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Theorem hlatexch4 30292
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 985 . . . . . . 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 981 . . . . . . 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 982 . . . . . . 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 2296 . . . . . . . 8  |-  ( le
`  K )  =  ( le `  K
)
5 hlatexch4.j . . . . . . . 8  |-  .\/  =  ( join `  K )
6 hlatexch4.a . . . . . . . 8  |-  A  =  ( Atoms `  K )
74, 5, 6hlatlej2 30187 . . . . . . 7  |-  ( ( K  e.  HL  /\  R  e.  A  /\  S  e.  A )  ->  S ( le `  K ) ( R 
.\/  S ) )
81, 2, 3, 7syl3anc 1182 . . . . . 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 993 . . . . . 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 4065 . . . . 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 986 . . . . . 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 987 . . . . . 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 992 . . . . . . 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 2542 . . . . . 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 30207 . . . . . 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 1195 . . . . 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 14 . . . 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 30179 . . . . 5  |-  ( ( K  e.  HL  /\  S  e.  A  /\  Q  e.  A )  ->  ( S  .\/  Q
)  =  ( Q 
.\/  S ) )
191, 3, 12, 18syl3anc 1182 . . . 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 4063 . . 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 30187 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  Q ( le `  K ) ( P 
.\/  Q ) )
221, 11, 12, 21syl3anc 1182 . . . . 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 4063 . . . 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 30207 . . . . 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 1195 . . . 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 14 . . 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 30175 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
281, 27syl 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 ) ) )  ->  K  e.  Lat )
29 eqid 2296 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
3029, 6atbase 30101 . . . . 5  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
3111, 30syl 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  e.  ( Base `  K
) )
3229, 6atbase 30101 . . . . 5  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
332, 32syl 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 ) ) )  ->  R  e.  ( Base `  K
) )
3429, 5, 6hlatjcl 30178 . . . . 5  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  S  e.  A )  ->  ( Q  .\/  S
)  e.  ( Base `  K ) )
351, 12, 3, 34syl3anc 1182 . . . 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 14184 . . . 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 1184 . . 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 887 . 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 991 . . 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 30288 . . 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 1200 . 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 201 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   joincjn 14094   Latclat 14167   Atomscatm 30075   HLchlt 30162
This theorem is referenced by:  cdlemg18a  31489
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-plt 14108  df-lub 14124  df-join 14126  df-lat 14168  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163
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