Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dihmeetlem5 Structured version   Unicode version

Theorem dihmeetlem5 32043
Description: Part of proof that isomorphism H is order-preserving . (Contributed by NM, 6-Apr-2014.)
Hypotheses
Ref Expression
dihmeetlem5.b  |-  B  =  ( Base `  K
)
dihmeetlem5.l  |-  .<_  =  ( le `  K )
dihmeetlem5.j  |-  .\/  =  ( join `  K )
dihmeetlem5.m  |-  ./\  =  ( meet `  K )
dihmeetlem5.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
dihmeetlem5  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( Q  e.  A  /\  Q  .<_  X ) )  ->  ( X  ./\  ( Y  .\/  Q
) )  =  ( ( X  ./\  Y
)  .\/  Q )
)

Proof of Theorem dihmeetlem5
StepHypRef Expression
1 simpl1 960 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( Q  e.  A  /\  Q  .<_  X ) )  ->  K  e.  HL )
2 simprl 733 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( Q  e.  A  /\  Q  .<_  X ) )  ->  Q  e.  A )
3 simpl2 961 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( Q  e.  A  /\  Q  .<_  X ) )  ->  X  e.  B )
4 simpl3 962 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( Q  e.  A  /\  Q  .<_  X ) )  ->  Y  e.  B )
5 simprr 734 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( Q  e.  A  /\  Q  .<_  X ) )  ->  Q  .<_  X )
6 dihmeetlem5.b . . . 4  |-  B  =  ( Base `  K
)
7 dihmeetlem5.l . . . 4  |-  .<_  =  ( le `  K )
8 dihmeetlem5.j . . . 4  |-  .\/  =  ( join `  K )
9 dihmeetlem5.m . . . 4  |-  ./\  =  ( meet `  K )
10 dihmeetlem5.a . . . 4  |-  A  =  ( Atoms `  K )
116, 7, 8, 9, 10atmod2i1 30595 . . 3  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  Q  .<_  X )  ->  ( ( X  ./\  Y )  .\/  Q )  =  ( X 
./\  ( Y  .\/  Q ) ) )
121, 2, 3, 4, 5, 11syl131anc 1197 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( Q  e.  A  /\  Q  .<_  X ) )  ->  ( ( X  ./\  Y )  .\/  Q )  =  ( X 
./\  ( Y  .\/  Q ) ) )
1312eqcomd 2440 1  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( Q  e.  A  /\  Q  .<_  X ) )  ->  ( X  ./\  ( Y  .\/  Q
) )  =  ( ( X  ./\  Y
)  .\/  Q )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   Basecbs 13461   lecple 13528   joincjn 14393   meetcmee 14394   Atomscatm 29998   HLchlt 30085
This theorem is referenced by:  dihmeetlem6  32044  dihjatc1  32046  dihmeetlem10N  32051
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-poset 14395  df-plt 14407  df-lub 14423  df-glb 14424  df-join 14425  df-meet 14426  df-p0 14460  df-lat 14467  df-clat 14529  df-oposet 29911  df-ol 29913  df-oml 29914  df-covers 30001  df-ats 30002  df-atl 30033  df-cvlat 30057  df-hlat 30086  df-psubsp 30237  df-pmap 30238  df-padd 30530
  Copyright terms: Public domain W3C validator