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Theorem 2dim 30267
Description: Generate a height-3 element (2-dimensional plane) from an atom. (Contributed by NM, 3-May-2012.)
Hypotheses
Ref Expression
2dim.j  |-  .\/  =  ( join `  K )
2dim.c  |-  C  =  (  <o  `  K )
2dim.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
2dim  |-  ( ( K  e.  HL  /\  P  e.  A )  ->  E. q  e.  A  E. r  e.  A  ( P C ( P 
.\/  q )  /\  ( P  .\/  q ) C ( ( P 
.\/  q )  .\/  r ) ) )
Distinct variable groups:    r, q, A    .\/ , q, r    K, q, r    P, q, r
Allowed substitution hints:    C( r, q)

Proof of Theorem 2dim
Dummy variable  s is distinct from all other variables.
StepHypRef Expression
1 2dim.j . . 3  |-  .\/  =  ( join `  K )
2 eqid 2436 . . 3  |-  ( le
`  K )  =  ( le `  K
)
3 2dim.a . . 3  |-  A  =  ( Atoms `  K )
41, 2, 33dim1 30264 . 2  |-  ( ( K  e.  HL  /\  P  e.  A )  ->  E. q  e.  A  E. r  e.  A  E. s  e.  A  ( P  =/=  q  /\  -.  r ( le
`  K ) ( P  .\/  q )  /\  -.  s ( le `  K ) ( ( P  .\/  q )  .\/  r
) ) )
5 df-3an 938 . . . . . . . 8  |-  ( ( P  =/=  q  /\  -.  r ( le `  K ) ( P 
.\/  q )  /\  -.  s ( le `  K ) ( ( P  .\/  q ) 
.\/  r ) )  <-> 
( ( P  =/=  q  /\  -.  r
( le `  K
) ( P  .\/  q ) )  /\  -.  s ( le `  K ) ( ( P  .\/  q ) 
.\/  r ) ) )
65rexbii 2730 . . . . . . 7  |-  ( E. s  e.  A  ( P  =/=  q  /\  -.  r ( le `  K ) ( P 
.\/  q )  /\  -.  s ( le `  K ) ( ( P  .\/  q ) 
.\/  r ) )  <->  E. s  e.  A  ( ( P  =/=  q  /\  -.  r
( le `  K
) ( P  .\/  q ) )  /\  -.  s ( le `  K ) ( ( P  .\/  q ) 
.\/  r ) ) )
7 r19.42v 2862 . . . . . . 7  |-  ( E. s  e.  A  ( ( P  =/=  q  /\  -.  r ( le
`  K ) ( P  .\/  q ) )  /\  -.  s
( le `  K
) ( ( P 
.\/  q )  .\/  r ) )  <->  ( ( P  =/=  q  /\  -.  r ( le `  K ) ( P 
.\/  q ) )  /\  E. s  e.  A  -.  s ( le `  K ) ( ( P  .\/  q )  .\/  r
) ) )
86, 7bitri 241 . . . . . 6  |-  ( E. s  e.  A  ( P  =/=  q  /\  -.  r ( le `  K ) ( P 
.\/  q )  /\  -.  s ( le `  K ) ( ( P  .\/  q ) 
.\/  r ) )  <-> 
( ( P  =/=  q  /\  -.  r
( le `  K
) ( P  .\/  q ) )  /\  E. s  e.  A  -.  s ( le `  K ) ( ( P  .\/  q ) 
.\/  r ) ) )
98simplbi 447 . . . . 5  |-  ( E. s  e.  A  ( P  =/=  q  /\  -.  r ( le `  K ) ( P 
.\/  q )  /\  -.  s ( le `  K ) ( ( P  .\/  q ) 
.\/  r ) )  ->  ( P  =/=  q  /\  -.  r
( le `  K
) ( P  .\/  q ) ) )
10 simplll 735 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  P  e.  A )  /\  q  e.  A )  /\  r  e.  A )  ->  K  e.  HL )
11 hlatl 30158 . . . . . . . . . 10  |-  ( K  e.  HL  ->  K  e.  AtLat )
1210, 11syl 16 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  P  e.  A )  /\  q  e.  A )  /\  r  e.  A )  ->  K  e.  AtLat )
13 simplr 732 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  P  e.  A )  /\  q  e.  A )  /\  r  e.  A )  ->  q  e.  A )
14 simpllr 736 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  P  e.  A )  /\  q  e.  A )  /\  r  e.  A )  ->  P  e.  A )
152, 3atncmp 30110 . . . . . . . . 9  |-  ( ( K  e.  AtLat  /\  q  e.  A  /\  P  e.  A )  ->  ( -.  q ( le `  K ) P  <->  q  =/=  P ) )
1612, 13, 14, 15syl3anc 1184 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  P  e.  A )  /\  q  e.  A )  /\  r  e.  A )  ->  ( -.  q ( le `  K ) P  <->  q  =/=  P ) )
17 necom 2685 . . . . . . . 8  |-  ( q  =/=  P  <->  P  =/=  q )
1816, 17syl6rbb 254 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  P  e.  A )  /\  q  e.  A )  /\  r  e.  A )  ->  ( P  =/=  q  <->  -.  q
( le `  K
) P ) )
19 eqid 2436 . . . . . . . . . 10  |-  ( Base `  K )  =  (
Base `  K )
2019, 3atbase 30087 . . . . . . . . 9  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
