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Theorem hlatexch4 33444
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 1018 . . . . . . 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 1014 . . . . . . 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 1015 . . . . . . 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 2452 . . . . . . . 8  |-  ( le
`  K )  =  ( le `  K
)
5 hlatexch4.j . . . . . . . 8  |-  .\/  =  ( join `  K )
6 hlatexch4.a . . . . . . . 8  |-  A  =  ( Atoms `  K )
74, 5, 6hlatlej2 33339 . . . . . . 7  |-  ( ( K  e.  HL  /\  R  e.  A  /\  S  e.  A )  ->  S ( le `  K ) ( R 
.\/  S ) )
81, 2, 3, 7syl3anc 1219 . . . . . 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 1026 . . . . . 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 4421 . . . . 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 1019 . . . . . 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 1020 . . . . . 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 1025 . . . . . . 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 2720 . . . . . 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 33359 . . . . . 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 1232 . . . . 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 33331 . . . . 5  |-  ( ( K  e.  HL  /\  S  e.  A  /\  Q  e.  A )  ->  ( S  .\/  Q
)  =  ( Q 
.\/  S ) )
191, 3, 12, 18syl3anc 1219 . . . 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 4419 . . 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 33339 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  Q ( le `  K ) ( P 
.\/  Q ) )
221, 11, 12, 21syl3anc 1219 . . . . 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 4419 . . . 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 33359 . . . . 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 1232 . . . 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 33327 . . . . 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 2452 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
3029, 6atbase 33253 . . . . 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 33253 . . . . 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 33330 . . . . 5  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  S  e.  A )  ->  ( Q  .\/  S
)  e.  ( Base `  K ) )
351, 12, 3, 34syl3anc 1219 . . . 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 15346 . . . 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 1221 . . 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 912 . 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 1024 . . 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 33440 . . 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 1237 . 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 210 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 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2645   class class class wbr 4395   ` cfv 5521  (class class class)co 6195   Basecbs 14287   lecple 14359   joincjn 15228   Latclat 15329   Atomscatm 33227   HLchlt 33314
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-poset 15230  df-plt 15242  df-lub 15258  df-glb 15259  df-join 15260  df-meet 15261  df-p0 15323  df-lat 15330  df-covers 33230  df-ats 33231  df-atl 33262  df-cvlat 33286  df-hlat 33315
This theorem is referenced by:  cdlemg18a  34641
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