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Theorem dalemswapyzps 30487
Description: Lemma for dath 30533. Swap the  Y and 
Z planes, along with dummy concurrency (center of perspectivity) atoms  c and  d, to allow reuse of analogous proofs. (Contributed by NM, 17-Aug-2012.)
Hypotheses
Ref Expression
dalem.ph  |-  ( ph  <->  ( ( ( K  e.  HL  /\  C  e.  ( Base `  K
) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A
) )  /\  ( Y  e.  O  /\  Z  e.  O )  /\  ( ( -.  C  .<_  ( P  .\/  Q
)  /\  -.  C  .<_  ( Q  .\/  R
)  /\  -.  C  .<_  ( R  .\/  P
) )  /\  ( -.  C  .<_  ( S 
.\/  T )  /\  -.  C  .<_  ( T 
.\/  U )  /\  -.  C  .<_  ( U 
.\/  S ) )  /\  ( C  .<_  ( P  .\/  S )  /\  C  .<_  ( Q 
.\/  T )  /\  C  .<_  ( R  .\/  U ) ) ) ) )
dalem.l  |-  .<_  =  ( le `  K )
dalem.j  |-  .\/  =  ( join `  K )
dalem.a  |-  A  =  ( Atoms `  K )
dalem.ps  |-  ( ps  <->  ( ( c  e.  A  /\  d  e.  A
)  /\  -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
) ) ) )
Assertion
Ref Expression
dalemswapyzps  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( d  e.  A  /\  c  e.  A )  /\  -.  d  .<_  Z  /\  (
c  =/=  d  /\  -.  c  .<_  Z  /\  C  .<_  ( d  .\/  c ) ) ) )

Proof of Theorem dalemswapyzps
StepHypRef Expression
1 dalem.ps . . . . 5  |-  ( ps  <->  ( ( c  e.  A  /\  d  e.  A
)  /\  -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
) ) ) )
21dalemddea 30481 . . . 4  |-  ( ps 
->  d  e.  A
)
31dalemccea 30480 . . . 4  |-  ( ps 
->  c  e.  A
)
42, 3jca 519 . . 3  |-  ( ps 
->  ( d  e.  A  /\  c  e.  A
) )
543ad2ant3 980 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( d  e.  A  /\  c  e.  A
) )
61dalem-ddly 30483 . . . 4  |-  ( ps 
->  -.  d  .<_  Y )
763ad2ant3 980 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  d  .<_  Y )
8 simp2 958 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  Y  =  Z )
98breq2d 4224 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( d  .<_  Y  <->  d  .<_  Z ) )
107, 9mtbid 292 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  d  .<_  Z )
111dalemccnedd 30484 . . . 4  |-  ( ps 
->  c  =/=  d
)
12113ad2ant3 980 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  =/=  d )
131dalem-ccly 30482 . . . . 5  |-  ( ps 
->  -.  c  .<_  Y )
14133ad2ant3 980 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  c  .<_  Y )
158breq2d 4224 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( c  .<_  Y  <->  c  .<_  Z ) )
1614, 15mtbid 292 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  c  .<_  Z )
171dalemclccjdd 30485 . . . . 5  |-  ( ps 
->  C  .<_  ( c 
.\/  d ) )
18173ad2ant3 980 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  C  .<_  ( c  .\/  d ) )
19 dalem.ph . . . . . . 7  |-  ( ph  <->  ( ( ( K  e.  HL  /\  C  e.  ( Base `  K
) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A
) )  /\  ( Y  e.  O  /\  Z  e.  O )  /\  ( ( -.  C  .<_  ( P  .\/  Q
)  /\  -.  C  .<_  ( Q  .\/  R
)  /\  -.  C  .<_  ( R  .\/  P
) )  /\  ( -.  C  .<_  ( S 
.\/  T )  /\  -.  C  .<_  ( T 
.\/  U )  /\  -.  C  .<_  ( U 
.\/  S ) )  /\  ( C  .<_  ( P  .\/  S )  /\  C  .<_  ( Q 
.\/  T )  /\  C  .<_  ( R  .\/  U ) ) ) ) )
2019dalemkehl 30420 . . . . . 6  |-  ( ph  ->  K  e.  HL )
21203ad2ant1 978 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  HL )
2233ad2ant3 980 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  e.  A )
2323ad2ant3 980 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
d  e.  A )
24 dalem.j . . . . . 6  |-  .\/  =  ( join `  K )
25 dalem.a . . . . . 6  |-  A  =  ( Atoms `  K )
2624, 25hlatjcom 30165 . . . . 5  |-  ( ( K  e.  HL  /\  c  e.  A  /\  d  e.  A )  ->  ( c  .\/  d
)  =  ( d 
.\/  c ) )
2721, 22, 23, 26syl3anc 1184 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( c  .\/  d
)  =  ( d 
.\/  c ) )
2818, 27breqtrd 4236 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  C  .<_  ( d  .\/  c ) )
2912, 16, 283jca 1134 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( c  =/=  d  /\  -.  c  .<_  Z  /\  C  .<_  ( d  .\/  c ) ) )
305, 10, 293jca 1134 1  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( d  e.  A  /\  c  e.  A )  /\  -.  d  .<_  Z  /\  (
c  =/=  d  /\  -.  c  .<_  Z  /\  C  .<_  ( d  .\/  c ) ) ) )
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   Basecbs 13469   lecple 13536   joincjn 14401   Atomscatm 30061   HLchlt 30148
This theorem is referenced by:  dalem56  30525
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-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-join 14433  df-lat 14475  df-ats 30065  df-atl 30096  df-cvlat 30120  df-hlat 30149
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