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Theorem dalemcea 33270
Description: Lemma for dath 33346. 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 33235 . . 3  |-  ( ph  ->  K  e.  OP )
3 dalemc.a . . . 4  |-  A  =  ( Atoms `  K )
41, 3dalemceb 33248 . . 3  |-  ( ph  ->  C  e.  ( Base `  K ) )
51dalemkehl 33233 . . . 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 33263 . . . 4  |-  ( ph  ->  ( P  .\/  S
)  e.  ( LLines `  K ) )
111dalemqea 33237 . . . . 5  |-  ( ph  ->  Q  e.  A )
121dalemtea 33240 . . . . 5  |-  ( ph  ->  T  e.  A )
131, 6, 7, 3, 8, 9dalemqnet 33262 . . . . 5  |-  ( ph  ->  Q  =/=  T )
14 eqid 2462 . . . . . 6  |-  ( LLines `  K )  =  (
LLines `  K )
157, 3, 14llni2 33122 . . . . 5  |-  ( ( ( K  e.  HL  /\  Q  e.  A  /\  T  e.  A )  /\  Q  =/=  T
)  ->  ( Q  .\/  T )  e.  (
LLines `  K ) )
165, 11, 12, 13, 15syl31anc 1279 . . . 4  |-  ( ph  ->  ( Q  .\/  T
)  e.  ( LLines `  K ) )
171, 6, 7, 3, 8, 9dalem1 33269 . . . 4  |-  ( ph  ->  ( P  .\/  S
)  =/=  ( Q 
.\/  T ) )
181dalem-clpjq 33247 . . . . . . . 8  |-  ( ph  ->  -.  C  .<_  ( P 
.\/  Q ) )
191, 7, 3dalempjqeb 33255 . . . . . . . . . . 11  |-  ( ph  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
20 eqid 2462 . . . . . . . . . . . 12  |-  ( Base `  K )  =  (
Base `  K )
21 eqid 2462 . . . . . . . . . . . 12  |-  ( 0.
`  K )  =  ( 0. `  K
)
2220, 6, 21op0le 32797 . . . . . . . . . . 11  |-  ( ( K  e.  OP  /\  ( P  .\/  Q )  e.  ( Base `  K
) )  ->  ( 0. `  K )  .<_  ( P  .\/  Q ) )
232, 19, 22syl2anc 671 . . . . . . . . . 10  |-  ( ph  ->  ( 0. `  K
)  .<_  ( P  .\/  Q ) )
24 breq1 4419 . . . . . . . . . 10  |-  ( C  =  ( 0. `  K )  ->  ( C  .<_  ( P  .\/  Q )  <->  ( 0. `  K )  .<_  ( P 
.\/  Q ) ) )
2523, 24syl5ibrcom 230 . . . . . . . . 9  |-  ( ph  ->  ( C  =  ( 0. `  K )  ->  C  .<_  ( P 
.\/  Q ) ) )
2625necon3bd 2650 . . . . . . . 8  |-  ( ph  ->  ( -.  C  .<_  ( P  .\/  Q )  ->  C  =/=  ( 0. `  K ) ) )
2718, 26mpd 15 . . . . . . 7  |-  ( ph  ->  C  =/=  ( 0.
`  K ) )
28 eqid 2462 . . . . . . . . 9  |-  ( lt
`  K )  =  ( lt `  K
)
2920, 28, 21opltn0 32801 . . . . . . . 8  |-  ( ( K  e.  OP  /\  C  e.  ( Base `  K ) )  -> 
( ( 0. `  K ) ( lt
`  K ) C  <-> 
C  =/=  ( 0.
`  K ) ) )
302, 4, 29syl2anc 671 . . . . . . 7  |-  ( ph  ->  ( ( 0. `  K ) ( lt
`  K ) C  <-> 
C  =/=  ( 0.
