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Theorem dalem21 33012
Description: Lemma for dath 33054. Show that lines  c d and  P S intersect at an atom. (Contributed by NM, 2-Aug-2012.)
Hypotheses
Ref Expression
dalem.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 ) ) ) ) )
dalem.l  |-  .<_  =  ( le `  K )
dalem.j  |-  .\/  =  ( join `  K )
dalem.a  |-  A  =  ( Atoms `  K )
dalem.ps  |-  ( ps  <->  ( ( c  e.  A  /\  d  e.  A
)  /\  -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
) ) ) )
dalem21.m  |-  ./\  =  ( meet `  K )
dalem21.o  |-  O  =  ( LPlanes `  K )
dalem21.y  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
dalem21.z  |-  Z  =  ( ( S  .\/  T )  .\/  U )
Assertion
Ref Expression
dalem21  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( c  .\/  d )  ./\  ( P  .\/  S ) )  e.  A )

Proof of Theorem dalem21
StepHypRef Expression
1 dalem.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 ) ) ) ) )
21dalemkehl 32941 . . 3  |-  ( ph  ->  K  e.  HL )
323ad2ant1 1026 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  HL )
4 dalem.l . . . 4  |-  .<_  =  ( le `  K )
5 dalem.j . . . 4  |-  .\/  =  ( join `  K )
6 dalem.a . . . 4  |-  A  =  ( Atoms `  K )
7 dalem.ps . . . 4  |-  ( ps  <->  ( ( c  e.  A  /\  d  e.  A
)  /\  -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
) ) ) )
81, 4, 5, 6, 7dalemcjden 33010 . . 3  |-  ( (
ph  /\  ps )  ->  ( c  .\/  d
)  e.  ( LLines `  K ) )
983adant2 1024 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( c  .\/  d
)  e.  ( LLines `  K ) )
10 dalem21.o . . . 4  |-  O  =  ( LPlanes `  K )
11 dalem21.y . . . 4  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
121, 4, 5, 6, 10, 11dalempjsen 32971 . . 3  |-  ( ph  ->  ( P  .\/  S
)  e.  ( LLines `  K ) )
13123ad2ant1 1026 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( P  .\/  S
)  e.  ( LLines `  K ) )
141, 4, 5, 6, 10, 11dalemply 32972 . . . . . . 7  |-  ( ph  ->  P  .<_  Y )
1514adantr 466 . . . . . 6  |-  ( (
ph  /\  Y  =  Z )  ->  P  .<_  Y )
16 dalem21.z . . . . . . 7  |-  Z  =  ( ( S  .\/  T )  .\/  U )
171, 4, 5, 6, 16dalemsly 32973 . . . . . 6  |-  ( (
ph  /\  Y  =  Z )  ->  S  .<_  Y )
181dalemkelat 32942 . . . . . . . 8  |-  ( ph  ->  K  e.  Lat )
191, 6dalempeb 32957 . . . . . . . 8  |-  ( ph  ->  P  e.  ( Base `  K ) )
201, 6dalemseb 32960 . . . . . . . 8  |-  ( ph  ->  S  e.  ( Base `  K ) )
211, 10dalemyeb 32967 . . . . . . . 8  |-  ( ph  ->  Y  e.  ( Base `  K ) )
22 eqid 2420 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
2322, 4, 5latjle12 16260 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( P  e.  ( Base `  K )  /\  S  e.  ( Base `  K )  /\  Y  e.  ( Base `  K
) ) )  -> 
( ( P  .<_  Y  /\  S  .<_  Y )  <-> 
( P  .\/  S
)  .<_  Y ) )
2418, 19, 20, 21, 23syl13anc 1266 . . . . . . 7  |-  ( ph  ->  ( ( P  .<_  Y  /\  S  .<_  Y )  <-> 
( P  .\/  S
)  .<_  Y ) )
2524adantr 466 . . . . . 6  |-  ( (
ph  /\  Y  =  Z )  ->  (
( P  .<_  Y  /\  S  .<_  Y )  <->  ( P  .\/  S )  .<_  Y ) )
2615, 17, 25mpbi2and 929 . . . . 5  |-  ( (
ph  /\  Y  =  Z )  ->  ( P  .\/  S )  .<_  Y )
27263adant3 1025 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( P  .\/  S
