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Theorem 3dimlem2 29987
Description: Lemma for 3dim1 29995. (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 29896 . . . . . 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 29921 . . . . . . 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 2462 . . . 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 4211 . . 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 6082 . . . 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 4211 . . 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 1652    e. wcel 1725    =/= wne 2593   class class class wbr 4199   ` cfv 5440  (class class class)co 6067   lecple 13519   joincjn 14384   Atomscatm 29792   HLchlt 29879
This theorem is referenced by:  3dim1  29995  3dim2  29996
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411  ax-rep 4307  ax-sep 4317  ax-nul 4325  ax-pow 4364  ax-pr 4390  ax-un 4687
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-nel 2596  df-ral 2697  df-rex 2698  df-reu 2699  df-rab 2701  df-v 2945  df-sbc 3149  df-csb 3239  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-nul 3616  df-if 3727  df-pw 3788  df-sn 3807  df-pr 3808  df-op 3810  df-uni 4003  df-iun 4082  df-br 4200  df-opab 4254  df-mpt 4255  df-id 4485  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-rn 4875  df-res 4876  df-ima 4877  df-iota 5404  df-fun 5442  df-fn 5443  df-f 5444  df-f1 5445  df-fo 5446  df-f1o 5447  df-fv 5448  df-ov 6070  df-oprab 6071  df-mpt2 6072  df-1st 6335  df-2nd 6336  df-undef 6529  df-riota 6535  df-poset 14386  df-plt 14398  df-lub 14414  df-join 14416  df-lat 14458  df-covers 29795  df-ats 29796  df-atl 29827  df-cvlat 29851  df-hlat 29880
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