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Theorem meetcomALT 14452
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 3874 . . . 4  |-  { Y ,  X }  =  { X ,  Y }
21fveq2i 5723 . . 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 2435 . . . 4  |-  ( glb `  K )  =  ( glb `  K )
6 meetcom.m . . . 4  |-  ./\  =  ( meet `  K )
74, 5, 6meetval 14444 . . 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 14444 . 2  |-  ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  =  ( ( glb `  K ) `
 { X ,  Y } ) )
103, 8, 93eqtr4rd 2478 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 1652    e. wcel 1725   {cpr 3807   ` cfv 5446  (class class class)co 6073   Basecbs 13461   glbcglb 14392   meetcmee 14394
This theorem is referenced by:  meetcom  14453
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-meet 14426
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