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Theorem dihmeetlem5 35272
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 991 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( Q  e.  A  /\  Q  .<_  X ) )  ->  K  e.  HL )
2 simprl 755 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( Q  e.  A  /\  Q  .<_  X ) )  ->  Q  e.  A )
3 simpl2 992 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( Q  e.  A  /\  Q  .<_  X ) )  ->  X  e.  B )
4 simpl3 993 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( Q  e.  A  /\  Q  .<_  X ) )  ->  Y  e.  B )
5 simprr 756 . . 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 33824 . . 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 1232 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( Q  e.  A  /\  Q  .<_  X ) )  ->  ( ( X  ./\  Y )  .\/  Q )  =  ( X 
./\  ( Y  .\/  Q ) ) )
1312eqcomd 2460 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 965    = wceq 1370    e. wcel 1758   class class class wbr 4395   ` cfv 5521  (class class class)co 6195   Basecbs 14287   lecple 14359   joincjn 15228   meetcmee 15229   Atomscatm 33227   HLchlt 33314
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-iun 4276  df-iin 4277  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-1st 6682  df-2nd 6683  df-poset 15230  df-plt 15242  df-lub 15258  df-glb 15259  df-join 15260  df-meet 15261  df-p0 15323  df-lat 15330  df-clat 15392  df-oposet 33140  df-ol 33142  df-oml 33143  df-covers 33230  df-ats 33231  df-atl 33262  df-cvlat 33286  df-hlat 33315  df-psubsp 33466  df-pmap 33467  df-padd 33759
This theorem is referenced by:  dihmeetlem6  35273  dihjatc1  35275  dihmeetlem10N  35280
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