This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem osumcllem1N

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 osumcllem1N 
|- ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ p e. U ) -> ( U i^i M ) = 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 3 4 sspadd1 
 |-  ( ( K e. HL /\ X C_ A /\ Y C_ A ) -> X C_ ( X .+ Y ) )
10 9 adantr 
 |-  ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ p e. U ) -> X C_ ( X .+ Y ) )
11 simpl1 
 |-  ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ p e. U ) -> K e. HL )
12 3 4 paddssat 
 |-  ( ( K e. HL /\ X C_ A /\ Y C_ A ) -> ( X .+ Y ) C_ A )
13 12 adantr 
 |-  ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ p e. U ) -> ( X .+ Y ) C_ A )
14 3 5 2polssN 
 |-  ( ( K e. HL /\ ( X .+ Y ) C_ A ) -> ( X .+ Y ) C_ ( ._|_ ` ( ._|_ ` ( X .+ Y ) ) ) )
15 11 13 14 syl2anc 
 |-  ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ p e. U ) -> ( X .+ Y ) C_ ( ._|_ ` ( ._|_ ` ( X .+ Y ) ) ) )
16 15 8 sseqtrrdi 
 |-  ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ p e. U ) -> ( X .+ Y ) C_ U )
17 10 16 sstrd 
 |-  ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ p e. U ) -> X C_ U )
18 simpr 
 |-  ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ p e. U ) -> p e. U )
19 18 snssd 
 |-  ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ p e. U ) -> { p } C_ U )
20 simpl2 
 |-  ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ p e. U ) -> X C_ A )
21 3 5 polssatN 
 |-  ( ( K e. HL /\ ( X .+ Y ) C_ A ) -> ( ._|_ ` ( X .+ Y ) ) C_ A )
22 11 13 21 syl2anc 
 |-  ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ p e. U ) -> ( ._|_ ` ( X .+ Y ) ) C_ A )
23 3 5 polssatN 
 |-  ( ( K e. HL /\ ( ._|_ ` ( X .+ Y ) ) C_ A ) -> ( ._|_ ` ( ._|_ ` ( X .+ Y ) ) ) C_ A )
24 11 22 23 syl2anc 
 |-  ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ p e. U ) -> ( ._|_ ` ( ._|_ ` ( X .+ Y ) ) ) C_ A )
25 8 24 eqsstrid 
 |-  ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ p e. U ) -> U C_ A )
26 19 25 sstrd 
 |-  ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ p e. U ) -> { p } C_ A )
27 eqid 
 |-  ( PSubSp ` K ) = ( PSubSp ` K )
28 3 27 5 polsubN 
 |-  ( ( K e. HL /\ ( ._|_ ` ( X .+ Y ) ) C_ A ) -> ( ._|_ ` ( ._|_ ` ( X .+ Y ) ) ) e. ( PSubSp ` K ) )
29 11 22 28 syl2anc 
 |-  ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ p e. U ) -> ( ._|_ ` ( ._|_ ` ( X .+ Y ) ) ) e. ( PSubSp ` K ) )
30 8 29 eqeltrid 
 |-  ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ p e. U ) -> U e. ( PSubSp ` K ) )
31 3 27 4 paddss 
 |-  ( ( K e. HL /\ ( X C_ A /\ { p } C_ A /\ U e. ( PSubSp ` K ) ) ) -> ( ( X C_ U /\ { p } C_ U ) <-> ( X .+ { p } ) C_ U ) )
32 11 20 26 30 31 syl13anc 
 |-  ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ p e. U ) -> ( ( X C_ U /\ { p } C_ U ) <-> ( X .+ { p } ) C_ U ) )
33 17 19 32 mpbi2and 
 |-  ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ p e. U ) -> ( X .+ { p } ) C_ U )
34 7 33 eqsstrid 
 |-  ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ p e. U ) -> M C_ U )
35 sseqin2 
 |-  ( M C_ U <-> ( U i^i M ) = M )
36 34 35 sylib 
 |-  ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ p e. U ) -> ( U i^i M ) = M )