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Theorem 2llnmat 29713
Description: Two intersecting lines intersect at an atom. (Contributed by NM, 30-Apr-2012.)
Hypotheses
Ref Expression
2llnmat.m  |-  ./\  =  ( meet `  K )
2llnmat.z  |-  .0.  =  ( 0. `  K )
2llnmat.a  |-  A  =  ( Atoms `  K )
2llnmat.n  |-  N  =  ( LLines `  K )
Assertion
Ref Expression
2llnmat  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  Y  e.  N )  /\  ( X  =/=  Y  /\  ( X  ./\  Y
)  =/=  .0.  )
)  ->  ( X  ./\ 
Y )  e.  A
)

Proof of Theorem 2llnmat
Dummy variable  p is distinct from all other variables.
StepHypRef Expression
1 simpl1 958 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  Y  e.  N )  /\  ( X  =/=  Y  /\  ( X  ./\  Y
)  =/=  .0.  )
)  ->  K  e.  HL )
2 hlatl 29550 . . . . 5  |-  ( K  e.  HL  ->  K  e.  AtLat )
31, 2syl 15 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  Y  e.  N )  /\  ( X  =/=  Y  /\  ( X  ./\  Y
)  =/=  .0.  )
)  ->  K  e.  AtLat
)
4 hllat 29553 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
51, 4syl 15 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  Y  e.  N )  /\  ( X  =/=  Y  /\  ( X  ./\  Y
)  =/=  .0.  )
)  ->  K  e.  Lat )
6 simpl2 959 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  Y  e.  N )  /\  ( X  =/=  Y  /\  ( X  ./\  Y
)  =/=  .0.  )
)  ->  X  e.  N )
7 eqid 2283 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
8 2llnmat.n . . . . . . 7  |-  N  =  ( LLines `  K )
97, 8llnbase 29698 . . . . . 6  |-  ( X  e.  N  ->  X  e.  ( Base `  K
) )
106, 9syl 15 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  Y  e.  N )  /\  ( X  =/=  Y  /\  ( X  ./\  Y
)  =/=  .0.  )
)  ->  X  e.  ( Base `  K )
)
11 simpl3 960 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  Y  e.  N )  /\  ( X  =/=  Y  /\  ( X  ./\  Y
)  =/=  .0.  )
)  ->  Y  e.  N )
127, 8llnbase 29698 . . . . . 6  |-  ( Y  e.  N  ->  Y  e.  ( Base `  K
) )
1311, 12syl 15 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  Y  e.  N )  /\  ( X  =/=  Y  /\  ( X  ./\  Y
)  =/=  .0.  )
)  ->  Y  e.  ( Base `  K )
)
14 2llnmat.m . . . . . 6  |-  ./\  =  ( meet `  K )
157, 14latmcl 14157 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  ( Base `  K )  /\  Y  e.  ( Base `  K
) )  ->  ( X  ./\  Y )  e.  ( Base `  K
) )
165, 10, 13, 15syl3anc 1182 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  Y  e.  N )  /\  ( X  =/=  Y  /\  ( X  ./\  Y
)  =/=  .0.  )
)  ->  ( X  ./\ 
Y )  e.  (
Base `  K )
)
17 simprr 733 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  Y  e.  N )  /\  ( X  =/=  Y  /\  ( X  ./\  Y
)  =/=  .0.  )
)  ->  ( X  ./\ 
Y )  =/=  .0.  )
18 eqid 2283 . . . . 5  |-  ( le
`  K )  =  ( le `  K
)
19 2llnmat.z . . . . 5  |-  .0.  =  ( 0. `  K )
20 2llnmat.a . . . . 5  |-  A  =  ( Atoms `  K )
217, 18, 19, 20atlex 29506 . . . 4  |-  ( ( K  e.  AtLat  /\  ( X  ./\  Y )  e.  ( Base `  K
)  /\  ( X  ./\ 
Y )  =/=  .0.  )  ->  E. p  e.  A  p ( le `  K ) ( X 
./\  Y ) )
