h if ( C e. ~H , C , 0h ) ) ) < ( if ( D e. RR , D , 2 ) / 2 ) /\ ( normh ` ( if ( C e. ~H , C , 0h ) -h if ( B e. ~H , B , 0h ) ) ) < ( if ( D e. RR , D , 2 ) / 2 ) ) -> ( normh ` ( if ( A e. ~H , A , 0h ) -h if ( B e. ~H , B , 0h ) ) ) < if ( D e. RR , D , 2 ) )
34 6 14 21 27 33 dedth4h 
 |-  ( ( ( A e. ~H /\ B e. ~H ) /\ ( C e. ~H /\ D e. RR ) ) -> ( ( ( normh ` ( A -h C ) ) < ( D / 2 ) /\ ( normh ` ( C -h B ) ) < ( D / 2 ) ) -> ( normh ` ( A -h B ) ) < D ) )