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Theorem 3dim0 29939
Description: There exists a 3-dimensional (height-4) element i.e. a volume. (Contributed by NM, 25-Jul-2012.)
Hypotheses
Ref Expression
3dim0.j  |-  .\/  =  ( join `  K )
3dim0.l  |-  .<_  =  ( le `  K )
3dim0.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
3dim0  |-  ( K  e.  HL  ->  E. p  e.  A  E. q  e.  A  E. r  e.  A  E. s  e.  A  ( p  =/=  q  /\  -.  r  .<_  ( p  .\/  q
)  /\  -.  s  .<_  ( ( p  .\/  q )  .\/  r
) ) )
Distinct variable groups:    q, p, r, s, A    .\/ , r, s    K, p, q, r, s
Allowed substitution hints:    .\/ ( q, p)    .<_ ( s, r, q, p)

Proof of Theorem 3dim0
StepHypRef Expression
1 3dim0.j . . 3  |-  .\/  =  ( join `  K )
2 eqid 2404 . . 3  |-  (  <o  `  K )  =  ( 
<o  `  K )
3 3dim0.a . . 3  |-  A  =  ( Atoms `  K )
41, 2, 3athgt 29938 . 2  |-  ( K  e.  HL  ->  E. p  e.  A  E. q  e.  A  ( p
(  <o  `  K )
( p  .\/  q
)  /\  E. r  e.  A  ( (
p  .\/  q )
(  <o  `  K )
( ( p  .\/  q )  .\/  r
)  /\  E. s  e.  A  ( (
p  .\/  q )  .\/  r ) (  <o  `  K ) ( ( ( p  .\/  q
)  .\/  r )  .\/  s ) ) ) )
5 df-3an 938 . . . . . . . . . 10  |-  ( ( p  =/=  q  /\  -.  r  .<_  ( p 
.\/  q )  /\  -.  s  .<_  ( ( p  .\/  q ) 
.\/  r ) )  <-> 
( ( p  =/=  q  /\  -.  r  .<_  ( p  .\/  q
) )  /\  -.  s  .<_  ( ( p 
.\/  q )  .\/  r ) ) )
6 simpll1 996 . . . . . . . . . . . . 13  |-  ( ( ( ( K  e.  HL  /\  p  e.  A  /\  q  e.  A )  /\  r  e.  A )  /\  s  e.  A )  ->  K  e.  HL )
7 eqid 2404 . . . . . . . . . . . . . . 15  |-  ( Base `  K )  =  (
Base `  K )
87, 1, 3hlatjcl 29849 . . . . . . . . . . . . . 14  |-  ( ( K  e.  HL  /\  p  e.  A  /\  q  e.  A )  ->  ( p  .\/  q
)  e.  ( Base `  K ) )
98ad2antrr 707 . . . . . . . . . . . . 13  |-  ( ( ( ( K  e.  HL  /\  p  e.  A  /\  q  e.  A )  /\  r  e.  A )  /\  s  e.  A )  ->  (
p  .\/  q )  e.  ( Base `  K
) )
10 simplr 732 . . . . . . . . . . . . 13  |-  ( ( ( ( K  e.  HL  /\  p  e.  A  /\  q  e.  A )  /\  r  e.  A )  /\  s  e.  A )  ->  r  e.  A )
11 3dim0.l . . . . . . . . . . . . . 14  |-  .<_  =  ( le `  K )
127, 11, 1, 2, 3cvr1 29892 . . . . . . . . . . . . 13  |-  ( ( K  e.  HL  /\  ( p  .\/  q )  e.  ( Base `  K
)  /\  r  e.  A )  ->  ( -.  r  .<_  ( p 
.\/  q )  <->  ( p  .\/  q ) (  <o  `  K ) ( ( p  .\/  q ) 
.\/  r ) ) )
136, 9, 10, 12syl3anc 1184 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  p  e.  A  /\  q  e.  A )  /\  r  e.  A )  /\  s  e.  A )  ->  ( -.  r  .<_  ( p 
.\/  q )  <->  ( p  .\/  q ) (  <o  `  K ) ( ( p  .\/  q ) 
.\/  r ) ) )
1413anbi2d 685 . . . . . . . . . . 11  |-  ( ( ( ( K  e.  HL  /\  p  e.  A  /\  q  e.  A )  /\  r  e.  A )  /\  s  e.  A )  ->  (
( p  =/=  q  /\  -.  r  .<_  ( p 
.\/  q ) )  <-> 
( p  =/=  q  /\  ( p  .\/  q
