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Theorem meetcomALT 14387
Description: The meet of a poset commutes. (This may not be a theorem under other definitions of meet.) (Contributed by NM, 17-Sep-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
meetcom.b  |-  B  =  ( Base `  K
)
meetcom.m  |-  ./\  =  ( meet `  K )
Assertion
Ref Expression
meetcomALT  |-  ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  =  ( Y 
./\  X ) )

Proof of Theorem meetcomALT
StepHypRef Expression
1 prcom 3825 . . . 4  |-  { Y ,  X }  =  { X ,  Y }
21fveq2i 5671 . . 3  |-  ( ( glb `  K ) `
 { Y ,  X } )  =  ( ( glb `  K
) `  { X ,  Y } )
32a1i 11 . 2  |-  ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( glb `  K
) `  { Y ,  X } )  =  ( ( glb `  K
) `  { X ,  Y } ) )
4 meetcom.b . . . 4  |-  B  =  ( Base `  K
)
5 eqid 2387 . . . 4  |-  ( glb `  K )  =  ( glb `  K )
6 meetcom.m . . . 4  |-  ./\  =  ( meet `  K )
74, 5, 6meetval 14379 . . 3  |-  ( ( K  e.  A  /\  Y  e.  B  /\  X  e.  B )  ->  ( Y  ./\  X
)  =  ( ( glb `  K ) `
 { Y ,  X } ) )
873com23 1159 . 2  |-  ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  B )  ->  ( Y  ./\  X
)  =  ( ( glb `  K ) `
 { Y ,  X } ) )
94, 5, 6meetval 14379 . 2  |-  ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  =  ( ( glb `  K ) `
 { X ,  Y } ) )
103, 8, 93eqtr4rd 2430 1  |-  ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  =  ( Y 
./\  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1649    e. wcel 1717   {cpr 3758   ` cfv 5394  (class class class)co 6020   Basecbs 13396   glbcglb 14327   meetcmee 14329
This theorem is referenced by:  meetcom  14388
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-meet 14361
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