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Theorem cdlemefrs29pre00 34033
Description: ***START OF VALUE AT ATOM STUFF TO REPLACE ONES BELOW*** FIX COMMENT. TODO: see if this is the optimal utility theorem using lhpmat 33666. (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 1035 . . . . 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 650 . . . . . 6  |-  ( (
ph  /\  s  =  R )  <->  ( ps  /\  s  =  R ) )
43baibr 920 . . . . 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 2471 . . . . . . . . . 10  |-  ( 0.
`  K )  =  ( 0. `  K
)
9 cdlemefrs29.a . . . . . . . . . 10  |-  A  =  ( Atoms `  K )
10 cdlemefrs29.h . . . . . . . . . 10  |-  H  =  ( LHyp `  K
)
116, 7, 8, 9, 10lhpmat 33666 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  -> 
( R  ./\  W
)  =  ( 0.
`  K ) )
12113adant3 1050 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ps )  ->  ( R  ./\  W )  =  ( 0.
`  K ) )
1312adantr 472 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ps )  /\  s  e.  A )  ->  ( R  ./\  W
)  =  ( 0.
`  K ) )
1413oveq2d 6324 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ps )  /\  s  e.  A )  ->  ( s  .\/  ( R  ./\  W ) )  =  ( s  .\/  ( 0. `  K ) ) )
15 simpl1l 1081 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ps )  /\  s  e.  A )  ->  K  e.  HL )
16 hlol 32998 . . . . . . . 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 32926 . . . . . . . 8  |-  ( s  e.  A  ->  s  e.  B )
2019adantl 473 . . . . . . 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 32862 . . . . . . 7  |-  ( ( K  e.  OL  /\  s  e.  B )  ->  ( s  .\/  ( 0. `  K ) )  =  s )
2317, 20, 22syl2anc 673 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ps )  /\  s  e.  A )  ->  ( s  .\/  ( 0. `  K ) )  =  s )
2414, 23eqtrd 2505 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ps )  /\  s  e.  A )  ->  ( s  .\/  ( R  ./\  W ) )  =  s )
2524eqeq1d 2473 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ps )  /\  s  e.  A )  ->  ( ( s  .\/  ( R  ./\  W ) )  =  R  <->  s  =  R ) )
2625anbi2d 718 . . . 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 293 . . 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 718 . 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 661 . 2  |-  ( ( ( -.  s  .<_  W  /\  ph )  /\  ( s  .\/  ( R  ./\  W ) )  =  R )  <->  ( -.  s  .<_  W  /\  ( ph  /\  ( s  .\/  ( R  ./\  W ) )  =  R ) ) )
3028, 29syl6rbbr 272 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 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   class class class wbr 4395   ` cfv 5589  (class class class)co 6308   Basecbs 15199   lecple 15275   joincjn 16267   meetcmee 16268   0.cp0 16361   OLcol 32811   Atomscatm 32900   HLchlt 32987   LHypclh 33620
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-preset 16251  df-poset 16269  df-plt 16282  df-lub 16298  df-glb 16299  df-join 16300  df-meet 16301  df-p0 16363  df-lat 16370  df-oposet 32813  df-ol 32815  df-oml 32816  df-covers 32903  df-ats 32904  df-atl 32935  df-cvlat 32959  df-hlat 32988  df-lhyp 33624
This theorem is referenced by:  cdlemefrs29clN  34037  cdlemefrs32fva  34038  cdlemefs29pre00N  34050
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