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Theorem hlatexch4 32755
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 1035 . . . . . . 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 1031 . . . . . . 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 1032 . . . . . . 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 2429 . . . . . . . 8  |-  ( le
`  K )  =  ( le `  K
)
5 hlatexch4.j . . . . . . . 8  |-  .\/  =  ( join `  K )
6 hlatexch4.a . . . . . . . 8  |-  A  =  ( Atoms `  K )
74, 5, 6hlatlej2 32650 . . . . . . 7  |-  ( ( K  e.  HL  /\  R  e.  A  /\  S  e.  A )  ->  S ( le `  K ) ( R 
.\/  S ) )
81, 2, 3, 7syl3anc 1264 . . . . . 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 1043 . . . . . 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 4452 . . . . 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 1036 . . . . . 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 1037 . . . . . 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 1042 . . . . . . 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 2702 . . . . . 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 32670 . . . . . 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 1277 . . . . 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 32642 . . . . 5  |-  ( ( K  e.  HL  /\  S  e.  A  /\  Q  e.  A )  ->  ( S  .\/  Q
)  =  ( Q 
.\/  S ) )
191, 3, 12, 18syl3anc 1264 . . . 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 4450 . . 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 32650 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  Q ( le `  K ) ( P 
.\/  Q ) )
221, 11, 12, 21syl3anc 1264 . . . . 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 4450 . . . 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 32670 . . . . 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 1277 . . . 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 32638 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
281, 27syl 17 . . . 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 2429 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
3029, 6atbase 32564 . . . . 5  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
3111, 30syl 17 . . . 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 32564 . . . . 5  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
332, 32syl 17 . . . 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 32641 . . . . 5  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  S  e.  A )  ->  ( Q  .\/  S
)  e.  ( Base `  K ) )
351, 12, 3, 34syl3anc 1264 . . . 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 16259 . . . 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 1266 . . 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 929 . 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 1041 . . 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 32751 . . 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 1282 . 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 213 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 setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625   class class class wbr 4426   ` cfv 5601  (class class class)co 6305   Basecbs 15084   lecple 15159   joincjn 16140   Latclat 16242   Atomscatm 32538   HLchlt 32625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-preset 16124  df-poset 16142  df-plt 16155  df-lub 16171  df-glb 16172  df-join 16173  df-meet 16174  df-p0 16236  df-lat 16243  df-covers 32541  df-ats 32542  df-atl 32573  df-cvlat 32597  df-hlat 32626
This theorem is referenced by:  cdlemg18a  33954
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