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Theorem hlatexch4 29646
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 2380 . . . . . . . 8  |-  ( le
`  K )  =  ( le `  K
)
5 hlatexch4.j . . . . . . . 8  |-  .\/  =  ( join `  K )
6 hlatexch4.a . . . . . . . 8  |-  A  =  ( Atoms `  K )
74, 5, 6hlatlej2 29541 . . . . . . 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 4172 . . . . 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 2626 . . . . . 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 29561 . . . . . 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 29533 . . . . 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 4170 . . 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 29541 . . . . . 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 4170 . . . 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 29561 . . . . 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 29529 . . . . 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 2380 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
3029, 6atbase 29455 . . . . 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 29455 . . . . 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 29532 . . . . 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 14411 . . . 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 29642 . . 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 1649    e. wcel 1717    =/= wne 2543   class class class wbr 4146   ` cfv 5387  (class class class)co 6013   Basecbs 13389   lecple 13456   joincjn 14321   Latclat 14394   Atomscatm 29429   HLchlt 29516
This theorem is referenced by:  cdlemg18a  30843
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-undef 6472  df-riota 6478  df-poset 14323  df-plt 14335  df-lub 14351  df-join 14353  df-lat 14395  df-covers 29432  df-ats 29433  df-atl 29464  df-cvlat 29488  df-hlat 29517
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