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Theorem atexchcvrN 33089
Description: Atom exchange property. Version of hlatexch2 33045 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 991 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  R )  /\  P C ( Q  .\/  R
) )  ->  K  e.  HL )
2 simpl21 1066 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  R )  /\  P C ( Q  .\/  R
) )  ->  P  e.  A )
3 eqid 2443 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
4 atexchcvr.a . . . . . . 7  |-  A  =  ( Atoms `  K )
53, 4atbase 32939 . . . . . 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 33013 . . . . . . 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 1067 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  R )  /\  P C ( Q  .\/  R
) )  ->  Q  e.  A )
103, 4atbase 32939 . . . . . . 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 1068 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  R )  /\  P C ( Q  .\/  R
) )  ->  R  e.  A )
133, 4atbase 32939 . . . . . . 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 15226 . . . . . 6  |-  ( ( K  e.  Lat  /\  Q  e.  ( Base `  K )  /\  R  e.  ( Base `  K
) )  ->  ( Q  .\/  R )  e.  ( Base `  K
) )
178, 11, 14, 16syl3anc 1218 . . . . 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 1168 . . . 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 2443 . . . . 5  |-  ( le
`  K )  =  ( le `  K
)
20 atexchcvr.c . . . . 5  |-  C  =  (  <o  `  K )
213, 19, 20cvrle 32928 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  ( Base `  K )  /\  ( Q  .\/  R )  e.  ( Base `  K
) )  /\  P C ( Q  .\/  R ) )  ->  P
( le `  K
) ( Q  .\/  R ) )
2218, 21sylancom 667 . . 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 434 . 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 33045 . 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 991 . . . 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 1067 . . . 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 1066 . . . 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 1068 . . . 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 993 . . . 4  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  R )  /\  Q ( le `  K ) ( P  .\/  R
) )  ->  P  =/=  R )
30 simpr 461 . . . 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 33082 . . . 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 1236 . . 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 434 . 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 55 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 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2611   class class class wbr 4297   ` cfv 5423  (class class class)co 6096   Basecbs 14179   lecple 14250   joincjn 15119   Latclat 15220    <o ccvr 32912   Atomscatm 32913   HLchlt 33000
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-poset 15121  df-plt 15133  df-lub 15149  df-glb 15150  df-join 15151  df-meet 15152  df-p0 15214  df-lat 15221  df-clat 15283  df-oposet 32826  df-ol 32828  df-oml 32829  df-covers 32916  df-ats 32917  df-atl 32948  df-cvlat 32972  df-hlat 33001
This theorem is referenced by:  atexchltN  33090
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