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Theorem dalem28 30424
Description: Lemma for dath 30460. Lemma dalem27 30423 expressed differently. (Contributed by NM, 4-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
) ) ) )
dalem23.m  |-  ./\  =  ( meet `  K )
dalem23.o  |-  O  =  ( LPlanes `  K )
dalem23.y  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
dalem23.z  |-  Z  =  ( ( S  .\/  T )  .\/  U )
dalem23.g  |-  G  =  ( ( c  .\/  P )  ./\  ( d  .\/  S ) )
Assertion
Ref Expression
dalem28  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  P  .<_  ( G  .\/  c ) )

Proof of Theorem dalem28
StepHypRef Expression
1 dalem.ph . . 3  |-  ( 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 ) ) ) ) )
2 dalem.l . . 3  |-  .<_  =  ( le `  K )
3 dalem.j . . 3  |-  .\/  =  ( join `  K )
4 dalem.a . . 3  |-  A  =  ( Atoms `  K )
5 dalem.ps . . 3  |-  ( ps  <->  ( ( c  e.  A  /\  d  e.  A
)  /\  -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
) ) ) )
6 dalem23.m . . 3  |-  ./\  =  ( meet `  K )
7 dalem23.o . . 3  |-  O  =  ( LPlanes `  K )
8 dalem23.y . . 3  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
9 dalem23.z . . 3  |-  Z  =  ( ( S  .\/  T )  .\/  U )
10 dalem23.g . . 3  |-  G  =  ( ( c  .\/  P )  ./\  ( d  .\/  S ) )
111, 2, 3, 4, 5, 6, 7, 8, 9, 10dalem27 30423 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  .<_  ( G  .\/  P ) )
121dalemkehl 30347 . . . 4  |-  ( ph  ->  K  e.  HL )
13123ad2ant1 978 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  HL )
145dalemccea 30407 . . . 4  |-  ( ps 
->  c  e.  A
)
15143ad2ant3 980 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  e.  A )
161dalempea 30350 . . . 4  |-  ( ph  ->  P  e.  A )
17163ad2ant1 978 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  P  e.  A )
181, 2, 3, 4, 5, 6, 7, 8, 9, 10dalem23 30420 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  G  e.  A )
191, 2, 3, 4, 5, 6, 7, 8, 9, 10dalem25 30422 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  =/=  G )
202, 3, 4hlatexch1 30119 . . 3  |-  ( ( K  e.  HL  /\  ( c  e.  A  /\  P  e.  A  /\  G  e.  A
)  /\  c  =/=  G )  ->  ( c  .<_  ( G  .\/  P
)  ->  P  .<_  ( G  .\/  c ) ) )
2113, 15, 17, 18, 19, 20syl131anc 1197 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( c  .<_  ( G 
.\/  P )  ->  P  .<_  ( G  .\/  c ) ) )
2211, 21mpd 15 1  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  P  .<_  ( G  .\/  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 2598   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   Basecbs 13461   lecple 13528   joincjn 14393   meetcmee 14394   Atomscatm 29988   HLchlt 30075   LPlanesclpl 30216
This theorem is referenced by:  dalem33  30429  dalem38  30434  dalem44  30440
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-poset 14395  df-plt 14407  df-lub 14423  df-glb 14424  df-join 14425  df-meet 14426  df-p0 14460  df-lat 14467  df-clat 14529  df-oposet 29901  df-ol 29903  df-oml 29904  df-covers 29991  df-ats 29992  df-atl 30023  df-cvlat 30047  df-hlat 30076  df-llines 30222  df-lplanes 30223
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