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Theorem 3dim0 30268
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 2296 . . 3  |-  (  <o  `  K )  =  ( 
<o  `  K )
3 3dim0.a . . 3  |-  A  =  ( Atoms `  K )
41, 2, 3athgt 30267 . 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 936 . . . . . . . . . 10  |-  ( ( p  =/=  q  /\  -.  r  .<_  ( p 
.\/  q )  /\  -.  s  .<_  ( ( p  .\/  q ) 
.\/  r ) )  <-> 
( ( p  =/=  q  /\  -.  r  .<_  ( p  .\/  q
) )  /\  -.  s  .<_  ( ( p 
.\/  q )  .\/  r ) ) )
6 simpll1 994 . . . . . . . . . . . . 13  |-  ( ( ( ( K  e.  HL  /\  p  e.  A  /\  q  e.  A )  /\  r  e.  A )  /\  s  e.  A )  ->  K  e.  HL )
7 eqid 2296 . . . . . . . . . . . . . . 15  |-  ( Base `  K )  =  (
Base `  K )
87, 1, 3hlatjcl 30178 . . . . . . . . . . . . . 14  |-  ( ( K  e.  HL  /\  p  e.  A  /\  q  e.  A )  ->  ( p  .\/  q
)  e.  ( Base `  K ) )
98ad2antrr 706 . . . . . . . . . . . . 13  |-  ( ( ( ( K  e.  HL  /\  p  e.  A  /\  q  e.  A )  /\  r  e.  A )  /\  s  e.  A )  ->  (
p  .\/  q )  e.  ( Base `  K
) )
10 simplr 731 . . . . . . . . . . . . 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 30221 . . . . . . . . . . . . 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 1182 . . . . . . . . . . . 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 684 . . . . . . . . . . 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 30175 . . . . . . . . . . . . . 14  |-  ( K  e.  HL  ->  K  e.  Lat )
166, 15syl 15 . . . . . . . . . . . . 13  |-  ( ( ( ( K  e.  HL  /\  p  e.  A  /\  q  e.  A )  /\  r  e.  A )  /\  s  e.  A )  ->  K  e.  Lat )
177, 3atbase 30101 . . . . . . . . . . . . . 14  |-  ( r  e.  A  ->  r  e.  ( Base `  K
) )
1817ad2antlr 707 . . . . . . . . . . . . 13  |-  ( ( ( ( K  e.  HL  /\  p  e.  A  /\  q  e.  A )  /\  r  e.  A )  /\  s  e.  A )  ->  r  e.  ( Base `  K
) )
197, 1latjcl 14172 . . . . . . . . . . . . 13  |-  ( ( K  e.  Lat  /\  ( p  .\/  q )  e.  ( Base `  K
)  /\  r  e.  ( Base `  K )
)  ->  ( (
p  .\/  q )  .\/  r )  e.  (
Base `  K )
)
2016, 9, 18, 19syl3anc 1182 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  p  e.  A  /\  q  e.  A )  /\  r  e.  A )  /\  s  e.  A )  ->  (
( p  .\/  q
)  .\/  r )  e.  ( Base `  K
) )
21 simpr 447 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  p  e.  A  /\  q  e.  A )  /\  r  e.  A )  /\  s  e.  A )  ->  s  e.  A )
227, 11, 1, 2, 3cvr1 30221 . . . . . . . . . . . 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 1182 . . . . . . . . . . 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 691 . . . . . . . . . 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 248 . . . . . . . . 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 2573 . . . . . . . 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 2707 . . . . . . . . 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 630 . . . . . . . . 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 240 . . . . . . . 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 252 . . . . . . 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 2573 . . . . . 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 2707 . . . . . 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 252 . . . . 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 30228 . . . . . 6  |-  ( ( K  e.  HL  /\  p  e.  A  /\  q  e.  A )  ->  ( p  =/=  q  <->  p (  <o  `  K )
( p  .\/  q
) ) )
3534anbi1d 685 . . . . 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 244 . . . 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 1152 . . 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 2597 . 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 223 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   joincjn 14094   Latclat 14167    <o ccvr 30074   Atomscatm 30075   HLchlt 30162
This theorem is referenced by:  3dim1  30278
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163
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