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Theorem lhpmcvr3 33666
Description: Specialization of lhpmcvr2 33665. TODO: Use this to simplify many uses of  ( P  .\/  ( X  ./\  W ) )  =  X to become  P  .<_  X. (Contributed by NM, 6-Apr-2014.)
Hypotheses
Ref Expression
lhpmcvr2.b  |-  B  =  ( Base `  K
)
lhpmcvr2.l  |-  .<_  =  ( le `  K )
lhpmcvr2.j  |-  .\/  =  ( join `  K )
lhpmcvr2.m  |-  ./\  =  ( meet `  K )
lhpmcvr2.a  |-  A  =  ( Atoms `  K )
lhpmcvr2.h  |-  H  =  ( LHyp `  K
)
Assertion
Ref Expression
lhpmcvr3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( P  .<_  X  <->  ( P  .\/  ( X  ./\  W ) )  =  X ) )

Proof of Theorem lhpmcvr3
StepHypRef Expression
1 simpl1l 1039 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  P  .<_  X )  ->  K  e.  HL )
2 simpl3l 1043 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  P  .<_  X )  ->  P  e.  A )
3 simpl2l 1041 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  P  .<_  X )  ->  X  e.  B )
4 simpl1r 1040 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  P  .<_  X )  ->  W  e.  H )
5 lhpmcvr2.b . . . . . 6  |-  B  =  ( Base `  K
)
6 lhpmcvr2.h . . . . . 6  |-  H  =  ( LHyp `  K
)
75, 6lhpbase 33639 . . . . 5  |-  ( W  e.  H  ->  W  e.  B )
84, 7syl 16 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  P  .<_  X )  ->  W  e.  B )
9 simpr 461 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  P  .<_  X )  ->  P  .<_  X )
10 lhpmcvr2.l . . . . 5  |-  .<_  =  ( le `  K )
11 lhpmcvr2.j . . . . 5  |-  .\/  =  ( join `  K )
12 lhpmcvr2.m . . . . 5  |-  ./\  =  ( meet `  K )
13 lhpmcvr2.a . . . . 5  |-  A  =  ( Atoms `  K )
145, 10, 11, 12, 13atmod3i1 33505 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  W  e.  B
)  /\  P  .<_  X )  ->  ( P  .\/  ( X  ./\  W
) )  =  ( X  ./\  ( P  .\/  W ) ) )
151, 2, 3, 8, 9, 14syl131anc 1231 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  P  .<_  X )  -> 
( P  .\/  ( X  ./\  W ) )  =  ( X  ./\  ( P  .\/  W ) ) )
16 simpl1 991 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  P  .<_  X )  -> 
( K  e.  HL  /\  W  e.  H ) )
17 simpl3 993 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  P  .<_  X )  -> 
( P  e.  A  /\  -.  P  .<_  W ) )
18 eqid 2441 . . . . . 6  |-  ( 1.
`  K )  =  ( 1. `  K
)
1910, 11, 18, 13, 6lhpjat2 33662 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  -> 
( P  .\/  W
)  =  ( 1.
`  K ) )
2016, 17, 19syl2anc 661 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  P  .<_  X )  -> 
( P  .\/  W
)  =  ( 1.
`  K ) )
2120oveq2d 6105 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  P  .<_  X )  -> 
( X  ./\  ( P  .\/  W ) )  =  ( X  ./\  ( 1. `  K ) ) )
22 hlol 33003 . . . . 5  |-  ( K  e.  HL  ->  K  e.  OL )
231, 22syl 16 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  P  .<_  X )  ->  K  e.  OL )
245, 12, 18olm11 32869 . . . 4  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  ( X  ./\  ( 1. `  K ) )  =  X )
2523, 3, 24syl2anc 661 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  P  .<_  X )  -> 
( X  ./\  ( 1. `  K ) )  =  X )
2615, 21, 253eqtrd 2477 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  P  .<_  X )  -> 
( P  .\/  ( X  ./\  W ) )  =  X )
27 simpl1l 1039 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( P  .\/  ( X 
./\  W ) )  =  X )  ->  K  e.  HL )
28 hllat 33005 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
2927, 28syl 16 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( P  .\/  ( X 
./\  W ) )  =  X )  ->  K  e.  Lat )
30 simpl3l 1043 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( P  .\/  ( X 
./\  W ) )  =  X )  ->  P  e.  A )
315, 13atbase 32931 . . . . 5  |-  ( P  e.  A  ->  P  e.  B )
3230, 31syl 16 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( P  .\/  ( X 
./\  W ) )  =  X )  ->  P  e.  B )
33 simpl2l 1041 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( P  .\/  ( X 
./\  W ) )  =  X )  ->  X  e.  B )
34 simpl1r 1040 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( P  .\/  ( X 
./\  W ) )  =  X )  ->  W  e.  H )
3534, 7syl 16 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( P  .\/  ( X 
./\  W ) )  =  X )  ->  W  e.  B )
365, 12latmcl 15220 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  W  e.  B )  ->  ( X  ./\  W
)  e.  B )
3729, 33, 35, 36syl3anc 1218 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( P  .\/  ( X 
./\  W ) )  =  X )  -> 
( X  ./\  W
)  e.  B )
385, 10, 11latlej1 15228 . . . 4  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  ( X  ./\  W )  e.  B )  ->  P  .<_  ( P  .\/  ( X  ./\  W ) ) )
3929, 32, 37, 38syl3anc 1218 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( P  .\/  ( X 
./\  W ) )  =  X )  ->  P  .<_  ( P  .\/  ( X  ./\  W ) ) )
40 simpr 461 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( P  .\/  ( X 
./\  W ) )  =  X )  -> 
( P  .\/  ( X  ./\  W ) )  =  X )
4139, 40breqtrd 4314 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( P  .\/  ( X 
./\  W ) )  =  X )  ->  P  .<_  X )
4226, 41impbida 828 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( P  .<_  X  <->  ( P  .\/  ( X  ./\  W ) )  =  X ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   class class class wbr 4290   ` cfv 5416  (class class class)co 6089   Basecbs 14172   lecple 14243   joincjn 15112   meetcmee 15113   1.cp1 15206   Latclat 15213   OLcol 32816   Atomscatm 32905   HLchlt 32992   LHypclh 33625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-iun 4171  df-iin 4172  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-1st 6575  df-2nd 6576  df-poset 15114  df-plt 15126  df-lub 15142  df-glb 15143  df-join 15144  df-meet 15145  df-p0 15207  df-p1 15208  df-lat 15214  df-clat 15276  df-oposet 32818  df-ol 32820  df-oml 32821  df-covers 32908  df-ats 32909  df-atl 32940  df-cvlat 32964  df-hlat 32993  df-psubsp 33144  df-pmap 33145  df-padd 33437  df-lhyp 33629
This theorem is referenced by:  dihvalcq2  34889
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