223, 16, 17, 21syl3anc 1182 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  Y  e.  N )  /\  ( X  =/=  Y  /\  ( X  ./\  Y
)  =/=  .0.  )
)  ->  E. p  e.  A  p ( le `  K ) ( X  ./\  Y )
)
23 simp1rl 1020 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  N  /\  Y  e.  N )  /\  ( X  =/=  Y  /\  ( X  ./\  Y )  =/= 
.0.  ) )  /\  p  e.  A  /\  p ( le `  K ) ( X 
./\  Y ) )  ->  X  =/=  Y
)
24 simp1l 979 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  X  e.  N  /\  Y  e.  N )  /\  ( X  =/=  Y  /\  ( X  ./\  Y )  =/= 
.0.  ) )  /\  p  e.  A  /\  p ( le `  K ) ( X 
./\  Y ) )  ->  ( K  e.  HL  /\  X  e.  N  /\  Y  e.  N ) )
2518, 8llncmp 29711 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  X  e.  N  /\  Y  e.  N )  ->  ( X ( le
`  K ) Y  <-> 
X  =  Y ) )
2624, 25syl 15 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  X  e.  N  /\  Y  e.  N )  /\  ( X  =/=  Y  /\  ( X  ./\  Y )  =/= 
.0.  ) )  /\  p  e.  A  /\  p ( le `  K ) ( X 
./\  Y ) )  ->  ( X ( le `  K ) Y  <->  X  =  Y
) )
27 simp1l1 1048 . . . . . . . . . . 11  |-  ( ( ( ( K  e.  HL  /\  X  e.  N  /\  Y  e.  N )  /\  ( X  =/=  Y  /\  ( X  ./\  Y )  =/= 
.0.  ) )  /\  p  e.  A  /\  p ( le `  K ) ( X 
./\  Y ) )  ->  K  e.  HL )
2827, 4syl 15 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  X  e.  N  /\  Y  e.  N )  /\  ( X  =/=  Y  /\  ( X  ./\  Y )  =/= 
.0.  ) )  /\  p  e.  A  /\  p ( le `  K ) ( X 
./\  Y ) )  ->  K  e.  Lat )
29 simp1l2 1049 . . . . . . . . . . 11  |-  ( ( ( ( K  e.  HL  /\  X  e.  N  /\  Y  e.  N )  /\  ( X  =/=  Y  /\  ( X  ./\  Y )  =/= 
.0.  ) )  /\  p  e.  A  /\  p ( le `  K ) ( X 
./\  Y ) )  ->  X  e.  N
)
3029, 9syl 15 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  X  e.  N  /\  Y  e.  N )  /\  ( X  =/=  Y  /\  ( X  ./\  Y )  =/= 
.0.  ) )  /\  p  e.  A  /\  p ( le `  K ) ( X 
./\  Y ) )  ->  X  e.  (
Base `  K )
)
31 simp1l3 1050 . . . . . . . . . . 11  |-  ( ( ( ( K  e.  HL  /\  X  e.  N  /\  Y  e.  N )  /\  ( X  =/=  Y  /\  ( X  ./\  Y )  =/= 
.0.  ) )  /\  p  e.  A  /\  p ( le `  K ) ( X 
./\  Y ) )  ->  Y  e.  N
)
3231, 12syl 15 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  X  e.  N  /\  Y  e.  N )  /\  ( X  =/=  Y  /\  ( X  ./\  Y )  =/= 
.0.  ) )  /\  p  e.  A  /\  p ( le `  K ) ( X 
./\  Y ) )  ->  Y  e.  (
Base `  K )
)
337, 18, 14latleeqm1 14185 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  X  e.  ( Base `  K )  /\  Y  e.  ( Base `  K
) )  ->  ( X ( le `  K ) Y  <->  ( X  ./\ 
Y )  =  X ) )
3428, 30, 32, 33syl3anc 1182 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  X  e.  N  /\  Y  e.  N )  /\  ( X  =/=  Y  /\  ( X  ./\  Y )  =/= 
.0.  ) )  /\  p  e.  A  /\  p ( le `  K ) ( X 
./\  Y ) )  ->  ( X ( le `  K ) Y  <->  ( X  ./\  Y )  =  X ) )
3526, 34bitr3d 246 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  N  /\  Y  e.  N )  /\  ( X  =/=  Y  /\  ( X  ./\  Y )  =/= 
.0.  ) )  /\  p  e.  A  /\  p ( le `  K ) ( X 
./\  Y ) )  ->  ( X  =  Y  <->  ( X  ./\  Y )  =  X ) )
