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Theorem lemeet2 14343
Description: A meet's second argument is greater than or equal to the meet. (Contributed by NM, 19-Jul-2012.)
Hypotheses
Ref Expression
meetval2.b  |-  B  =  ( Base `  K
)
meetval2.s  |-  .<_  =  ( le `  K )
meetval2.m  |-  ./\  =  ( meet `  K )
Assertion
Ref Expression
lemeet2  |-  ( ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  ( X  ./\ 
Y )  e.  B
)  ->  ( X  ./\ 
Y )  .<_  Y )

Proof of Theorem lemeet2
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 meetval2.b . . . 4  |-  B  =  ( Base `  K
)
2 meetval2.s . . . 4  |-  .<_  =  ( le `  K )
3 meetval2.m . . . 4  |-  ./\  =  ( meet `  K )
41, 2, 3meetlem 14341 . . 3  |-  ( ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  ( X  ./\ 
Y )  e.  B
)  ->  ( (
( X  ./\  Y
)  .<_  X  /\  ( X  ./\  Y )  .<_  Y )  /\  A. z  e.  B  (
( z  .<_  X  /\  z  .<_  Y )  -> 
z  .<_  ( X  ./\  Y ) ) ) )
54simpld 445 . 2  |-  ( ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  ( X  ./\ 
Y )  e.  B
)  ->  ( ( X  ./\  Y )  .<_  X  /\  ( X  ./\  Y )  .<_  Y )
)
65simprd 449 1  |-  ( ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  ( X  ./\ 
Y )  e.  B
)  ->  ( X  ./\ 
Y )  .<_  Y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 935    = wceq 1647    e. wcel 1715   A.wral 2628   class class class wbr 4125   ` cfv 5358  (class class class)co 5981   Basecbs 13356   lecple 13423   meetcmee 14289
This theorem is referenced by:  meetle  14344  latmle2  14393
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-undef 6440  df-riota 6446  df-glb 14319  df-meet 14321
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