) (  <o  `  K
) ( ( p 
.\/  q )  .\/  r ) ) ) )
15 hllat 29846 . . . . . . . . . . . . . 14  |-  ( K  e.  HL  ->  K  e.  Lat )
166, 15syl 16 . . . . . . . . . . . . 13  |-  ( ( ( ( K  e.  HL  /\  p  e.  A  /\  q  e.  A )  /\  r  e.  A )  /\  s  e.  A )  ->  K  e.  Lat )
177, 3atbase 29772 . . . . . . . . . . . . . 14  |-  ( r  e.  A  ->  r  e.  ( Base `  K
) )
1817ad2antlr 708 . . . . . . . . . . . . 13  |-  ( ( ( ( K  e.  HL  /\  p  e.  A  /\  q  e.  A )  /\  r  e.  A )  /\  s  e.  A )  ->  r  e.  ( Base `  K
) )
197, 1latjcl 14434 . . . . . . . . . . . . 13  |-  ( ( K  e.  Lat  /\  ( p  .\/  q )  e.  ( Base `  K
)  /\  r  e.  ( Base `  K )
)  ->  ( (
p  .\/  q )  .\/  r )  e.  (
Base `  K )
)
2016, 9, 18, 19syl3anc 1184 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  p  e.  A  /\  q  e.  A )  /\  r  e.  A )  /\  s  e.  A )  ->  (
( p  .\/  q
)  .\/  r )  e.  ( Base `  K
) )
21 simpr 448 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  p  e.  A  /\  q  e.  A )  /\  r  e.  A )  /\  s  e.  A )  ->  s  e.  A )
227, 11, 1, 2, 3cvr1 29892 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  ( ( p  .\/  q )  .\/  r
)  e.  ( Base `  K )  /\  s  e.  A )  ->  ( -.  s  .<_  ( ( p  .\/  q ) 
.\/  r )  <->  ( (
p  .\/  q )  .\/  r ) (  <o  `  K ) ( ( ( p  .\/  q
)  .\/  r )  .\/  s ) ) )
236, 20, 21, 22syl3anc 1184 . . . . . . . . . . 11  |-  ( ( ( ( K  e.  HL  /\  p  e.  A  /\  q  e.  A )  /\  r  e.  A )  /\  s  e.  A )  ->  ( -.  s  .<_  ( ( p  .\/  q ) 
.\/  r )  <->  ( (
p  .\/  q )  .\/  r ) (  <o  `  K ) ( ( ( p  .\/  q
)  .\/  r )  .\/  s ) ) )
2414, 23anbi12d 692 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  p  e.  A  /\  q  e.  A )  /\  r  e.  A )  /\  s  e.  A )  ->  (
( ( p  =/=  q  /\  -.  r  .<_  ( p  .\/  q
) )  /\  -.  s  .<_  ( ( p 
.\/  q )  .\/  r ) )  <->  ( (
p  =/=  q  /\  ( p  .\/  q ) (  <o  `  K )
( ( p  .\/  q )  .\/  r
) )  /\  (
( p  .\/  q
)  .\/  r )
(  <o  `  K )
( ( ( p 
.\/  q )  .\/  r )  .\/  s
) ) ) )
255, 24syl5bb 249 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  p  e.  A  /\  q  e.  A )  /\  r  e.  A )  /\  s  e.  A )  ->  (
( p  =/=  q  /\  -.  r  .<_  ( p 
.\/  q )  /\  -.  s  .<_  ( ( p  .\/  q ) 
.\/  r ) )  <-> 
( ( p  =/=  q  /\  ( p 
.\/  q ) ( 
<o  `  K ) ( ( p  .\/  q
)  .\/  r )
)  /\  ( (
p  .\/  q )  .\/  r ) (  <o  `  K ) ( ( ( p  .\/  q
)  .\/  r )  .\/  s ) ) ) )
2625rexbidva 2683 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  p  e.  A  /\  q  e.  A )  /\  r  e.  A
)  ->  ( E. s  e.  A  (
p  =/=  q  /\  -.  r  .<_  ( p 
.\/  q )  /\  -.  s  .<_  ( ( p  .\/  q ) 
.\/  r ) )  <->  E. s  e.  A  ( ( p  =/=  q  /\  ( p 
.\/  q ) ( 
<o  `  K ) ( ( p  .\/  q
)  .\/  r )
)  /\  ( (
