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Theorem 2dim 30281
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 2296 . . 3  |-  ( le
`  K )  =  ( le `  K
)
3 2dim.a . . 3  |-  A  =  ( Atoms `  K )
41, 2, 33dim1 30278 . 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 936 . . . . . . . 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 2581 . . . . . . 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 2707 . . . . . . 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 240 . . . . . 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 446 . . . . 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 734 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  P  e.  A )  /\  q  e.  A )  /\  r  e.  A )  ->  K  e.  HL )
11 hlatl 30172 . . . . . . . . . 10  |-  ( K  e.  HL  ->  K  e.  AtLat )
1210, 11syl 15 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  P  e.  A )  /\  q  e.  A )  /\  r  e.  A )  ->  K  e.  AtLat )
13 simplr 731 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  P  e.  A )  /\  q  e.  A )  /\  r  e.  A )  ->  q  e.  A )
14 simpllr 735 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  P  e.  A )  /\  q  e.  A )  /\  r  e.  A )  ->  P  e.  A )
152, 3atncmp 30124 . . . . . . . . 9  |-  ( ( K  e.  AtLat  /\  q  e.  A  /\  P  e.  A )  ->  ( -.  q ( le `  K ) P  <->  q  =/=  P ) )
1612, 13, 14, 15syl3anc 1182 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  P  e.  A )  /\  q  e.  A )  /\  r  e.  A )  ->  ( -.  q ( le `  K ) P  <->  q  =/=  P ) )
17 necom 2540 . . . . . . . 8  |-  ( q  =/=  P  <->  P  =/=  q )
1816, 17syl6rbb 253 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  P  e.  A )  /\  q  e.  A )  /\  r  e.  A )  ->  ( P  =/=  q  <->  -.  q
( le `  K
) P ) )
19 eqid 2296 . . . . . . . . . 10  |-  ( Base `  K )  =  (
Base `  K )
2019, 3atbase 30101 . . . . . . . . 9  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
2114, 20syl 15 . . . . . . . 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 30221 . . . . . . . 8  |-  ( ( K  e.  HL  /\  P  e.  ( Base `  K )  /\  q  e.  A )  ->  ( -.  q ( le `  K ) P  <->  P C
( P  .\/  q
) ) )
2410, 21, 13, 23syl3anc 1182 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  P  e.  A )  /\  q  e.  A )  /\  r  e.  A )  ->  ( -.  q ( le `  K ) P  <->  P C
( P  .\/  q
) ) )
2518, 24bitrd 244 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A )  /\  q  e.  A )  /\  r  e.  A )  ->  ( P  =/=  q  <->  P C
( P  .\/  q
) ) )
2619, 1, 3hlatjcl 30178 . . . . . . . 8  |-  ( ( K  e.  HL  /\  P  e.  A  /\  q  e.  A )  ->  ( P  .\/  q
)  e.  ( Base `  K ) )
2710, 14, 13, 26syl3anc 1182 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  P  e.  A )  /\  q  e.  A )  /\  r  e.  A )  ->  ( P  .\/  q )  e.  ( Base `  K
) )
28 simpr 447 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  P  e.  A )  /\  q  e.  A )  /\  r  e.  A )  ->  r  e.  A )
2919, 2, 1, 22, 3cvr1 30221 . . . . . . 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 1182 . . . . . 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 691 . . . . 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 210 . . . 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 2668 . . 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 2668 . 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 14 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   joincjn 14094    <o ccvr 30074   Atomscatm 30075   AtLatcal 30076   HLchlt 30162
This theorem is referenced by:  1dimN  30282  1cvratex  30284
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163
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