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Theorem 4atexlemswapqr 30860
Description: Lemma for 4atexlem7 30872. Swap  Q and  R, so that theorems involving  C can be reused for  D. Note that  U must be expanded because it involves  Q. (Contributed by NM, 25-Nov-2012.)
Hypotheses
Ref Expression
4thatlem.ph  |-  ( ph  <->  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( S  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W  /\  ( P  .\/  R )  =  ( Q  .\/  R ) )  /\  ( T  e.  A  /\  ( U  .\/  T )  =  ( V  .\/  T ) ) )  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P 
.\/  Q ) ) ) )
4thatlemslps.l  |-  .<_  =  ( le `  K )
4thatlemslps.j  |-  .\/  =  ( join `  K )
4thatlemslps.a  |-  A  =  ( Atoms `  K )
4thatlemsw.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
Assertion
Ref Expression
4atexlemswapqr  |-  ( ph  ->  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( S  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W  /\  ( P  .\/  Q )  =  ( R  .\/  Q ) )  /\  ( T  e.  A  /\  ( ( ( P 
.\/  R )  ./\  W )  .\/  T )  =  ( V  .\/  T ) ) )  /\  ( P  =/=  R  /\  -.  S  .<_  ( P 
.\/  R ) ) ) )

Proof of Theorem 4atexlemswapqr
StepHypRef Expression
1 4thatlem.ph . . . 4  |-  ( ph  <->  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( S  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W  /\  ( P  .\/  R )  =  ( Q  .\/  R ) )  /\  ( T  e.  A  /\  ( U  .\/  T )  =  ( V  .\/  T ) ) )  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P 
.\/  Q ) ) ) )
2 simp11 987 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( S  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W  /\  ( P  .\/  R )  =  ( Q  .\/  R ) )  /\  ( T  e.  A  /\  ( U  .\/  T )  =  ( V  .\/  T ) ) )  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
31, 2sylbi 188 . . 3  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
414atexlempw 30846 . . 3  |-  ( ph  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
5 simp22 991 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( S  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W  /\  ( P  .\/  R )  =  ( Q  .\/  R ) )  /\  ( T  e.  A  /\  ( U  .\/  T )  =  ( V  .\/  T ) ) )  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  ( R  e.  A  /\  -.  R  .<_  W  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) ) )
6 3simpa 954 . . . . 5  |-  ( ( R  e.  A  /\  -.  R  .<_  W  /\  ( P  .\/  R )  =  ( Q  .\/  R ) )  ->  ( R  e.  A  /\  -.  R  .<_  W ) )
75, 6syl 16 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( S  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W  /\  ( P  .\/  R )  =  ( Q  .\/  R ) )  /\  ( T  e.  A  /\  ( U  .\/  T )  =  ( V  .\/  T ) ) )  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  ( R  e.  A  /\  -.  R  .<_  W ) )
81, 7sylbi 188 . . 3  |-  ( ph  ->  ( R  e.  A  /\  -.  R  .<_  W ) )
93, 4, 83jca 1134 . 2  |-  ( ph  ->  ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )
1014atexlems 30849 . . 3  |-  ( ph  ->  S  e.  A )
1114atexlemq 30848 . . . 4  |-  ( ph  ->  Q  e.  A )
12 simp13r 1073 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( S  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W  /\  ( P  .\/  R )  =  ( Q  .\/  R ) )  /\  ( T  e.  A  /\  ( U  .\/  T )  =  ( V  .\/  T ) ) )  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  -.  Q  .<_  W )
131, 12sylbi 188 . . . 4  |-  ( ph  ->  -.  Q  .<_  W )
1414atexlemkc 30855 . . . . 5  |-  ( ph  ->  K  e.  CvLat )
1514atexlemp 30847 . . . . 5  |-  ( ph  ->  P  e.  A )
168simpld 446 . . . . 5  |-  ( ph  ->  R  e.  A )
1714atexlempnq 30852 . . . . 5  |-  ( ph  ->  P  =/=  Q )
