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Theorem meetcomALT 14153
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 3718 . . . 4  |-  { Y ,  X }  =  { X ,  Y }
21fveq2i 5544 . . 3  |-  ( ( glb `  K ) `
 { Y ,  X } )  =  ( ( glb `  K
) `  { X ,  Y } )
32a1i 10 . 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 2296 . . . 4  |-  ( glb `  K )  =  ( glb `  K )
6 meetcom.m . . . 4  |-  ./\  =  ( meet `  K )
74, 5, 6meetval 14145 . . 3  |-  ( ( K  e.  A  /\  Y  e.  B  /\  X  e.  B )  ->  ( Y  ./\  X
)  =  ( ( glb `  K ) `
 { Y ,  X } ) )
873com23 1157 . 2  |-  ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  B )  ->  ( Y  ./\  X
)  =  ( ( glb `  K ) `
 { Y ,  X } ) )
94, 5, 6meetval 14145 . 2  |-  ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  =  ( ( glb `  K ) `
 { X ,  Y } ) )
103, 8, 93eqtr4rd 2339 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 934    = wceq 1632    e. wcel 1696   {cpr 3654   ` cfv 5271  (class class class)co 5874   Basecbs 13164   glbcglb 14093   meetcmee 14095
This theorem is referenced by:  meetcom  14154
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-meet 14127
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