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Theorem lecmtN 29751
Description: Ordered elements commute. (lecmi 23065 analog.) (Contributed by NM, 10-Nov-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
lecmt.b  |-  B  =  ( Base `  K
)
lecmt.l  |-  .<_  =  ( le `  K )
lecmt.c  |-  C  =  ( cm `  K
)
Assertion
Ref Expression
lecmtN  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  ->  X C Y ) )

Proof of Theorem lecmtN
StepHypRef Expression
1 omllat 29737 . . . . 5  |-  ( K  e.  OML  ->  K  e.  Lat )
213ad2ant1 978 . . . 4  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  Lat )
3 simp2 958 . . . 4  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  X  e.  B )
4 omlop 29736 . . . . . . 7  |-  ( K  e.  OML  ->  K  e.  OP )
543ad2ant1 978 . . . . . 6  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  OP )
6 lecmt.b . . . . . . 7  |-  B  =  ( Base `  K
)
7 eqid 2412 . . . . . . 7  |-  ( oc
`  K )  =  ( oc `  K
)
86, 7opoccl 29689 . . . . . 6  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( ( oc `  K ) `  X
)  e.  B )
95, 3, 8syl2anc 643 . . . . 5  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( oc `  K ) `  X
)  e.  B )
10 simp3 959 . . . . 5  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  Y  e.  B )
11 eqid 2412 . . . . . 6  |-  ( join `  K )  =  (
join `  K )
126, 11latjcl 14442 . . . . 5  |-  ( ( K  e.  Lat  /\  ( ( oc `  K ) `  X
)  e.  B  /\  Y  e.  B )  ->  ( ( ( oc
`  K ) `  X ) ( join `  K ) Y )  e.  B )
132, 9, 10, 12syl3anc 1184 . . . 4  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( ( oc
`  K ) `  X ) ( join `  K ) Y )  e.  B )
14 lecmt.l . . . . 5  |-  .<_  =  ( le `  K )
15 eqid 2412 . . . . 5  |-  ( meet `  K )  =  (
meet `  K )
166, 14, 15latmle1 14468 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  ( ( ( oc
`  K ) `  X ) ( join `  K ) Y )  e.  B )  -> 
( X ( meet `  K ) ( ( ( oc `  K
) `  X )
( join `  K ) Y ) )  .<_  X )
172, 3, 13, 16syl3anc 1184 . . 3  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X ( meet `  K ) ( ( ( oc `  K
) `  X )
( join `  K ) Y ) )  .<_  X )
186, 15latmcl 14443 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  ( ( ( oc
`  K ) `  X ) ( join `  K ) Y )  e.  B )  -> 
( X ( meet `  K ) ( ( ( oc `  K
) `  X )
( join `  K ) Y ) )  e.  B )
192, 3, 13, 18syl3anc 1184 . . . 4  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X ( meet `  K ) ( ( ( oc `  K
) `  X )
( join `  K ) Y ) )  e.  B )
206, 14lattr 14448 . . . 4  |-  ( ( K  e.  Lat  /\  ( ( X (
meet `  K )
( ( ( oc
`  K ) `  X ) ( join `  K ) Y ) )  e.  B  /\  X  e.  B  /\  Y  e.  B )
)  ->  ( (
( X ( meet `  K ) ( ( ( oc `  K
) `  X )
( join `  K ) Y ) )  .<_  X  /\  X  .<_  Y )  ->  ( X (
meet `  K )
( ( ( oc
`  K ) `  X ) ( join `  K ) Y ) )  .<_  Y )
)
212, 19, 3, 10, 20syl13anc 1186 . . 3  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( ( X ( meet `  K
) ( ( ( oc `  K ) `
 X ) (
join `  K ) Y ) )  .<_  X  /\  X  .<_  Y )  ->  ( X (
meet `  K )
( ( ( oc
`  K ) `  X ) ( join `  K ) Y ) )  .<_  Y )
)
2217, 21mpand 657 . 2  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  -> 
( X ( meet `  K ) ( ( ( oc `  K
) `  X )
( join `  K ) Y ) )  .<_  Y ) )
23 lecmt.c . . 3  |-  C  =  ( cm `  K
)
246, 14, 11, 15, 7, 23cmtbr4N 29750 . 2  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X C Y  <-> 
( X ( meet `  K ) ( ( ( oc `  K
) `  X )
( join `  K ) Y ) )  .<_  Y ) )
2522, 24sylibrd 226 1  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  ->  X C Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   class class class wbr 4180   ` cfv 5421  (class class class)co 6048   Basecbs 13432   lecple 13499   occoc 13500   joincjn 14364   meetcmee 14365   Latclat 14437   OPcops 29667   cmccmtN 29668   OMLcoml 29670
This theorem is referenced by:  cmtidN  29752  omlmod1i2N  29755
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 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-undef 6510  df-riota 6516  df-poset 14366  df-lub 14394  df-glb 14395  df-join 14396  df-meet 14397  df-lat 14438  df-oposet 29671  df-cmtN 29672  df-ol 29673  df-oml 29674
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