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Theorem 3dimlem3a 29649
Description: Lemma for 3dim3 29658. (Contributed by NM, 27-Jul-2012.)
Hypotheses
Ref Expression
3dim0.j  |-  .\/  =  ( join `  K )
3dim0.l  |-  .<_  =  ( le `  K )
3dim0.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
3dimlem3a  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( -.  T  .<_  ( ( Q 
.\/  R )  .\/  S )  /\  -.  P  .<_  ( Q  .\/  R
)  /\  P  .<_  ( ( Q  .\/  R
)  .\/  S )
) )  ->  -.  T  .<_  ( ( P 
.\/  Q )  .\/  R ) )

Proof of Theorem 3dimlem3a
StepHypRef Expression
1 simp31 991 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( -.  T  .<_  ( ( Q 
.\/  R )  .\/  S )  /\  -.  P  .<_  ( Q  .\/  R
)  /\  P  .<_  ( ( Q  .\/  R
)  .\/  S )
) )  ->  -.  T  .<_  ( ( Q 
.\/  R )  .\/  S ) )
2 simp11 985 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( -.  T  .<_  ( ( Q 
.\/  R )  .\/  S )  /\  -.  P  .<_  ( Q  .\/  R
)  /\  P  .<_  ( ( Q  .\/  R
)  .\/  S )
) )  ->  K  e.  HL )
3 hllat 29553 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
42, 3syl 15 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( -.  T  .<_  ( ( Q 
.\/  R )  .\/  S )  /\  -.  P  .<_  ( Q  .\/  R
)  /\  P  .<_  ( ( Q  .\/  R
)  .\/  S )
) )  ->  K  e.  Lat )
5 simp13 987 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( -.  T  .<_  ( ( Q 
.\/  R )  .\/  S )  /\  -.  P  .<_  ( Q  .\/  R
)  /\  P  .<_  ( ( Q  .\/  R
)  .\/  S )
) )  ->  Q  e.  A )
6 eqid 2283 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
7 3dim0.a . . . . . . 7  |-  A  =  ( Atoms `  K )
86, 7atbase 29479 . . . . . 6  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
95, 8syl 15 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( -.  T  .<_  ( ( Q 
.\/  R )  .\/  S )  /\  -.  P  .<_  ( Q  .\/  R
)  /\  P  .<_  ( ( Q  .\/  R
)  .\/  S )
) )  ->  Q  e.  ( Base `  K
) )
10 simp2l 981 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( -.  T  .<_  ( ( Q 
.\/  R )  .\/  S )  /\  -.  P  .<_  ( Q  .\/  R
)  /\  P  .<_  ( ( Q  .\/  R
)  .\/  S )
) )  ->  R  e.  A )
116, 7atbase 29479 . . . . . 6  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
1210, 11syl 15 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( -.  T  .<_  ( ( Q 
.\/  R )  .\/  S )  /\  -.  P  .<_  ( Q  .\/  R
)  /\  P  .<_  ( ( Q  .\/  R
)  .\/  S )
) )  ->  R  e.  ( Base `  K
) )
13 simp12 986 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( -.  T  .<_  ( ( Q 
.\/  R )  .\/  S )  /\  -.  P  .<_  ( Q  .\/  R
)  /\  P  .<_  ( ( Q  .\/  R
)  .\/  S )
) )  ->  P  e.  A )
146, 7atbase 29479 . . . . . 6  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
1513, 14syl 15 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( -.  T  .<_  ( ( Q 
.\/  R )  .\/  S )  /\  -.  P  .<_  ( Q  .\/  R
)  /\  P  .<_  ( ( Q  .\/  R
)  .\/  S )
) )  ->  P  e.  ( Base `  K
) )
16 3dim0.j . . . . . 6  |-  .\/  =  ( join `  K )
176, 16latjrot 14206 . . . . 5  |-  ( ( K  e.  Lat  /\  ( Q  e.  ( Base `  K )  /\  R  e.  ( Base `  K )  /\  P  e.  ( Base `  K
) ) )  -> 
( ( Q  .\/  R )  .\/  P )  =  ( ( P 
.\/  Q )  .\/  R ) )
184, 9, 12, 15, 17syl13anc 1184 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( -.  T  .<_  ( ( Q 
