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Theorem dalemcea 34857
Description: Lemma for dath 34933. Frequently-used utility lemma. Here we show that  C must be an atom. This is an assumption in most presentations of Desargue's theorem; instead, we assume only the  C is a lattice element, in order to make later substitutions for  C easier. (Contributed by NM, 23-Sep-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 )
dalem1.o  |-  O  =  ( LPlanes `  K )
dalem1.y  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
Assertion
Ref Expression
dalemcea  |-  ( ph  ->  C  e.  A )

Proof of Theorem dalemcea
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 ) ) ) ) )
21dalemkeop 34822 . . 3  |-  ( ph  ->  K  e.  OP )
3 dalemc.a . . . 4  |-  A  =  ( Atoms `  K )
41, 3dalemceb 34835 . . 3  |-  ( ph  ->  C  e.  ( Base `  K ) )
51dalemkehl 34820 . . . 4  |-  ( ph  ->  K  e.  HL )
6 dalemc.l . . . . 5  |-  .<_  =  ( le `  K )
7 dalemc.j . . . . 5  |-  .\/  =  ( join `  K )
8 dalem1.o . . . . 5  |-  O  =  ( LPlanes `  K )
9 dalem1.y . . . . 5  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
101, 6, 7, 3, 8, 9dalempjsen 34850 . . . 4  |-  ( ph  ->  ( P  .\/  S
)  e.  ( LLines `  K ) )
111dalemqea 34824 . . . . 5  |-  ( ph  ->  Q  e.  A )
121dalemtea 34827 . . . . 5  |-  ( ph  ->  T  e.  A )
131, 6, 7, 3, 8, 9dalemqnet 34849 . . . . 5  |-  ( ph  ->  Q  =/=  T )
14 eqid 2467 . . . . . 6  |-  ( LLines `  K )  =  (
LLines `  K )
157, 3, 14llni2 34709 . . . . 5  |-  ( ( ( K  e.  HL  /\  Q  e.  A  /\  T  e.  A )  /\  Q  =/=  T
)  ->  ( Q  .\/  T )  e.  (
LLines `  K ) )
165, 11, 12, 13, 15syl31anc 1231 . . . 4  |-  ( ph  ->  ( Q  .\/  T
)  e.  ( LLines `  K ) )
171, 6, 7, 3, 8, 9dalem1 34856 . . . 4  |-  ( ph  ->  ( P  .\/  S
)  =/=  ( Q 
.\/  T ) )
181dalem-clpjq 34834 . . . . . . . 8  |-  ( ph  ->  -.  C  .<_  ( P 
.\/  Q ) )
191, 7, 3dalempjqeb 34842 . . . . . . . . . . 11  |-  ( ph  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
20 eqid 2467 . . . . . . . . . . . 12  |-  ( Base `  K )  =  (
Base `  K )
21 eqid 2467 . . . . . . . . . . . 12  |-  ( 0.
`  K )  =  ( 0. `  K
)
2220, 6, 21op0le 34384 . . . . . . . . . . 11  |-  ( ( K  e.  OP  /\  ( P  .\/  Q )  e.  ( Base `  K
) )  ->  ( 0. `  K )  .<_  ( P  .\/  Q ) )
232, 19, 22syl2anc 661 . . . . . . . . . 10  |-  ( ph  ->  ( 0. `  K
)  .<_  ( P  .\/  Q ) )
24 breq1 4456 . . . . . . . . . 10  |-  ( C  =  ( 0. `  K )  ->  ( C  .<_  ( P  .\/  Q )  <->  ( 0. `  K )  .<_  ( P 
.\/  Q ) ) )
2523, 24syl5ibrcom 222 . . . . . . . . 9  |-  ( ph  ->  ( C  =  ( 0. `  K )  ->  C  .<_  ( P 
.\/  Q ) ) )
2625necon3bd 2679 . . . . . . . 8  |-  ( ph  ->  ( -.  C  .<_  ( P  .\/  Q )  ->  C  =/=  ( 0. `  K ) ) )
2718, 26mpd 15 . . . . . . 7  |-  ( ph  ->  C  =/=  ( 0.
`  K ) )
28 eqid 2467 . . . . . . . . 9  |-  ( lt
`  K )  =  ( lt `  K
)
2920, 28, 21opltn0 34388 . . . . . . . 8  |-  ( ( K  e.  OP  /\  C  e.  ( Base `  K ) )  -> 
( ( 0. `  K ) ( lt
`  K ) C  <-> 
C  =/=  ( 0.