3635necon3bid 2481 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  N  /\  Y  e.  N )  /\  ( X  =/=  Y  /\  ( X  ./\  Y )  =/= 
.0.  ) )  /\  p  e.  A  /\  p ( le `  K ) ( X 
./\  Y ) )  ->  ( X  =/= 
Y  <->  ( X  ./\  Y )  =/=  X ) )
3723, 36mpbid 201 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  e.  N  /\  Y  e.  N )  /\  ( X  =/=  Y  /\  ( X  ./\  Y )  =/= 
.0.  ) )  /\  p  e.  A  /\  p ( le `  K ) ( X 
./\  Y ) )  ->  ( X  ./\  Y )  =/=  X )
38 simp3 957 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  X  e.  N  /\  Y  e.  N )  /\  ( X  =/=  Y  /\  ( X  ./\  Y )  =/= 
.0.  ) )  /\  p  e.  A  /\  p ( le `  K ) ( X 
./\  Y ) )  ->  p ( le
`  K ) ( X  ./\  Y )
)
397, 18, 14latmle1 14182 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  X  e.  ( Base `  K )  /\  Y  e.  ( Base `  K
) )  ->  ( X  ./\  Y ) ( le `  K ) X )
4028, 30, 32, 39syl3anc 1182 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  X  e.  N  /\  Y  e.  N )  /\  ( X  =/=  Y  /\  ( X  ./\  Y )  =/= 
.0.  ) )  /\  p  e.  A  /\  p ( le `  K ) ( X 
./\  Y ) )  ->  ( X  ./\  Y ) ( le `  K ) X )
41 hlpos 29555 . . . . . . . . . . 11  |-  ( K  e.  HL  ->  K  e.  Poset )
4227, 41syl 15 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  X  e.  N  /\  Y  e.  N )  /\  ( X  =/=  Y  /\  ( X  ./\  Y )  =/= 
.0.  ) )  /\  p  e.  A  /\  p ( le `  K ) ( X 
./\  Y ) )  ->  K  e.  Poset )
437, 20atbase 29479 . . . . . . . . . . 11  |-  ( p  e.  A  ->  p  e.  ( Base `  K
) )
44433ad2ant2 977 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  X  e.  N  /\  Y  e.  N )  /\  ( X  =/=  Y  /\  ( X  ./\  Y )  =/= 
.0.  ) )  /\  p  e.  A  /\  p ( le `  K ) ( X 
./\  Y ) )  ->  p  e.  (
Base `  K )
)
4528, 30, 32, 15syl3anc 1182 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  X  e.  N  /\  Y  e.  N )  /\  ( X  =/=  Y  /\  ( X  ./\  Y )  =/= 
.0.  ) )  /\  p  e.  A  /\  p ( le `  K ) ( X 
./\  Y ) )  ->  ( X  ./\  Y )  e.  ( Base `  K ) )
46 simp2 956 . . . . . . . . . . 11  |-  ( ( ( ( K  e.  HL  /\  X  e.  N  /\  Y  e.  N )  /\  ( X  =/=  Y  /\  ( X  ./\  Y )  =/= 
.0.  ) )  /\  p  e.  A  /\  p ( le `  K ) ( X 
./\  Y ) )  ->  p  e.  A
)
477, 18, 28, 44, 45, 30, 38, 40lattrd 14164 . . . . . . . . . . 11  |-  ( ( ( ( K  e.  HL  /\  X  e.  N  /\  Y  e.  N )  /\  ( X  =/=  Y  /\  ( X  ./\  Y )  =/= 
.0.  ) )  /\  p  e.  A  /\  p ( le `  K ) ( X 
./\  Y ) )  ->  p ( le
`  K ) X )
48 eqid 2283 . . . . . . . . . . . 12  |-  (  <o  `  K )  =  ( 
<o  `  K )
4918, 48, 20, 8atcvrlln2 29708 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  p  e.  A  /\  X  e.  N )  /\  p ( le `  K ) X )  ->  p (  <o  `  K ) X )
5027, 46, 29, 47, 49syl31anc 1185 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  X  e.  N  /\  Y  e.  N )  /\  ( X  =/=  Y  /\  ( X  ./\  Y )  =/= 
.0.  ) )  /\  p  e.  A  /\  p ( le `  K ) ( X 
./\  Y ) )  ->  p (  <o  `  K ) X )
517, 18, 48cvrnbtwn4 29469 . . . . . . . . . 10  |-  ( ( K  e.  Poset  /\  (
p  e.  ( Base `  K )  /\  X  e.  ( Base `  K
)  /\  ( X  ./\ 
