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Theorem 2dim 29964
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 2412 . . 3  |-  ( le
`  K )  =  ( le `  K
)
3 2dim.a . . 3  |-  A  =  ( Atoms `  K )
41, 2, 33dim1 29961 . 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 2699 . . . . . . 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 2830 . . . . . . 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 29855 . . . . . . . . . 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 29807 . . . . . . . . 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 2656 . . . . . . . 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 2412 . . . . . . . . . 10  |-  ( Base `  K )  =  (
Base `  K )
2019, 3atbase 29784 . . . . . . . . 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 29904 . . . . . . . 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 29861 . . . . . . . 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 29904 . . . . . . 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 2786 . . 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 2786 . 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 1649    e. wcel 1721    =/= wne 2575   E.wrex 2675   class class class wbr 4180   ` cfv 5421  (class class class)co 6048   Basecbs 13432   lecple 13499   joincjn 14364    <o ccvr 29757   Atomscatm 29758   AtLatcal 29759   HLchlt 29845
This theorem is referenced by:  1dimN  29965  1cvratex  29967
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  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 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-undef 6510  df-riota 6516  df-poset 14366  df-plt 14378  df-lub 14394  df-glb 14395  df-join 14396  df-meet 14397  df-p0 14431  df-lat 14438  df-clat 14500  df-oposet 29671  df-ol 29673  df-oml 29674  df-covers 29761  df-ats 29762  df-atl 29793  df-cvlat 29817  df-hlat 29846
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