`  K ) ) )
302, 4, 29syl2anc 661 . . . . . . 7  |-  ( ph  ->  ( ( 0. `  K ) ( lt
`  K ) C  <-> 
C  =/=  ( 0.
`  K ) ) )
3127, 30mpbird 232 . . . . . 6  |-  ( ph  ->  ( 0. `  K
) ( lt `  K ) C )
321dalemclpjs 34831 . . . . . . 7  |-  ( ph  ->  C  .<_  ( P  .\/  S ) )
331dalemclqjt 34832 . . . . . . 7  |-  ( ph  ->  C  .<_  ( Q  .\/  T ) )
341dalemkelat 34821 . . . . . . . 8  |-  ( ph  ->  K  e.  Lat )
351dalempea 34823 . . . . . . . . 9  |-  ( ph  ->  P  e.  A )
361dalemsea 34826 . . . . . . . . 9  |-  ( ph  ->  S  e.  A )
3720, 7, 3hlatjcl 34564 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  P  e.  A  /\  S  e.  A )  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
385, 35, 36, 37syl3anc 1228 . . . . . . . 8  |-  ( ph  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
3920, 7, 3hlatjcl 34564 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  T  e.  A )  ->  ( Q  .\/  T
)  e.  ( Base `  K ) )
405, 11, 12, 39syl3anc 1228 . . . . . . . 8  |-  ( ph  ->  ( Q  .\/  T
)  e.  ( Base `  K ) )
41 eqid 2467 . . . . . . . . 9  |-  ( meet `  K )  =  (
meet `  K )
4220, 6, 41latlem12 15582 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( C  e.  ( Base `  K )  /\  ( P  .\/  S )  e.  ( Base `  K
)  /\  ( Q  .\/  T )  e.  (
Base `  K )
) )  ->  (
( C  .<_  ( P 
.\/  S )  /\  C  .<_  ( Q  .\/  T ) )  <->  C  .<_  ( ( P  .\/  S
) ( meet `  K
) ( Q  .\/  T ) ) ) )
4334, 4, 38, 40, 42syl13anc 1230 . . . . . . 7  |-  ( ph  ->  ( ( C  .<_  ( P  .\/  S )  /\  C  .<_  ( Q 
.\/  T ) )  <-> 
C  .<_  ( ( P 
.\/  S ) (
meet `  K )
( Q  .\/  T
) ) ) )
4432, 33, 43mpbi2and 919 . . . . . 6  |-  ( ph  ->  C  .<_  ( ( P  .\/  S ) (
meet `  K )
( Q  .\/  T
) ) )
45 opposet 34379 . . . . . . . 8  |-  ( K  e.  OP  ->  K  e.  Poset )
462, 45syl 16 . . . . . . 7  |-  ( ph  ->  K  e.  Poset )
4720, 21op0cl 34382 . . . . . . . 8  |-  ( K  e.  OP  ->  ( 0. `  K )  e.  ( Base `  K
) )
482, 47syl 16 . . . . . . 7  |-  ( ph  ->  ( 0. `  K
)  e.  ( Base `  K ) )
4920, 41latmcl 15556 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( P  .\/  S )  e.  ( Base `  K
)  /\  ( Q  .\/  T )  e.  (
Base `  K )
)  ->  ( ( P  .\/  S ) (
meet `  K )
( Q  .\/  T
) )  e.  (
Base `  K )
)
5034, 38, 40, 49syl3anc 1228 . . . . . . 7  |-  ( ph  ->  ( ( P  .\/  S ) ( meet `  K
) ( Q  .\/  T ) )  e.  (
Base `  K )
)
5120, 6, 28pltletr 15475 . . . . . . 7  |-  ( ( K  e.  Poset  /\  (
( 0. `  K
)  e.  ( Base `  K )  /\  C  e.  ( Base `  K
)  /\  ( ( P  .\/  S ) (
meet `  K )
( Q  .\/  T
) )  e.  (
Base `  K )
) )  ->  (
( ( 0. `  K ) ( lt
`  K ) C  /\  C  .<_  ( ( P  .\/  S ) ( meet `  K
) ( Q  .\/  T ) ) )  -> 
( 0. `  K
) ( lt `  K ) ( ( P  .\/  S ) ( meet `  K
) ( Q  .\/  T ) ) ) )
5246, 48, 4, 50, 51syl13anc 1230 . . . . . 6  |-  ( ph  ->  ( ( ( 0.
