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Theorem cdlemefrs29cpre1 30880
Description: TODO FIX COMMENT (Contributed by NM, 29-Mar-2013.)
Hypotheses
Ref Expression
cdlemefrs27.b  |-  B  =  ( Base `  K
)
cdlemefrs27.l  |-  .<_  =  ( le `  K )
cdlemefrs27.j  |-  .\/  =  ( join `  K )
cdlemefrs27.m  |-  ./\  =  ( meet `  K )
cdlemefrs27.a  |-  A  =  ( Atoms `  K )
cdlemefrs27.h  |-  H  =  ( LHyp `  K
)
cdlemefrs27.eq  |-  ( s  =  R  ->  ( ph 
<->  ps ) )
cdlemefrs27.nb  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q  /\  (
s  e.  A  /\  ( -.  s  .<_  W  /\  ph ) ) )  ->  N  e.  B )
cdlemefrs27.rnb  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  ->  [_ R  /  s ]_ N  e.  B
)
Assertion
Ref Expression
cdlemefrs29cpre1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  ->  E! z  e.  B  A. s  e.  A  ( ( ( -.  s  .<_  W  /\  ph )  /\  ( s 
.\/  ( R  ./\  W ) )  =  R )  ->  z  =  ( N  .\/  ( R 
./\  W ) ) ) )
Distinct variable groups:    z, s, A    H, s    .\/ , s    K, s    .<_ , s    P, s    Q, s    R, s    W, s    ps, s    z, A    z, B    z, H    z, K    z, 
.<_    z, N    z, P    z, Q    z, R    z, W    ps, z    B, s   
z,  .\/    ./\ , s, z    ph, z
Allowed substitution hints:    ph( s)    N( s)

Proof of Theorem cdlemefrs29cpre1
StepHypRef Expression
1 cdlemefrs27.b . . 3  |-  B  =  ( Base `  K
)
2 cdlemefrs27.l . . 3  |-  .<_  =  ( le `  K )
3 cdlemefrs27.j . . 3  |-  .\/  =  ( join `  K )
4 cdlemefrs27.m . . 3  |-  ./\  =  ( meet `  K )
5 cdlemefrs27.a . . 3  |-  A  =  ( Atoms `  K )
6 cdlemefrs27.h . . 3  |-  H  =  ( LHyp `  K
)
7 cdlemefrs27.eq . . 3  |-  ( s  =  R  ->  ( ph 
<->  ps ) )
8 cdlemefrs27.nb . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q  /\  (
s  e.  A  /\  ( -.  s  .<_  W  /\  ph ) ) )  ->  N  e.  B )
9 cdlemefrs27.rnb . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  ->  [_ R  /  s ]_ N  e.  B
)
101, 2, 3, 4, 5, 6, 7, 8, 9cdlemefrs29bpre1 30879 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  ->  E. z  e.  B  A. s  e.  A  ( ( ( -.  s  .<_  W  /\  ph )  /\  ( s 
.\/  ( R  ./\  W ) )  =  R )  ->  z  =  ( N  .\/  ( R 
./\  W ) ) ) )
11 simp11 987 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  ->  ( K  e.  HL  /\  W  e.  H ) )
12 simp2rl 1026 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  ->  R  e.  A )
131, 5atbase 29772 . . . . . 6  |-  ( R  e.  A  ->  R  e.  B )
1412, 13syl 16 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  ->  R  e.  B )
15 simp2rr 1027 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  ->  -.  R  .<_  W )
161, 2, 3, 4, 5, 6lhpmcvr2 30506 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  B  /\  -.  R  .<_  W ) )  ->  E. s  e.  A  ( -.  s  .<_  W  /\  ( s  .\/  ( R  ./\  W ) )  =  R ) )
1711, 14, 15, 16syl12anc 1182 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  ->  E. s  e.  A  ( -.  s  .<_  W  /\  ( s  .\/  ( R  ./\  W ) )  =  R ) )
18 simpl3 962 . . . . . . . . 9  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  /\  s  e.  A
)  ->  ps )
197pm5.32ri 620 . . . . . . . . . 10  |-  ( (
ph  /\  s  =  R )  <->  ( ps  /\  s  =  R ) )
2019baibr 873 . . . . . . . . 9  |-  ( ps 
->  ( s  =  R  <-> 
( ph  /\  s  =  R ) ) )
2118, 20syl 16 . . . . . . . 8  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  /\  s  e.  A
)  ->  ( s  =  R  <->  ( ph  /\  s  =  R )
) )
22 simp2r 984 . . . . . . . . . . . . 13  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  ->  ( R  e.  A  /\  -.  R  .<_  W ) )
23 eqid 2404 . . . . . . . . . . . . . 14  |-  ( 0.
`  K )  =  ( 0. `  K
)
242, 4, 23, 5, 6lhpmat 30512 . . . . . . . . . . . . 13  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  -> 
( R  ./\  W
)  =  ( 0.
`  K ) )
2511, 22, 24syl2anc 643 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  ->  ( R  ./\  W
)  =  ( 0.
`  K ) )
2625adantr 452 . . . . . . . . . . 11  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  /\  s  e.  A
)  ->  ( R  ./\ 
W )  =  ( 0. `  K ) )
2726oveq2d 6056 . . . . . . . . . 10  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  /\  s  e.  A
)  ->  ( s  .\/  ( R  ./\  W
) )  =  ( s  .\/  ( 0.