`  K ) ) )
3127, 30mpbird 240 . . . . . 6  |-  ( ph  ->  ( 0. `  K
) ( lt `  K ) C )
321dalemclpjs 33244 . . . . . . 7  |-  ( ph  ->  C  .<_  ( P  .\/  S ) )
331dalemclqjt 33245 . . . . . . 7  |-  ( ph  ->  C  .<_  ( Q  .\/  T ) )
341dalemkelat 33234 . . . . . . . 8  |-  ( ph  ->  K  e.  Lat )
351dalempea 33236 . . . . . . . . 9  |-  ( ph  ->  P  e.  A )
361dalemsea 33239 . . . . . . . . 9  |-  ( ph  ->  S  e.  A )
3720, 7, 3hlatjcl 32977 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  P  e.  A  /\  S  e.  A )  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
385, 35, 36, 37syl3anc 1276 . . . . . . . 8  |-  ( ph  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
3920, 7, 3hlatjcl 32977 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  T  e.  A )  ->  ( Q  .\/  T
)  e.  ( Base `  K ) )
405, 11, 12, 39syl3anc 1276 . . . . . . . 8  |-  ( ph  ->  ( Q  .\/  T
)  e.  ( Base `  K ) )
41 eqid 2462 . . . . . . . . 9  |-  ( meet `  K )  =  (
meet `  K )
4220, 6, 41latlem12 16373 . . . . . . . 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 1278 . . . . . . 7  |-  ( ph  ->  ( ( C  .<_  ( P  .\/  S )  /\  C  .<_  ( Q 
.\/  T ) )  <-> 
C  .<_  ( ( P 
.\/  S ) (
meet `  K )
( Q  .\/  T
) ) ) )
4432, 33, 43mpbi2and 937 . . . . . 6  |-  ( ph  ->  C  .<_  ( ( P  .\/  S ) (
meet `  K )
( Q  .\/  T
) ) )
45 opposet 32792 . . . . . . . 8  |-  ( K  e.  OP  ->  K  e.  Poset )
462, 45syl 17 . . . . . . 7  |-  ( ph  ->  K  e.  Poset )
4720, 21op0cl 32795 . . . . . . . 8  |-  ( K  e.  OP  ->  ( 0. `  K )  e.  ( Base `  K
) )
482, 47syl 17 . . . . . . 7  |-  ( ph  ->  ( 0. `  K
)  e.  ( Base `  K ) )
4920, 41latmcl 16347 . . . . . . . 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 1276 . . . . . . 7  |-  ( ph  ->  ( ( P  .\/  S ) ( meet `  K
) ( Q  .\/  T ) )  e.  (
Base `  K )
)
5120, 6, 28pltletr 16266 . . . . . . 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 1278 . . . . . 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 690 . . . . 5  |-  ( ph  ->  ( 0. `  K
) ( lt `  K ) ( ( P  .\/  S ) ( meet `  K
) ( Q  .\/  T ) ) )
5420, 28, 21opltn0 32801 . . . . . 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 671 . . . . 5  |-  ( ph  ->  ( ( 0. `  K ) ( lt
`  K ) ( ( P  .\/  S
) ( meet `  K
) ( Q  .\/  T ) )  <->  ( ( P  .\/  S ) (
meet `  K )
( Q  .\/  T
) )  =/=  ( 0. `  K ) ) )
5653, 55mpbid 215 . . . 4  |-  ( ph  ->  ( ( P  .\/  S ) ( meet `  K
) ( Q  .\/  T ) )  =/=  ( 0. `  K ) )
5741, 21, 3, 142llnmat 33134 . . . 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 1284 . . 3  |-  ( ph  ->  ( ( P  .\/  S ) ( meet `  K
) ( Q  .\/  T ) )  e.  A
)
5920, 6, 21, 3leat2 32905 . . 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 1284 . 2  |-  ( ph  ->  C  =  ( ( P  .\/  S ) ( meet `  K
) ( Q  .\/  T ) ) )
6160, 58eqeltrd 2540 1  |-  ( ph  ->  C  e.  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 375    /\ w3a 991    = wceq 1455    e. wcel 1898    =/= wne 2633   class class class wbr 4416   ` cfv 5601  (class class class)co 6315   Basecbs 15170   lecple 15246   Posetcpo 16234   ltcplt 16235   joincjn 16238   meetcmee 16239   0.cp0 16332   Latclat 16340   OPcops 32783   Atomscatm 32874   HLchlt 32961   LLinesclln 33101   LPlanesclpl 33102
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4529  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-reu 2756  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-iun 4294  df-br 4417  df-opab 4476  df-mpt 4477  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6277  df-ov 6318  df-oprab 6319  df-preset 16222  df-poset 16240  df-plt 16253  df-lub 16269  df-glb 16270  df-join 16271  df-meet 16272  df-p0 16334  df-lat 16341  df-clat 16403  df-oposet 32787  df-ol 32789  df-oml 32790  df-covers 32877  df-ats 32878  df-atl 32909  df-cvlat 32933  df-hlat 32962  df-llines 33108  df-lplanes 33109
This theorem is referenced by:  dalem2  33271  dalem5  33277  dalem-cly  33281  dalem9  33282  dalem19  33292  dalem21  33304  dalem25  33308
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