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Theorem 3atlem7 30017
Description: Lemma for 3at 30018. (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
3atlem7  |-  ( ( ( 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  =/=  Q
)  /\  ( ( P  .\/  Q )  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  (
( P  .\/  Q
)  .\/  R )  =  ( ( S 
.\/  T )  .\/  U ) )

Proof of Theorem 3atlem7
StepHypRef Expression
1 simpl1 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  =/= 
Q )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  /\  Q  .<_  ( P  .\/  U ) )  ->  ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
) )
2 simpl2l 1010 . . 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  =/= 
Q )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  /\  Q  .<_  ( P  .\/  U ) )  ->  -.  R  .<_  ( P  .\/  Q
) )
3 simpl2r 1011 . . 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  =/= 
Q )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  /\  Q  .<_  ( P  .\/  U ) )  ->  P  =/=  Q )
4 simpr 448 . . 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  =/= 
Q )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  /\  Q  .<_  ( P  .\/  U ) )  ->  Q  .<_  ( P  .\/  U ) )
5 simpl3 962 . . 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  =/= 
Q )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  /\  Q  .<_  ( P  .\/  U ) )  ->  ( ( P  .\/  Q )  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )
6 3at.l . . . 4  |-  .<_  =  ( le `  K )
7 3at.j . . . 4  |-  .\/  =  ( join `  K )
8 3at.a . . . 4  |-  A  =  ( Atoms `  K )
96, 7, 83atlem6 30016 . . 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  =/=  Q  /\  Q  .<_  ( P 
.\/  U ) )  /\  ( ( P 
.\/  Q )  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  (
( P  .\/  Q
)  .\/  R )  =  ( ( S 
.\/  T )  .\/  U ) )
101, 2, 3, 4, 5, 9syl131anc 1197 . 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  =/= 
Q )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  /\  Q  .<_  ( P  .\/  U ) )  ->  ( ( P  .\/  Q )  .\/  R )  =  ( ( S  .\/  T ) 
.\/  U ) )
11 simpl1 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  =/= 
Q )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  /\  -.  Q  .<_  ( P  .\/  U
) )  ->  ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A
) ) )
12 simpl2l 1010 . . 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  =/= 
Q )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  /\  -.  Q  .<_  ( P  .\/  U
) )  ->  -.  R  .<_  ( P  .\/  Q ) )
13 simpl2r 1011 . . 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  =/= 
Q )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  /\  -.  Q  .<_  ( P  .\/  U
) )  ->  P  =/=  Q )
14 simpr 448 . . 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  =/= 
Q )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  /\  -.  Q  .<_  ( P  .\/  U
) )  ->  -.  Q  .<_  ( P  .\/  U ) )
15 simpl3 962 . . 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  =/= 
Q )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  /\  -.  Q  .<_  ( P  .\/  U
) )  ->  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )
166, 7, 83atlem5 30015 . . 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  =/=  Q  /\  -.  Q  .<_  ( P 
.\/  U ) )  /\  ( ( P 
.\/  Q )  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  (
( P  .\/  Q
)  .\/  R )  =  ( ( S 
.\/  T )  .\/  U ) )
1711, 12, 13, 14, 15, 16syl131anc 1197 . 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  =/= 
Q )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  /\  -.  Q  .<_  ( P  .\/  U
) )  ->  (
( P  .\/  Q
)  .\/  R )  =  ( ( S 
.\/  T )  .\/  U ) )
1810, 17pm2.61dan 767 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  =/=  Q
)  /\  ( ( 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 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:  3at  30018
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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