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Theorem 4atexlemswapqr 33061
Description: Lemma for 4atexlem7 33073. 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 1027 . . . 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 195 . . 3  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
414atexlempw 33047 . . 3  |-  ( ph  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
5 simp22 1031 . . . . 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 994 . . . . 5  |-  ( ( R  e.  A  /\  -.  R  .<_  W  /\  ( P  .\/  R )  =  ( Q  .\/  R ) )  ->  ( R  e.  A  /\  -.  R  .<_  W ) )
75, 6syl 17 . . . 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 195 . . 3  |-  ( ph  ->  ( R  e.  A  /\  -.  R  .<_  W ) )
93, 4, 83jca 1177 . 2  |-  ( ph  ->  ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )
1014atexlems 33050 . . 3  |-  ( ph  ->  S  e.  A )
1114atexlemq 33049 . . . 4  |-  ( ph  ->  Q  e.  A )
12 simp13r 1113 . . . . 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 195 . . . 4  |-  ( ph  ->  -.  Q  .<_  W )
1414atexlemkc 33056 . . . . 5  |-  ( ph  ->  K  e.  CvLat )
1514atexlemp 33048 . . . . 5  |-  ( ph  ->  P  e.  A )
168simpld 457 . . . . 5  |-  ( ph  ->  R  e.  A )
1714atexlempnq 33053 . . . . 5  |-  ( ph  ->  P  =/=  Q )
18 simp223 1140 . . . . . 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 195 . . . . 5  |-  ( ph  ->  ( P  .\/  R
)  =  ( Q 
.\/  R ) )
20 4thatlemslps.a . . . . . 6  |-  A  =  ( Atoms `  K )
21 4thatlemslps.j . . . . . 6  |-  .\/  =  ( join `  K )
2220, 21cvlsupr7 32347 . . . . 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 1248 . . . 4  |-  ( ph  ->  ( P  .\/  Q
)  =  ( R 
.\/  Q ) )
2411, 13, 233jca 1177 . . 3  |-  ( ph  ->  ( Q  e.  A  /\  -.  Q  .<_  W  /\  ( P  .\/  Q )  =  ( R  .\/  Q ) ) )
2514atexlemt 33051 . . . 4  |-  ( ph  ->  T  e.  A )
26 4thatlemsw.u . . . . . . 7  |-  U  =  ( ( P  .\/  Q )  ./\  W )
2720, 21cvlsupr8 32348 . . . . . . . . 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 1248 . . . . . . . 8  |-  ( ph  ->  ( P  .\/  Q
)  =  ( P 
.\/  R ) )
2928oveq1d 6249 . . . . . . 7  |-  ( ph  ->  ( ( P  .\/  Q )  ./\  W )  =  ( ( P 
.\/  R )  ./\  W ) )
3026, 29syl5eq 2455 . . . . . 6  |-  ( ph  ->  U  =  ( ( P  .\/  R ) 
./\  W ) )
3130oveq1d 6249 . . . . 5  |-  ( ph  ->  ( U  .\/  T
)  =  ( ( ( P  .\/  R
)  ./\  W )  .\/  T ) )
3214atexlemutvt 33052 . . . . 5  |-  ( ph  ->  ( U  .\/  T
)  =  ( V 
.\/  T ) )
3331, 32eqtr3d 2445 . . . 4  |-  ( ph  ->  ( ( ( P 
.\/  R )  ./\  W )  .\/  T )  =  ( V  .\/  T ) )
3425, 33jca 530 . . 3  |-  ( ph  ->  ( T  e.  A  /\  ( ( ( P 
.\/  R )  ./\  W )  .\/  T )  =  ( V  .\/  T ) ) )
3510, 24, 343jca 1177 . 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 32345 . . . . 5  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  ( P  .\/  R
)  =  ( Q 
.\/  R ) ) )  ->  R  =/=  P )
3736necomd 2674 . . . 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 1248 . . 3  |-  ( ph  ->  P  =/=  R )
3914atexlemnslpq 33054 . . . 4  |-  ( ph  ->  -.  S  .<_  ( P 
.\/  Q ) )
4028eqcomd 2410 . . . . 5  |-  ( ph  ->  ( P  .\/  R
)  =  ( P 
.\/  Q ) )
4140breq2d 4406 . . . 4  |-  ( ph  ->  ( S  .<_  ( P 
.\/  R )  <->  S  .<_  ( P  .\/  Q ) ) )
4239, 41mtbird 299 . . 3  |-  ( ph  ->  -.  S  .<_  ( P 
.\/  R ) )
4338, 42jca 530 . 2  |-  ( ph  ->  ( P  =/=  R  /\  -.  S  .<_  ( P 
.\/  R ) ) )
449, 35, 433jca 1177 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   class class class wbr 4394   ` cfv 5525  (class class class)co 6234   lecple 14808   joincjn 15789   Atomscatm 32262   CvLatclc 32264   HLchlt 32349
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6196  df-ov 6237  df-oprab 6238  df-preset 15773  df-poset 15791  df-plt 15804  df-lub 15820  df-glb 15821  df-join 15822  df-meet 15823  df-p0 15885  df-lat 15892  df-covers 32265  df-ats 32266  df-atl 32297  df-cvlat 32321  df-hlat 32350
This theorem is referenced by:  4atexlemex4  33071
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