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Theorem llnexch2N 35991
Description: Line exchange property (compare cvlatexch2 35459 for atoms). (Contributed by NM, 18-Nov-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
llnexch.l  |-  .<_  =  ( le `  K )
llnexch.j  |-  .\/  =  ( join `  K )
llnexch.m  |-  ./\  =  ( meet `  K )
llnexch.a  |-  A  =  ( Atoms `  K )
llnexch.n  |-  N  =  ( LLines `  K )
Assertion
Ref Expression
llnexch2N  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  Z  e.  N
)  /\  ( ( X  ./\  Y )  e.  A  /\  X  =/= 
Z ) )  -> 
( ( X  ./\  Y )  .<_  Z  ->  ( X  ./\  Z )  .<_  Y ) )

Proof of Theorem llnexch2N
StepHypRef Expression
1 llnexch.l . . 3  |-  .<_  =  ( le `  K )
2 llnexch.j . . 3  |-  .\/  =  ( join `  K )
3 llnexch.m . . 3  |-  ./\  =  ( meet `  K )
4 llnexch.a . . 3  |-  A  =  ( Atoms `  K )
5 llnexch.n . . 3  |-  N  =  ( LLines `  K )
61, 2, 3, 4, 5llnexchb2 35990 . 2  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  Z  e.  N
)  /\  ( ( X  ./\  Y )  e.  A  /\  X  =/= 
Z ) )  -> 
( ( X  ./\  Y )  .<_  Z  <->  ( X  ./\ 
Y )  =  ( X  ./\  Z )
) )
7 hllat 35485 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
873ad2ant1 1015 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  Z  e.  N
)  /\  ( ( X  ./\  Y )  e.  A  /\  X  =/= 
Z ) )  ->  K  e.  Lat )
9 simp21 1027 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  Z  e.  N
)  /\  ( ( X  ./\  Y )  e.  A  /\  X  =/= 
Z ) )  ->  X  e.  N )
10 eqid 2454 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
1110, 5llnbase 35630 . . . . 5  |-  ( X  e.  N  ->  X  e.  ( Base `  K
) )
129, 11syl 16 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  Z  e.  N
)  /\  ( ( X  ./\  Y )  e.  A  /\  X  =/= 
Z ) )  ->  X  e.  ( Base `  K ) )
13 simp22 1028 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  Z  e.  N
)  /\  ( ( X  ./\  Y )  e.  A  /\  X  =/= 
Z ) )  ->  Y  e.  N )
1410, 5llnbase 35630 . . . . 5  |-  ( Y  e.  N  ->  Y  e.  ( Base `  K
) )
1513, 14syl 16 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  Z  e.  N
)  /\  ( ( X  ./\  Y )  e.  A  /\  X  =/= 
Z ) )  ->  Y  e.  ( Base `  K ) )
1610, 1, 3latmle2 15906 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  ( Base `  K )  /\  Y  e.  ( Base `  K
) )  ->  ( X  ./\  Y )  .<_  Y )
178, 12, 15, 16syl3anc 1226 . . 3  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  Z  e.  N
)  /\  ( ( X  ./\  Y )  e.  A  /\  X  =/= 
Z ) )  -> 
( X  ./\  Y
)  .<_  Y )
18 breq1 4442 . . 3  |-  ( ( X  ./\  Y )  =  ( X  ./\  Z )  ->  ( ( X  ./\  Y )  .<_  Y 
<->  ( X  ./\  Z
)  .<_  Y ) )
1917, 18syl5ibcom 220 . 2  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  Z  e.  N
)  /\  ( ( X  ./\  Y )  e.  A  /\  X  =/= 
Z ) )  -> 
( ( X  ./\  Y )  =  ( X 
./\  Z )  -> 
( X  ./\  Z
)  .<_  Y ) )
206, 19sylbid 215 1  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  Z  e.  N
)  /\  ( ( X  ./\  Y )  e.  A  /\  X  =/= 
Z ) )  -> 
( ( X  ./\  Y )  .<_  Z  ->  ( X  ./\  Z )  .<_  Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   Basecbs 14716   lecple 14791   joincjn 15772   meetcmee 15773   Latclat 15874   Atomscatm 35385   HLchlt 35472   LLinesclln 35612
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 15756  df-poset 15774  df-plt 15787  df-lub 15803  df-glb 15804  df-join 15805  df-meet 15806  df-p0 15868  df-lat 15875  df-clat 15937  df-oposet 35298  df-ol 35300  df-oml 35301  df-covers 35388  df-ats 35389  df-atl 35420  df-cvlat 35444  df-hlat 35473  df-llines 35619  df-psubsp 35624  df-pmap 35625  df-padd 35917
This theorem is referenced by: (None)
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