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Theorem lemeet2 14419
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 14417 . . 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 446 . 2  |-  ( ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  ( X  ./\ 
Y )  e.  B
)  ->  ( ( X  ./\  Y )  .<_  X  /\  ( X  ./\  Y )  .<_  Y )
)
65simprd 450 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 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2674   class class class wbr 4180   ` cfv 5421  (class class class)co 6048   Basecbs 13432   lecple 13499   meetcmee 14365
This theorem is referenced by:  meetle  14420  latmle2  14469
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-undef 6510  df-riota 6516  df-glb 14395  df-meet 14397
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