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Theorem 3dimlem2 29624
Description: Lemma for 3dim1 29632. (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
3dimlem2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  S  .<_  ( Q 
.\/  R )  /\  -.  T  .<_  ( ( Q  .\/  R ) 
.\/  S ) )  /\  ( P  =/= 
Q  /\  P  .<_  ( Q  .\/  R ) ) )  ->  ( P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q )  /\  -.  T  .<_  ( ( P  .\/  Q )  .\/  S ) ) )

Proof of Theorem 3dimlem2
StepHypRef Expression
1 simp3l 985 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  S  .<_  ( Q 
.\/  R )  /\  -.  T  .<_  ( ( Q  .\/  R ) 
.\/  S ) )  /\  ( P  =/= 
Q  /\  P  .<_  ( Q  .\/  R ) ) )  ->  P  =/=  Q )
2 simp22 991 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  S  .<_  ( Q 
.\/  R )  /\  -.  T  .<_  ( ( Q  .\/  R ) 
.\/  S ) )  /\  ( P  =/= 
Q  /\  P  .<_  ( Q  .\/  R ) ) )  ->  -.  S  .<_  ( Q  .\/  R ) )
3 3dim0.j . . . . . . 7  |-  .\/  =  ( join `  K )
4 3dim0.a . . . . . . 7  |-  A  =  ( Atoms `  K )
53, 4hlatjcom 29533 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  =  ( Q 
.\/  P ) )
653ad2ant1 978 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  S  .<_  ( Q 
.\/  R )  /\  -.  T  .<_  ( ( Q  .\/  R ) 
.\/  S ) )  /\  ( P  =/= 
Q  /\  P  .<_  ( Q  .\/  R ) ) )  ->  ( P  .\/  Q )  =  ( Q  .\/  P
) )
7 simp3r 986 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  S  .<_  ( Q 
.\/  R )  /\  -.  T  .<_  ( ( Q  .\/  R ) 
.\/  S ) )  /\  ( P  =/= 
Q  /\  P  .<_  ( Q  .\/  R ) ) )  ->  P  .<_  ( Q  .\/  R
) )
8 simp11 987 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  S  .<_  ( Q 
.\/  R )  /\  -.  T  .<_  ( ( Q  .\/  R ) 
.\/  S ) )  /\  ( P  =/= 
Q  /\  P  .<_  ( Q  .\/  R ) ) )  ->  K  e.  HL )
9 simp12 988 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  S  .<_  ( Q 
.\/  R )  /\  -.  T  .<_  ( ( Q  .\/  R ) 
.\/  S ) )  /\  ( P  =/= 
Q  /\  P  .<_  ( Q  .\/  R ) ) )  ->  P  e.  A )
10 simp21 990 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  S  .<_  ( Q 
.\/  R )  /\  -.  T  .<_  ( ( Q  .\/  R ) 
.\/  S ) )  /\  ( P  =/= 
Q  /\  P  .<_  ( Q  .\/  R ) ) )  ->  R  e.  A )
11 simp13 989 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  S  .<_  ( Q 
.\/  R )  /\  -.  T  .<_  ( ( Q  .\/  R ) 
.\/  S ) )  /\  ( P  =/= 
Q  /\  P  .<_  ( Q  .\/  R ) ) )  ->  Q  e.  A )
12 3dim0.l . . . . . . . 8  |-  .<_  =  ( le `  K )
1312, 3, 4hlatexchb1 29558 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  R  e.  A  /\  Q  e.  A
)  /\  P  =/=  Q )  ->  ( P  .<_  ( Q  .\/  R
)  <->  ( Q  .\/  P )  =  ( Q 
.\/  R ) ) )
148, 9, 10, 11, 1, 13syl131anc 1197 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  S  .<_  ( Q 
.\/  R )  /\  -.  T  .<_  ( ( Q  .\/  R ) 
.\/  S ) )  /\  ( P  =/= 
