Theorem cdleme20y

Description: Part of proof of Lemma E in Crawley p. 113. Utility lemma. (Contributed by NM, 17-Nov-2012) (Proof shortened by OpenAI, 25-Mar-2020)

	Ref	Expression
Hypotheses	cdleme20z.l	|- .<_ = ( le ` K )
	cdleme20z.j	|- .\/ = ( join ` K )
	cdleme20z.m	|- ./\ = ( meet ` K )
	cdleme20z.a	|- A = ( Atoms ` K )
Assertion	cdleme20y	|- ( ( K e. HL /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( S =/= T /\ -. R .<_ ( S .\/ T ) ) ) -> ( ( S .\/ R ) ./\ ( T .\/ R ) ) = R )

Proof

Step	Hyp	Ref	Expression
1		cdleme20z.l	|- .<_ = ( le ` K )
2		cdleme20z.j	|- .\/ = ( join ` K )
3		cdleme20z.m	|- ./\ = ( meet ` K )
4		cdleme20z.a	|- A = ( Atoms ` K )
5		simp1	|- ( ( K e. HL /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( S =/= T /\ -. R .<_ ( S .\/ T ) ) ) -> K e. HL )
6		simp22	|- ( ( K e. HL /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( S =/= T /\ -. R .<_ ( S .\/ T ) ) ) -> S e. A )
7		simp23	|- ( ( K e. HL /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( S =/= T /\ -. R .<_ ( S .\/ T ) ) ) -> T e. A )
8		simp21	|- ( ( K e. HL /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( S =/= T /\ -. R .<_ ( S .\/ T ) ) ) -> R e. A )
9		simp3	|- ( ( K e. HL /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( S =/= T /\ -. R .<_ ( S .\/ T ) ) ) -> ( S =/= T /\ -. R .<_ ( S .\/ T ) ) )
10	1 2 3 4	2llnma2rN	|- ( ( K e. HL /\ ( S e. A /\ T e. A /\ R e. A ) /\ ( S =/= T /\ -. R .<_ ( S .\/ T ) ) ) -> ( ( S .\/ R ) ./\ ( T .\/ R ) ) = R )
11	5 6 7 8 9 10	syl131anc	|- ( ( K e. HL /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( S =/= T /\ -. R .<_ ( S .\/ T ) ) ) -> ( ( S .\/ R ) ./\ ( T .\/ R ) ) = R )