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Metamath Proof Explorer

Theorem latcl2

Description: The join and meet of any two elements exist. (Contributed by NM, 14-Sep-2018)

Ref Expression
Hypotheses latcl2.b 
|- B = ( Base ` K )
latcl2.j 
|- .\/ = ( join ` K )
latcl2.m 
|- ./\ = ( meet ` K )
latcl2.k 
|- ( ph -> K e. Lat )
latcl2.x 
|- ( ph -> X e. B )
latcl2.y 
|- ( ph -> Y e. B )
Assertion latcl2 
|- ( ph -> ( <. X , Y >. e. dom .\/ /\ <. X , Y >. e. dom ./\ ) )

Proof

Step Hyp Ref Expression
1 latcl2.b 
 |-  B = ( Base ` K )
2 latcl2.j 
 |-  .\/ = ( join ` K )
3 latcl2.m 
 |-  ./\ = ( meet ` K )
4 latcl2.k 
 |-  ( ph -> K e. Lat )
5 latcl2.x 
 |-  ( ph -> X e. B )
6 latcl2.y 
 |-  ( ph -> Y e. B )
7 5 6 opelxpd 
 |-  ( ph -> <. X , Y >. e. ( B X. B ) )
8 1 2 3 islat 
 |-  ( K e. Lat <-> ( K e. Poset /\ ( dom .\/ = ( B X. B ) /\ dom ./\ = ( B X. B ) ) ) )
9 4 8 sylib 
 |-  ( ph -> ( K e. Poset /\ ( dom .\/ = ( B X. B ) /\ dom ./\ = ( B X. B ) ) ) )
10 9 simprld 
 |-  ( ph -> dom .\/ = ( B X. B ) )
11 7 10 eleqtrrd 
 |-  ( ph -> <. X , Y >. e. dom .\/ )
12 9 simprrd 
 |-  ( ph -> dom ./\ = ( B X. B ) )
13 7 12 eleqtrrd 
 |-  ( ph -> <. X , Y >. e. dom ./\ )
14 11 13 jca 
 |-  ( ph -> ( <. X , Y >. e. dom .\/ /\ <. X , Y >. e. dom ./\ ) )