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Theorem atexchcvrN 29554
Description: Atom exchange property. Version of hlatexch2 29510 with covers relation. (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
atexchcvr.j  |-  .\/  =  ( join `  K )
atexchcvr.a  |-  A  =  ( Atoms `  K )
atexchcvr.c  |-  C  =  (  <o  `  K )
Assertion
Ref Expression
atexchcvrN  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  R )  ->  ( P C ( Q  .\/  R )  ->  Q C
( P  .\/  R
) ) )

Proof of Theorem atexchcvrN
StepHypRef Expression
1 simpl1 960 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  R )  /\  P C ( Q  .\/  R
) )  ->  K  e.  HL )
2 simpl21 1035 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  R )  /\  P C ( Q  .\/  R
) )  ->  P  e.  A )
3 eqid 2387 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
4 atexchcvr.a . . . . . . 7  |-  A  =  ( Atoms `  K )
53, 4atbase 29404 . . . . . 6  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
62, 5syl 16 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  R )  /\  P C ( Q  .\/  R
) )  ->  P  e.  ( Base `  K
) )
7 hllat 29478 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  Lat )
81, 7syl 16 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  R )  /\  P C ( Q  .\/  R
) )  ->  K  e.  Lat )
9 simpl22 1036 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  R )  /\  P C ( Q  .\/  R
) )  ->  Q  e.  A )
103, 4atbase 29404 . . . . . . 7  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
119, 10syl 16 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  R )  /\  P C ( Q  .\/  R
) )  ->  Q  e.  ( Base `  K
) )
12 simpl23 1037 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  R )  /\  P C ( Q  .\/  R
) )  ->  R  e.  A )
133, 4atbase 29404 . . . . . . 7  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
1412, 13syl 16 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  R )  /\  P C ( Q  .\/  R
) )  ->  R  e.  ( Base `  K
) )
15 atexchcvr.j . . . . . . 7  |-  .\/  =  ( join `  K )
163, 15latjcl 14406 . . . . . 6  |-  ( ( K  e.  Lat  /\  Q  e.  ( Base `  K )  /\  R  e.  ( Base `  K
) )  ->  ( Q  .\/  R )  e.  ( Base `  K
) )
178, 11, 14, 16syl3anc 1184 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  R )  /\  P C ( Q  .\/  R
) )  ->  ( Q  .\/  R )  e.  ( Base `  K
) )
181, 6, 173jca 1134 . . . 4  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  R )  /\  P C ( Q  .\/  R
) )  ->  ( K  e.  HL  /\  P  e.  ( Base `  K
)  /\  ( Q  .\/  R )  e.  (
Base `  K )
) )
19 eqid 2387 . . . . 5  |-  ( le
`  K )  =  ( le `  K
)
20 atexchcvr.c . . . . 5  |-  C  =  (  <o  `  K )
213, 19, 20cvrle 29393 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  ( Base `  K )  /\  ( Q  .\/  R )  e.  ( Base `  K
) )  /\  P C ( Q  .\/  R ) )  ->  P
( le `  K
) ( Q  .\/  R ) )
2218, 21sylancom 649 . . 3  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  R )  /\  P C ( Q  .\/  R
) )  ->  P
( le `  K
) ( Q  .\/  R ) )
2322ex 424 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  R )  ->  ( P C ( Q  .\/  R )  ->  P ( le `  K ) ( Q  .\/  R ) ) )
2419, 15, 4hlatexch2 29510 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  R )  ->  ( P
( le `  K
) ( Q  .\/  R )  ->  Q ( le `  K ) ( P  .\/  R ) ) )
25 simpl1 960 . . . 4  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  R )  /\  Q ( le `  K ) ( P  .\/  R
) )  ->  K  e.  HL )
26 simpl22 1036 . . . 4  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  R )  /\  Q ( le `  K ) ( P  .\/  R
) )  ->  Q  e.  A )
27 simpl21 1035 . . . 4  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  R )  /\  Q ( le `  K ) ( P  .\/  R
) )  ->  P  e.  A )
28 simpl23 1037 . . . 4  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  R )  /\  Q ( le `  K ) ( P  .\/  R
) )  ->  R  e.  A )
29 simpl3 962 . . . 4  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  R )  /\  Q ( le `  K ) ( P  .\/  R
) )  ->  P  =/=  R )
30 simpr 448 . . . 4  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  R )  /\  Q ( le `  K ) ( P  .\/  R
) )  ->  Q
( le `  K
) ( P  .\/  R ) )
3119, 15, 20, 4atcvrj2 29547 . . . 4  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  P  e.  A  /\  R  e.  A
)  /\  ( P  =/=  R  /\  Q ( le `  K ) ( P  .\/  R
) ) )  ->  Q C ( P  .\/  R ) )
3225, 26, 27, 28, 29, 30, 31syl132anc 1202 . . 3  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  R )  /\  Q ( le `  K ) ( P  .\/  R
) )  ->  Q C ( P  .\/  R ) )
3332ex 424 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  R )  ->  ( Q
( le `  K
) ( P  .\/  R )  ->  Q C
( P  .\/  R
) ) )
3423, 24, 333syld 53 1  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  R )  ->  ( P C ( Q  .\/  R )  ->  Q C
( P  .\/  R
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2550   class class class wbr 4153   ` cfv 5394  (class class class)co 6020   Basecbs 13396   lecple 13463   joincjn 14328   Latclat 14401    <o ccvr 29377   Atomscatm 29378   HLchlt 29465
This theorem is referenced by:  atexchltN  29555
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-undef 6479  df-riota 6485  df-poset 14330  df-plt 14342  df-lub 14358  df-glb 14359  df-join 14360  df-meet 14361  df-p0 14395  df-lat 14402  df-clat 14464  df-oposet 29291  df-ol 29293  df-oml 29294  df-covers 29381  df-ats 29382  df-atl 29413  df-cvlat 29437  df-hlat 29466
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