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Theorem cdlemefrs29pre00 30560
Description: ***START OF VALUE AT ATOM STUFF TO REPLACE ONES BELOW*** FIX COMMENT. TODO: see if this is the optimal utility theorem using lhpmat 30195. (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 962 . . . . 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 620 . . . . . 6  |-  ( (
ph  /\  s  =  R )  <->  ( ps  /\  s  =  R ) )
43baibr 873 . . . . 5  |-  ( ps 
->  ( s  =  R  <-> 
( ph  /\  s  =  R ) ) )
51, 4syl 16 . . . 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 2380 . . . . . . . . . 10  |-  ( 0.
`  K )  =  ( 0. `  K
)
9 cdlemefrs29.a . . . . . . . . . 10  |-  A  =  ( Atoms `  K )
10 cdlemefrs29.h . . . . . . . . . 10  |-  H  =  ( LHyp `  K
)
116, 7, 8, 9, 10lhpmat 30195 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  -> 
( R  ./\  W
)  =  ( 0.
`  K ) )
12113adant3 977 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ps )  ->  ( R  ./\  W )  =  ( 0.
`  K ) )
1312adantr 452 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ps )  /\  s  e.  A )  ->  ( R  ./\  W
)  =  ( 0.
`  K ) )
1413oveq2d 6029 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ps )  /\  s  e.  A )  ->  ( s  .\/  ( R  ./\  W ) )  =  ( s  .\/  ( 0. `  K ) ) )
15 simpl1l 1008 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ps )  /\  s  e.  A )  ->  K  e.  HL )
16 hlol 29527 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  OL )
1715, 16syl 16 . . . . . . 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 29455 . . . . . . . 8  |-  ( s  e.  A  ->  s  e.  B )
2019adantl 453 . . . . . . 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 29391 . . . . . . 7  |-  ( ( K  e.  OL  /\  s  e.  B )  ->  ( s  .\/  ( 0. `  K ) )  =  s )
2317, 20, 22syl2anc 643 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ps )  /\  s  e.  A )  ->  ( s  .\/  ( 0. `  K ) )  =  s )
2414, 23eqtrd 2412 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ps )  /\  s  e.  A )  ->  ( s  .\/  ( R  ./\  W ) )  =  s )
2524eqeq1d 2388 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ps )  /\  s  e.  A )  ->  ( ( s  .\/  ( R  ./\  W ) )  =  R  <->  s  =  R ) )
2625anbi2d 685 . . . 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 277 . . 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 685 . 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 631 . 2  |-  ( ( ( -.  s  .<_  W  /\  ph )  /\  ( s  .\/  ( R  ./\  W ) )  =  R )  <->  ( -.  s  .<_  W  /\  ( ph  /\  ( s  .\/  ( R  ./\  W ) )  =  R ) ) )
3028, 29syl6rbbr 256 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 set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   class class class wbr 4146   ` cfv 5387  (class class class)co 6013   Basecbs 13389   lecple 13456   joincjn 14321   meetcmee 14322   0.cp0 14386   OLcol 29340   Atomscatm 29429   HLchlt 29516   LHypclh 30149
This theorem is referenced by:  cdlemefrs29clN  30564  cdlemefrs32fva  30565  cdlemefs29pre00N  30577
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-undef 6472  df-riota 6478  df-poset 14323  df-plt 14335  df-lub 14351  df-glb 14352  df-join 14353  df-meet 14354  df-p0 14388  df-lat 14395  df-oposet 29342  df-ol 29344  df-oml 29345  df-covers 29432  df-ats 29433  df-atl 29464  df-cvlat 29488  df-hlat 29517  df-lhyp 30153
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