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Theorem 3dim0 34128
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 2460 . . 3  |-  (  <o  `  K )  =  ( 
<o  `  K )
3 3dim0.a . . 3  |-  A  =  ( Atoms `  K )
41, 2, 3athgt 34127 . 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 970 . . . . . . . . . 10  |-  ( ( p  =/=  q  /\  -.  r  .<_  ( p 
.\/  q )  /\  -.  s  .<_  ( ( p  .\/  q ) 
.\/  r ) )  <-> 
( ( p  =/=  q  /\  -.  r  .<_  ( p  .\/  q
) )  /\  -.  s  .<_  ( ( p 
.\/  q )  .\/  r ) ) )
6 simpll1 1030 . . . . . . . . . . . . 13  |-  ( ( ( ( K  e.  HL  /\  p  e.  A  /\  q  e.  A )  /\  r  e.  A )  /\  s  e.  A )  ->  K  e.  HL )
7 eqid 2460 . . . . . . . . . . . . . . 15  |-  ( Base `  K )  =  (
Base `  K )
87, 1, 3hlatjcl 34038 . . . . . . . . . . . . . 14  |-  ( ( K  e.  HL  /\  p  e.  A  /\  q  e.  A )  ->  ( p  .\/  q
)  e.  ( Base `  K ) )
98ad2antrr 725 . . . . . . . . . . . . 13  |-  ( ( ( ( K  e.  HL  /\  p  e.  A  /\  q  e.  A )  /\  r  e.  A )  /\  s  e.  A )  ->  (
p  .\/  q )  e.  ( Base `  K
) )
10 simplr 754 . . . . . . . . . . . . 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 34081 . . . . . . . . . . . . 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 1223 . . . . . . . . . . . 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 703 . . . . . . . . . . 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 34035 . . . . . . . . . . . . . 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 33961 . . . . . . . . . . . . . 14  |-  ( r  e.  A  ->  r  e.  ( Base `  K
) )
1817ad2antlr 726 . . . . . . . . . . . . 13  |-  ( ( ( ( K  e.  HL  /\  p  e.  A  /\  q  e.  A )  /\  r  e.  A )  /\  s  e.  A )  ->  r  e.  ( Base `  K
) )
197, 1latjcl 15527 . . . . . . . . . . . . 13  |-  ( ( K  e.  Lat  /\  ( p  .\/  q )  e.  ( Base `  K
)  /\  r  e.  ( Base `  K )
)  ->  ( (
p  .\/  q )  .\/  r )  e.  (
Base `  K )
)
2016, 9, 18, 19syl3anc 1223 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  p  e.  A  /\  q  e.  A )  /\  r  e.  A )  /\  s  e.  A )  ->  (
( p  .\/  q
)  .\/  r )  e.  ( Base `  K
) )
21 simpr 461 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  p  e.  A  /\  q  e.  A )  /\  r  e.  A )  /\  s  e.  A )  ->  s  e.  A )
227, 11, 1, 2, 3cvr1 34081 . . . . . . . . . . . 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 1223 . . . . . . . . . . 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 710 . . . . . . . . . 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 257 . . . . . . . . 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 2963 . . . . . . . 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 3009 . . . . . . . . 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 649 . . . . . . . . 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 249 . . . . . . . 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 261 . . . . . . 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 2963 . . . . . 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 3009 . . . . . 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 261 . . . . 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 34088 . . . . . 6  |-  ( ( K  e.  HL  /\  p  e.  A  /\  q  e.  A )  ->  ( p  =/=  q  <->  p (  <o  `  K )
( p  .\/  q
) ) )
3534anbi1d 704 . . . . 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 253 . . . 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 1192 . . 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 2972 . 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 232 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762    =/= wne 2655   E.wrex 2808   class class class wbr 4440   ` cfv 5579  (class class class)co 6275   Basecbs 14479   lecple 14551   joincjn 15420   Latclat 15521    <o ccvr 33934   Atomscatm 33935   HLchlt 34022
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-poset 15422  df-plt 15434  df-lub 15450  df-glb 15451  df-join 15452  df-meet 15453  df-p0 15515  df-p1 15516  df-lat 15522  df-clat 15584  df-oposet 33848  df-ol 33850  df-oml 33851  df-covers 33938  df-ats 33939  df-atl 33970  df-cvlat 33994  df-hlat 34023
This theorem is referenced by:  3dim1  34138
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