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Theorem 3dim0 29622
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 2380 . . 3  |-  (  <o  `  K )  =  ( 
<o  `  K )
3 3dim0.a . . 3  |-  A  =  ( Atoms `  K )
41, 2, 3athgt 29621 . 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 2380 . . . . . . . . . . . . . . 15  |-  ( Base `  K )  =  (
Base `  K )
87, 1, 3hlatjcl 29532 . . . . . . . . . . . . . 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 29575 . . . . . . . . . . . . 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 29529 . . . . . . . . . . . . . 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 29455 . . . . . . . . . . . . . 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 14399 . . . . . . . . . . . . 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 29575 . . . . . . . . . . . 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 2659 . . . . . . . 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 2798 . . . . . . . . 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 2659 . . . . . 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 2798 . . . . . 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 29582 . . . . . 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 2683 . 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 1717    =/= wne 2543   E.wrex 2643   class class class wbr 4146   ` cfv 5387  (class class class)co 6013   Basecbs 13389   lecple 13456   joincjn 14321   Latclat 14394    <o ccvr 29428   Atomscatm 29429   HLchlt 29516
This theorem is referenced by:  3dim1  29632
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-undef 6472  df-riota 6478  df-poset 14323  df-plt 14335  df-lub 14351  df-glb 14352  df-join 14353  df-meet 14354  df-p0 14388  df-lat 14395  df-clat 14457  df-oposet 29342  df-ol 29344  df-oml 29345  df-covers 29432  df-ats 29433  df-atl 29464  df-cvlat 29488  df-hlat 29517
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