Metamath Proof Explorer

Theorem mdsymi

Description: M-symmetry of the Hilbert lattice. Lemma 5 of Maeda p. 168. (Contributed by NM, 3-Jul-2004) (New usage is discouraged.)

Ref Expression
Hypotheses mdsym.1 
|- A e. CH
mdsym.2 
|- B e. CH
Assertion mdsymi 
|- ( A MH B <-> B MH A )

Proof

Step Hyp Ref Expression
1 mdsym.1 
 |-  A e. CH
2 mdsym.2 
 |-  B e. CH
3 2 choccli 
 |-  ( _|_ ` B ) e. CH
4 1 choccli 
 |-  ( _|_ ` A ) e. CH
5 eqid 
 |-  ( ( _|_ ` B ) vH x ) = ( ( _|_ ` B ) vH x )
6 3 4 5 mdsymlem8 
 |-  ( ( ( _|_ ` B ) =/= 0H /\ ( _|_ ` A ) =/= 0H ) -> ( ( _|_ ` A ) MH* ( _|_ ` B ) <-> ( _|_ ` B ) MH* ( _|_ ` A ) ) )
7 mddmd 
 |-  ( ( A e. CH /\ B e. CH ) -> ( A MH B <-> ( _|_ ` A ) MH* ( _|_ ` B ) ) )
8 1 2 7 mp2an 
 |-  ( A MH B <-> ( _|_ ` A ) MH* ( _|_ ` B ) )
9 mddmd 
 |-  ( ( B e. CH /\ A e. CH ) -> ( B MH A <-> ( _|_ ` B ) MH* ( _|_ ` A ) ) )
10 2 1 9 mp2an 
 |-  ( B MH A <-> ( _|_ ` B ) MH* ( _|_ ` A ) )
11 6 8 10 3bitr4g 
 |-  ( ( ( _|_ ` B ) =/= 0H /\ ( _|_ ` A ) =/= 0H ) -> ( A MH B <-> B MH A ) )
12 1 chssii 
 |-  A C_ ~H
13 fveq2 
 |-  ( ( _|_ ` B ) = 0H -> ( _|_ ` ( _|_ ` B ) ) = ( _|_ ` 0H ) )
14 2 pjococi 
 |-  ( _|_ ` ( _|_ ` B ) ) = B
15 choc0 
 |-  ( _|_ ` 0H ) = ~H
16 13 14 15 3eqtr3g 
 |-  ( ( _|_ ` B ) = 0H -> B = ~H )
17 12 16 sseqtrrid 
 |-  ( ( _|_ ` B ) = 0H -> A C_ B )
18 ssmd1 
 |-  ( ( A e. CH /\ B e. CH /\ A C_ B ) -> A MH B )
19 1 2 18 mp3an12 
 |-  ( A C_ B -> A MH B )
20 ssmd2 
 |-  ( ( A e. CH /\ B e. CH /\ A C_ B ) -> B MH A )
21 1 2 20 mp3an12 
 |-  ( A C_ B -> B MH A )
22 19 21 jca 
 |-  ( A C_ B -> ( A MH B /\ B MH A ) )
23 pm5.1 
 |-  ( ( A MH B /\ B MH A ) -> ( A MH B <-> B MH A ) )
24 17 22 23 3syl 
 |-  ( ( _|_ ` B ) = 0H -> ( A MH B <-> B MH A ) )
25 2 chssii 
 |-  B C_ ~H
26 fveq2 
 |-  ( ( _|_ ` A ) = 0H -> ( _|_ ` ( _|_ ` A ) ) = ( _|_ ` 0H ) )
27 1 pjococi 
 |-  ( _|_ ` ( _|_ ` A ) ) = A
28 26 27 15 3eqtr3g 
 |-  ( ( _|_ ` A ) = 0H -> A = ~H )
29 25 28 sseqtrrid 
 |-  ( ( _|_ ` A ) = 0H -> B C_ A )
30 ssmd2 
 |-  ( ( B e. CH /\ A e. CH /\ B C_ A ) -> A MH B )
31 2 1 30 mp3an12 
 |-  ( B C_ A -> A MH B )
32 ssmd1 
 |-  ( ( B e. CH /\ A e. CH /\ B C_ A ) -> B MH A )
33 2 1 32 mp3an12 
 |-  ( B C_ A -> B MH A )
34 31 33 jca 
 |-  ( B C_ A -> ( A MH B /\ B MH A ) )
35 29 34 23 3syl 
 |-  ( ( _|_ ` A ) = 0H -> ( A MH B <-> B MH A ) )
36 11 24 35 pm2.61iine 
 |-  ( A MH B <-> B MH A )