)  .<_  Y )
287dalem-ccly 33003 . . . . . . 7  |-  ( ps 
->  -.  c  .<_  Y )
2928adantl 467 . . . . . 6  |-  ( (
ph  /\  ps )  ->  -.  c  .<_  Y )
3018adantr 466 . . . . . . . 8  |-  ( (
ph  /\  ps )  ->  K  e.  Lat )
317, 6dalemcceb 33007 . . . . . . . . 9  |-  ( ps 
->  c  e.  ( Base `  K ) )
3231adantl 467 . . . . . . . 8  |-  ( (
ph  /\  ps )  ->  c  e.  ( Base `  K ) )
337dalemddea 33002 . . . . . . . . . 10  |-  ( ps 
->  d  e.  A
)
3422, 6atbase 32608 . . . . . . . . . 10  |-  ( d  e.  A  ->  d  e.  ( Base `  K
) )
3533, 34syl 17 . . . . . . . . 9  |-  ( ps 
->  d  e.  ( Base `  K ) )
3635adantl 467 . . . . . . . 8  |-  ( (
ph  /\  ps )  ->  d  e.  ( Base `  K ) )
3722, 4, 5latlej1 16258 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  c  e.  ( Base `  K )  /\  d  e.  ( Base `  K
) )  ->  c  .<_  ( c  .\/  d
) )
3830, 32, 36, 37syl3anc 1264 . . . . . . 7  |-  ( (
ph  /\  ps )  ->  c  .<_  ( c  .\/  d ) )
39 eqid 2420 . . . . . . . . . 10  |-  ( LLines `  K )  =  (
LLines `  K )
4022, 39llnbase 32827 . . . . . . . . 9  |-  ( ( c  .\/  d )  e.  ( LLines `  K
)  ->  ( c  .\/  d )  e.  (
Base `  K )
)
418, 40syl 17 . . . . . . . 8  |-  ( (
ph  /\  ps )  ->  ( c  .\/  d
)  e.  ( Base `  K ) )
4221adantr 466 . . . . . . . 8  |-  ( (
ph  /\  ps )  ->  Y  e.  ( Base `  K ) )
4322, 4lattr 16254 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( c  e.  (
Base `  K )  /\  ( c  .\/  d
)  e.  ( Base `  K )  /\  Y  e.  ( Base `  K
) ) )  -> 
( ( c  .<_  ( c  .\/  d
)  /\  ( c  .\/  d )  .<_  Y )  ->  c  .<_  Y ) )
4430, 32, 41, 42, 43syl13anc 1266 . . . . . . 7  |-  ( (
ph  /\  ps )  ->  ( ( c  .<_  ( c  .\/  d
)  /\  ( c  .\/  d )  .<_  Y )  ->  c  .<_  Y ) )
4538, 44mpand 679 . . . . . 6  |-  ( (
ph  /\  ps )  ->  ( ( c  .\/  d )  .<_  Y  -> 
c  .<_  Y ) )
4629, 45mtod 180 . . . . 5  |-  ( (
ph  /\  ps )  ->  -.  ( c  .\/  d )  .<_  Y )
47463adant2 1024 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  ( c  .\/  d
)  .<_  Y )
48 nbrne2 4435 . . . 4  |-  ( ( ( P  .\/  S
)  .<_  Y  /\  -.  ( c  .\/  d
)  .<_  Y )  -> 
( P  .\/  S
)  =/=  ( c 
.\/  d ) )
4927, 47, 48syl2anc 665 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( P  .\/  S
)  =/=  ( c 
.\/  d ) )
5049necomd 2693 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( c  .\/  d
)  =/=  ( P 
.\/  S ) )
51 hlatl 32679 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  AtLat )
522, 51syl 17 . . . . 5  |-  ( ph  ->  K  e.  AtLat )
5352adantr 466 . . . 4  |-  ( (
ph  /\  ps )  ->  K  e.  AtLat )
541dalempea 32944 . . . . . . 7  |-  ( ph  ->  P  e.  A )
551dalemsea 32947 . . . . . . 7  |-  ( ph  ->  S  e.  A )
5622, 5, 6hlatjcl 32685 . . . . . . 7  |-  ( ( K  e.  HL  /\  P  e.  A  /\  S  e.  A )  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
572, 54, 55, 56syl3anc 1264 . . . . . 6  |-  ( ph  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
5857adantr 466 . . . . 5  |-  ( (
ph  /\  ps )  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
59 dalem21.m . . . . . 6  |-  ./\  =  ( meet `  K )
6022, 59latmcl 16250 . . . . 5  |-  ( ( K  e.  Lat  /\  ( c  .\/  d
