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Theorem dalemcea 30358
Description: Lemma for dath 30434. 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 30323 . . 3  |-  ( ph  ->  K  e.  OP )
3 dalemc.a . . . 4  |-  A  =  ( Atoms `  K )
41, 3dalemceb 30336 . . 3  |-  ( ph  ->  C  e.  ( Base `  K ) )
51dalemkehl 30321 . . . 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 30351 . . . 4  |-  ( ph  ->  ( P  .\/  S
)  e.  ( LLines `  K ) )
111dalemqea 30325 . . . . 5  |-  ( ph  ->  Q  e.  A )
121dalemtea 30328 . . . . 5  |-  ( ph  ->  T  e.  A )
131, 6, 7, 3, 8, 9dalemqnet 30350 . . . . 5  |-  ( ph  ->  Q  =/=  T )
14 eqid 2435 . . . . . 6  |-  ( LLines `  K )  =  (
LLines `  K )
157, 3, 14llni2 30210 . . . . 5  |-  ( ( ( K  e.  HL  /\  Q  e.  A  /\  T  e.  A )  /\  Q  =/=  T
)  ->  ( Q  .\/  T )  e.  (
LLines `  K ) )
165, 11, 12, 13, 15syl31anc 1187 . . . 4  |-  ( ph  ->  ( Q  .\/  T
)  e.  ( LLines `  K ) )
171, 6, 7, 3, 8, 9dalem1 30357 . . . 4  |-  ( ph  ->  ( P  .\/  S
)  =/=  ( Q 
.\/  T ) )
181dalem-clpjq 30335 . . . . . . . 8  |-  ( ph  ->  -.  C  .<_  ( P 
.\/  Q ) )
191, 7, 3dalempjqeb 30343 . . . . . . . . . . 11  |-  ( ph  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
20 eqid 2435 . . . . . . . . . . . 12  |-  ( Base `  K )  =  (
Base `  K )
21 eqid 2435 . . . . . . . . . . . 12  |-  ( 0.
`  K )  =  ( 0. `  K
)
2220, 6, 21op0le 29885 . . . . . . . . . . 11  |-  ( ( K  e.  OP  /\  ( P  .\/  Q )  e.  ( Base `  K
) )  ->  ( 0. `  K )  .<_  ( P  .\/  Q ) )
232, 19, 22syl2anc 643 . . . . . . . . . 10  |-  ( ph  ->  ( 0. `  K
)  .<_  ( P  .\/  Q ) )
24 breq1 4207 . . . . . . . . . 10  |-  ( C  =  ( 0. `  K )  ->  ( C  .<_  ( P  .\/  Q )  <->  ( 0. `  K )  .<_  ( P 
.\/  Q ) ) )
2523, 24syl5ibrcom 214 . . . . . . . . 9  |-  ( ph  ->  ( C  =  ( 0. `  K )  ->  C  .<_  ( P 
.\/  Q ) ) )
2625necon3bd 2635 . . . . . . . 8  |-  ( ph  ->  ( -.  C  .<_  ( P  .\/  Q )  ->  C  =/=  ( 0. `  K ) ) )
2718, 26mpd 15 . . . . . . 7  |-  ( ph  ->  C  =/=  ( 0.
`  K ) )
28 eqid 2435 . . . . . . . . 9  |-  ( lt
`  K )  =  ( lt `  K
)
2920, 28, 21opltn0 29889 . . . . . . . 8  |-  ( ( K  e.  OP  /\  C  e.  ( Base `  K ) )  -> 
( ( 0. `  K ) ( lt
`  K ) C  <-> 
C  =/=  ( 0.
`  K ) ) )
302, 4, 29syl2anc 643 . . . . . . 7  |-  ( ph  ->  ( ( 0. `  K ) ( lt
`  K ) C  <-> 
C  =/=  ( 0.
