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Theorem dalemswapyzps 33657
Description: Lemma for dath 33703. 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 33651 . . . 4  |-  ( ps 
->  d  e.  A
)
31dalemccea 33650 . . . 4  |-  ( ps 
->  c  e.  A
)
42, 3jca 532 . . 3  |-  ( ps 
->  ( d  e.  A  /\  c  e.  A
) )
543ad2ant3 1011 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( d  e.  A  /\  c  e.  A
) )
61dalem-ddly 33653 . . . 4  |-  ( ps 
->  -.  d  .<_  Y )
763ad2ant3 1011 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  d  .<_  Y )
8 simp2 989 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  Y  =  Z )
98breq2d 4411 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( d  .<_  Y  <->  d  .<_  Z ) )
107, 9mtbid 300 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  d  .<_  Z )
111dalemccnedd 33654 . . . 4  |-  ( ps 
->  c  =/=  d
)
12113ad2ant3 1011 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  =/=  d )
131dalem-ccly 33652 . . . . 5  |-  ( ps 
->  -.  c  .<_  Y )
14133ad2ant3 1011 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  c  .<_  Y )
158breq2d 4411 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( c  .<_  Y  <->  c  .<_  Z ) )
1614, 15mtbid 300 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  c  .<_  Z )
171dalemclccjdd 33655 . . . . 5  |-  ( ps 
->  C  .<_  ( c 
.\/  d ) )
18173ad2ant3 1011 . . . 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 33590 . . . . . 6  |-  ( ph  ->  K  e.  HL )
21203ad2ant1 1009 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  HL )
2233ad2ant3 1011 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  e.  A )
2323ad2ant3 1011 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
d  e.  A )
24 dalem.j . . . . . 6  |-  .\/  =  ( join `  K )
25 dalem.a . . . . . 6  |-  A  =  ( Atoms `  K )
2624, 25hlatjcom 33335 . . . . 5  |-  ( ( K  e.  HL  /\  c  e.  A  /\  d  e.  A )  ->  ( c  .\/  d
)  =  ( d 
.\/  c ) )
2721, 22, 23, 26syl3anc 1219 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( c  .\/  d
)  =  ( d 
.\/  c ) )
2818, 27breqtrd 4423 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  C  .<_  ( d  .\/  c ) )
2912, 16, 283jca 1168 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( c  =/=  d  /\  -.  c  .<_  Z  /\  C  .<_  ( d  .\/  c ) ) )
305, 10, 293jca 1168 1  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( d  e.  A  /\  c  e.  A )  /\  -.  d  .<_  Z  /\  (
c  =/=  d  /\  -.  c  .<_  Z  /\  C  .<_  ( d  .\/  c ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2647   class class class wbr 4399   ` cfv 5525  (class class class)co 6199   Basecbs 14291   lecple 14363   joincjn 15232   Atomscatm 33231   HLchlt 33318
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 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-reu 2805  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-id 4743  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6160  df-ov 6202  df-oprab 6203  df-lub 15262  df-join 15264  df-lat 15334  df-ats 33235  df-atl 33266  df-cvlat 33290  df-hlat 33319
This theorem is referenced by:  dalem56  33695
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