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Theorem 3atlem3 29674
Description: Lemma for 3at 29679. (Contributed by NM, 23-Jun-2012.)
Hypotheses
Ref Expression
3at.l  |-  .<_  =  ( le `  K )
3at.j  |-  .\/  =  ( join `  K )
3at.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
3atlem3  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/=  U  /\  -.  Q  .<_  ( P 
.\/  U ) )  /\  ( ( P 
.\/  Q )  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  (
( P  .\/  Q
)  .\/  R )  =  ( ( S 
.\/  T )  .\/  U ) )

Proof of Theorem 3atlem3
StepHypRef Expression
1 simpl1 958 . . 3  |-  ( ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/= 
U  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  /\  P  .<_  ( T  .\/  U ) )  ->  ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
) )
2 simpl21 1033 . . 3  |-  ( ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/= 
U  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  /\  P  .<_  ( T  .\/  U ) )  ->  -.  R  .<_  ( P  .\/  Q
) )
3 simpl22 1034 . . . 4  |-  ( ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/= 
U  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  /\  P  .<_  ( T  .\/  U ) )  ->  P  =/=  U )
4 simpr 447 . . . 4  |-  ( ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/= 
U  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  /\  P  .<_  ( T  .\/  U ) )  ->  P  .<_  ( T  .\/  U ) )
53, 4jca 518 . . 3  |-  ( ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/= 
U  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  /\  P  .<_  ( T  .\/  U ) )  ->  ( P  =/=  U  /\  P  .<_  ( T  .\/  U ) ) )
6 simpl23 1035 . . 3  |-  ( ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/= 
U  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  /\  P  .<_  ( T  .\/  U ) )  ->  -.  Q  .<_  ( P  .\/  U
) )
7 simpl3 960 . . 3  |-  ( ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/= 
U  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  /\  P  .<_  ( T  .\/  U ) )  ->  ( ( P  .\/  Q )  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )
8 3at.l . . . 4  |-  .<_  =  ( le `  K )
9 3at.j . . . 4  |-  .\/  =  ( join `  K )
10 3at.a . . . 4  |-  A  =  ( Atoms `  K )
118, 9, 103atlem2 29673 . . 3  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  ( P  =/= 
U  /\  P  .<_  ( T  .\/  U ) )  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  ( ( P  .\/  Q )  .\/  R )  =  ( ( S  .\/  T ) 
.\/  U ) )
121, 2, 5, 6, 7, 11syl131anc 1195 . 2  |-  ( ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/= 
U  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  /\  P  .<_  ( T  .\/  U ) )  ->  ( ( P  .\/  Q )  .\/  R )  =  ( ( S  .\/  T ) 
.\/  U ) )
13 simpl1 958 . . 3  |-  ( ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/= 
U  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  /\  -.  P  .<_  ( T  .\/  U
) )  ->  ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A
) ) )
14 simpl21 1033 . . 3  |-  ( ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/= 
U  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  /\  -.  P  .<_  ( T  .\/  U
) )  ->  -.  R  .<_  ( P  .\/  Q ) )
15 simpr 447 . . 3  |-  ( ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/= 
U  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  /\  -.  P  .<_  ( T  .\/  U
) )  ->  -.  P  .<_  ( T  .\/  U ) )
16 simpl23 1035 . . 3  |-  ( ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/= 
U  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  /\  -.  P  .<_  ( T  .\/  U
) )  ->  -.  Q  .<_  ( P  .\/  U ) )
17 simpl3 960 . . 3  |-  ( ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/= 
U  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  /\  -.  P  .<_  ( T  .\/  U
) )  ->  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )
188, 9, 103atlem1 29672 . . 3  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  -.  P  .<_  ( T  .\/  U )  /\  -.  Q  .<_  ( P  .\/  U ) )  /\  ( ( P  .\/  Q ) 
.\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  ( ( P  .\/  Q )  .\/  R )  =  ( ( S  .\/  T ) 
.\/  U ) )
1913, 14, 15, 16, 17, 18syl131anc 1195 . 2  |-  ( ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/= 
U  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  /\  -.  P  .<_  ( T  .\/  U
) )  ->  (
( P  .\/  Q
)  .\/  R )  =  ( ( S 
.\/  T )  .\/  U ) )
2012, 19pm2.61dan 766 1  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/=  U  /\  -.  Q  .<_  ( P 
.\/  U ) )  /\  ( ( P 
.\/  Q )  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  (
( P  .\/  Q
)  .\/  R )  =  ( ( S 
.\/  T )  .\/  U ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   lecple 13215   joincjn 14078   Atomscatm 29453   HLchlt 29540
This theorem is referenced by:  3atlem4  29675  3atlem5  29676
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-lub 14108  df-join 14110  df-lat 14152  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541
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