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Theorem dalemswapyzps 33300
Description: Lemma for dath 33346. 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 33294 . . . 4  |-  ( ps 
->  d  e.  A
)
31dalemccea 33293 . . . 4  |-  ( ps 
->  c  e.  A
)
42, 3jca 539 . . 3  |-  ( ps 
->  ( d  e.  A  /\  c  e.  A
) )
543ad2ant3 1037 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( d  e.  A  /\  c  e.  A
) )
61dalem-ddly 33296 . . . 4  |-  ( ps 
->  -.  d  .<_  Y )
763ad2ant3 1037 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  d  .<_  Y )
8 simp2 1015 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  Y  =  Z )
98breq2d 4428 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( d  .<_  Y  <->  d  .<_  Z ) )
107, 9mtbid 306 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  d  .<_  Z )
111dalemccnedd 33297 . . . 4  |-  ( ps 
->  c  =/=  d
)
12113ad2ant3 1037 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  =/=  d )
131dalem-ccly 33295 . . . . 5  |-  ( ps 
->  -.  c  .<_  Y )
14133ad2ant3 1037 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  c  .<_  Y )
158breq2d 4428 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( c  .<_  Y  <->  c  .<_  Z ) )
1614, 15mtbid 306 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  c  .<_  Z )
171dalemclccjdd 33298 . . . . 5  |-  ( ps 
->  C  .<_  ( c 
.\/  d ) )
18173ad2ant3 1037 . . . 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 33233 . . . . . 6  |-  ( ph  ->  K  e.  HL )
21203ad2ant1 1035 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  HL )
2233ad2ant3 1037 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  e.  A )
2323ad2ant3 1037 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
d  e.  A )
24 dalem.j . . . . . 6  |-  .\/  =  ( join `  K )
25 dalem.a . . . . . 6  |-  A  =  ( Atoms `  K )
2624, 25hlatjcom 32978 . . . . 5  |-  ( ( K  e.  HL  /\  c  e.  A  /\  d  e.  A )  ->  ( c  .\/  d
)  =  ( d 
.\/  c ) )
2721, 22, 23, 26syl3anc 1276 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( c  .\/  d
)  =  ( d 
.\/  c ) )
2818, 27breqtrd 4441 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  C  .<_  ( d  .\/  c ) )
2912, 16, 283jca 1194 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( c  =/=  d  /\  -.  c  .<_  Z  /\  C  .<_  ( d  .\/  c ) ) )
305, 10, 293jca 1194 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 189    /\ wa 375    /\ w3a 991    = wceq 1455    e. wcel 1898    =/= wne 2633   class class class wbr 4416   ` cfv 5601  (class class class)co 6315   Basecbs 15170   lecple 15246   joincjn 16238   Atomscatm 32874   HLchlt 32961
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4529  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-reu 2756  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-iun 4294  df-br 4417  df-opab 4476  df-mpt 4477  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  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 6277  df-ov 6318  df-oprab 6319  df-lub 16269  df-join 16271  df-lat 16341  df-ats 32878  df-atl 32909  df-cvlat 32933  df-hlat 32962
This theorem is referenced by:  dalem56  33338
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