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Theorem cdlemefrs29pre00 31206
Description: ***START OF VALUE AT ATOM STUFF TO REPLACE ONES BELOW*** FIX COMMENT. TODO: see if this is the optimal utility theorem using lhpmat 30841. (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 2296 . . . . . . . . . 10  |-  ( 0.
`  K )  =  ( 0. `  K
)
9 cdlemefrs29.a . . . . . . . . . 10  |-  A  =  ( Atoms `  K )
10 cdlemefrs29.h . . . . . . . . . 10  |-  H  =  ( LHyp `  K
)
116, 7, 8, 9, 10lhpmat 30841 . . . . . . . . 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 5890 . . . . . 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 30173 . . . . . . . 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 30101 . . . . . . . 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 30037 . . . . . . 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 2328 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ps )  /\  s  e.  A )  ->  ( s  .\/  ( R  ./\  W ) )  =  s )
2524eqeq1d 2304 . . . 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 1632    e. wcel 1696   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   joincjn 14094   meetcmee 14095   0.cp0 14159   OLcol 29986   Atomscatm 30075   HLchlt 30162   LHypclh 30795
This theorem is referenced by:  cdlemefrs29clN  31210  cdlemefrs32fva  31211  cdlemefs29pre00N  31223
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-lat 14168  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-lhyp 30799
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