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Metamath Proof Explorer

Theorem lubss

Description: Subset law for least upper bounds. ( chsupss analog.) (Contributed by NM, 20-Oct-2011)

Ref Expression
Hypotheses lublem.b 
|- B = ( Base ` K )
lublem.l 
|- .<_ = ( le ` K )
lublem.u 
|- U = ( lub ` K )
Assertion lubss 
|- ( ( K e. CLat /\ T C_ B /\ S C_ T ) -> ( U ` S ) .<_ ( U ` T ) )

Proof

Step Hyp Ref Expression
1 lublem.b 
 |-  B = ( Base ` K )
2 lublem.l 
 |-  .<_ = ( le ` K )
3 lublem.u 
 |-  U = ( lub ` K )
4 simp1 
 |-  ( ( K e. CLat /\ T C_ B /\ S C_ T ) -> K e. CLat )
5 sstr2 
 |-  ( S C_ T -> ( T C_ B -> S C_ B ) )
6 5 impcom 
 |-  ( ( T C_ B /\ S C_ T ) -> S C_ B )
7 6 3adant1 
 |-  ( ( K e. CLat /\ T C_ B /\ S C_ T ) -> S C_ B )
8 1 3 clatlubcl 
 |-  ( ( K e. CLat /\ T C_ B ) -> ( U ` T ) e. B )
9 8 3adant3 
 |-  ( ( K e. CLat /\ T C_ B /\ S C_ T ) -> ( U ` T ) e. B )
10 4 7 9 3jca 
 |-  ( ( K e. CLat /\ T C_ B /\ S C_ T ) -> ( K e. CLat /\ S C_ B /\ ( U ` T ) e. B ) )
11 simpl1 
 |-  ( ( ( K e. CLat /\ T C_ B /\ S C_ T ) /\ y e. S ) -> K e. CLat )
12 simpl2 
 |-  ( ( ( K e. CLat /\ T C_ B /\ S C_ T ) /\ y e. S ) -> T C_ B )
13 ssel2 
 |-  ( ( S C_ T /\ y e. S ) -> y e. T )
14 13 3ad2antl3 
 |-  ( ( ( K e. CLat /\ T C_ B /\ S C_ T ) /\ y e. S ) -> y e. T )
15 1 2 3 lubub 
 |-  ( ( K e. CLat /\ T C_ B /\ y e. T ) -> y .<_ ( U ` T ) )
16 11 12 14 15 syl3anc 
 |-  ( ( ( K e. CLat /\ T C_ B /\ S C_ T ) /\ y e. S ) -> y .<_ ( U ` T ) )
17 16 ralrimiva 
 |-  ( ( K e. CLat /\ T C_ B /\ S C_ T ) -> A. y e. S y .<_ ( U ` T ) )
18 1 2 3 lubl 
 |-  ( ( K e. CLat /\ S C_ B /\ ( U ` T ) e. B ) -> ( A. y e. S y .<_ ( U ` T ) -> ( U ` S ) .<_ ( U ` T ) ) )
19 10 17 18 sylc 
 |-  ( ( K e. CLat /\ T C_ B /\ S C_ T ) -> ( U ` S ) .<_ ( U ` T ) )