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Theorem atexchcvrN 34236
Description: Atom exchange property. Version of hlatexch2 34192 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 999 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  R )  /\  P C ( Q  .\/  R
) )  ->  K  e.  HL )
2 simpl21 1074 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  R )  /\  P C ( Q  .\/  R
) )  ->  P  e.  A )
3 eqid 2467 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
4 atexchcvr.a . . . . . . 7  |-  A  =  ( Atoms `  K )
53, 4atbase 34086 . . . . . 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 34160 . . . . . . 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 1075 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  R )  /\  P C ( Q  .\/  R
) )  ->  Q  e.  A )
103, 4atbase 34086 . . . . . . 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 1076 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P  =/=  R )  /\  P C ( Q  .\/  R
) )  ->  R  e.  A )
133, 4atbase 34086 . . . . . . 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 15531 . . . . . 6  |-  ( ( K  e.  Lat  /\  Q  e.  ( Base `  K )  /\  R  e.  ( Base `  K
) )  ->  ( Q  .\/  R )  e.  ( Base `  K
) )
178, 11, 14, 16syl3anc 1228 . . . . 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 1176 . . . 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 2467 . . . . 5  |-  ( le
`  K )  =  ( le `  K
)
20 atexchcvr.c . . . . 5  |-  C  =  (  <o  `  K )
213, 19, 20cvrle 34075 . . . 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 34192 . 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 999 . . . 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 1075 . . . 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 1074 . . . 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 1076 . . . 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 1001 . . . 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 34229 . . . 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 1246 . . 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   Basecbs 14483   lecple 14555   joincjn 15424   Latclat 15525    <o ccvr 34059   Atomscatm 34060   HLchlt 34147
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-poset 15426  df-plt 15438  df-lub 15454  df-glb 15455  df-join 15456  df-meet 15457  df-p0 15519  df-lat 15526  df-clat 15588  df-oposet 33973  df-ol 33975  df-oml 33976  df-covers 34063  df-ats 34064  df-atl 34095  df-cvlat 34119  df-hlat 34148
This theorem is referenced by:  atexchltN  34237
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