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Theorem atexchcvrN 32457
Description: Atom exchange property. Version of hlatexch2 32413 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 1000 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  R )  /\  P C ( Q  .\/  R
) )  ->  K  e.  HL )
2 simpl21 1075 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  R )  /\  P C ( Q  .\/  R
) )  ->  P  e.  A )
3 eqid 2402 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
4 atexchcvr.a . . . . . . 7  |-  A  =  ( Atoms `  K )
53, 4atbase 32307 . . . . . 6  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
62, 5syl 17 . . . . 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 32381 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  Lat )
81, 7syl 17 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  R )  /\  P C ( Q  .\/  R
) )  ->  K  e.  Lat )
9 simpl22 1076 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  R )  /\  P C ( Q  .\/  R
) )  ->  Q  e.  A )
103, 4atbase 32307 . . . . . . 7  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
119, 10syl 17 . . . . . 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 1077 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  R )  /\  P C ( Q  .\/  R
) )  ->  R  e.  A )
133, 4atbase 32307 . . . . . . 7  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
1412, 13syl 17 . . . . . 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 16005 . . . . . 6  |-  ( ( K  e.  Lat  /\  Q  e.  ( Base `  K )  /\  R  e.  ( Base `  K
) )  ->  ( Q  .\/  R )  e.  ( Base `  K
) )
178, 11, 14, 16syl3anc 1230 . . . . 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 1177 . . . 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 2402 . . . . 5  |-  ( le
`  K )  =  ( le `  K
)
20 atexchcvr.c . . . . 5  |-  C  =  (  <o  `  K )
213, 19, 20cvrle 32296 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  ( Base `  K )  /\  ( Q  .\/  R )  e.  ( Base `  K
) )  /\  P C ( Q  .\/  R ) )  ->  P
( le `  K
) ( Q  .\/  R ) )
2218, 21sylancom 665 . . 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 432 . 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 32413 . 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 1000 . . . 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 1076 . . . 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 1075 . . . 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 1077 . . . 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 1002 . . . 4  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  R )  /\  Q ( le `  K ) ( P  .\/  R
) )  ->  P  =/=  R )
30 simpr 459 . . . 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 32450 . . . 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 1248 . . 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 432 . 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 54 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 setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   class class class wbr 4395   ` cfv 5569  (class class class)co 6278   Basecbs 14841   lecple 14916   joincjn 15897   Latclat 15999    <o ccvr 32280   Atomscatm 32281   HLchlt 32368
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-preset 15881  df-poset 15899  df-plt 15912  df-lub 15928  df-glb 15929  df-join 15930  df-meet 15931  df-p0 15993  df-lat 16000  df-clat 16062  df-oposet 32194  df-ol 32196  df-oml 32197  df-covers 32284  df-ats 32285  df-atl 32316  df-cvlat 32340  df-hlat 32369
This theorem is referenced by:  atexchltN  32458
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