Metamath Proof Explorer

Theorem osumcllem2N

Description: Lemma for osumclN . (Contributed by NM, 25-Mar-2012) (New usage is discouraged.)

Ref Expression
Hypotheses osumcllem.l 
|- .<_ = ( le ` K )
osumcllem.j 
|- .\/ = ( join ` K )
osumcllem.a 
|- A = ( Atoms ` K )
osumcllem.p 
|- .+ = ( +P ` K )
osumcllem.o 
|- ._|_ = ( _|_P ` K )
osumcllem.c 
|- C = ( PSubCl ` K )
osumcllem.m 
|- M = ( X .+ { p } )
osumcllem.u 
|- U = ( ._|_ ` ( ._|_ ` ( X .+ Y ) ) )
Assertion osumcllem2N 
|- ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ p e. U ) -> X C_ ( U i^i M ) )

Proof

Step Hyp Ref Expression
1 osumcllem.l 
 |-  .<_ = ( le ` K )
2 osumcllem.j 
 |-  .\/ = ( join ` K )
3 osumcllem.a 
 |-  A = ( Atoms ` K )
4 osumcllem.p 
 |-  .+ = ( +P ` K )
5 osumcllem.o 
 |-  ._|_ = ( _|_P ` K )
6 osumcllem.c 
 |-  C = ( PSubCl ` K )
7 osumcllem.m 
 |-  M = ( X .+ { p } )
8 osumcllem.u 
 |-  U = ( ._|_ ` ( ._|_ ` ( X .+ Y ) ) )
9 simpl1 
 |-  ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ p e. U ) -> K e. HL )
10 simpl2 
 |-  ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ p e. U ) -> X C_ A )
11 simpr 
 |-  ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ p e. U ) -> p e. U )
12 11 snssd 
 |-  ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ p e. U ) -> { p } C_ U )
13 3 4 paddssat 
 |-  ( ( K e. HL /\ X C_ A /\ Y C_ A ) -> ( X .+ Y ) C_ A )
14 13 adantr 
 |-  ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ p e. U ) -> ( X .+ Y ) C_ A )
15 3 5 polssatN 
 |-  ( ( K e. HL /\ ( X .+ Y ) C_ A ) -> ( ._|_ ` ( X .+ Y ) ) C_ A )
16 9 14 15 syl2anc 
 |-  ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ p e. U ) -> ( ._|_ ` ( X .+ Y ) ) C_ A )
17 3 5 polssatN 
 |-  ( ( K e. HL /\ ( ._|_ ` ( X .+ Y ) ) C_ A ) -> ( ._|_ ` ( ._|_ ` ( X .+ Y ) ) ) C_ A )
18 9 16 17 syl2anc 
 |-  ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ p e. U ) -> ( ._|_ ` ( ._|_ ` ( X .+ Y ) ) ) C_ A )
19 8 18 eqsstrid 
 |-  ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ p e. U ) -> U C_ A )
20 12 19 sstrd 
 |-  ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ p e. U ) -> { p } C_ A )
21 3 4 sspadd1 
 |-  ( ( K e. HL /\ X C_ A /\ { p } C_ A ) -> X C_ ( X .+ { p } ) )
22 9 10 20 21 syl3anc 
 |-  ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ p e. U ) -> X C_ ( X .+ { p } ) )
23 22 7 sseqtrrdi 
 |-  ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ p e. U ) -> X C_ M )
24 1 2 3 4 5 6 7 8 osumcllem1N 
 |-  ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ p e. U ) -> ( U i^i M ) = M )
25 23 24 sseqtrrd 
 |-  ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ p e. U ) -> X C_ ( U i^i M ) )