Metamath Proof Explorer

Theorem cvrletrN

Description: Property of an element above a covering. (Contributed by NM, 7-Dec-2012) (New usage is discouraged.)

Ref Expression
Hypotheses cvrletr.b 
|- B = ( Base ` K )
cvrletr.l 
|- .<_ = ( le ` K )
cvrletr.s 
|- .< = ( lt ` K )
cvrletr.c 
|- C = (
Assertion cvrletrN 
|- ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X C Y /\ Y .<_ Z ) -> X .< Z ) )

Proof

Step Hyp Ref Expression
1 cvrletr.b 
 |-  B = ( Base ` K )
2 cvrletr.l 
 |-  .<_ = ( le ` K )
3 cvrletr.s 
 |-  .< = ( lt ` K )
4 cvrletr.c 
 |-  C = (
5 simpll 
 |-  ( ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) ) /\ X C Y ) -> K e. Poset )
6 simplr1 
 |-  ( ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) ) /\ X C Y ) -> X e. B )
7 simplr2 
 |-  ( ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) ) /\ X C Y ) -> Y e. B )
8 simpr 
 |-  ( ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) ) /\ X C Y ) -> X C Y )
9 1 3 4 cvrlt 
 |-  ( ( ( K e. Poset /\ X e. B /\ Y e. B ) /\ X C Y ) -> X .< Y )
10 5 6 7 8 9 syl31anc 
 |-  ( ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) ) /\ X C Y ) -> X .< Y )
11 1 2 3 pltletr 
 |-  ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .< Y /\ Y .<_ Z ) -> X .< Z ) )
12 11 adantr 
 |-  ( ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) ) /\ X C Y ) -> ( ( X .< Y /\ Y .<_ Z ) -> X .< Z ) )
13 10 12 mpand 
 |-  ( ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) ) /\ X C Y ) -> ( Y .<_ Z -> X .< Z ) )
14 13 expimpd 
 |-  ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X C Y /\ Y .<_ Z ) -> X .< Z ) )