Metamath Proof Explorer

Theorem sshhococi

Description: The join of two Hilbert space subsets (not necessarily closed subspaces) equals the join of their closures (double orthocomplements). (Contributed by NM, 1-Jun-2004) (Revised by Mario Carneiro, 15-May-2014) (New usage is discouraged.)

Ref Expression
Hypotheses sshjococ.1 
|- A C_ ~H
sshjococ.2 
|- B C_ ~H
Assertion sshhococi 
|- ( A vH B ) = ( ( _|_ ` ( _|_ ` A ) ) vH ( _|_ ` ( _|_ ` B ) ) )

Proof

Step Hyp Ref Expression
1 sshjococ.1 
 |-  A C_ ~H
2 sshjococ.2 
 |-  B C_ ~H
3 ococss 
 |-  ( A C_ ~H -> A C_ ( _|_ ` ( _|_ ` A ) ) )
4 1 3 ax-mp 
 |-  A C_ ( _|_ ` ( _|_ ` A ) )
5 ococss 
 |-  ( B C_ ~H -> B C_ ( _|_ ` ( _|_ ` B ) ) )
6 2 5 ax-mp 
 |-  B C_ ( _|_ ` ( _|_ ` B ) )
7 unss12 
 |-  ( ( A C_ ( _|_ ` ( _|_ ` A ) ) /\ B C_ ( _|_ ` ( _|_ ` B ) ) ) -> ( A u. B ) C_ ( ( _|_ ` ( _|_ ` A ) ) u. ( _|_ ` ( _|_ ` B ) ) ) )
8 4 6 7 mp2an 
 |-  ( A u. B ) C_ ( ( _|_ ` ( _|_ ` A ) ) u. ( _|_ ` ( _|_ ` B ) ) )
9 1 2 unssi 
 |-  ( A u. B ) C_ ~H
10 occl 
 |-  ( A C_ ~H -> ( _|_ ` A ) e. CH )
11 1 10 ax-mp 
 |-  ( _|_ ` A ) e. CH
12 11 choccli 
 |-  ( _|_ ` ( _|_ ` A ) ) e. CH
13 12 chssii 
 |-  ( _|_ ` ( _|_ ` A ) ) C_ ~H
14 occl 
 |-  ( B C_ ~H -> ( _|_ ` B ) e. CH )
15 2 14 ax-mp 
 |-  ( _|_ ` B ) e. CH
16 15 choccli 
 |-  ( _|_ ` ( _|_ ` B ) ) e. CH
17 16 chssii 
 |-  ( _|_ ` ( _|_ ` B ) ) C_ ~H
18 13 17 unssi 
 |-  ( ( _|_ ` ( _|_ ` A ) ) u. ( _|_ ` ( _|_ ` B ) ) ) C_ ~H
19 9 18 occon2i 
 |-  ( ( A u. B ) C_ ( ( _|_ ` ( _|_ ` A ) ) u. ( _|_ ` ( _|_ ` B ) ) ) -> ( _|_ ` ( _|_ ` ( A u. B ) ) ) C_ ( _|_ ` ( _|_ ` ( ( _|_ ` ( _|_ ` A ) ) u. ( _|_ ` ( _|_ ` B ) ) ) ) ) )
20 8 19 ax-mp 
 |-  ( _|_ ` ( _|_ ` ( A u. B ) ) ) C_ ( _|_ ` ( _|_ ` ( ( _|_ ` ( _|_ ` A ) ) u. ( _|_ ` ( _|_ ` B ) ) ) ) )
21 sshjval 
 |-  ( ( A C_ ~H /\ B C_ ~H ) -> ( A vH B ) = ( _|_ ` ( _|_ ` ( A u. B ) ) ) )
22 1 2 21 mp2an 
