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Metamath Proof Explorer

Theorem paddasslem11

Description: Lemma for paddass . The case when p = z . (Contributed by NM, 11-Jan-2012)

Ref Expression
Hypotheses paddasslem.l 
|- .<_ = ( le ` K )
paddasslem.j 
|- .\/ = ( join ` K )
paddasslem.a 
|- A = ( Atoms ` K )
paddasslem.p 
|- .+ = ( +P ` K )
Assertion paddasslem11 
|- ( ( ( ( K e. HL /\ p = z ) /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) ) /\ z e. Z ) -> p e. ( ( X .+ Y ) .+ Z ) )

Proof

Step Hyp Ref Expression
1 paddasslem.l 
 |-  .<_ = ( le ` K )
2 paddasslem.j 
 |-  .\/ = ( join ` K )
3 paddasslem.a 
 |-  A = ( Atoms ` K )
4 paddasslem.p 
 |-  .+ = ( +P ` K )
5 simplll 
 |-  ( ( ( ( K e. HL /\ p = z ) /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) ) /\ z e. Z ) -> K e. HL )
6 simplr3 
 |-  ( ( ( ( K e. HL /\ p = z ) /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) ) /\ z e. Z ) -> Z C_ A )
7 simplr1 
 |-  ( ( ( ( K e. HL /\ p = z ) /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) ) /\ z e. Z ) -> X C_ A )
8 simplr2 
 |-  ( ( ( ( K e. HL /\ p = z ) /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) ) /\ z e. Z ) -> Y C_ A )
9 3 4 paddssat 
 |-  ( ( K e. HL /\ X C_ A /\ Y C_ A ) -> ( X .+ Y ) C_ A )
10 5 7 8 9 syl3anc 
 |-  ( ( ( ( K e. HL /\ p = z ) /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) ) /\ z e. Z ) -> ( X .+ Y ) C_ A )
11 3 4 sspadd2 
 |-  ( ( K e. HL /\ Z C_ A /\ ( X .+ Y ) C_ A ) -> Z C_ ( ( X .+ Y ) .+ Z ) )
12 5 6 10 11 syl3anc 
 |-  ( ( ( ( K e. HL /\ p = z ) /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) ) /\ z e. Z ) -> Z C_ ( ( X .+ Y ) .+ Z ) )
13 simpllr 
 |-  ( ( ( ( K e. HL /\ p = z ) /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) ) /\ z e. Z ) -> p = z )
14 simpr 
 |-  ( ( ( ( K e. HL /\ p = z ) /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) ) /\ z e. Z ) -> z e. Z )
15 13 14 eqeltrd 
 |-  ( ( ( ( K e. HL /\ p = z ) /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) ) /\ z e. Z ) -> p e. Z )
16 12 15 sseldd 
 |-  ( ( ( ( K e. HL /\ p = z ) /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) ) /\ z e. Z ) -> p e. ( ( X .+ Y ) .+ Z ) )