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Theorem 4atexlemswapqr 34736
Description: Lemma for 4atexlem7 34748. 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 1021 . . . 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 34722 . . 3  |-  ( ph  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
5 simp22 1025 . . . . 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 988 . . . . 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 1171 . 2  |-  ( ph  ->  ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )
1014atexlems 34725 . . 3  |-  ( ph  ->  S  e.  A )
1114atexlemq 34724 . . . 4  |-  ( ph  ->  Q  e.  A )
12 simp13r 1107 . . . . 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 34731 . . . . 5  |-  ( ph  ->  K  e.  CvLat )
1514atexlemp 34723 . . . . 5  |-  ( ph  ->  P  e.  A )
168simpld 459 . . . . 5  |-  ( ph  ->  R  e.  A )
1714atexlempnq 34728 . . . . 5  |-  ( ph  ->  P  =/=  Q )
18 simp223 1134 . . . . . 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 34022 . . . . 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 1241 . . . 4  |-  ( ph  ->  ( P  .\/  Q
)  =  ( R 
.\/  Q ) )
2411, 13, 233jca 1171 . . 3  |-  ( ph  ->  ( Q  e.  A  /\  -.  Q  .<_  W  /\  ( P  .\/  Q )  =  ( R  .\/  Q ) ) )
2514atexlemt 34726 . . . 4  |-  ( ph  ->  T  e.  A )
26 4thatlemsw.u . . . . . . 7  |-  U  =  ( ( P  .\/  Q )  ./\  W )
2720, 21cvlsupr8 34023 . . . . . . . . 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 1241 . . . . . . . 8  |-  ( ph  ->  ( P  .\/  Q
)  =  ( P 
.\/  R ) )
2928oveq1d 6292 . . . . . . 7  |-  ( ph  ->  ( ( P  .\/  Q )  ./\  W )  =  ( ( P 
.\/  R )  ./\  W ) )
3026, 29syl5eq 2515 . . . . . 6  |-  ( ph  ->  U  =  ( ( P  .\/  R ) 
./\  W ) )
3130oveq1d 6292 . . . . 5  |-  ( ph  ->  ( U  .\/  T
)  =  ( ( ( P  .\/  R
)  ./\  W )  .\/  T ) )
3214atexlemutvt 34727 . . . . 5  |-  ( ph  ->  ( U  .\/  T
)  =  ( V 
.\/  T ) )
3331, 32eqtr3d 2505 . . . 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 1171 . 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 34020 . . . . 5  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  ( P  .\/  R
)  =  ( Q 
.\/  R ) ) )  ->  R  =/=  P )
3736necomd 2733 . . . 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 1241 . . 3  |-  ( ph  ->  P  =/=  R )
3914atexlemnslpq 34729 . . . 4  |-  ( ph  ->  -.  S  .<_  ( P 
.\/  Q ) )
4028eqcomd 2470 . . . . 5  |-  ( ph  ->  ( P  .\/  R
)  =  ( P 
.\/  Q ) )
4140breq2d 4454 . . . 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 1171 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 968    = wceq 1374    e. wcel 1762    =/= wne 2657   class class class wbr 4442   ` cfv 5581  (class class class)co 6277   lecple 14553   joincjn 15422   Atomscatm 33937   CvLatclc 33939   HLchlt 34024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-poset 15424  df-plt 15436  df-lub 15452  df-glb 15453  df-join 15454  df-meet 15455  df-p0 15517  df-lat 15524  df-covers 33940  df-ats 33941  df-atl 33972  df-cvlat 33996  df-hlat 34025
This theorem is referenced by:  4atexlemex4  34746
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