`  K ) ) )
28 simp11l 1068 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  ->  K  e.  HL )
29 hlol 29844 . . . . . . . . . . . 12  |-  ( K  e.  HL  ->  K  e.  OL )
3028, 29syl 16 . . . . . . . . . . 11  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  ->  K  e.  OL )
311, 5atbase 29772 . . . . . . . . . . 11  |-  ( s  e.  A  ->  s  e.  B )
321, 3, 23olj01 29708 . . . . . . . . . . 11  |-  ( ( K  e.  OL  /\  s  e.  B )  ->  ( s  .\/  ( 0. `  K ) )  =  s )
3330, 31, 32syl2an 464 . . . . . . . . . 10  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  /\  s  e.  A
)  ->  ( s  .\/  ( 0. `  K
) )  =  s )
3427, 33eqtrd 2436 . . . . . . . . 9  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  /\  s  e.  A
)  ->  ( s  .\/  ( R  ./\  W
) )  =  s )
3534eqeq1d 2412 . . . . . . . 8  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  /\  s  e.  A
)  ->  ( (
s  .\/  ( R  ./\ 
W ) )  =  R  <->  s  =  R ) )
3635anbi2d 685 . . . . . . . 8  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  /\  s  e.  A
)  ->  ( ( ph  /\  ( s  .\/  ( R  ./\  W ) )  =  R )  <-> 
( ph  /\  s  =  R ) ) )
3721, 35, 363bitr4d 277 . . . . . . 7  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  /\  s  e.  A
)  ->  ( (
s  .\/  ( R  ./\ 
W ) )  =  R  <->  ( ph  /\  ( s  .\/  ( R  ./\  W ) )  =  R ) ) )
3837anbi2d 685 . . . . . 6  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  /\  s  e.  A
)  ->  ( ( -.  s  .<_  W  /\  ( s  .\/  ( R  ./\  W ) )  =  R )  <->  ( -.  s  .<_  W  /\  ( ph  /\  ( s  .\/  ( R  ./\  W ) )  =  R ) ) ) )
39 anass 631 . . . . . 6  |-  ( ( ( -.  s  .<_  W  /\  ph )  /\  ( s  .\/  ( R  ./\  W ) )  =  R )  <->  ( -.  s  .<_  W  /\  ( ph  /\  ( s  .\/  ( R  ./\  W ) )  =  R ) ) )
4038, 39syl6bbr 255 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  /\  s  e.  A
)  ->  ( ( -.  s  .<_  W  /\  ( s  .\/  ( R  ./\  W ) )  =  R )  <->  ( ( -.  s  .<_  W  /\  ph )  /\  ( s 
.\/  ( R  ./\  W ) )  =  R ) ) )
4140rexbidva 2683 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  ->  ( E. s  e.  A  ( -.  s  .<_  W  /\  ( s 
.\/  ( R  ./\  W ) )  =  R )  <->  E. s  e.  A  ( ( -.  s  .<_  W  /\  ph )  /\  ( s  .\/  ( R  ./\  W ) )  =  R ) ) )
4217, 41mpbid 202 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  ->  E. s  e.  A  ( ( -.  s  .<_  W  /\  ph )  /\  ( s  .\/  ( R  ./\  W ) )  =  R ) )
43 reusv1 4682 . . 3  |-  ( E. s  e.  A  ( ( -.  s  .<_  W  /\  ph )  /\  ( s  .\/  ( R  ./\  W ) )  =  R )  -> 
( E! z  e.  B  A. s  e.  A  ( ( ( -.  s  .<_  W  /\  ph )  /\  ( s 
.\/  ( R  ./\  W ) )  =  R )  ->  z  =  ( N  .\/  ( R 
./\  W ) ) )  <->  E. z  e.  B  A. s  e.  A  ( ( ( -.  s  .<_  W  /\  ph )  /\  ( s 
.\/  ( R  ./\  W ) )  =  R )  ->  z  =  ( N  .\/  ( R 
./\  W ) ) ) ) )
4442, 43syl 16 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  ->  ( E! z  e.  B  A. s  e.  A  ( ( ( -.  s  .<_  W  /\  ph )  /\  ( s 
.\/  ( R  ./\  W ) )  =  R )  ->  z  =  ( N  .\/  ( R 
./\  W ) ) )  <->  E. z  e.  B  A. s  e.  A  ( ( ( -.  s  .<_  W  /\  ph )  /\  ( s 
.\/  ( R  ./\  W ) )  =  R )  ->  z  =  ( N  .\/  ( R 
./\  W ) ) ) ) )
4510, 44mpbird 224 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  ->  E! z  e.  B  A. s  e.  A  ( ( ( -.  s  .<_  W  /\  ph )  /\  ( s 
.\/  ( R  ./\  W ) )  =  R )  ->  z  =  ( N  .\/  ( R 
./\  W ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   A.wral 2666   E.wrex 2667   E!wreu 2668   [_csb 3211   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   Basecbs 13424   lecple 13491   joincjn 14356   meetcmee 14357   0.cp0 14421   OLcol 29657   Atomscatm 29746   HLchlt 29833   LHypclh 30466
This theorem is referenced by:  cdlemefrs29clN  30881  cdlemefrs32fva  30882  cdlemefs29cpre1N  30900
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-p1 14424  df-lat 14430  df-clat 14492  df-oposet 29659  df-ol 29661  df-oml 29662  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834  df-lhyp 30470
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