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Theorem 2dim 35591
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 2454 . . 3  |-  ( le
`  K )  =  ( le `  K
)
3 2dim.a . . 3  |-  A  =  ( Atoms `  K )
41, 2, 33dim1 35588 . 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 973 . . . . . . . 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 2956 . . . . . . 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 3009 . . . . . . 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 249 . . . . . 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 458 . . . . 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 757 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  P  e.  A )  /\  q  e.  A )  /\  r  e.  A )  ->  K  e.  HL )
11 hlatl 35482 . . . . . . . . . 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 753 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  P  e.  A )  /\  q  e.  A )  /\  r  e.  A )  ->  q  e.  A )
14 simpllr 758 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  P  e.  A )  /\  q  e.  A )  /\  r  e.  A )  ->  P  e.  A )
152, 3atncmp 35434 . . . . . . . . 9  |-  ( ( K  e.  AtLat  /\  q  e.  A  /\  P  e.  A )  ->  ( -.  q ( le `  K ) P  <->  q  =/=  P ) )
1612, 13, 14, 15syl3anc 1226 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  P  e.  A )  /\  q  e.  A )  /\  r  e.  A )  ->  ( -.  q ( le `  K ) P  <->  q  =/=  P ) )
17 necom 2723 . . . . . . . 8  |-  ( q  =/=  P  <->  P  =/=  q )
1816, 17syl6rbb 262 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  P  e.  A )  /\  q  e.  A )  /\  r  e.  A )  ->  ( P  =/=  q  <->  -.  q
( le `  K
) P ) )
19 eqid 2454 . . . . . . . . . 10  |-  ( Base `  K )  =  (
Base `  K )
2019, 3atbase 35411 . . . . . . . . 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 35531 . . . . . . . 8  |-  ( ( K  e.  HL  /\  P  e.  ( Base `  K )  /\  q  e.  A )  ->  ( -.  q ( le `  K ) P  <->  P C
( P  .\/  q
) ) )
2410, 21, 13, 23syl3anc 1226 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  P  e.  A )  /\  q  e.  A )  /\  r  e.  A )  ->  ( -.  q ( le `  K ) P  <->  P C
( P  .\/  q
) ) )
2518, 24bitrd 253 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A )  /\  q  e.  A )  /\  r  e.  A )  ->  ( P  =/=  q  <->  P C
( P  .\/  q
) ) )
2619, 1, 3hlatjcl 35488 . . . . . . . 8  |-  ( ( K  e.  HL  /\  P  e.  A  /\  q  e.  A )  ->  ( P  .\/  q
)  e.  ( Base `  K ) )
2710, 14, 13, 26syl3anc 1226 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  P  e.  A )  /\  q  e.  A )  /\  r  e.  A )  ->  ( P  .\/  q )  e.  ( Base `  K
) )
28 simpr 459 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  P  e.  A )  /\  q  e.  A )  /\  r  e.  A )  ->  r  e.  A )
2919, 2, 1, 22, 3cvr1 35531 . . . . . . 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 1226 . . . . . 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 708 . . . . 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 219 . . . 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 2929 . . 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 2929 . 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   E.wrex 2805   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   Basecbs 14716   lecple 14791   joincjn 15772    <o ccvr 35384   Atomscatm 35385   AtLatcal 35386   HLchlt 35472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-preset 15756  df-poset 15774  df-plt 15787  df-lub 15803  df-glb 15804  df-join 15805  df-meet 15806  df-p0 15868  df-p1 15869  df-lat 15875  df-clat 15937  df-oposet 35298  df-ol 35300  df-oml 35301  df-covers 35388  df-ats 35389  df-atl 35420  df-cvlat 35444  df-hlat 35473
This theorem is referenced by:  1dimN  35592  1cvratex  35594
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