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Theorem hlatexch4 30215
Description: Exchange 2 atoms. (Contributed by NM, 13-May-2013.)
Hypotheses
Ref Expression
hlatexch4.j  |-  .\/  =  ( join `  K )
hlatexch4.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
hlatexch4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( P  =/=  R  /\  Q  =/= 
S  /\  ( P  .\/  Q )  =  ( R  .\/  S ) ) )  ->  ( P  .\/  R )  =  ( Q  .\/  S
) )

Proof of Theorem hlatexch4
StepHypRef Expression
1 simp11 987 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( P  =/=  R  /\  Q  =/= 
S  /\  ( P  .\/  Q )  =  ( R  .\/  S ) ) )  ->  K  e.  HL )
2 simp2l 983 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( P  =/=  R  /\  Q  =/= 
S  /\  ( P  .\/  Q )  =  ( R  .\/  S ) ) )  ->  R  e.  A )
3 simp2r 984 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( P  =/=  R  /\  Q  =/= 
S  /\  ( P  .\/  Q )  =  ( R  .\/  S ) ) )  ->  S  e.  A )
4 eqid 2435 . . . . . . . 8  |-  ( le
`  K )  =  ( le `  K
)
5 hlatexch4.j . . . . . . . 8  |-  .\/  =  ( join `  K )
6 hlatexch4.a . . . . . . . 8  |-  A  =  ( Atoms `  K )
74, 5, 6hlatlej2 30110 . . . . . . 7  |-  ( ( K  e.  HL  /\  R  e.  A  /\  S  e.  A )  ->  S ( le `  K ) ( R 
.\/  S ) )
81, 2, 3, 7syl3anc 1184 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( P  =/=  R  /\  Q  =/= 
S  /\  ( P  .\/  Q )  =  ( R  .\/  S ) ) )  ->  S
( le `  K
) ( R  .\/  S ) )
9 simp33 995 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( P  =/=  R  /\  Q  =/= 
S  /\  ( P  .\/  Q )  =  ( R  .\/  S ) ) )  ->  ( P  .\/  Q )  =  ( R  .\/  S
) )
108, 9breqtrrd 4230 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( P  =/=  R  /\  Q  =/= 
S  /\  ( P  .\/  Q )  =  ( R  .\/  S ) ) )  ->  S
( le `  K
) ( P  .\/  Q ) )
11 simp12 988 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( P  =/=  R  /\  Q  =/= 
S  /\  ( P  .\/  Q )  =  ( R  .\/  S ) ) )  ->  P  e.  A )
12 simp13 989 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( P  =/=  R  /\  Q  =/= 
S  /\  ( P  .\/  Q )  =  ( R  .\/  S ) ) )  ->  Q  e.  A )
13 simp32 994 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( P  =/=  R  /\  Q  =/= 
S  /\  ( P  .\/  Q )  =  ( R  .\/  S ) ) )  ->  Q  =/=  S )
1413necomd 2681 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( P  =/=  R  /\  Q  =/= 
S  /\  ( P  .\/  Q )  =  ( R  .\/  S ) ) )  ->  S  =/=  Q )
154, 5, 6hlatexch2 30130 . . . . . 6  |-  ( ( K  e.  HL  /\  ( S  e.  A  /\  P  e.  A  /\  Q  e.  A
)  /\  S  =/=  Q )  ->  ( S
( le `  K
) ( P  .\/  Q )  ->  P ( le `  K ) ( S  .\/  Q ) ) )
161, 3, 11, 12, 14, 15syl131anc 1197 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( P  =/=  R  /\  Q  =/= 
S  /\  ( P  .\/  Q )  =  ( R  .\/  S ) ) )  ->  ( S ( le `  K ) ( P 
.\/  Q )  ->  P ( le `  K ) ( S 
.\/  Q ) ) )
1710, 16mpd 15 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( P  =/=  R  /\  Q  =/= 
S  /\  ( P  .\/  Q )  =  ( R  .\/  S ) ) )  ->  P
( le `  K
) ( S  .\/  Q ) )
185, 6hlatjcom 30102 . . . . 5  |-  ( ( K  e.  HL  /\  S  e.  A  /\  Q  e.  A )  ->  ( S  .\/  Q
)  =  ( Q 
.\/  S ) )
191, 3, 12, 18syl3anc 1184 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( P  =/=  R  /\  Q  =/= 
S  /\  ( P  .\/  Q )  =  ( R  .\/  S ) ) )  ->  ( S  .\/  Q )  =  ( Q  .\/  S
) )
2017, 19breqtrd 4228 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( P  =/=  R  /\  Q  =/= 
S  /\  ( P  .\/  Q )  =  ( R  .\/  S ) ) )  ->  P
( le `  K
) ( Q  .\/  S ) )
214, 5, 6hlatlej2 30110 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  Q ( le `  K ) ( P 
.\/  Q ) )
221, 11, 12, 21syl3anc 1184 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( P  =/=  R  /\  Q  =/= 
S  /\  ( P  .\/  Q )  =  ( R  .\/  S ) ) )  ->  Q
( le `  K
) ( P  .\/  Q ) )
2322, 9breqtrd 4228 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( P  =/=  R  /\  Q  =/= 