18 simp223 1100 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( S  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W  /\  ( P  .\/  R )  =  ( Q  .\/  R ) )  /\  ( T  e.  A  /\  ( U  .\/  T )  =  ( V  .\/  T ) ) )  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  ( P  .\/  R )  =  ( Q  .\/  R ) )
191, 18sylbi 188 . . . . 5  |-  ( ph  ->  ( P  .\/  R
)  =  ( Q 
.\/  R ) )
20 4thatlemslps.a . . . . . 6  |-  A  =  ( Atoms `  K )
21 4thatlemslps.j . . . . . 6  |-  .\/  =  ( join `  K )
2220, 21cvlsupr7 30146 . . . . 5  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  ( P  .\/  R
)  =  ( Q 
.\/  R ) ) )  ->  ( P  .\/  Q )  =  ( R  .\/  Q ) )
2314, 15, 11, 16, 17, 19, 22syl132anc 1202 . . . 4  |-  ( ph  ->  ( P  .\/  Q
)  =  ( R 
.\/  Q ) )
2411, 13, 233jca 1134 . . 3  |-  ( ph  ->  ( Q  e.  A  /\  -.  Q  .<_  W  /\  ( P  .\/  Q )  =  ( R  .\/  Q ) ) )
2514atexlemt 30850 . . . 4  |-  ( ph  ->  T  e.  A )
26 4thatlemsw.u . . . . . . 7  |-  U  =  ( ( P  .\/  Q )  ./\  W )
2720, 21cvlsupr8 30147 . . . . . . . . 9  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  ( P  .\/  R
)  =  ( Q 
.\/  R ) ) )  ->  ( P  .\/  Q )  =  ( P  .\/  R ) )
2814, 15, 11, 16, 17, 19, 27syl132anc 1202 . . . . . . . 8  |-  ( ph  ->  ( P  .\/  Q
)  =  ( P 
.\/  R ) )
2928oveq1d 6096 . . . . . . 7  |-  ( ph  ->  ( ( P  .\/  Q )  ./\  W )  =  ( ( P 
.\/  R )  ./\  W ) )
3026, 29syl5eq 2480 . . . . . 6  |-  ( ph  ->  U  =  ( ( P  .\/  R ) 
./\  W ) )
3130oveq1d 6096 . . . . 5  |-  ( ph  ->  ( U  .\/  T
)  =  ( ( ( P  .\/  R
)  ./\  W )  .\/  T ) )
3214atexlemutvt 30851 . . . . 5  |-  ( ph  ->  ( U  .\/  T
)  =  ( V 
.\/  T ) )
3331, 32eqtr3d 2470 . . . 4  |-  ( ph  ->  ( ( ( P 
.\/  R )  ./\  W )  .\/  T )  =  ( V  .\/  T ) )
3425, 33jca 519 . . 3  |-  ( ph  ->  ( T  e.  A  /\  ( ( ( P 
.\/  R )  ./\  W )  .\/  T )  =  ( V  .\/  T ) ) )
3510, 24, 343jca 1134 . 2  |-  ( ph  ->  ( S  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W  /\  ( P  .\/  Q )  =  ( R  .\/  Q ) )  /\  ( T  e.  A  /\  ( ( ( P 
.\/  R )  ./\  W )  .\/  T )  =  ( V  .\/  T ) ) ) )
3620, 21cvlsupr5 30144 . . . . 5  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  ( P  .\/  R
)  =  ( Q 
.\/  R ) ) )  ->  R  =/=  P )
3736necomd 2687 . . . 4  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  ( P  .\/  R
)  =  ( Q 
.\/  R ) ) )  ->  P  =/=  R )
3814, 15, 11, 16, 17, 19, 37syl132anc 1202 . . 3  |-  ( ph  ->  P  =/=  R )
3914atexlemnslpq 30853 . . . 4  |-  ( ph  ->  -.  S  .<_  ( P 
.\/  Q ) )
4028eqcomd 2441 . . . . 5  |-  ( ph  ->  ( P  .\/  R
)  =  ( P 
.\/  Q ) )
4140breq2d 4224 . . . 4  |-  ( ph  ->  ( S  .<_  ( P 
.\/  R )  <->  S  .<_  ( P  .\/  Q ) ) )
4239, 41mtbird 293 . . 3  |-  ( ph  ->  -.  S  .<_  ( P 
.\/  R ) )
4338, 42jca 519 . 2  |-  ( ph  ->  ( P  =/=  R  /\  -.  S  .<_  ( P 
.\/  R ) ) )
449, 35, 433jca 1134 1  |-  ( ph  ->  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( S  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W  /\  ( P  .\/  Q )  =  ( R  .\/  Q ) )  /\  ( T  e.  A  /\  ( ( ( P 
.\/  R )  ./\  W )  .\/  T )  =  ( V  .\/  T ) ) )  /\  ( P  =/=  R  /\  -.  S  .<_  ( P 
.\/  R ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   lecple 13536   joincjn 14401   Atomscatm 30061   CvLatclc 30063   HLchlt 30148
This theorem is referenced by:  4atexlemex4  30870
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-undef 6543  df-riota 6549  df-poset 14403  df-plt 14415  df-lub 14431  df-join 14433  df-lat 14475  df-covers 30064  df-ats 30065  df-atl 30096  df-cvlat 30120  df-hlat 30149
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