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Theorem dihmeetlem5 34675
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 986 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( Q  e.  A  /\  Q  .<_  X ) )  ->  K  e.  HL )
2 simprl 750 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( Q  e.  A  /\  Q  .<_  X ) )  ->  Q  e.  A )
3 simpl2 987 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( Q  e.  A  /\  Q  .<_  X ) )  ->  X  e.  B )
4 simpl3 988 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( Q  e.  A  /\  Q  .<_  X ) )  ->  Y  e.  B )
5 simprr 751 . . 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 33227 . . 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 1226 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( Q  e.  A  /\  Q  .<_  X ) )  ->  ( ( X  ./\  Y )  .\/  Q )  =  ( X 
./\  ( Y  .\/  Q ) ) )
1312eqcomd 2446 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761   class class class wbr 4289   ` cfv 5415  (class class class)co 6090   Basecbs 14170   lecple 14241   joincjn 15110   meetcmee 15111   Atomscatm 32630   HLchlt 32717
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-1st 6576  df-2nd 6577  df-poset 15112  df-plt 15124  df-lub 15140  df-glb 15141  df-join 15142  df-meet 15143  df-p0 15205  df-lat 15212  df-clat 15274  df-oposet 32543  df-ol 32545  df-oml 32546  df-covers 32633  df-ats 32634  df-atl 32665  df-cvlat 32689  df-hlat 32718  df-psubsp 32869  df-pmap 32870  df-padd 33162
This theorem is referenced by:  dihmeetlem6  34676  dihjatc1  34678  dihmeetlem10N  34683
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