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Theorem dalemswapyzps 29879
Description: Lemma for dath 29925. 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 29873 . . . 4  |-  ( ps 
->  d  e.  A
)
31dalemccea 29872 . . . 4  |-  ( ps 
->  c  e.  A
)
42, 3jca 518 . . 3  |-  ( ps 
->  ( d  e.  A  /\  c  e.  A
) )
543ad2ant3 978 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( d  e.  A  /\  c  e.  A
) )
61dalem-ddly 29875 . . . 4  |-  ( ps 
->  -.  d  .<_  Y )
763ad2ant3 978 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  d  .<_  Y )
8 simp2 956 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  Y  =  Z )
98breq2d 4035 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( d  .<_  Y  <->  d  .<_  Z ) )
107, 9mtbid 291 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  d  .<_  Z )
111dalemccnedd 29876 . . . 4  |-  ( ps 
->  c  =/=  d
)
12113ad2ant3 978 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  =/=  d )
131dalem-ccly 29874 . . . . 5  |-  ( ps 
->  -.  c  .<_  Y )
14133ad2ant3 978 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  c  .<_  Y )
158breq2d 4035 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( c  .<_  Y  <->  c  .<_  Z ) )
1614, 15mtbid 291 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  c  .<_  Z )
171dalemclccjdd 29877 . . . . 5  |-  ( ps 
->  C  .<_  ( c 
.\/  d ) )
18173ad2ant3 978 . . . 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 29812 . . . . . 6  |-  ( ph  ->  K  e.  HL )
21203ad2ant1 976 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  HL )
2233ad2ant3 978 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  e.  A )
2323ad2ant3 978 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
d  e.  A )
24 dalem.j . . . . . 6  |-  .\/  =  ( join `  K )
25 dalem.a . . . . . 6  |-  A  =  ( Atoms `  K )
2624, 25hlatjcom 29557 . . . . 5  |-  ( ( K  e.  HL  /\  c  e.  A  /\  d  e.  A )  ->  ( c  .\/  d
)  =  ( d 
.\/  c ) )
2721, 22, 23, 26syl3anc 1182 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( c  .\/  d
)  =  ( d 
.\/  c ) )
2818, 27breqtrd 4047 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  C  .<_  ( d  .\/  c ) )
2912, 16, 283jca 1132 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( c  =/=  d  /\  -.  c  .<_  Z  /\  C  .<_  ( d  .\/  c ) ) )
305, 10, 293jca 1132 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   joincjn 14078   Atomscatm 29453   HLchlt 29540
This theorem is referenced by:  dalem56  29917
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-join 14110  df-lat 14152  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541
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