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Theorem cdleme22aa 29797
Description: Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph, 3rd line on p. 115. Show that t 
\/ v = p  \/ q implies v = u. (Contributed by NM, 2-Dec-2012.)
Hypotheses
Ref Expression
cdleme22.l  |-  .<_  =  ( le `  K )
cdleme22.j  |-  .\/  =  ( join `  K )
cdleme22.m  |-  ./\  =  ( meet `  K )
cdleme22.a  |-  A  =  ( Atoms `  K )
cdleme22.h  |-  H  =  ( LHyp `  K
)
cdleme22.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
Assertion
Ref Expression
cdleme22aa  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( V  e.  A  /\  V  .<_  W  /\  V  .<_  ( P  .\/  Q ) ) )  ->  V  =  U )

Proof of Theorem cdleme22aa
StepHypRef Expression
1 simp33 995 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( V  e.  A  /\  V  .<_  W  /\  V  .<_  ( P  .\/  Q ) ) )  ->  V  .<_  ( P  .\/  Q ) )
2 simp32 994 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( V  e.  A  /\  V  .<_  W  /\  V  .<_  ( P  .\/  Q ) ) )  ->  V  .<_  W )
3 simp1l 981 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( V  e.  A  /\  V  .<_  W  /\  V  .<_  ( P  .\/  Q ) ) )  ->  K  e.  HL )
4 hllat 28822 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
53, 4syl 17 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( V  e.  A  /\  V  .<_  W  /\  V  .<_  ( P  .\/  Q ) ) )  ->  K  e.  Lat )
6 simp31 993 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( V  e.  A  /\  V  .<_  W  /\  V  .<_  ( P  .\/  Q ) ) )  ->  V  e.  A )
7 eqid 2286 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
8 cdleme22.a . . . . . . 7  |-  A  =  ( Atoms `  K )
97, 8atbase 28748 . . . . . 6  |-  ( V  e.  A  ->  V  e.  ( Base `  K
) )
106, 9syl 17 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( V  e.  A  /\  V  .<_  W  /\  V  .<_  ( P  .\/  Q ) ) )  ->  V  e.  ( Base `  K ) )
11 simp21l 1074 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( V  e.  A  /\  V  .<_  W  /\  V  .<_  ( P  .\/  Q ) ) )  ->  P  e.  A )
12 simp22 991 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( V  e.  A  /\  V  .<_  W  /\  V  .<_  ( P  .\/  Q ) ) )  ->  Q  e.  A )
13 cdleme22.j . . . . . . 7  |-  .\/  =  ( join `  K )
147, 13, 8hlatjcl 28825 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
153, 11, 12, 14syl3anc 1184 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( V  e.  A  /\  V  .<_  W  /\  V  .<_  ( P  .\/  Q ) ) )  -> 
( P  .\/  Q
)  e.  ( Base `  K ) )
16 simp1r 982 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( V  e.  A  /\  V  .<_  W  /\  V  .<_  ( P  .\/  Q ) ) )  ->  W  e.  H )
17 cdleme22.h . . . . . . 7  |-  H  =  ( LHyp `  K
)
187, 17lhpbase 29456 . . . . . 6  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
1916, 18syl 17 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( V  e.  A  /\  V  .<_  W  /\  V  .<_  ( P  .\/  Q ) ) )  ->  W  e.  ( Base `  K ) )
20 cdleme22.l . . . . . 6  |-  .<_  =  ( le `  K )
21 cdleme22.m . . . . . 6  |-  ./\  =  ( meet `  K )
227, 20, 21latlem12 14180 . . . . 5  |-  ( ( K  e.  Lat  /\  ( V  e.  ( Base `  K )  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
) )  ->  (
( V  .<_  ( P 
.\/  Q )  /\  V  .<_  W )  <->  V  .<_  ( ( P  .\/  Q
)  ./\  W )
) )
235, 10, 15, 19, 22syl13anc 1186 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( V  e.  A  /\  V  .<_  W  /\  V  .<_  ( P  .\/  Q ) ) )  -> 