 |-  ( A vH B ) = ( _|_ ` ( _|_ ` ( A u. B ) ) )
23 12 16 chjvali 
 |-  ( ( _|_ ` ( _|_ ` A ) ) vH ( _|_ ` ( _|_ ` B ) ) ) = ( _|_ ` ( _|_ ` ( ( _|_ ` ( _|_ ` A ) ) u. ( _|_ ` ( _|_ ` B ) ) ) ) )
24 20 22 23 3sstr4i 
 |-  ( A vH B ) C_ ( ( _|_ ` ( _|_ ` A ) ) vH ( _|_ ` ( _|_ ` B ) ) )
25 ssun1 
 |-  A C_ ( A u. B )
26 ococss 
 |-  ( ( A u. B ) C_ ~H -> ( A u. B ) C_ ( _|_ ` ( _|_ ` ( A u. B ) ) ) )
27 9 26 ax-mp 
 |-  ( A u. B ) C_ ( _|_ ` ( _|_ ` ( A u. B ) ) )
28 25 27 sstri 
 |-  A C_ ( _|_ ` ( _|_ ` ( A u. B ) ) )
29 28 22 sseqtrri 
 |-  A C_ ( A vH B )
30 sshjcl 
 |-  ( ( A C_ ~H /\ B C_ ~H ) -> ( A vH B ) e. CH )
31 1 2 30 mp2an 
 |-  ( A vH B ) e. CH
32 31 chssii 
 |-  ( A vH B ) C_ ~H
33 1 32 occon2i 
 |-  ( A C_ ( A vH B ) -> ( _|_ ` ( _|_ ` A ) ) C_ ( _|_ ` ( _|_ ` ( A vH B ) ) ) )
34 29 33 ax-mp 
 |-  ( _|_ ` ( _|_ ` A ) ) C_ ( _|_ ` ( _|_ ` ( A vH B ) ) )
35 ssun2 
 |-  B C_ ( A u. B )
36 35 27 sstri 
 |-  B C_ ( _|_ ` ( _|_ ` ( A u. B ) ) )
37 36 22 sseqtrri 
 |-  B C_ ( A vH B )
38 2 32 occon2i 
 |-  ( B C_ ( A vH B ) -> ( _|_ ` ( _|_ ` B ) ) C_ ( _|_ ` ( _|_ ` ( A vH B ) ) ) )
39 37 38 ax-mp 
 |-  ( _|_ ` ( _|_ ` B ) ) C_ ( _|_ ` ( _|_ ` ( A vH B ) ) )
40 31 choccli 
 |-  ( _|_ ` ( A vH B ) ) e. CH
41 40 choccli 
 |-  ( _|_ ` ( _|_ ` ( A vH B ) ) ) e. CH
42 12 16 41 chlubii 
 |-  ( ( ( _|_ ` ( _|_ ` A ) ) C_ ( _|_ ` ( _|_ ` ( A vH B ) ) ) /\ ( _|_ ` ( _|_ ` B ) ) C_ ( _|_ ` ( _|_ ` ( A vH B ) ) ) ) -> ( ( _|_ ` ( _|_ ` A ) ) vH ( _|_ ` ( _|_ ` B ) ) ) C_ ( _|_ ` ( _|_ ` ( A vH B ) ) ) )
43 34 39 42 mp2an 
 |-  ( ( _|_ ` ( _|_ ` A ) ) vH ( _|_ ` ( _|_ ` B ) ) ) C_ ( _|_ ` ( _|_ ` ( A vH B ) ) )
44 31 ococi 
 |-  ( _|_ ` ( _|_ ` ( A vH B ) ) ) = ( A vH B )
45 43 44 sseqtri 
 |-  ( ( _|_ ` ( _|_ ` A ) ) vH ( _|_ ` ( _|_ ` B ) ) ) C_ ( A vH B )
46 24 45 eqssi 
 |-  ( A vH B ) = ( ( _|_ ` ( _|_ ` A ) ) vH ( _|_ ` ( _|_ ` B ) ) )