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Theorem dalempnes 35115
Description: Lemma for dath 35200. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)
Hypotheses
Ref Expression
dalema.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 ) ) ) ) )
dalemc.l  |-  .<_  =  ( le `  K )
dalemc.j  |-  .\/  =  ( join `  K )
dalemc.a  |-  A  =  ( Atoms `  K )
dalempnes.o  |-  O  =  ( LPlanes `  K )
dalempnes.y  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
Assertion
Ref Expression
dalempnes  |-  ( ph  ->  P  =/=  S )

Proof of Theorem dalempnes
StepHypRef Expression
1 dalema.ph . . . 4  |-  ( 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 ) ) ) ) )
21dalemkelat 35088 . . 3  |-  ( ph  ->  K  e.  Lat )
3 dalemc.a . . . 4  |-  A  =  ( Atoms `  K )
41, 3dalemceb 35102 . . 3  |-  ( ph  ->  C  e.  ( Base `  K ) )
51, 3dalemseb 35106 . . 3  |-  ( ph  ->  S  e.  ( Base `  K ) )
61, 3dalemteb 35107 . . 3  |-  ( ph  ->  T  e.  ( Base `  K ) )
7 simp321 1147 . . . 4  |-  ( ( ( ( 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 ) ) ) )  ->  -.  C  .<_  ( S  .\/  T ) )
81, 7sylbi 195 . . 3  |-  ( ph  ->  -.  C  .<_  ( S 
.\/  T ) )
9 eqid 2443 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
10 dalemc.l . . . 4  |-  .<_  =  ( le `  K )
11 dalemc.j . . . 4  |-  .\/  =  ( join `  K )
129, 10, 11latnlej2l 15576 . . 3  |-  ( ( K  e.  Lat  /\  ( C  e.  ( Base `  K )  /\  S  e.  ( Base `  K )  /\  T  e.  ( Base `  K
) )  /\  -.  C  .<_  ( S  .\/  T ) )  ->  -.  C  .<_  S )
132, 4, 5, 6, 8, 12syl131anc 1242 . 2  |-  ( ph  ->  -.  C  .<_  S )
141dalemclpjs 35098 . . . . 5  |-  ( ph  ->  C  .<_  ( P  .\/  S ) )
15 oveq1 6288 . . . . . 6  |-  ( P  =  S  ->  ( P  .\/  S )  =  ( S  .\/  S
) )
1615breq2d 4449 . . . . 5  |-  ( P  =  S  ->  ( C  .<_  ( P  .\/  S )  <->  C  .<_  ( S 
.\/  S ) ) )
1714, 16syl5ibcom 220 . . . 4  |-  ( ph  ->  ( P  =  S  ->  C  .<_  ( S 
.\/  S ) ) )
181dalemkehl 35087 . . . . . 6  |-  ( ph  ->  K  e.  HL )
191dalemsea 35093 . . . . . 6  |-  ( ph  ->  S  e.  A )
2011, 3hlatjidm 34833 . . . . . 6  |-  ( ( K  e.  HL  /\  S  e.  A )  ->  ( S  .\/  S
)  =  S )
2118, 19, 20syl2anc 661 . . . . 5  |-  ( ph  ->  ( S  .\/  S
)  =  S )
2221breq2d 4449 . . . 4  |-  ( ph  ->  ( C  .<_  ( S 
.\/  S )  <->  C  .<_  S ) )
2317, 22sylibd 214 . . 3  |-  ( ph  ->  ( P  =  S  ->  C  .<_  S ) )
2423necon3bd 2655 . 2  |-  ( ph  ->  ( -.  C  .<_  S  ->  P  =/=  S
) )
2513, 24mpd 15 1  |-  ( ph  ->  P  =/=  S )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804    =/= wne 2638   class class class wbr 4437   ` cfv 5578  (class class class)co 6281   Basecbs 14509   lecple 14581   joincjn 15447   Latclat 15549   Atomscatm 34728   HLchlt 34815   LPlanesclpl 34956
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-preset 15431  df-poset 15449  df-lub 15478  df-glb 15479  df-join 15480  df-meet 15481  df-lat 15550  df-ats 34732  df-atl 34763  df-cvlat 34787  df-hlat 34816
This theorem is referenced by:  dalempjsen  35117  dalem24  35161
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