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Theorem dalemcea 35506
Description: Lemma for dath 35582. 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 35471 . . 3  |-  ( ph  ->  K  e.  OP )
3 dalemc.a . . . 4  |-  A  =  ( Atoms `  K )
41, 3dalemceb 35484 . . 3  |-  ( ph  ->  C  e.  ( Base `  K ) )
51dalemkehl 35469 . . . 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 35499 . . . 4  |-  ( ph  ->  ( P  .\/  S
)  e.  ( LLines `  K ) )
111dalemqea 35473 . . . . 5  |-  ( ph  ->  Q  e.  A )
121dalemtea 35476 . . . . 5  |-  ( ph  ->  T  e.  A )
131, 6, 7, 3, 8, 9dalemqnet 35498 . . . . 5  |-  ( ph  ->  Q  =/=  T )
14 eqid 2457 . . . . . 6  |-  ( LLines `  K )  =  (
LLines `  K )
157, 3, 14llni2 35358 . . . . 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 35505 . . . 4  |-  ( ph  ->  ( P  .\/  S
)  =/=  ( Q 
.\/  T ) )
181dalem-clpjq 35483 . . . . . . . 8  |-  ( ph  ->  -.  C  .<_  ( P 
.\/  Q ) )
191, 7, 3dalempjqeb 35491 . . . . . . . . . . 11  |-  ( ph  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
20 eqid 2457 . . . . . . . . . . . 12  |-  ( Base `  K )  =  (
Base `  K )
21 eqid 2457 . . . . . . . . . . . 12  |-  ( 0.
`  K )  =  ( 0. `  K
)
2220, 6, 21op0le 35033 . . . . . . . . . . 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 4459 . . . . . . . . . 10  |-  ( C  =  ( 0. `  K )  ->  ( C  .<_  ( P  .\/  Q )  <->  ( 0. `  K )  .<_  ( P 
.\/  Q ) ) )
2523, 24syl5ibrcom 222 . . . . . . . . 9  |-  ( ph  ->  ( C  =  ( 0. `  K )  ->  C  .<_  ( P 
.\/  Q ) ) )
2625necon3bd 2669 . . . . . . . 8  |-  ( ph  ->  ( -.  C  .<_  ( P  .\/  Q )  ->  C  =/=  ( 0. `  K ) ) )
2718, 26mpd 15 . . . . . . 7  |-  ( ph  ->  C  =/=  ( 0.
`  K ) )
28 eqid 2457 . . . . . . . . 9  |-  ( lt
`  K )  =  ( lt `  K
)
2920, 28, 21opltn0 35037 . . . . . . . 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 35480 . . . . . . 7  |-  ( ph  ->  C  .<_  ( P  .\/  S ) )
331dalemclqjt 35481 . . . . . . 7  |-  ( ph  ->  C  .<_  ( Q  .\/  T ) )
341dalemkelat 35470 . . . . . . . 8  |-  ( ph  ->  K  e.  Lat )
351dalempea 35472 . . . . . . . . 9  |-  ( ph  ->  P  e.  A )
361dalemsea 35475 . . . . . . . . 9  |-  ( ph  ->  S  e.  A )
3720, 7, 3hlatjcl 35213 . . . . . . . . 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 35213 . . . . . . . . 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 2457 . . . . . . . . 9  |-  ( meet `  K )  =  (
meet `  K )
4220, 6, 41latlem12 15835 . . . . . . . 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 921 . . . . . 6  |-  ( ph  ->  C  .<_  ( ( P  .\/  S ) (
meet `  K )
( Q  .\/  T
) ) )
45 opposet 35028 . . . . . . . 8  |-  ( K  e.  OP  ->  K  e.  Poset )
462, 45syl 16 . . . . . . 7  |-  ( ph  ->  K  e.  Poset )
4720, 21op0cl 35031 . . . . . . . 8  |-  ( K  e.  OP  ->  ( 0. `  K )  e.  ( Base `  K
) )
482, 47syl 16 . . . . . . 7  |-  ( ph  ->  ( 0. `  K
)  e.  ( Base `  K ) )
4920, 41latmcl 15809 . . . . . . . 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 15728 . . . . . . 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 35037 . . . . . 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 35370 . . . 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 35141 . . 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 2545 1  |-  ( ph  ->  C  e.  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    =/= wne 2652   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   Basecbs 14644   lecple 14719   Posetcpo 15696   ltcplt 15697   joincjn 15700   meetcmee 15701   0.cp0 15794   Latclat 15802   OPcops 35019   Atomscatm 35110   HLchlt 35197   LLinesclln 35337   LPlanesclpl 35338
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  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 6258  df-ov 6299  df-oprab 6300  df-preset 15684  df-poset 15702  df-plt 15715  df-lub 15731  df-glb 15732  df-join 15733  df-meet 15734  df-p0 15796  df-lat 15803  df-clat 15865  df-oposet 35023  df-ol 35025  df-oml 35026  df-covers 35113  df-ats 35114  df-atl 35145  df-cvlat 35169  df-hlat 35198  df-llines 35344  df-lplanes 35345
This theorem is referenced by:  dalem2  35507  dalem5  35513  dalem-cly  35517  dalem9  35518  dalem19  35528  dalem21  35540  dalem25  35544
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