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Theorem lemeet2 14487
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 14485 . . 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 447 . 2  |-  ( ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  ( X  ./\ 
Y )  e.  B
)  ->  ( ( X  ./\  Y )  .<_  X  /\  ( X  ./\  Y )  .<_  Y )
)
65simprd 451 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 360    /\ w3a 937    = wceq 1653    e. wcel 1727   A.wral 2711   class class class wbr 4237   ` cfv 5483  (class class class)co 6110   Basecbs 13500   lecple 13567   meetcmee 14433
This theorem is referenced by:  meetle  14488  latmle2  14537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6378  df-2nd 6379  df-undef 6572  df-riota 6578  df-glb 14463  df-meet 14465
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