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Theorem cmtcomN 29736
Description: Commutation is symmetric. Theorem 2(v) in [Kalmbach] p. 22. (cmcmi 23051 analog.) (Contributed by NM, 7-Nov-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
cmtcom.b  |-  B  =  ( Base `  K
)
cmtcom.c  |-  C  =  ( cm `  K
)
Assertion
Ref Expression
cmtcomN  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X C Y  <-> 
Y C X ) )

Proof of Theorem cmtcomN
StepHypRef Expression
1 cmtcom.b . . 3  |-  B  =  ( Base `  K
)
2 cmtcom.c . . 3  |-  C  =  ( cm `  K
)
31, 2cmtcomlemN 29735 . 2  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X C Y  ->  Y C X ) )
41, 2cmtcomlemN 29735 . . 3  |-  ( ( K  e.  OML  /\  Y  e.  B  /\  X  e.  B )  ->  ( Y C X  ->  X C Y ) )
543com23 1159 . 2  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( Y C X  ->  X C Y ) )
63, 5impbid 184 1  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X C Y  <-> 
Y C X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ w3a 936    = wceq 1649    e. wcel 1721   class class class wbr 4176   ` cfv 5417   Basecbs 13428   cmccmtN 29660   OMLcoml 29662
This theorem is referenced by:  cmt3N  29738  cmtbr3N  29741  omlmod1i2N  29747
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
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 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-undef 6506  df-riota 6512  df-poset 14362  df-lub 14390  df-glb 14391  df-join 14392  df-meet 14393  df-lat 14434  df-oposet 29663  df-cmtN 29664  df-ol 29665  df-oml 29666
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