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Theorem dihmeetlem5 37451
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 997 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( Q  e.  A  /\  Q  .<_  X ) )  ->  K  e.  HL )
2 simprl 754 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( Q  e.  A  /\  Q  .<_  X ) )  ->  Q  e.  A )
3 simpl2 998 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( Q  e.  A  /\  Q  .<_  X ) )  ->  X  e.  B )
4 simpl3 999 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( Q  e.  A  /\  Q  .<_  X ) )  ->  Y  e.  B )
5 simprr 755 . . 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 36001 . . 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 1239 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( Q  e.  A  /\  Q  .<_  X ) )  ->  ( ( X  ./\  Y )  .\/  Q )  =  ( X 
./\  ( Y  .\/  Q ) ) )
1312eqcomd 2462 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   Basecbs 14719   lecple 14794   joincjn 15775   meetcmee 15776   Atomscatm 35404   HLchlt 35491
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-preset 15759  df-poset 15777  df-plt 15790  df-lub 15806  df-glb 15807  df-join 15808  df-meet 15809  df-p0 15871  df-lat 15878  df-clat 15940  df-oposet 35317  df-ol 35319  df-oml 35320  df-covers 35407  df-ats 35408  df-atl 35439  df-cvlat 35463  df-hlat 35492  df-psubsp 35643  df-pmap 35644  df-padd 35936
This theorem is referenced by:  dihmeetlem6  37452  dihjatc1  37454  dihmeetlem10N  37459
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