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Theorem lecmtN 29446
Description: Ordered elements commute. (lecmi 22181 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 29432 . . . . 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 29431 . . . . . . 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 2283 . . . . . . 7  |-  ( oc
`  K )  =  ( oc `  K
)
86, 7opoccl 29384 . . . . . 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 2283 . . . . . 6  |-  ( join `  K )  =  (
join `  K )
126, 11latjcl 14156 . . . . 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 2283 . . . . 5  |-  ( meet `  K )  =  (
meet `  K )
166, 14, 15latmle1 14182 . . . 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 14157 . . . . 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 14162 . . . 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 29445 . 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 1623    e. wcel 1684   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   occoc 13216   joincjn 14078   meetcmee 14079   Latclat 14151   OPcops 29362   cmccmtN 29363   OMLcoml 29365
This theorem is referenced by:  cmtidN  29447  omlmod1i2N  29450
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-lat 14152  df-oposet 29366  df-cmtN 29367  df-ol 29368  df-oml 29369
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