p  .\/  q )  .\/  r ) (  <o  `  K ) ( ( ( p  .\/  q
)  .\/  r )  .\/  s ) ) ) )
27 r19.42v 2822 . . . . . . . . 9  |-  ( E. s  e.  A  ( ( p  =/=  q  /\  ( p  .\/  q
) (  <o  `  K
) ( ( p 
.\/  q )  .\/  r ) )  /\  ( ( p  .\/  q )  .\/  r
) (  <o  `  K
) ( ( ( p  .\/  q ) 
.\/  r )  .\/  s ) )  <->  ( (
p  =/=  q  /\  ( p  .\/  q ) (  <o  `  K )
( ( p  .\/  q )  .\/  r
) )  /\  E. s  e.  A  (
( p  .\/  q
)  .\/  r )
(  <o  `  K )
( ( ( p 
.\/  q )  .\/  r )  .\/  s
) ) )
28 anass 631 . . . . . . . . 9  |-  ( ( ( p  =/=  q  /\  ( p  .\/  q
) (  <o  `  K
) ( ( p 
.\/  q )  .\/  r ) )  /\  E. s  e.  A  ( ( p  .\/  q
)  .\/  r )
(  <o  `  K )
( ( ( p 
.\/  q )  .\/  r )  .\/  s
) )  <->  ( p  =/=  q  /\  (
( p  .\/  q
) (  <o  `  K
) ( ( p 
.\/  q )  .\/  r )  /\  E. s  e.  A  (
( p  .\/  q
)  .\/  r )
(  <o  `  K )
( ( ( p 
.\/  q )  .\/  r )  .\/  s
) ) ) )
2927, 28bitri 241 . . . . . . . 8  |-  ( E. s  e.  A  ( ( p  =/=  q  /\  ( p  .\/  q
) (  <o  `  K
) ( ( p 
.\/  q )  .\/  r ) )  /\  ( ( p  .\/  q )  .\/  r
) (  <o  `  K
) ( ( ( p  .\/  q ) 
.\/  r )  .\/  s ) )  <->  ( p  =/=  q  /\  (
( p  .\/  q
) (  <o  `  K
) ( ( p 
.\/  q )  .\/  r )  /\  E. s  e.  A  (
( p  .\/  q
)  .\/  r )
(  <o  `  K )
( ( ( p 
.\/  q )  .\/  r )  .\/  s
) ) ) )
3026, 29syl6bb 253 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  p  e.  A  /\  q  e.  A )  /\  r  e.  A
)  ->  ( E. s  e.  A  (
p  =/=  q  /\  -.  r  .<_  ( p 
.\/  q )  /\  -.  s  .<_  ( ( p  .\/  q ) 
.\/  r ) )  <-> 
( p  =/=  q  /\  ( ( p  .\/  q ) (  <o  `  K ) ( ( p  .\/  q ) 
.\/  r )  /\  E. s  e.  A  ( ( p  .\/  q
)  .\/  r )
(  <o  `  K )
( ( ( p 
.\/  q )  .\/  r )  .\/  s
) ) ) ) )
3130rexbidva 2683 . . . . . 6  |-  ( ( K  e.  HL  /\  p  e.  A  /\  q  e.  A )  ->  ( E. r  e.  A  E. s  e.  A  ( p  =/=  q  /\  -.  r  .<_  ( p  .\/  q
)  /\  -.  s  .<_  ( ( p  .\/  q )  .\/  r
) )  <->  E. r  e.  A  ( p  =/=  q  /\  (
( p  .\/  q
) (  <o  `  K
) ( ( p 
.\/  q )  .\/  r )  /\  E. s  e.  A  (
( p  .\/  q
)  .\/  r )
(  <o  `  K )
( ( ( p 
.\/  q )  .\/  r )  .\/  s
) ) ) ) )
32 r19.42v 2822 . . . . . 6  |-  ( E. r  e.  A  ( p  =/=  q  /\  ( ( p  .\/  q ) (  <o  `  K ) ( ( p  .\/  q ) 
.\/  r )  /\  E. s  e.  A  ( ( p  .\/  q
)  .\/  r )
(  <o  `  K )
( ( ( p 
.\/  q )  .\/  r )  .\/  s
) ) )  <->  ( p  =/=  q  /\  E. r  e.  A  ( (
p  .\/  q )
(  <o  `  K )
( ( p  .\/  q )  .\/  r
)  /\  E. s  e.  A  ( (
p  .\/  q )  .\/  r ) (  <o  `  K ) ( ( ( p  .\/  q
)  .\/  r )  .\/  s ) ) ) )
3331, 32syl6bb 253 . . . . 5  |-  ( ( K  e.  HL  /\  p  e.  A  /\  q  e.  A )  ->  ( E. r  e.  A  E. s  e.  A  ( p  =/=  q  /\  -.  r  .<_  ( p  .\/  q
)  /\  -.  s  .<_  ( ( p  .\/  q )  .\/  r
) )  <->  ( p  =/=  q  /\  E. r  e.  A  ( (
p  .\/  q )
(  <o  `  K )
