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Theorem atexchcvrN 30175
Description: Atom exchange property. Version of hlatexch2 30131 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 2436 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
4 atexchcvr.a . . . . . . 7  |-  A  =  ( Atoms `  K )
53, 4atbase 30025 . . . . . 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 30099 . . . . . . 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 30025 . . . . . . 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 30025 . . . . . . 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 14472 . . . . . 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 2436 . . . . 5  |-  ( le
`  K )  =  ( le `  K
)
20 atexchcvr.c . . . . 5  |-  C  =  (  <o  `  K )
213, 19, 20cvrle 30014 . . . 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 30131 . 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 30168 . . . 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 1652    e. wcel 1725    =/= wne 2599   class class class wbr 4205   ` cfv 5447  (class class class)co 6074   Basecbs 13462   lecple 13529   joincjn 14394   Latclat 14467    <o ccvr 29998   Atomscatm 29999   HLchlt 30086
This theorem is referenced by:  atexchltN  30176
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4313  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2703  df-rex 2704  df-reu 2705  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-iun 4088  df-br 4206  df-opab 4260  df-mpt 4261  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-1st 6342  df-2nd 6343  df-undef 6536  df-riota 6542  df-poset 14396  df-plt 14408  df-lub 14424  df-glb 14425  df-join 14426  df-meet 14427  df-p0 14461  df-lat 14468  df-clat 14530  df-oposet 29912  df-ol 29914  df-oml 29915  df-covers 30002  df-ats 30003  df-atl 30034  df-cvlat 30058  df-hlat 30087
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