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Theorem lhpmcvr3 30214
Description: Specialization of lhpmcvr2 30213. 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 1006 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  P  .<_  X )  ->  K  e.  HL )
2 simpl3l 1010 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  P  .<_  X )  ->  P  e.  A )
3 simpl2l 1008 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  P  .<_  X )  ->  X  e.  B )
4 simpl1r 1007 . . . . 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 30187 . . . . 5  |-  ( W  e.  H  ->  W  e.  B )
84, 7syl 15 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  P  .<_  X )  ->  W  e.  B )
9 simpr 447 . . . 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 30053 . . . 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 1195 . . 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 958 . . . . 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 960 . . . . 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 2283 . . . . . 6  |-  ( 1.
`  K )  =  ( 1. `  K
)
1910, 11, 18, 13, 6lhpjat2 30210 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  -> 
( P  .\/  W
)  =  ( 1.
`  K ) )
2016, 17, 19syl2anc 642 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  P  .<_  X )  -> 
( P  .\/  W
)  =  ( 1.
`  K ) )
2120oveq2d 5874 . . 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 29551 . . . . 5  |-  ( K  e.  HL  ->  K  e.  OL )
231, 22syl 15 . . . 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 29417 . . . 4  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  ( X  ./\  ( 1. `  K ) )  =  X )
2523, 3, 24syl2anc 642 . . 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 2319 . 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 1006 . . . . 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 29553 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
2927, 28syl 15 . . . 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 1010 . . . . 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 29479 . . . . 5  |-  ( P  e.  A  ->  P  e.  B )
3230, 31syl 15 . . . 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 1008 . . . . 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 1007 . . . . . 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 15 . . . . 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 14157 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  W  e.  B )  ->  ( X  ./\  W
)  e.  B )
3729, 33, 35, 36syl3anc 1182 . . . 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 14166 . . . 4  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  ( X  ./\  W )  e.  B )  ->  P  .<_  ( P  .\/  ( X  ./\  W ) ) )
3929, 32, 37, 38syl3anc 1182 . . 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 447 . . 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 4047 . 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 805 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 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   meetcmee 14079   1.cp1 14144   Latclat 14151   OLcol 29364   Atomscatm 29453   HLchlt 29540   LHypclh 30173
This theorem is referenced by:  dihvalcq2  31437
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-iin 3908  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-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-psubsp 29692  df-pmap 29693  df-padd 29985  df-lhyp 30177
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