Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  omlmod1i2N Unicode version

Theorem omlmod1i2N 29743
Description: Analog of modular law atmod1i2 30341 that holds in any OML. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
omlmod.b  |-  B  =  ( Base `  K
)
omlmod.l  |-  .<_  =  ( le `  K )
omlmod.j  |-  .\/  =  ( join `  K )
omlmod.m  |-  ./\  =  ( meet `  K )
omlmod.c  |-  C  =  ( cm `  K
)
Assertion
Ref Expression
omlmod1i2N  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  -> 
( X  .\/  ( Y  ./\  Z ) )  =  ( ( X 
.\/  Y )  ./\  Z ) )

Proof of Theorem omlmod1i2N
StepHypRef Expression
1 simp1 957 . . 3  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  ->  K  e.  OML )
2 simp23 992 . . 3  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  ->  Z  e.  B )
3 simp21 990 . . 3  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  ->  X  e.  B )
4 simp22 991 . . 3  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  ->  Y  e.  B )
5 simp3l 985 . . . . 5  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  ->  X  .<_  Z )
6 omlmod.b . . . . . . 7  |-  B  =  ( Base `  K
)
7 omlmod.l . . . . . . 7  |-  .<_  =  ( le `  K )
8 omlmod.c . . . . . . 7  |-  C  =  ( cm `  K
)
96, 7, 8lecmtN 29739 . . . . . 6  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Z  e.  B )  ->  ( X  .<_  Z  ->  X C Z ) )
101, 3, 2, 9syl3anc 1184 . . . . 5  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  -> 
( X  .<_  Z  ->  X C Z ) )
115, 10mpd 15 . . . 4  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  ->  X C Z )
126, 8cmtcomN 29732 . . . . 5  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Z  e.  B )  ->  ( X C Z  <-> 
Z C X ) )
131, 3, 2, 12syl3anc 1184 . . . 4  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  -> 
( X C Z  <-> 
Z C X ) )
1411, 13mpbid 202 . . 3  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  ->  Z C X )
15 simp3r 986 . . . 4  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  ->  Y C Z )
166, 8cmtcomN 29732 . . . . 5  |-  ( ( K  e.  OML  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Y C Z  <-> 
Z C Y ) )
171, 4, 2, 16syl3anc 1184 . . . 4  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  -> 
( Y C Z  <-> 
Z C Y ) )
1815, 17mpbid 202 . . 3  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  ->  Z C Y )
19 omlmod.j . . . 4  |-  .\/  =  ( join `  K )
20 omlmod.m . . . 4  |-  ./\  =  ( meet `  K )
216, 19, 20, 8omlfh1N 29741 . . 3  |-  ( ( K  e.  OML  /\  ( Z  e.  B  /\  X  e.  B  /\  Y  e.  B
)  /\  ( Z C X  /\  Z C Y ) )  -> 
( Z  ./\  ( X  .\/  Y ) )  =  ( ( Z 
./\  X )  .\/  ( Z  ./\  Y ) ) )
221, 2, 3, 4, 14, 18, 21syl132anc 1202 . 2  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  -> 
( Z  ./\  ( X  .\/  Y ) )  =  ( ( Z 
./\  X )  .\/  ( Z  ./\  Y ) ) )
23 omllat 29725 . . . 4  |-  ( K  e.  OML  ->  K  e.  Lat )
24233ad2ant1 978 . . 3  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  ->  K  e.  Lat )
256, 19latjcl 14434 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .\/  Y
)  e.  B )
2624, 3, 4, 25syl3anc 1184 . . 3  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  -> 
( X  .\/  Y
)  e.  B )
276, 20latmcom 14459 . . 3  |-  ( ( K  e.  Lat  /\  Z  e.  B  /\  ( X  .\/  Y )  e.  B )  -> 
( Z  ./\  ( X  .\/  Y ) )  =  ( ( X 
.\/  Y )  ./\  Z ) )
2824, 2, 26, 27syl3anc 1184 . 2  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  -> 
( Z  ./\  ( X  .\/  Y ) )  =  ( ( X 
.\/  Y )  ./\  Z ) )
296, 7, 20latleeqm2 14464 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Z  e.  B )  ->  ( X  .<_  Z  <->  ( Z  ./\ 
X )  =  X ) )
3024, 3, 2, 29syl3anc 1184 . . . 4  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  -> 
( X  .<_  Z  <->  ( Z  ./\ 
X )  =  X ) )
315, 30mpbid 202 . . 3  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  -> 
( Z  ./\  X
)  =  X )
326, 20latmcom 14459 . . . 4  |-  ( ( K  e.  Lat  /\  Z  e.  B  /\  Y  e.  B )  ->  ( Z  ./\  Y
)  =  ( Y 
./\  Z ) )
3324, 2, 4, 32syl3anc 1184 . . 3  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  -> 
( Z  ./\  Y
)  =  ( Y 
./\  Z ) )
3431, 33oveq12d 6058 . 2  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  -> 
( ( Z  ./\  X )  .\/  ( Z 
./\  Y ) )  =  ( X  .\/  ( Y  ./\  Z ) ) )
3522, 28, 343eqtr3rd 2445 1  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  -> 
( X  .\/  ( Y  ./\  Z ) )  =  ( ( X 
.\/  Y )  ./\  Z ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   Basecbs 13424   lecple 13491   joincjn 14356   meetcmee 14357   Latclat 14429   cmccmtN 29656   OMLcoml 29658
This theorem is referenced by:  omlspjN  29744
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 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-poset 14358  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-lat 14430  df-oposet 29659  df-cmtN 29660  df-ol 29661  df-oml 29662
  Copyright terms: Public domain W3C validator