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Theorem cdlemefrs29pre00 30584
Description: ***START OF VALUE AT ATOM STUFF TO REPLACE ONES BELOW*** FIX COMMENT. TODO: see if this is the optimal utility theorem using lhpmat 30219. (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 960 . . . . 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 619 . . . . . 6  |-  ( (
ph  /\  s  =  R )  <->  ( ps  /\  s  =  R ) )
43baibr 872 . . . . 5  |-  ( ps 
->  ( s  =  R  <-> 
( ph  /\  s  =  R ) ) )
51, 4syl 15 . . . 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 2283 . . . . . . . . . 10  |-  ( 0.
`  K )  =  ( 0. `  K
)
9 cdlemefrs29.a . . . . . . . . . 10  |-  A  =  ( Atoms `  K )
10 cdlemefrs29.h . . . . . . . . . 10  |-  H  =  ( LHyp `  K
)
116, 7, 8, 9, 10lhpmat 30219 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  -> 
( R  ./\  W
)  =  ( 0.
`  K ) )
12113adant3 975 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ps )  ->  ( R  ./\  W )  =  ( 0.
`  K ) )
1312adantr 451 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ps )  /\  s  e.  A )  ->  ( R  ./\  W
)  =  ( 0.
`  K ) )
1413oveq2d 5874 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ps )  /\  s  e.  A )  ->  ( s  .\/  ( R  ./\  W ) )  =  ( s  .\/  ( 0. `  K ) ) )
15 simpl1l 1006 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ps )  /\  s  e.  A )  ->  K  e.  HL )
16 hlol 29551 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  OL )
1715, 16syl 15 . . . . . . 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 29479 . . . . . . . 8  |-  ( s  e.  A  ->  s  e.  B )
2019adantl 452 . . . . . . 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 29415 . . . . . . 7  |-  ( ( K  e.  OL  /\  s  e.  B )  ->  ( s  .\/  ( 0. `  K ) )  =  s )
2317, 20, 22syl2anc 642 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ps )  /\  s  e.  A )  ->  ( s  .\/  ( 0. `  K ) )  =  s )
2414, 23eqtrd 2315 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ps )  /\  s  e.  A )  ->  ( s  .\/  ( R  ./\  W ) )  =  s )
2524eqeq1d 2291 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ps )  /\  s  e.  A )  ->  ( ( s  .\/  ( R  ./\  W ) )  =  R  <->  s  =  R ) )
2625anbi2d 684 . . . 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 276 . . 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 684 . 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 630 . 2  |-  ( ( ( -.  s  .<_  W  /\  ph )  /\  ( s  .\/  ( R  ./\  W ) )  =  R )  <->  ( -.  s  .<_  W  /\  ( ph  /\  ( s  .\/  ( R  ./\  W ) )  =  R ) ) )
3028, 29syl6rbbr 255 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 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   0.cp0 14143   OLcol 29364   Atomscatm 29453   HLchlt 29540   LHypclh 30173
This theorem is referenced by:  cdlemefrs29clN  30588  cdlemefrs32fva  30589  cdlemefs29pre00N  30601
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-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-lat 14152  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-lhyp 30177
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