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 ) )