( ( V  .<_  ( P  .\/  Q )  /\  V  .<_  W )  <-> 
V  .<_  ( ( P 
.\/  Q )  ./\  W ) ) )
241, 2, 23mpbi2and 889 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( V  e.  A  /\  V  .<_  W  /\  V  .<_  ( P  .\/  Q ) ) )  ->  V  .<_  ( ( P 
.\/  Q )  ./\  W ) )
25 cdleme22.u . . 3  |-  U  =  ( ( P  .\/  Q )  ./\  W )
2624, 25syl6breqr 4066 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( V  e.  A  /\  V  .<_  W  /\  V  .<_  ( P  .\/  Q ) ) )  ->  V  .<_  U )
27 hlatl 28819 . . . 4  |-  ( K  e.  HL  ->  K  e.  AtLat )
283, 27syl 17 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( V  e.  A  /\  V  .<_  W  /\  V  .<_  ( P  .\/  Q ) ) )  ->  K  e.  AtLat )
29 simp21r 1075 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( V  e.  A  /\  V  .<_  W  /\  V  .<_  ( P  .\/  Q ) ) )  ->  -.  P  .<_  W )
30 simp23 992 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( V  e.  A  /\  V  .<_  W  /\  V  .<_  ( P  .\/  Q ) ) )  ->  P  =/=  Q )
3120, 13, 21, 8, 17, 25cdleme0a 29669 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q ) )  ->  U  e.  A
)
323, 16, 11, 29, 12, 30, 31syl222anc 1200 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( V  e.  A  /\  V  .<_  W  /\  V  .<_  ( P  .\/  Q ) ) )  ->  U  e.  A )
3320, 8atcmp 28770 . . 3  |-  ( ( K  e.  AtLat  /\  V  e.  A  /\  U  e.  A )  ->  ( V  .<_  U  <->  V  =  U ) )
3428, 6, 32, 33syl3anc 1184 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( V  e.  A  /\  V  .<_  W  /\  V  .<_  ( P  .\/  Q ) ) )  -> 
( V  .<_  U  <->  V  =  U ) )
3526, 34mpbid 203 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( V  e.  A  /\  V  .<_  W  /\  V  .<_  ( P  .\/  Q ) ) )  ->  V  =  U )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 936    = wceq 1625    e. wcel 1687    =/= wne 2449   class class class wbr 4026   ` cfv 5223  (class class class)co 5821   Basecbs 13144   lecple 13211   joincjn 14074   meetcmee 14075   Latclat 14147   Atomscatm 28722   AtLatcal 28723   HLchlt 28809   LHypclh 29442
This theorem is referenced by:  cdleme22a  29798  cdleme22cN  29800  cdleme22f  29804
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1638  ax-8 1646  ax-13 1689  ax-14 1691  ax-6 1706  ax-7 1711  ax-11 1718  ax-12 1870  ax-ext 2267  ax-rep 4134  ax-sep 4144  ax-nul 4152  ax-pow 4189  ax-pr 4215  ax-un 4513
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1531  df-nf 1534  df-sb 1633  df-eu 2150  df-mo 2151  df-clab 2273  df-cleq 2279  df-clel 2282  df-nfc 2411  df-ne 2451  df-nel 2452  df-ral 2551  df-rex 2552  df-reu 2553  df-rab 2555  df-v 2793  df-sbc 2995  df-csb 3085  df-dif 3158  df-un 3160  df-in 3162  df-ss 3169  df-nul 3459  df-if 3569  df-pw 3630  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3831  df-iun 3910  df-br 4027  df-opab 4081  df-mpt 4082  df-id 4310  df-xp 4696  df-rel 4697  df-cnv 4698  df-co 4699  df-dm 4700  df-rn 4701  df-res 4702  df-ima 4703  df-fun 5225  df-fn 5226  df-f 5227  df-f1 5228  df-fo 5229  df-f1o 5230  df-fv 5231  df-ov 5824  df-oprab 5825  df-mpt2 5826  df-1st 6085  df-2nd 6086  df-iota 6254  df-undef 6293  df-riota 6301  df-poset 14076  df-plt 14088  df-lub 14104  df-glb 14105  df-join 14106  df-meet 14107  df-p0 14141  df-p1 14142  df-lat 14148  df-clat 14210  df-oposet 28635  df-ol 28637  df-oml 28638  df-covers 28725  df-ats 28726  df-atl 28757  df-cvlat 28781  df-hlat 28810  df-lhyp 29446
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