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Theorem cdlemefrs29pre00 33925
Description: ***START OF VALUE AT ATOM STUFF TO REPLACE ONES BELOW*** FIX COMMENT. TODO: see if this is the optimal utility theorem using lhpmat 33558. (Contributed by NM, 29-Mar-2013.)
Hypotheses
Ref Expression
cdlemefrs29.b  |-  B  =  ( Base `  K
)
cdlemefrs29.l  |-  .<_  =  ( le `  K )
cdlemefrs29.j  |-  .\/  =  ( join `  K )
cdlemefrs29.m  |-  ./\  =  ( meet `  K )
cdlemefrs29.a  |-  A  =  ( Atoms `  K )
cdlemefrs29.h  |-  H  =  ( LHyp `  K
)
cdlemefrs29.eq  |-  ( s  =  R  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
cdlemefrs29pre00  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ps )  /\  s  e.  A )  ->  ( ( ( -.  s  .<_  W  /\  ph )  /\  ( s 
.\/  ( R  ./\  W ) )  =  R )  <->  ( -.  s  .<_  W  /\  ( s 
.\/  ( R  ./\  W ) )  =  R ) ) )

Proof of Theorem cdlemefrs29pre00
StepHypRef Expression
1 simpl3 1011 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ps )  /\  s  e.  A )  ->  ps )
2 cdlemefrs29.eq . . . . . . 7  |-  ( s  =  R  ->  ( ph 
<->  ps ) )
32pm5.32ri 643 . . . . . 6  |-  ( (
ph  /\  s  =  R )  <->  ( ps  /\  s  =  R ) )
43baibr 913 . . . . 5  |-  ( ps 
->  ( s  =  R  <-> 
( ph  /\  s  =  R ) ) )
51, 4syl 17 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ps )  /\  s  e.  A )  ->  ( s  =  R  <-> 
( ph  /\  s  =  R ) ) )
6 cdlemefrs29.l . . . . . . . . . 10  |-  .<_  =  ( le `  K )
7 cdlemefrs29.m . . . . . . . . . 10  |-  ./\  =  ( meet `  K )
8 eqid 2423 . . . . . . . . . 10  |-  ( 0.
`  K )  =  ( 0. `  K
)
9 cdlemefrs29.a . . . . . . . . . 10  |-  A  =  ( Atoms `  K )
10 cdlemefrs29.h . . . . . . . . . 10  |-  H  =  ( LHyp `  K
)
116, 7, 8, 9, 10lhpmat 33558 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  -> 
( R  ./\  W
)  =  ( 0.
`  K ) )
12113adant3 1026 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ps )  ->  ( R  ./\  W )  =  ( 0.
`  K ) )
1312adantr 467 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ps )  /\  s  e.  A )  ->  ( R  ./\  W
)  =  ( 0.
`  K ) )
1413oveq2d 6319 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ps )  /\  s  e.  A )  ->  ( s  .\/  ( R  ./\  W ) )  =  ( s  .\/  ( 0. `  K ) ) )
15 simpl1l 1057 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ps )  /\  s  e.  A )  ->  K  e.  HL )
16 hlol 32890 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  OL )
1715, 16syl 17 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ps )  /\  s  e.  A )  ->  K  e.  OL )
18 cdlemefrs29.b . . . . . . . . 9  |-  B  =  ( Base `  K
)
1918, 9atbase 32818 . . . . . . . 8  |-  ( s  e.  A  ->  s  e.  B )
2019adantl 468 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ps )  /\  s  e.  A )  ->  s  e.  B )
21 cdlemefrs29.j . . . . . . . 8  |-  .\/  =  ( join `  K )
2218, 21, 8olj01 32754 . . . . . . 7  |-  ( ( K  e.  OL  /\  s  e.  B )  ->  ( s  .\/  ( 0. `  K ) )  =  s )
2317, 20, 22syl2anc 666 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ps )  /\  s  e.  A )  ->  ( s  .\/  ( 0. `  K ) )  =  s )
2414, 23eqtrd 2464 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ps )  /\  s  e.  A )  ->  ( s  .\/  ( R  ./\  W ) )  =  s )
2524eqeq1d 2425 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ps )  /\  s  e.  A )  ->  ( ( s  .\/  ( R  ./\  W ) )  =  R  <->  s  =  R ) )
2625anbi2d 709 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ps )  /\  s  e.  A )  ->  ( ( ph  /\  ( s  .\/  ( R  ./\  W ) )  =  R )  <->  ( ph  /\  s  =  R ) ) )
275, 25, 263bitr4d 289 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ps )  /\  s  e.  A )  ->  ( ( s  .\/  ( R  ./\  W ) )  =  R  <->  ( ph  /\  ( s  .\/  ( R  ./\  W ) )  =  R ) ) )
2827anbi2d 709 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ps )  /\  s  e.  A )  ->  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( R  ./\  W ) )  =  R )  <->  ( -.  s  .<_  W  /\  ( ph  /\  ( s  .\/  ( R  ./\  W ) )  =  R ) ) ) )
29 anass 654 . 2  |-  ( ( ( -.  s  .<_  W  /\  ph )  /\  ( s  .\/  ( R  ./\  W ) )  =  R )  <->  ( -.  s  .<_  W  /\  ( ph  /\  ( s  .\/  ( R  ./\  W ) )  =  R ) ) )
3028, 29syl6rbbr 268 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ps )  /\  s  e.  A )  ->  ( ( ( -.  s  .<_  W  /\  ph )  /\  ( s 
.\/  ( R  ./\  W ) )  =  R )  <->  ( -.  s  .<_  W  /\  ( s 
.\/  ( R  ./\  W ) )  =  R ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 983    = wceq 1438    e. wcel 1869   class class class wbr 4421   ` cfv 5599  (class class class)co 6303   Basecbs 15114   lecple 15190   joincjn 16182   meetcmee 16183   0.cp0 16276   OLcol 32703   Atomscatm 32792   HLchlt 32879   LHypclh 33512
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-reu 2783  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-id 4766  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-riota 6265  df-ov 6306  df-oprab 6307  df-preset 16166  df-poset 16184  df-plt 16197  df-lub 16213  df-glb 16214  df-join 16215  df-meet 16216  df-p0 16278  df-lat 16285  df-oposet 32705  df-ol 32707  df-oml 32708  df-covers 32795  df-ats 32796  df-atl 32827  df-cvlat 32851  df-hlat 32880  df-lhyp 33516
This theorem is referenced by:  cdlemefrs29clN  33929  cdlemefrs32fva  33930  cdlemefs29pre00N  33942
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