S  /\  ( P  .\/  Q )  =  ( R  .\/  S ) ) )  ->  Q
( le `  K
) ( R  .\/  S ) )
244, 5, 6hlatexch2 30130 . . . . 5  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  Q  =/=  S )  ->  ( Q
( le `  K
) ( R  .\/  S )  ->  R ( le `  K ) ( Q  .\/  S ) ) )
251, 12, 2, 3, 13, 24syl131anc 1197 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( P  =/=  R  /\  Q  =/= 
S  /\  ( P  .\/  Q )  =  ( R  .\/  S ) ) )  ->  ( Q ( le `  K ) ( R 
.\/  S )  ->  R ( le `  K ) ( Q 
.\/  S ) ) )
2623, 25mpd 15 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( P  =/=  R  /\  Q  =/= 
S  /\  ( P  .\/  Q )  =  ( R  .\/  S ) ) )  ->  R
( le `  K
) ( Q  .\/  S ) )
27 hllat 30098 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
281, 27syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( P  =/=  R  /\  Q  =/= 
S  /\  ( P  .\/  Q )  =  ( R  .\/  S ) ) )  ->  K  e.  Lat )
29 eqid 2435 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
3029, 6atbase 30024 . . . . 5  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
3111, 30syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( P  =/=  R  /\  Q  =/= 
S  /\  ( P  .\/  Q )  =  ( R  .\/  S ) ) )  ->  P  e.  ( Base `  K
) )
3229, 6atbase 30024 . . . . 5  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
332, 32syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( P  =/=  R  /\  Q  =/= 
S  /\  ( P  .\/  Q )  =  ( R  .\/  S ) ) )  ->  R  e.  ( Base `  K
) )
3429, 5, 6hlatjcl 30101 . . . . 5  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  S  e.  A )  ->  ( Q  .\/  S
)  e.  ( Base `  K ) )
351, 12, 3, 34syl3anc 1184 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( P  =/=  R  /\  Q  =/= 
S  /\  ( P  .\/  Q )  =  ( R  .\/  S ) ) )  ->  ( Q  .\/  S )  e.  ( Base `  K
) )
3629, 4, 5latjle12 14483 . . . 4  |-  ( ( K  e.  Lat  /\  ( P  e.  ( Base `  K )  /\  R  e.  ( Base `  K )  /\  ( Q  .\/  S )  e.  ( Base `  K
) ) )  -> 
( ( P ( le `  K ) ( Q  .\/  S
)  /\  R ( le `  K ) ( Q  .\/  S ) )  <->  ( P  .\/  R ) ( le `  K ) ( Q 
.\/  S ) ) )
3728, 31, 33, 35, 36syl13anc 1186 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( P  =/=  R  /\  Q  =/= 
S  /\  ( P  .\/  Q )  =  ( R  .\/  S ) ) )  ->  (
( P ( le
`  K ) ( Q  .\/  S )  /\  R ( le
`  K ) ( Q  .\/  S ) )  <->  ( P  .\/  R ) ( le `  K ) ( Q 
.\/  S ) ) )
3820, 26, 37mpbi2and 888 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( P  =/=  R  /\  Q  =/= 
S  /\  ( P  .\/  Q )  =  ( R  .\/  S ) ) )  ->  ( P  .\/  R ) ( le `  K ) ( Q  .\/  S
) )
39 simp31 993 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( P  =/=  R  /\  Q  =/= 
S  /\  ( P  .\/  Q )  =  ( R  .\/  S ) ) )  ->  P  =/=  R )
404, 5, 6ps-1 30211 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  R  e.  A  /\  P  =/=  R
)  /\  ( Q  e.  A  /\  S  e.  A ) )  -> 
( ( P  .\/  R ) ( le `  K ) ( Q 
.\/  S )  <->  ( P  .\/  R )  =  ( Q  .\/  S ) ) )
411, 11, 2, 39, 12, 3, 40syl132anc 1202 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( P  =/=  R  /\  Q  =/= 
S  /\  ( P  .\/  Q )  =  ( R  .\/  S ) ) )  ->  (
( P  .\/  R
) ( le `  K ) ( Q 
.\/  S )  <->  ( P  .\/  R )  =  ( Q  .\/  S ) ) )
4238, 41mpbid 202 1  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( P  =/=  R  /\  Q  =/= 
S  /\  ( P  .\/  Q )  =  ( R  .\/  S ) ) )  ->  ( P  .\/  R )  =  ( Q  .\/  S
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   Basecbs 13461   lecple 13528   joincjn 14393   Latclat 14466   Atomscatm 29998   HLchlt 30085
This theorem is referenced by:  cdlemg18a  31412
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-poset 14395  df-plt 14407  df-lub 14423  df-join 14425  df-lat 14467  df-covers 30001  df-ats 30002  df-atl 30033  df-cvlat 30057  df-hlat 30086
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