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Theorem llnexch2N 30729
Description: Line exchange property (compare cvlatexch2 30197 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 30728 . 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 30223 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
873ad2ant1 979 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  Z  e.  N
)  /\  ( ( X  ./\  Y )  e.  A  /\  X  =/= 
Z ) )  ->  K  e.  Lat )
9 simp21 991 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  Z  e.  N
)  /\  ( ( X  ./\  Y )  e.  A  /\  X  =/= 
Z ) )  ->  X  e.  N )
10 eqid 2438 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
1110, 5llnbase 30368 . . . . 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 992 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  Z  e.  N
)  /\  ( ( X  ./\  Y )  e.  A  /\  X  =/= 
Z ) )  ->  Y  e.  N )
1410, 5llnbase 30368 . . . . 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 14508 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  ( Base `  K )  /\  Y  e.  ( Base `  K
) )  ->  ( X  ./\  Y )  .<_  Y )
178, 12, 15, 16syl3anc 1185 . . 3  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  Z  e.  N
)  /\  ( ( X  ./\  Y )  e.  A  /\  X  =/= 
Z ) )  -> 
( X  ./\  Y
)  .<_  Y )
18 breq1 4217 . . 3  |-  ( ( X  ./\  Y )  =  ( X  ./\  Z )  ->  ( ( X  ./\  Y )  .<_  Y 
<->  ( X  ./\  Z
)  .<_  Y ) )
1917, 18syl5ibcom 213 . 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 208 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 set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   class class class wbr 4214   ` cfv 5456  (class class class)co 6083   Basecbs 13471   lecple 13538   joincjn 14403   meetcmee 14404   Latclat 14476   Atomscatm 30123   HLchlt 30210   LLinesclln 30350
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-iin 4098  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-undef 6545  df-riota 6551  df-poset 14405  df-plt 14417  df-lub 14433  df-glb 14434  df-join 14435  df-meet 14436  df-p0 14470  df-lat 14477  df-clat 14539  df-oposet 30036  df-ol 30038  df-oml 30039  df-covers 30126  df-ats 30127  df-atl 30158  df-cvlat 30182  df-hlat 30211  df-llines 30357  df-psubsp 30362  df-pmap 30363  df-padd 30655
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