Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dalemcea Unicode version

Theorem dalemcea 29849
Description: Lemma for dath 29925. 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 29814 . . 3  |-  ( ph  ->  K  e.  OP )
3 dalemc.a . . . 4  |-  A  =  ( Atoms `  K )
41, 3dalemceb 29827 . . 3  |-  ( ph  ->  C  e.  ( Base `  K ) )
51dalemkehl 29812 . . . 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 29842 . . . 4  |-  ( ph  ->  ( P  .\/  S
)  e.  ( LLines `  K ) )
111dalemqea 29816 . . . . 5  |-  ( ph  ->  Q  e.  A )
121dalemtea 29819 . . . . 5  |-  ( ph  ->  T  e.  A )
131, 6, 7, 3, 8, 9dalemqnet 29841 . . . . 5  |-  ( ph  ->  Q  =/=  T )
14 eqid 2283 . . . . . 6  |-  ( LLines `  K )  =  (
LLines `  K )
157, 3, 14llni2 29701 . . . . 5  |-  ( ( ( K  e.  HL  /\  Q  e.  A  /\  T  e.  A )  /\  Q  =/=  T
)  ->  ( Q  .\/  T )  e.  (
LLines `  K ) )
165, 11, 12, 13, 15syl31anc 1185 . . . 4  |-  ( ph  ->  ( Q  .\/  T
)  e.  ( LLines `  K ) )
171, 6, 7, 3, 8, 9dalem1 29848 . . . 4  |-  ( ph  ->  ( P  .\/  S
)  =/=  ( Q 
.\/  T ) )
181dalem-clpjq 29826 . . . . . . . 8  |-  ( ph  ->  -.  C  .<_  ( P 
.\/  Q ) )
191, 7, 3dalempjqeb 29834 . . . . . . . . . . 11  |-  ( ph  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
20 eqid 2283 . . . . . . . . . . . 12  |-  ( Base `  K )  =  (
Base `  K )
21 eqid 2283 . . . . . . . . . . . 12  |-  ( 0.
`  K )  =  ( 0. `  K
)
2220, 6, 21op0le 29376 . . . . . . . . . . 11  |-  ( ( K  e.  OP  /\  ( P  .\/  Q )  e.  ( Base `  K
) )  ->  ( 0. `  K )  .<_  ( P  .\/  Q ) )
232, 19, 22syl2anc 642 . . . . . . . . . 10  |-  ( ph  ->  ( 0. `  K
)  .<_  ( P  .\/  Q ) )
24 breq1 4026 . . . . . . . . . 10  |-  ( C  =  ( 0. `  K )  ->  ( C  .<_  ( P  .\/  Q )  <->  ( 0. `  K )  .<_  ( P 
.\/  Q ) ) )
2523, 24syl5ibrcom 213 . . . . . . . . 9  |-  ( ph  ->  ( C  =  ( 0. `  K )  ->  C  .<_  ( P 
.\/  Q ) ) )
2625necon3bd 2483 . . . . . . . 8  |-  ( ph  ->  ( -.  C  .<_  ( P  .\/  Q )  ->  C  =/=  ( 0. `  K ) ) )
2718, 26mpd 14 . . . . . . 7  |-  ( ph  ->  C  =/=  ( 0.
`  K ) )
28 eqid 2283 . . . . . . . . 9  |-  ( lt
`  K )  =  ( lt `  K
)
2920, 28, 21opltn0 29380 . . . . . . . 8  |-  ( ( K  e.  OP  /\  C  e.  ( Base `  K ) )  -> 
( ( 0. `  K ) ( lt
`  K ) C  <-> 
C  =/=  ( 0.
`  K ) ) )
302, 4, 29syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( ( 0. `  K ) ( lt
`  K ) C  <-> 
C  =/=  ( 0.
