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Theorem 4atexlemswapqr 34013
Description: Lemma for 4atexlem7 34025. 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 1018 . . . 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 33999 . . 3  |-  ( ph  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
5 simp22 1022 . . . . 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 985 . . . . 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 195 . . 3  |-  ( ph  ->  ( R  e.  A  /\  -.  R  .<_  W ) )
93, 4, 83jca 1168 . 2  |-  ( ph  ->  ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )
1014atexlems 34002 . . 3  |-  ( ph  ->  S  e.  A )
1114atexlemq 34001 . . . 4  |-  ( ph  ->  Q  e.  A )
12 simp13r 1104 . . . . 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 34008 . . . . 5  |-  ( ph  ->  K  e.  CvLat )
1514atexlemp 34000 . . . . 5  |-  ( ph  ->  P  e.  A )
168simpld 459 . . . . 5  |-  ( ph  ->  R  e.  A )
1714atexlempnq 34005 . . . . 5  |-  ( ph  ->  P  =/=  Q )
18 simp223 1131 . . . . . 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 33299 . . . . 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 1237 . . . 4  |-  ( ph  ->  ( P  .\/  Q
)  =  ( R 
.\/  Q ) )
2411, 13, 233jca 1168 . . 3  |-  ( ph  ->  ( Q  e.  A  /\  -.  Q  .<_  W  /\  ( P  .\/  Q )  =  ( R  .\/  Q ) ) )
2514atexlemt 34003 . . . 4  |-  ( ph  ->  T  e.  A )
26 4thatlemsw.u . . . . . . 7  |-  U  =  ( ( P  .\/  Q )  ./\  W )
2720, 21cvlsupr8 33300 . . . . . . . . 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 1237 . . . . . . . 8  |-  ( ph  ->  ( P  .\/  Q
)  =  ( P 
.\/  R ) )
2928oveq1d 6205 . . . . . . 7  |-  ( ph  ->  ( ( P  .\/  Q )  ./\  W )  =  ( ( P 
.\/  R )  ./\  W ) )
3026, 29syl5eq 2504 . . . . . 6  |-  ( ph  ->  U  =  ( ( P  .\/  R ) 
./\  W ) )
3130oveq1d 6205 . . . . 5  |-  ( ph  ->  ( U  .\/  T
)  =  ( ( ( P  .\/  R
)  ./\  W )  .\/  T ) )
3214atexlemutvt 34004 . . . . 5  |-  ( ph  ->  ( U  .\/  T
)  =  ( V 
.\/  T ) )
3331, 32eqtr3d 2494 . . . 4  |-  ( ph  ->  ( ( ( P 
.\/  R )  ./\  W )  .\/  T )  =  ( V  .\/  T ) )
3425, 33jca 532 . . 3  |-  ( ph  ->  ( T  e.  A  /\  ( ( ( P 
.\/  R )  ./\  W )  .\/  T )  =  ( V  .\/  T ) ) )
3510, 24, 343jca 1168 . 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 33297 . . . . 5  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  ( P  .\/  R
)  =  ( Q 
.\/  R ) ) )  ->  R  =/=  P )
3736necomd 2719 . . . 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 1237 . . 3  |-  ( ph  ->  P  =/=  R )
3914atexlemnslpq 34006 . . . 4  |-  ( ph  ->  -.  S  .<_  ( P 
.\/  Q ) )
4028eqcomd 2459 . . . . 5  |-  ( ph  ->  ( P  .\/  R
)  =  ( P 
.\/  Q ) )
4140breq2d 4402 . . . 4  |-  ( ph  ->  ( S  .<_  ( P 
.\/  R )  <->  S  .<_  ( P  .\/  Q ) ) )
4239, 41mtbird 301 . . 3  |-  ( ph  ->  -.  S  .<_  ( P 
.\/  R ) )
4338, 42jca 532 . 2  |-  ( ph  ->  ( P  =/=  R  /\  -.  S  .<_  ( P 
.\/  R ) ) )
449, 35, 433jca 1168 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 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2644   class class class wbr 4390   ` cfv 5516  (class class class)co 6190   lecple 14347   joincjn 15216   Atomscatm 33214   CvLatclc 33216   HLchlt 33301
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 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472
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 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-id 4734  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-poset 15218  df-plt 15230  df-lub 15246  df-glb 15247  df-join 15248  df-meet 15249  df-p0 15311  df-lat 15318  df-covers 33217  df-ats 33218  df-atl 33249  df-cvlat 33273  df-hlat 33302
This theorem is referenced by:  4atexlemex4  34023
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