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Theorem meetcomALT 14137
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 3705 . . . 4  |-  { Y ,  X }  =  { X ,  Y }
21fveq2i 5528 . . 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 2283 . . . 4  |-  ( glb `  K )  =  ( glb `  K )
6 meetcom.m . . . 4  |-  ./\  =  ( meet `  K )
74, 5, 6meetval 14129 . . 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 14129 . 2  |-  ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  =  ( ( glb `  K ) `
 { X ,  Y } ) )
103, 8, 93eqtr4rd 2326 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 1623    e. wcel 1684   {cpr 3641   ` cfv 5255  (class class class)co 5858   Basecbs 13148   glbcglb 14077   meetcmee 14079
This theorem is referenced by:  meetcom  14138
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-meet 14111
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