Y )  e.  (
Base `  K )
)  /\  p (  <o  `  K ) X )  ->  ( (
p ( le `  K ) ( X 
./\  Y )  /\  ( X  ./\  Y ) ( le `  K
) X )  <->  ( p  =  ( X  ./\  Y )  \/  ( X 
./\  Y )  =  X ) ) )
5242, 44, 30, 45, 50, 51syl131anc 1195 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  X  e.  N  /\  Y  e.  N )  /\  ( X  =/=  Y  /\  ( X  ./\  Y )  =/= 
.0.  ) )  /\  p  e.  A  /\  p ( le `  K ) ( X 
./\  Y ) )  ->  ( ( p ( le `  K
) ( X  ./\  Y )  /\  ( X 
./\  Y ) ( le `  K ) X )  <->  ( p  =  ( X  ./\  Y )  \/  ( X 
./\  Y )  =  X ) ) )
5338, 40, 52mpbi2and 887 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  N  /\  Y  e.  N )  /\  ( X  =/=  Y  /\  ( X  ./\  Y )  =/= 
.0.  ) )  /\  p  e.  A  /\  p ( le `  K ) ( X 
./\  Y ) )  ->  ( p  =  ( X  ./\  Y
)  \/  ( X 
./\  Y )  =  X ) )
54 neor 2530 . . . . . . . 8  |-  ( ( p  =  ( X 
./\  Y )  \/  ( X  ./\  Y
)  =  X )  <-> 
( p  =/=  ( X  ./\  Y )  -> 
( X  ./\  Y
)  =  X ) )
5553, 54sylib 188 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  N  /\  Y  e.  N )  /\  ( X  =/=  Y  /\  ( X  ./\  Y )  =/= 
.0.  ) )  /\  p  e.  A  /\  p ( le `  K ) ( X 
./\  Y ) )  ->  ( p  =/=  ( X  ./\  Y
)  ->  ( X  ./\ 
Y )  =  X ) )
5655necon1d 2515 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  e.  N  /\  Y  e.  N )  /\  ( X  =/=  Y  /\  ( X  ./\  Y )  =/= 
.0.  ) )  /\  p  e.  A  /\  p ( le `  K ) ( X 
./\  Y ) )  ->  ( ( X 
./\  Y )  =/= 
X  ->  p  =  ( X  ./\  Y ) ) )
5737, 56mpd 14 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  X  e.  N  /\  Y  e.  N )  /\  ( X  =/=  Y  /\  ( X  ./\  Y )  =/= 
.0.  ) )  /\  p  e.  A  /\  p ( le `  K ) ( X 
./\  Y ) )  ->  p  =  ( X  ./\  Y )
)
58573exp 1150 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  Y  e.  N )  /\  ( X  =/=  Y  /\  ( X  ./\  Y
)  =/=  .0.  )
)  ->  ( p  e.  A  ->  ( p ( le `  K
) ( X  ./\  Y )  ->  p  =  ( X  ./\  Y ) ) ) )
5958reximdvai 2653 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  Y  e.  N )  /\  ( X  =/=  Y  /\  ( X  ./\  Y
)  =/=  .0.  )
)  ->  ( E. p  e.  A  p
( le `  K
) ( X  ./\  Y )  ->  E. p  e.  A  p  =  ( X  ./\  Y ) ) )
6022, 59mpd 14 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  Y  e.  N )  /\  ( X  =/=  Y  /\  ( X  ./\  Y
)  =/=  .0.  )
)  ->  E. p  e.  A  p  =  ( X  ./\  Y ) )
61 risset 2590 . 2  |-  ( ( X  ./\  Y )  e.  A  <->  E. p  e.  A  p  =  ( X  ./\ 
Y ) )
6260, 61sylibr 203 1  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  Y  e.  N )  /\  ( X  =/=  Y  /\  ( X  ./\  Y
)  =/=  .0.  )
)  ->  ( X  ./\ 
Y )  e.  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   Posetcpo 14074   meetcmee 14079   0.cp0 14143   Latclat 14151    <o ccvr 29452   Atomscatm 29453   AtLatcal 29454   HLchlt 29540   LLinesclln 29680
This theorem is referenced by:  2at0mat0  29714  ps-2c  29717  2llnmeqat  29760  dalemcea  29849  dalem2  29850  dalem21  29883  dalem54  29915  cdlemc5  30384
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
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