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Theorem lecmtN 29515
Description: Ordered elements commute. (lecmi 22295 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 29501 . . . . 5  |-  ( K  e.  OML  ->  K  e.  Lat )
213ad2ant1 976 . . . 4  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  Lat )
3 simp2 956 . . . 4  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  X  e.  B )
4 omlop 29500 . . . . . . 7  |-  ( K  e.  OML  ->  K  e.  OP )
543ad2ant1 976 . . . . . 6  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  OP )
6 lecmt.b . . . . . . 7  |-  B  =  ( Base `  K
)
7 eqid 2358 . . . . . . 7  |-  ( oc
`  K )  =  ( oc `  K
)
86, 7opoccl 29453 . . . . . 6  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( ( oc `  K ) `  X
)  e.  B )
95, 3, 8syl2anc 642 . . . . 5  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( oc `  K ) `  X
)  e.  B )
10 simp3 957 . . . . 5  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  Y  e.  B )
11 eqid 2358 . . . . . 6  |-  ( join `  K )  =  (
join `  K )
126, 11latjcl 14255 . . . . 5  |-  ( ( K  e.  Lat  /\  ( ( oc `  K ) `  X
)  e.  B  /\  Y  e.  B )  ->  ( ( ( oc
`  K ) `  X ) ( join `  K ) Y )  e.  B )
132, 9, 10, 12syl3anc 1182 . . . 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 2358 . . . . 5  |-  ( meet `  K )  =  (
meet `  K )
166, 14, 15latmle1 14281 . . . 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 1182 . . 3  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X ( meet `  K ) ( ( ( oc `  K
) `  X )
( join `  K ) Y ) )  .<_  X )
186, 15latmcl 14256 . . . . 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 1182 . . . 4  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X ( meet `  K ) ( ( ( oc `  K
) `  X )
( join `  K ) Y ) )  e.  B )
206, 14lattr 14261 . . . 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 1184 . . 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 656 . 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 29514 . 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 225 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 358    /\ w3a 934    = wceq 1642    e. wcel 1710   class class class wbr 4104   ` cfv 5337  (class class class)co 5945   Basecbs 13245   lecple 13312   occoc 13313   joincjn 14177   meetcmee 14178   Latclat 14250   OPcops 29431   cmccmtN 29432   OMLcoml 29434
This theorem is referenced by:  cmtidN  29516  omlmod1i2N  29519
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-undef 6385  df-riota 6391  df-poset 14179  df-lub 14207  df-glb 14208  df-join 14209  df-meet 14210  df-lat 14251  df-oposet 29435  df-cmtN 29436  df-ol 29437  df-oml 29438
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