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Theorem dalem28 33647
Description: Lemma for dath 33683. Lemma dalem27 33646 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 33646 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  .<_  ( G  .\/  P ) )
121dalemkehl 33570 . . . 4  |-  ( ph  ->  K  e.  HL )
13123ad2ant1 1009 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  HL )
145dalemccea 33630 . . . 4  |-  ( ps 
->  c  e.  A
)
15143ad2ant3 1011 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  e.  A )
161dalempea 33573 . . . 4  |-  ( ph  ->  P  e.  A )
17163ad2ant1 1009 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  P  e.  A )
181, 2, 3, 4, 5, 6, 7, 8, 9, 10dalem23 33643 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  G  e.  A )
191, 2, 3, 4, 5, 6, 7, 8, 9, 10dalem25 33645 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  =/=  G )
202, 3, 4hlatexch1 33342 . . 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 1232 . 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2642   class class class wbr 4387   ` cfv 5513  (class class class)co 6187   Basecbs 14273   lecple 14344   joincjn 15213   meetcmee 15214   Atomscatm 33211   HLchlt 33298   LPlanesclpl 33439
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 1952  ax-ext 2430  ax-rep 4498  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469
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 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4187  df-iun 4268  df-br 4388  df-opab 4446  df-mpt 4447  df-id 4731  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6148  df-ov 6190  df-oprab 6191  df-poset 15215  df-plt 15227  df-lub 15243  df-glb 15244  df-join 15245  df-meet 15246  df-p0 15308  df-lat 15315  df-clat 15377  df-oposet 33124  df-ol 33126  df-oml 33127  df-covers 33214  df-ats 33215  df-atl 33246  df-cvlat 33270  df-hlat 33299  df-llines 33445  df-lplanes 33446
This theorem is referenced by:  dalem33  33652  dalem38  33657  dalem44  33663
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