( ( p  .\/  q )  .\/  r
)  /\  E. s  e.  A  ( (
p  .\/  q )  .\/  r ) (  <o  `  K ) ( ( ( p  .\/  q
)  .\/  r )  .\/  s ) ) ) ) )
341, 2, 3atcvr1 29899 . . . . . 6  |-  ( ( K  e.  HL  /\  p  e.  A  /\  q  e.  A )  ->  ( p  =/=  q  <->  p (  <o  `  K )
( p  .\/  q
) ) )
3534anbi1d 686 . . . . 5  |-  ( ( K  e.  HL  /\  p  e.  A  /\  q  e.  A )  ->  ( ( p  =/=  q  /\  E. r  e.  A  ( (
p  .\/  q )
(  <o  `  K )
( ( p  .\/  q )  .\/  r
)  /\  E. s  e.  A  ( (
p  .\/  q )  .\/  r ) (  <o  `  K ) ( ( ( p  .\/  q
)  .\/  r )  .\/  s ) ) )  <-> 
( p (  <o  `  K ) ( p 
.\/  q )  /\  E. r  e.  A  ( ( p  .\/  q
) (  <o  `  K
) ( ( p 
.\/  q )  .\/  r )  /\  E. s  e.  A  (
( p  .\/  q
)  .\/  r )
(  <o  `  K )
( ( ( p 
.\/  q )  .\/  r )  .\/  s
) ) ) ) )
3633, 35bitrd 245 . . . 4  |-  ( ( K  e.  HL  /\  p  e.  A  /\  q  e.  A )  ->  ( E. r  e.  A  E. s  e.  A  ( p  =/=  q  /\  -.  r  .<_  ( p  .\/  q
)  /\  -.  s  .<_  ( ( p  .\/  q )  .\/  r
) )  <->  ( p
(  <o  `  K )
( p  .\/  q
)  /\  E. r  e.  A  ( (
p  .\/  q )
(  <o  `  K )
( ( p  .\/  q )  .\/  r
)  /\  E. s  e.  A  ( (
p  .\/  q )  .\/  r ) (  <o  `  K ) ( ( ( p  .\/  q
)  .\/  r )  .\/  s ) ) ) ) )
37363expb 1154 . . 3  |-  ( ( K  e.  HL  /\  ( p  e.  A  /\  q  e.  A
) )  ->  ( E. r  e.  A  E. s  e.  A  ( p  =/=  q  /\  -.  r  .<_  ( p 
.\/  q )  /\  -.  s  .<_  ( ( p  .\/  q ) 
.\/  r ) )  <-> 
( p (  <o  `  K ) ( p 
.\/  q )  /\  E. r  e.  A  ( ( p  .\/  q
) (  <o  `  K
) ( ( p 
.\/  q )  .\/  r )  /\  E. s  e.  A  (
( p  .\/  q
)  .\/  r )
(  <o  `  K )
( ( ( p 
.\/  q )  .\/  r )  .\/  s
) ) ) ) )
38372rexbidva 2707 . 2  |-  ( K  e.  HL  ->  ( E. p  e.  A  E. q  e.  A  E. r  e.  A  E. s  e.  A  ( p  =/=  q  /\  -.  r  .<_  ( p 
.\/  q )  /\  -.  s  .<_  ( ( p  .\/  q ) 
.\/  r ) )  <->  E. p  e.  A  E. q  e.  A  ( p (  <o  `  K ) ( p 
.\/  q )  /\  E. r  e.  A  ( ( p  .\/  q
) (  <o  `  K
) ( ( p 
.\/  q )  .\/  r )  /\  E. s  e.  A  (
( p  .\/  q
)  .\/  r )
(  <o  `  K )
( ( ( p 
.\/  q )  .\/  r )  .\/  s
) ) ) ) )
394, 38mpbird 224 1  |-  ( K  e.  HL  ->  E. p  e.  A  E. q  e.  A  E. r  e.  A  E. s  e.  A  ( p  =/=  q  /\  -.  r  .<_  ( p  .\/  q
)  /\  -.  s  .<_  ( ( 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 2567   E.wrex 2667   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   Basecbs 13424   lecple 13491   joincjn 14356   Latclat 14429    <o ccvr 29745   Atomscatm 29746   HLchlt 29833
This theorem is referenced by:  3dim1  29949
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 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-lat 14430  df-clat 14492  df-oposet 29659  df-ol 29661  df-oml 29662  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834
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