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Theorem 3dim0 30328
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 2438 . . 3  |-  (  <o  `  K )  =  ( 
<o  `  K )
3 3dim0.a . . 3  |-  A  =  ( Atoms `  K )
41, 2, 3athgt 30327 . 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 939 . . . . . . . . . 10  |-  ( ( p  =/=  q  /\  -.  r  .<_  ( p 
.\/  q )  /\  -.  s  .<_  ( ( p  .\/  q ) 
.\/  r ) )  <-> 
( ( p  =/=  q  /\  -.  r  .<_  ( p  .\/  q
) )  /\  -.  s  .<_  ( ( p 
.\/  q )  .\/  r ) ) )
6 simpll1 997 . . . . . . . . . . . . 13  |-  ( ( ( ( K  e.  HL  /\  p  e.  A  /\  q  e.  A )  /\  r  e.  A )  /\  s  e.  A )  ->  K  e.  HL )
7 eqid 2438 . . . . . . . . . . . . . . 15  |-  ( Base `  K )  =  (
Base `  K )
87, 1, 3hlatjcl 30238 . . . . . . . . . . . . . 14  |-  ( ( K  e.  HL  /\  p  e.  A  /\  q  e.  A )  ->  ( p  .\/  q
)  e.  ( Base `  K ) )
98ad2antrr 708 . . . . . . . . . . . . 13  |-  ( ( ( ( K  e.  HL  /\  p  e.  A  /\  q  e.  A )  /\  r  e.  A )  /\  s  e.  A )  ->  (
p  .\/  q )  e.  ( Base `  K
) )
10 simplr 733 . . . . . . . . . . . . 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 30281 . . . . . . . . . . . . 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 1185 . . . . . . . . . . . 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 686 . . . . . . . . . . 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 30235 . . . . . . . . . . . . . 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 30161 . . . . . . . . . . . . . 14  |-  ( r  e.  A  ->  r  e.  ( Base `  K
) )
1817ad2antlr 709 . . . . . . . . . . . . 13  |-  ( ( ( ( K  e.  HL  /\  p  e.  A  /\  q  e.  A )  /\  r  e.  A )  /\  s  e.  A )  ->  r  e.  ( Base `  K
) )
197, 1latjcl 14484 . . . . . . . . . . . . 13  |-  ( ( K  e.  Lat  /\  ( p  .\/  q )  e.  ( Base `  K
)  /\  r  e.  ( Base `  K )
)  ->  ( (
p  .\/  q )  .\/  r )  e.  (
Base `  K )
)
2016, 9, 18, 19syl3anc 1185 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  p  e.  A  /\  q  e.  A )  /\  r  e.  A )  /\  s  e.  A )  ->  (
( p  .\/  q
)  .\/  r )  e.  ( Base `  K
) )
21 simpr 449 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  p  e.  A  /\  q  e.  A )  /\  r  e.  A )  /\  s  e.  A )  ->  s  e.  A )
227, 11, 1, 2, 3cvr1 30281 . . . . . . . . . . . 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 1185 . . . . . . . . . . 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 693 . . . . . . . . . 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 250 . . . . . . . . 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 2724 . . . . . . . 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 2864 . . . . . . . . 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 632 . . . . . . . . 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 242 . . . . . . . 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 254 . . . . . . 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 2724 . . . . . 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 2864 . . . . . 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 254 . . . . 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 30288 . . . . . 6  |-  ( ( K  e.  HL  /\  p  e.  A  /\  q  e.  A )  ->  ( p  =/=  q  <->  p (  <o  `  K )
( p  .\/  q
) ) )
3534anbi1d 687 . . . . 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 246 . . . 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 1155 . . 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 2748 . 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 225 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 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   E.wrex 2708   class class class wbr 4215   ` cfv 5457  (class class class)co 6084   Basecbs 13474   lecple 13541   joincjn 14406   Latclat 14479    <o ccvr 30134   Atomscatm 30135   HLchlt 30222
This theorem is referenced by:  3dim1  30338
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-undef 6546  df-riota 6552  df-poset 14408  df-plt 14420  df-lub 14436  df-glb 14437  df-join 14438  df-meet 14439  df-p0 14473  df-lat 14480  df-clat 14542  df-oposet 30048  df-ol 30050  df-oml 30051  df-covers 30138  df-ats 30139  df-atl 30170  df-cvlat 30194  df-hlat 30223
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