`  K ) ) )
3127, 30mpbird 224 . . . . . 6  |-  ( ph  ->  ( 0. `  K
) ( lt `  K ) C )
321dalemclpjs 30332 . . . . . . 7  |-  ( ph  ->  C  .<_  ( P  .\/  S ) )
331dalemclqjt 30333 . . . . . . 7  |-  ( ph  ->  C  .<_  ( Q  .\/  T ) )
341dalemkelat 30322 . . . . . . . 8  |-  ( ph  ->  K  e.  Lat )
351dalempea 30324 . . . . . . . . 9  |-  ( ph  ->  P  e.  A )
361dalemsea 30327 . . . . . . . . 9  |-  ( ph  ->  S  e.  A )
3720, 7, 3hlatjcl 30065 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  P  e.  A  /\  S  e.  A )  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
385, 35, 36, 37syl3anc 1184 . . . . . . . 8  |-  ( ph  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
3920, 7, 3hlatjcl 30065 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  T  e.  A )  ->  ( Q  .\/  T
)  e.  ( Base `  K ) )
405, 11, 12, 39syl3anc 1184 . . . . . . . 8  |-  ( ph  ->  ( Q  .\/  T
)  e.  ( Base `  K ) )
41 eqid 2435 . . . . . . . . 9  |-  ( meet `  K )  =  (
meet `  K )
4220, 6, 41latlem12 14497 . . . . . . . 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 1186 . . . . . . 7  |-  ( ph  ->  ( ( C  .<_  ( P  .\/  S )  /\  C  .<_  ( Q 
.\/  T ) )  <-> 
C  .<_  ( ( P 
.\/  S ) (
meet `  K )
( Q  .\/  T
) ) ) )
4432, 33, 43mpbi2and 888 . . . . . 6  |-  ( ph  ->  C  .<_  ( ( P  .\/  S ) (
meet `  K )
( Q  .\/  T
) ) )
45 opposet 29881 . . . . . . . 8  |-  ( K  e.  OP  ->  K  e.  Poset )
462, 45syl 16 . . . . . . 7  |-  ( ph  ->  K  e.  Poset )
4720, 21op0cl 29883 . . . . . . . 8  |-  ( K  e.  OP  ->  ( 0. `  K )  e.  ( Base `  K
) )
482, 47syl 16 . . . . . . 7  |-  ( ph  ->  ( 0. `  K
)  e.  ( Base `  K ) )
4920, 41latmcl 14470 . . . . . . . 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 1184 . . . . . . 7  |-  ( ph  ->  ( ( P  .\/  S ) ( meet `  K
) ( Q  .\/  T ) )  e.  (
Base `  K )
)
5120, 6, 28pltletr 14418 . . . . . . 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 1186 . . . . . 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 661 . . . . 5  |-  ( ph  ->  ( 0. `  K
) ( lt `  K ) ( ( P  .\/  S ) ( meet `  K
) ( Q  .\/  T ) ) )
5420, 28, 21opltn0 29889 . . . . . 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 643 . . . . 5  |-  ( ph  ->  ( ( 0. `  K ) ( lt
`  K ) ( ( P  .\/  S
) ( meet `  K
) ( Q  .\/  T ) )  <->  ( ( P  .\/  S ) (
meet `  K )
( Q  .\/  T
) )  =/=  ( 0. `  K ) ) )
5653, 55mpbid 202 . . . 4  |-  ( ph  ->  ( ( P  .\/  S ) ( meet `  K
) ( Q  .\/  T ) )  =/=  ( 0. `  K ) )
5741, 21, 3, 142llnmat 30222 . . . 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 1192 . . 3  |-  ( ph  ->  ( ( P  .\/  S ) ( meet `  K
) ( Q  .\/  T ) )  e.  A
)
5920, 6, 21, 3leat2 29993 . . 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 1192 . 2  |-  ( ph  ->  C  =  ( ( P  .\/  S ) ( meet `  K
) ( Q  .\/  T ) ) )
6160, 58eqeltrd 2509 1  |-  ( ph  ->  C  e.  A )
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 13459   lecple 13526   Posetcpo 14387   ltcplt 14388   joincjn 14391   meetcmee 14392   0.cp0 14456   Latclat 14464   OPcops 29871   Atomscatm 29962   HLchlt 30049   LLinesclln 30189   LPlanesclpl 30190
This theorem is referenced by:  dalem2  30359  dalem5  30365  dalem-cly  30369  dalem9  30370  dalem19  30380  dalem21  30392  dalem25  30396
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 14393  df-plt 14405  df-lub 14421  df-glb 14422  df-join 14423  df-meet 14424  df-p0 14458  df-lat 14465  df-clat 14527  df-oposet 29875  df-ol 29877  df-oml 29878  df-covers 29965  df-ats 29966  df-atl 29997  df-cvlat 30021  df-hlat 30050  df-llines 30196  df-lplanes 30197
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