`  K ) ( lt `  K ) C  /\  C  .<_  ( ( P  .\/  S
) ( meet `  K
) ( Q  .\/  T ) ) )  -> 
( 0. `  K
) ( lt `  K ) ( ( P  .\/  S ) ( meet `  K
) ( Q  .\/  T ) ) ) )
5331, 44, 52mp2and 679 . . . . 5  |-  ( ph  ->  ( 0. `  K
) ( lt `  K ) ( ( P  .\/  S ) ( meet `  K
) ( Q  .\/  T ) ) )
5420, 28, 21opltn0 34388 . . . . . 6  |-  ( ( K  e.  OP  /\  ( ( P  .\/  S ) ( meet `  K
) ( Q  .\/  T ) )  e.  (
Base `  K )
)  ->  ( ( 0. `  K ) ( lt `  K ) ( ( P  .\/  S ) ( meet `  K
) ( Q  .\/  T ) )  <->  ( ( P  .\/  S ) (
meet `  K )
( Q  .\/  T
) )  =/=  ( 0. `  K ) ) )
552, 50, 54syl2anc 661 . . . . 5  |-  ( ph  ->  ( ( 0. `  K ) ( lt
`  K ) ( ( P  .\/  S
) ( meet `  K
) ( Q  .\/  T ) )  <->  ( ( P  .\/  S ) (
meet `  K )
( Q  .\/  T
) )  =/=  ( 0. `  K ) ) )
5653, 55mpbid 210 . . . 4  |-  ( ph  ->  ( ( P  .\/  S ) ( meet `  K
) ( Q  .\/  T ) )  =/=  ( 0. `  K ) )
5741, 21, 3, 142llnmat 34721 . . . 4  |-  ( ( ( K  e.  HL  /\  ( P  .\/  S
)  e.  ( LLines `  K )  /\  ( Q  .\/  T )  e.  ( LLines `  K )
)  /\  ( ( P  .\/  S )  =/=  ( Q  .\/  T
)  /\  ( ( P  .\/  S ) (
meet `  K )
( Q  .\/  T
) )  =/=  ( 0. `  K ) ) )  ->  ( ( P  .\/  S ) (
meet `  K )
( Q  .\/  T
) )  e.  A
)
585, 10, 16, 17, 56, 57syl32anc 1236 . . 3  |-  ( ph  ->  ( ( P  .\/  S ) ( meet `  K
) ( Q  .\/  T ) )  e.  A
)
5920, 6, 21, 3leat2 34492 . . 3  |-  ( ( ( K  e.  OP  /\  C  e.  ( Base `  K )  /\  (
( P  .\/  S
) ( meet `  K
) ( Q  .\/  T ) )  e.  A
)  /\  ( C  =/=  ( 0. `  K
)  /\  C  .<_  ( ( P  .\/  S
) ( meet `  K
) ( Q  .\/  T ) ) ) )  ->  C  =  ( ( P  .\/  S
) ( meet `  K
) ( Q  .\/  T ) ) )
602, 4, 58, 27, 44, 59syl32anc 1236 . 2  |-  ( ph  ->  C  =  ( ( P  .\/  S ) ( meet `  K
) ( Q  .\/  T ) ) )
6160, 58eqeltrd 2555 1  |-  ( ph  ->  C  e.  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4453   ` cfv 5594  (class class class)co 6295   Basecbs 14507   lecple 14579   Posetcpo 15444   ltcplt 15445   joincjn 15448   meetcmee 15449   0.cp0 15541   Latclat 15549   OPcops 34370   Atomscatm 34461   HLchlt 34548   LLinesclln 34688   LPlanesclpl 34689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-poset 15450  df-plt 15462  df-lub 15478  df-glb 15479  df-join 15480  df-meet 15481  df-p0 15543  df-lat 15550  df-clat 15612  df-oposet 34374  df-ol 34376  df-oml 34377  df-covers 34464  df-ats 34465  df-atl 34496  df-cvlat 34520  df-hlat 34549  df-llines 34695  df-lplanes 34696
This theorem is referenced by:  dalem2  34858  dalem5  34864  dalem-cly  34868  dalem9  34869  dalem19  34879  dalem21  34891  dalem25  34895
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