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Theorem cmtidN 29423
Description: Any element commutes with itself. (cmidi 22953 analog.) (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
cmtid.b  |-  B  =  ( Base `  K
)
cmtid.c  |-  C  =  ( cm `  K
)
Assertion
Ref Expression
cmtidN  |-  ( ( K  e.  OML  /\  X  e.  B )  ->  X C X )

Proof of Theorem cmtidN
StepHypRef Expression
1 omllat 29408 . . 3  |-  ( K  e.  OML  ->  K  e.  Lat )
2 cmtid.b . . . 4  |-  B  =  ( Base `  K
)
3 eqid 2380 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
42, 3latref 14402 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B )  ->  X ( le `  K ) X )
51, 4sylan 458 . 2  |-  ( ( K  e.  OML  /\  X  e.  B )  ->  X ( le `  K ) X )
6 cmtid.c . . . 4  |-  C  =  ( cm `  K
)
72, 3, 6lecmtN 29422 . . 3  |-  ( ( K  e.  OML  /\  X  e.  B  /\  X  e.  B )  ->  ( X ( le
`  K ) X  ->  X C X ) )
873anidm23 1243 . 2  |-  ( ( K  e.  OML  /\  X  e.  B )  ->  ( X ( le
`  K ) X  ->  X C X ) )
95, 8mpd 15 1  |-  ( ( K  e.  OML  /\  X  e.  B )  ->  X C X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   class class class wbr 4146   ` cfv 5387   Basecbs 13389   lecple 13456   Latclat 14394   cmccmtN 29339   OMLcoml 29341
This theorem is referenced by:  omlspjN  29427
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 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-undef 6472  df-riota 6478  df-poset 14323  df-lub 14351  df-glb 14352  df-join 14353  df-meet 14354  df-lat 14395  df-oposet 29342  df-cmtN 29343  df-ol 29344  df-oml 29345
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