`  K ) ) )
3127, 30mpbird 223 . . . . . 6  |-  ( ph  ->  ( 0. `  K
) ( lt `  K ) C )
321dalemclpjs 29823 . . . . . . 7  |-  ( ph  ->  C  .<_  ( P  .\/  S ) )
331dalemclqjt 29824 . . . . . . 7  |-  ( ph  ->  C  .<_  ( Q  .\/  T ) )
341dalemkelat 29813 . . . . . . . 8  |-  ( ph  ->  K  e.  Lat )
351dalempea 29815 . . . . . . . . 9  |-  ( ph  ->  P  e.  A )
361dalemsea 29818 . . . . . . . . 9  |-  ( ph  ->  S  e.  A )
3720, 7, 3hlatjcl 29556 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  P  e.  A  /\  S  e.  A )  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
385, 35, 36, 37syl3anc 1182 . . . . . . . 8  |-  ( ph  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
3920, 7, 3hlatjcl 29556 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  T  e.  A )  ->  ( Q  .\/  T
)  e.  ( Base `  K ) )
405, 11, 12, 39syl3anc 1182 . . . . . . . 8  |-  ( ph  ->  ( Q  .\/  T
)  e.  ( Base `  K ) )
41 eqid 2283 . . . . . . . . 9  |-  ( meet `  K )  =  (
meet `  K )
4220, 6, 41latlem12 14184 . . . . . . . 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 1184 . . . . . . 7  |-  ( ph  ->  ( ( C  .<_  ( P  .\/  S )  /\  C  .<_  ( Q 
.\/  T ) )  <-> 
C  .<_  ( ( P 
.\/  S ) (
meet `  K )
( Q  .\/  T
) ) ) )
4432, 33, 43mpbi2and 887 . . . . . 6  |-  ( ph  ->  C  .<_  ( ( P  .\/  S ) (
meet `  K )
( Q  .\/  T
) ) )
45 opposet 29372 . . . . . . . 8  |-  ( K  e.  OP  ->  K  e.  Poset )
462, 45syl 15 . . . . . . 7  |-  ( ph  ->  K  e.  Poset )
4720, 21op0cl 29374 . . . . . . . 8  |-  ( K  e.  OP  ->  ( 0. `  K )  e.  ( Base `  K
) )
482, 47syl 15 . . . . . . 7  |-  ( ph  ->  ( 0. `  K
)  e.  ( Base `  K ) )
4920, 41latmcl 14157 . . . . . . . 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 1182 . . . . . . 7  |-  ( ph  ->  ( ( P  .\/  S ) ( meet `  K
) ( Q  .\/  T ) )  e.  (
Base `  K )
)
5120, 6, 28pltletr 14105 . . . . . . 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 1184 . . . . . 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 660 . . . . 5  |-  ( ph  ->  ( 0. `  K
) ( lt `  K ) ( ( P  .\/  S ) ( meet `  K
) ( Q  .\/  T ) ) )
5420, 28, 21opltn0 29380 . . . . . 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 642 . . . . 5  |-  ( ph  ->  ( ( 0. `  K ) ( lt
`  K ) ( ( P  .\/  S
) ( meet `  K
) ( Q  .\/  T ) )  <->  ( ( P  .\/  S ) (
meet `  K )
( Q  .\/  T
) )  =/=  ( 0. `  K ) ) )
5653, 55mpbid 201 . . . 4  |-  ( ph  ->  ( ( P  .\/  S ) ( meet `  K
) ( Q  .\/  T ) )  =/=  ( 0. `  K ) )
5741, 21, 3, 142llnmat 29713 . . . 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 1190 . . 3  |-  ( ph  ->  ( ( P  .\/  S ) ( meet `  K
) ( Q  .\/  T ) )  e.  A
)
5920, 6, 21, 3leat2 29484 . . 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 1190 . 2  |-  ( ph  ->  C  =  ( ( P  .\/  S ) ( meet `  K
) ( Q  .\/  T ) ) )
6160, 58eqeltrd 2357 1  |-  ( ph  ->  C  e.  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   Posetcpo 14074   ltcplt 14075   joincjn 14078   meetcmee 14079   0.cp0 14143   Latclat 14151   OPcops 29362   Atomscatm 29453   HLchlt 29540   LLinesclln 29680   LPlanesclpl 29681
This theorem is referenced by:  dalem2  29850  dalem5  29856  dalem-cly  29860  dalem9  29861  dalem19  29871  dalem21  29883  dalem25  29887
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-llines 29687  df-lplanes 29688
  Copyright terms: Public domain W3C validator