2114, 20syl 16 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  P  e.  A )  /\  q  e.  A )  /\  r  e.  A )  ->  P  e.  ( Base `  K
) )
22 2dim.c . . . . . . . . 9  |-  C  =  (  <o  `  K )
2319, 2, 1, 22, 3cvr1 30207 . . . . . . . 8  |-  ( ( K  e.  HL  /\  P  e.  ( Base `  K )  /\  q  e.  A )  ->  ( -.  q ( le `  K ) P  <->  P C
( P  .\/  q
) ) )
2410, 21, 13, 23syl3anc 1184 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  P  e.  A )  /\  q  e.  A )  /\  r  e.  A )  ->  ( -.  q ( le `  K ) P  <->  P C
( P  .\/  q
) ) )
2518, 24bitrd 245 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A )  /\  q  e.  A )  /\  r  e.  A )  ->  ( P  =/=  q  <->  P C
( P  .\/  q
) ) )
2619, 1, 3hlatjcl 30164 . . . . . . . 8  |-  ( ( K  e.  HL  /\  P  e.  A  /\  q  e.  A )  ->  ( P  .\/  q
)  e.  ( Base `  K ) )
2710, 14, 13, 26syl3anc 1184 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  P  e.  A )  /\  q  e.  A )  /\  r  e.  A )  ->  ( P  .\/  q )  e.  ( Base `  K
) )
28 simpr 448 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  P  e.  A )  /\  q  e.  A )  /\  r  e.  A )  ->  r  e.  A )
2919, 2, 1, 22, 3cvr1 30207 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( P  .\/  q )  e.  ( Base `  K
)  /\  r  e.  A )  ->  ( -.  r ( le `  K ) ( P 
.\/  q )  <->  ( P  .\/  q ) C ( ( P  .\/  q
)  .\/  r )
) )
3010, 27, 28, 29syl3anc 1184 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A )  /\  q  e.  A )  /\  r  e.  A )  ->  ( -.  r ( le `  K ) ( P 
.\/  q )  <->  ( P  .\/  q ) C ( ( P  .\/  q
)  .\/  r )
) )
3125, 30anbi12d 692 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A )  /\  q  e.  A )  /\  r  e.  A )  ->  (
( P  =/=  q  /\  -.  r ( le
`  K ) ( P  .\/  q ) )  <->  ( P C ( P  .\/  q
)  /\  ( P  .\/  q ) C ( ( P  .\/  q
)  .\/  r )
) ) )
329, 31syl5ib 211 . . . 4  |-  ( ( ( ( K  e.  HL  /\  P  e.  A )  /\  q  e.  A )  /\  r  e.  A )  ->  ( E. s  e.  A  ( P  =/=  q  /\  -.  r ( le
`  K ) ( P  .\/  q )  /\  -.  s ( le `  K ) ( ( P  .\/  q )  .\/  r
) )  ->  ( P C ( P  .\/  q )  /\  ( P  .\/  q ) C ( ( P  .\/  q )  .\/  r
) ) ) )
3332reximdva 2818 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A )  /\  q  e.  A
)  ->  ( E. r  e.  A  E. s  e.  A  ( P  =/=  q  /\  -.  r ( le `  K ) ( P 
.\/  q )  /\  -.  s ( le `  K ) ( ( P  .\/  q ) 
.\/  r ) )  ->  E. r  e.  A  ( P C ( P 
.\/  q )  /\  ( P  .\/  q ) C ( ( P 
.\/  q )  .\/  r ) ) ) )
3433reximdva 2818 . 2  |-  ( ( K  e.  HL  /\  P  e.  A )  ->  ( E. q  e.  A  E. r  e.  A  E. s  e.  A  ( P  =/=  q  /\  -.  r
( le `  K
) ( P  .\/  q )  /\  -.  s ( le `  K ) ( ( P  .\/  q ) 
.\/  r ) )  ->  E. q  e.  A  E. r  e.  A  ( P C ( P 
.\/  q )  /\  ( P  .\/  q ) C ( ( P 
.\/  q )  .\/  r ) ) ) )
354, 34mpd 15 1  |-  ( ( K  e.  HL  /\  P  e.  A )  ->  E. q  e.  A  E. r  e.  A  ( P C ( P 
.\/  q )  /\  ( P  .\/  q ) C ( ( P 
.\/  q )  .\/  r ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   E.wrex 2706   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   Basecbs 13469   lecple 13536   joincjn 14401    <o ccvr 30060   Atomscatm 30061   AtLatcal 30062   HLchlt 30148
This theorem is referenced by:  1dimN  30268  1cvratex  30270
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-undef 6543  df-riota 6549  df-poset 14403  df-plt 14415  df-lub 14431  df-glb 14432  df-join 14433  df-meet 14434  df-p0 14468  df-lat 14475  df-clat 14537  df-oposet 29974  df-ol 29976  df-oml 29977  df-covers 30064  df-ats 30065  df-atl 30096  df-cvlat 30120  df-hlat 30149
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