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Theorem dalemswapyzps 32964
Description: Lemma for dath 33010. 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 32958 . . . 4  |-  ( ps 
->  d  e.  A
)
31dalemccea 32957 . . . 4  |-  ( ps 
->  c  e.  A
)
42, 3jca 534 . . 3  |-  ( ps 
->  ( d  e.  A  /\  c  e.  A
) )
543ad2ant3 1028 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( d  e.  A  /\  c  e.  A
) )
61dalem-ddly 32960 . . . 4  |-  ( ps 
->  -.  d  .<_  Y )
763ad2ant3 1028 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  d  .<_  Y )
8 simp2 1006 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  Y  =  Z )
98breq2d 4438 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( d  .<_  Y  <->  d  .<_  Z ) )
107, 9mtbid 301 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  d  .<_  Z )
111dalemccnedd 32961 . . . 4  |-  ( ps 
->  c  =/=  d
)
12113ad2ant3 1028 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  =/=  d )
131dalem-ccly 32959 . . . . 5  |-  ( ps 
->  -.  c  .<_  Y )
14133ad2ant3 1028 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  c  .<_  Y )
158breq2d 4438 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( c  .<_  Y  <->  c  .<_  Z ) )
1614, 15mtbid 301 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  c  .<_  Z )
171dalemclccjdd 32962 . . . . 5  |-  ( ps 
->  C  .<_  ( c 
.\/  d ) )
18173ad2ant3 1028 . . . 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 32897 . . . . . 6  |-  ( ph  ->  K  e.  HL )
21203ad2ant1 1026 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  HL )
2233ad2ant3 1028 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  e.  A )
2323ad2ant3 1028 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
d  e.  A )
24 dalem.j . . . . . 6  |-  .\/  =  ( join `  K )
25 dalem.a . . . . . 6  |-  A  =  ( Atoms `  K )
2624, 25hlatjcom 32642 . . . . 5  |-  ( ( K  e.  HL  /\  c  e.  A  /\  d  e.  A )  ->  ( c  .\/  d
)  =  ( d 
.\/  c ) )
2721, 22, 23, 26syl3anc 1264 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( c  .\/  d
)  =  ( d 
.\/  c ) )
2818, 27breqtrd 4450 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  C  .<_  ( d  .\/  c ) )
2912, 16, 283jca 1185 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( c  =/=  d  /\  -.  c  .<_  Z  /\  C  .<_  ( d  .\/  c ) ) )
305, 10, 293jca 1185 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625   class class class wbr 4426   ` cfv 5601  (class class class)co 6305   Basecbs 15084   lecple 15159   joincjn 16140   Atomscatm 32538   HLchlt 32625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-lub 16171  df-join 16173  df-lat 16243  df-ats 32542  df-atl 32573  df-cvlat 32597  df-hlat 32626
This theorem is referenced by:  dalem56  33002
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