.\/  R )  .\/  S )  /\  -.  P  .<_  ( Q  .\/  R
)  /\  P  .<_  ( ( Q  .\/  R
)  .\/  S )
) )  ->  (
( Q  .\/  R
)  .\/  P )  =  ( ( P 
.\/  Q )  .\/  R ) )
19 simp33 993 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( -.  T  .<_  ( ( Q 
.\/  R )  .\/  S )  /\  -.  P  .<_  ( Q  .\/  R
)  /\  P  .<_  ( ( Q  .\/  R
)  .\/  S )
) )  ->  P  .<_  ( ( Q  .\/  R )  .\/  S ) )
20 simp2r 982 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( -.  T  .<_  ( ( Q 
.\/  R )  .\/  S )  /\  -.  P  .<_  ( Q  .\/  R
)  /\  P  .<_  ( ( Q  .\/  R
)  .\/  S )
) )  ->  S  e.  A )
216, 16, 7hlatjcl 29556 . . . . . . 7  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  R  e.  A )  ->  ( Q  .\/  R
)  e.  ( Base `  K ) )
222, 5, 10, 21syl3anc 1182 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( -.  T  .<_  ( ( Q 
.\/  R )  .\/  S )  /\  -.  P  .<_  ( Q  .\/  R
)  /\  P  .<_  ( ( Q  .\/  R
)  .\/  S )
) )  ->  ( Q  .\/  R )  e.  ( Base `  K
) )
23 simp32 992 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( -.  T  .<_  ( ( Q 
.\/  R )  .\/  S )  /\  -.  P  .<_  ( Q  .\/  R
)  /\  P  .<_  ( ( Q  .\/  R
)  .\/  S )
) )  ->  -.  P  .<_  ( Q  .\/  R ) )
24 3dim0.l . . . . . . 7  |-  .<_  =  ( le `  K )
256, 24, 16, 7hlexchb1 29573 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  S  e.  A  /\  ( Q  .\/  R
)  e.  ( Base `  K ) )  /\  -.  P  .<_  ( Q 
.\/  R ) )  ->  ( P  .<_  ( ( Q  .\/  R
)  .\/  S )  <->  ( ( Q  .\/  R
)  .\/  P )  =  ( ( Q 
.\/  R )  .\/  S ) ) )
262, 13, 20, 22, 23, 25syl131anc 1195 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( -.  T  .<_  ( ( Q 
.\/  R )  .\/  S )  /\  -.  P  .<_  ( Q  .\/  R
)  /\  P  .<_  ( ( Q  .\/  R
)  .\/  S )
) )  ->  ( P  .<_  ( ( Q 
.\/  R )  .\/  S )  <->  ( ( Q 
.\/  R )  .\/  P )  =  ( ( Q  .\/  R ) 
.\/  S ) ) )
2719, 26mpbid 201 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( -.  T  .<_  ( ( Q 
.\/  R )  .\/  S )  /\  -.  P  .<_  ( Q  .\/  R
)  /\  P  .<_  ( ( Q  .\/  R
)  .\/  S )
) )  ->  (
( Q  .\/  R
)  .\/  P )  =  ( ( Q 
.\/  R )  .\/  S ) )
2818, 27eqtr3d 2317 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( -.  T  .<_  ( ( Q 
.\/  R )  .\/  S )  /\  -.  P  .<_  ( Q  .\/  R
)  /\  P  .<_  ( ( Q  .\/  R
)  .\/  S )
) )  ->  (
( P  .\/  Q
)  .\/  R )  =  ( ( Q 
.\/  R )  .\/  S ) )
2928breq2d 4035 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( -.  T  .<_  ( ( Q 
.\/  R )  .\/  S )  /\  -.  P  .<_  ( Q  .\/  R
)  /\  P  .<_  ( ( Q  .\/  R
)  .\/  S )
) )  ->  ( T  .<_  ( ( P 
.\/  Q )  .\/  R )  <->  T  .<_  ( ( Q  .\/  R ) 
.\/  S ) ) )
301, 29mtbird 292 1  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( -.  T  .<_  ( ( Q 
.\/  R )  .\/  S )  /\  -.  P  .<_  ( Q  .\/  R
)  /\  P  .<_  ( ( Q  .\/  R
)  .\/  S )
) )  ->  -.  T  .<_  ( ( P 
.\/  Q )  .\/  R ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   joincjn 14078   Latclat 14151   Atomscatm 29453   HLchlt 29540
This theorem is referenced by:  3dimlem3  29650  3dim3  29658
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-lub 14108  df-join 14110  df-lat 14152  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541
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