Q  /\  P  .<_  ( Q  .\/  R ) ) )  ->  ( P  .<_  ( Q  .\/  R )  <->  ( Q  .\/  P )  =  ( Q 
.\/  R ) ) )
157, 14mpbid 202 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  S  .<_  ( Q 
.\/  R )  /\  -.  T  .<_  ( ( Q  .\/  R ) 
.\/  S ) )  /\  ( P  =/= 
Q  /\  P  .<_  ( Q  .\/  R ) ) )  ->  ( Q  .\/  P )  =  ( Q  .\/  R
) )
166, 15eqtrd 2412 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  S  .<_  ( Q 
.\/  R )  /\  -.  T  .<_  ( ( Q  .\/  R ) 
.\/  S ) )  /\  ( P  =/= 
Q  /\  P  .<_  ( Q  .\/  R ) ) )  ->  ( P  .\/  Q )  =  ( Q  .\/  R
) )
1716breq2d 4158 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  S  .<_  ( Q 
.\/  R )  /\  -.  T  .<_  ( ( Q  .\/  R ) 
.\/  S ) )  /\  ( P  =/= 
Q  /\  P  .<_  ( Q  .\/  R ) ) )  ->  ( S  .<_  ( P  .\/  Q )  <->  S  .<_  ( Q 
.\/  R ) ) )
182, 17mtbird 293 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  S  .<_  ( Q 
.\/  R )  /\  -.  T  .<_  ( ( Q  .\/  R ) 
.\/  S ) )  /\  ( P  =/= 
Q  /\  P  .<_  ( Q  .\/  R ) ) )  ->  -.  S  .<_  ( P  .\/  Q ) )
19 simp23 992 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  S  .<_  ( Q 
.\/  R )  /\  -.  T  .<_  ( ( Q  .\/  R ) 
.\/  S ) )  /\  ( P  =/= 
Q  /\  P  .<_  ( Q  .\/  R ) ) )  ->  -.  T  .<_  ( ( Q 
.\/  R )  .\/  S ) )
2016oveq1d 6028 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  S  .<_  ( Q 
.\/  R )  /\  -.  T  .<_  ( ( Q  .\/  R ) 
.\/  S ) )  /\  ( P  =/= 
Q  /\  P  .<_  ( Q  .\/  R ) ) )  ->  (
( P  .\/  Q
)  .\/  S )  =  ( ( Q 
.\/  R )  .\/  S ) )
2120breq2d 4158 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  S  .<_  ( Q 
.\/  R )  /\  -.  T  .<_  ( ( Q  .\/  R ) 
.\/  S ) )  /\  ( P  =/= 
Q  /\  P  .<_  ( Q  .\/  R ) ) )  ->  ( T  .<_  ( ( P 
.\/  Q )  .\/  S )  <->  T  .<_  ( ( Q  .\/  R ) 
.\/  S ) ) )
2219, 21mtbird 293 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  S  .<_  ( Q 
.\/  R )  /\  -.  T  .<_  ( ( Q  .\/  R ) 
.\/  S ) )  /\  ( P  =/= 
Q  /\  P  .<_  ( Q  .\/  R ) ) )  ->  -.  T  .<_  ( ( P 
.\/  Q )  .\/  S ) )
231, 18, 223jca 1134 1  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  S  .<_  ( Q 
.\/  R )  /\  -.  T  .<_  ( ( Q  .\/  R ) 
.\/  S ) )  /\  ( P  =/= 
Q  /\  P  .<_  ( Q  .\/  R ) ) )  ->  ( P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q )  /\  -.  T  .<_  ( ( P  .\/  Q )  .\/  S ) ) )
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   class class class wbr 4146   ` cfv 5387  (class class class)co 6013   lecple 13456   joincjn 14321   Atomscatm 29429   HLchlt 29516
This theorem is referenced by:  3dim1  29632  3dim2  29633
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-join 14353  df-lat 14395  df-covers 29432  df-ats 29433  df-atl 29464  df-cvlat 29488  df-hlat 29517
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