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Theorem lhpmcvr3 34830
Description: Specialization of lhpmcvr2 34829. 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 1047 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  P  .<_  X )  ->  K  e.  HL )
2 simpl3l 1051 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  P  .<_  X )  ->  P  e.  A )
3 simpl2l 1049 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  P  .<_  X )  ->  X  e.  B )
4 simpl1r 1048 . . . . 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 34803 . . . . 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 34669 . . . 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 1241 . . 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 999 . . . . 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 1001 . . . . 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 2467 . . . . . 6  |-  ( 1.
`  K )  =  ( 1. `  K
)
1910, 11, 18, 13, 6lhpjat2 34826 . . . . 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 6299 . . 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 34167 . . . . 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 34033 . . . 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 2512 . 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 1047 . . . . 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 34169 . . . . 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 1051 . . . . 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 34095 . . . . 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 1049 . . . . 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 1048 . . . . . 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 15538 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  W  e.  B )  ->  ( X  ./\  W
)  e.  B )
3729, 33, 35, 36syl3anc 1228 . . . 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 15546 . . . 4  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  ( X  ./\  W )  e.  B )  ->  P  .<_  ( P  .\/  ( X  ./\  W ) ) )
3929, 32, 37, 38syl3anc 1228 . . 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 4471 . 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 830 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 973    = wceq 1379    e. wcel 1767   class class class wbr 4447   ` cfv 5587  (class class class)co 6283   Basecbs 14489   lecple 14561   joincjn 15430   meetcmee 15431   1.cp1 15524   Latclat 15531   OLcol 33980   Atomscatm 34069   HLchlt 34156   LHypclh 34789
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-1st 6784  df-2nd 6785  df-poset 15432  df-plt 15444  df-lub 15460  df-glb 15461  df-join 15462  df-meet 15463  df-p0 15525  df-p1 15526  df-lat 15532  df-clat 15594  df-oposet 33982  df-ol 33984  df-oml 33985  df-covers 34072  df-ats 34073  df-atl 34104  df-cvlat 34128  df-hlat 34157  df-psubsp 34308  df-pmap 34309  df-padd 34601  df-lhyp 34793
This theorem is referenced by:  dihvalcq2  36053
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