)  e.  ( Base `  K )  /\  ( P  .\/  S )  e.  ( Base `  K
) )  ->  (
( c  .\/  d
)  ./\  ( P  .\/  S ) )  e.  ( Base `  K
) )
6130, 41, 58, 60syl3anc 1264 . . . 4  |-  ( (
ph  /\  ps )  ->  ( ( c  .\/  d )  ./\  ( P  .\/  S ) )  e.  ( Base `  K
) )
621, 4, 5, 6, 10, 11dalemcea 32978 . . . . 5  |-  ( ph  ->  C  e.  A )
6362adantr 466 . . . 4  |-  ( (
ph  /\  ps )  ->  C  e.  A )
647dalemclccjdd 33006 . . . . . 6  |-  ( ps 
->  C  .<_  ( c 
.\/  d ) )
6564adantl 467 . . . . 5  |-  ( (
ph  /\  ps )  ->  C  .<_  ( c  .\/  d ) )
661dalemclpjs 32952 . . . . . 6  |-  ( ph  ->  C  .<_  ( P  .\/  S ) )
6766adantr 466 . . . . 5  |-  ( (
ph  /\  ps )  ->  C  .<_  ( P  .\/  S ) )
681, 6dalemceb 32956 . . . . . . 7  |-  ( ph  ->  C  e.  ( Base `  K ) )
6968adantr 466 . . . . . 6  |-  ( (
ph  /\  ps )  ->  C  e.  ( Base `  K ) )
7022, 4, 59latlem12 16276 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( C  e.  ( Base `  K )  /\  ( c  .\/  d
)  e.  ( Base `  K )  /\  ( P  .\/  S )  e.  ( Base `  K
) ) )  -> 
( ( C  .<_  ( c  .\/  d )  /\  C  .<_  ( P 
.\/  S ) )  <-> 
C  .<_  ( ( c 
.\/  d )  ./\  ( P  .\/  S ) ) ) )
7130, 69, 41, 58, 70syl13anc 1266 . . . . 5  |-  ( (
ph  /\  ps )  ->  ( ( C  .<_  ( c  .\/  d )  /\  C  .<_  ( P 
.\/  S ) )  <-> 
C  .<_  ( ( c 
.\/  d )  ./\  ( P  .\/  S ) ) ) )
7265, 67, 71mpbi2and 929 . . . 4  |-  ( (
ph  /\  ps )  ->  C  .<_  ( (
c  .\/  d )  ./\  ( P  .\/  S
) ) )
73 eqid 2420 . . . . 5  |-  ( 0.
`  K )  =  ( 0. `  K
)
7422, 4, 73, 6atlen0 32629 . . . 4  |-  ( ( ( K  e.  AtLat  /\  ( ( c  .\/  d )  ./\  ( P  .\/  S ) )  e.  ( Base `  K
)  /\  C  e.  A )  /\  C  .<_  ( ( c  .\/  d )  ./\  ( P  .\/  S ) ) )  ->  ( (
c  .\/  d )  ./\  ( P  .\/  S
) )  =/=  ( 0. `  K ) )
7553, 61, 63, 72, 74syl31anc 1267 . . 3  |-  ( (
ph  /\  ps )  ->  ( ( c  .\/  d )  ./\  ( P  .\/  S ) )  =/=  ( 0. `  K ) )
76753adant2 1024 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( c  .\/  d )  ./\  ( P  .\/  S ) )  =/=  ( 0. `  K ) )
7759, 73, 6, 392llnmat 32842 . 2  |-  ( ( ( K  e.  HL  /\  ( c  .\/  d
)  e.  ( LLines `  K )  /\  ( P  .\/  S )  e.  ( LLines `  K )
)  /\  ( (
c  .\/  d )  =/=  ( P  .\/  S
)  /\  ( (
c  .\/  d )  ./\  ( P  .\/  S
) )  =/=  ( 0. `  K ) ) )  ->  ( (
c  .\/  d )  ./\  ( P  .\/  S
) )  e.  A
)
783, 9, 13, 50, 76, 77syl32anc 1272 1  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( c  .\/  d )  ./\  ( P  .\/  S ) )  e.  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1867    =/= wne 2616   class class class wbr 4417   ` cfv 5592  (class class class)co 6296   Basecbs 15081   lecple 15157   joincjn 16141   meetcmee 16142   0.cp0 16235   Latclat 16243   Atomscatm 32582   AtLatcal 32583   HLchlt 32669   LLinesclln 32809   LPlanesclpl 32810
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-preset 16125  df-poset 16143  df-plt 16156  df-lub 16172  df-glb 16173  df-join 16174  df-meet 16175  df-p0 16237  df-lat 16244  df-clat 16306  df-oposet 32495  df-ol 32497  df-oml 32498  df-covers 32585  df-ats 32586  df-atl 32617  df-cvlat 32641  df-hlat 32670  df-llines 32816  df-lplanes 32817
This theorem is referenced by:  dalem22  33013
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