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Theorem cdleme8 30364
Description: Part of proof of Lemma E in [Crawley] p. 113, 2nd paragraph on p. 114.  C represents s1. In their notation, we prove p  \/ s1 = p  \/ s. (Contributed by NM, 9-Jun-2012.)
Hypotheses
Ref Expression
cdleme8.l  |-  .<_  =  ( le `  K )
cdleme8.j  |-  .\/  =  ( join `  K )
cdleme8.m  |-  ./\  =  ( meet `  K )
cdleme8.a  |-  A  =  ( Atoms `  K )
cdleme8.h  |-  H  =  ( LHyp `  K
)
cdleme8.4  |-  C  =  ( ( P  .\/  S )  ./\  W )
Assertion
Ref Expression
cdleme8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  S  e.  A )  ->  ( P  .\/  C )  =  ( P  .\/  S
) )

Proof of Theorem cdleme8
StepHypRef Expression
1 cdleme8.4 . . 3  |-  C  =  ( ( P  .\/  S )  ./\  W )
21oveq2i 6031 . 2  |-  ( P 
.\/  C )  =  ( P  .\/  (
( P  .\/  S
)  ./\  W )
)
3 simp1l 981 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  S  e.  A )  ->  K  e.  HL )
4 simp2l 983 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  S  e.  A )  ->  P  e.  A )
5 hllat 29478 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
63, 5syl 16 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  S  e.  A )  ->  K  e.  Lat )
7 eqid 2387 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
8 cdleme8.a . . . . . . 7  |-  A  =  ( Atoms `  K )
97, 8atbase 29404 . . . . . 6  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
104, 9syl 16 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  S  e.  A )  ->  P  e.  ( Base `  K
) )
117, 8atbase 29404 . . . . . 6  |-  ( S  e.  A  ->  S  e.  ( Base `  K
) )
12113ad2ant3 980 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  S  e.  A )  ->  S  e.  ( Base `  K
) )
13 cdleme8.j . . . . . 6  |-  .\/  =  ( join `  K )
147, 13latjcl 14406 . . . . 5  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  S  e.  ( Base `  K
) )  ->  ( P  .\/  S )  e.  ( Base `  K
) )
156, 10, 12, 14syl3anc 1184 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  S  e.  A )  ->  ( P  .\/  S )  e.  ( Base `  K
) )
16 simp1r 982 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  S  e.  A )  ->  W  e.  H )
17 cdleme8.h . . . . . 6  |-  H  =  ( LHyp `  K
)
187, 17lhpbase 30112 . . . . 5  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
1916, 18syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  S  e.  A )  ->  W  e.  ( Base `  K
) )
20 cdleme8.l . . . . . 6  |-  .<_  =  ( le `  K )
217, 20, 13latlej1 14416 . . . . 5  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  S  e.  ( Base `  K
) )  ->  P  .<_  ( P  .\/  S
) )
226, 10, 12, 21syl3anc 1184 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  S  e.  A )  ->  P  .<_  ( P  .\/  S
) )
23 cdleme8.m . . . . 5  |-  ./\  =  ( meet `  K )
247, 20, 13, 23, 8atmod3i1 29978 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  ( P  .\/  S
)  e.  ( Base `  K )  /\  W  e.  ( Base `  K
) )  /\  P  .<_  ( P  .\/  S
) )  ->  ( P  .\/  ( ( P 
.\/  S )  ./\  W ) )  =  ( ( P  .\/  S
)  ./\  ( P  .\/  W ) ) )
253, 4, 15, 19, 22, 24syl131anc 1197 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  S  e.  A )  ->  ( P  .\/  ( ( P 
.\/  S )  ./\  W ) )  =  ( ( P  .\/  S
)  ./\  ( P  .\/  W ) ) )
26 eqid 2387 . . . . . 6  |-  ( 1.
`  K )  =  ( 1. `  K
)
2720, 13, 26, 8, 17lhpjat2 30135 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  -> 
( P  .\/  W
)  =  ( 1.
`  K ) )
28273adant3 977 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  S  e.  A )  ->  ( P  .\/  W )  =  ( 1. `  K
) )
2928oveq2d 6036 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  S  e.  A )  ->  (
( P  .\/  S
)  ./\  ( P  .\/  W ) )  =  ( ( P  .\/  S )  ./\  ( 1. `  K ) ) )
30 hlol 29476 . . . . 5  |-  ( K  e.  HL  ->  K  e.  OL )
313, 30syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  S  e.  A )  ->  K  e.  OL )
327, 23, 26olm11 29342 . . . 4  |-  ( ( K  e.  OL  /\  ( P  .\/  S )  e.  ( Base `  K
) )  ->  (
( P  .\/  S
)  ./\  ( 1. `  K ) )  =  ( P  .\/  S
) )
3331, 15, 32syl2anc 643 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  S  e.  A )  ->  (
( P  .\/  S
)  ./\  ( 1. `  K ) )  =  ( P  .\/  S
) )
3425, 29, 333eqtrd 2423 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  S  e.  A )  ->  ( P  .\/  ( ( P 
.\/  S )  ./\  W ) )  =  ( P  .\/  S ) )
352, 34syl5eq 2431 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  S  e.  A )  ->  ( P  .\/  C )  =  ( P  .\/  S
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   class class class wbr 4153   ` cfv 5394  (class class class)co 6020   Basecbs 13396   lecple 13463   joincjn 14328   meetcmee 14329   1.cp1 14394   Latclat 14401   OLcol 29289   Atomscatm 29378   HLchlt 29465   LHypclh 30098
This theorem is referenced by:  cdleme8tN  30369  cdleme15a  30388  cdleme17b  30401  cdlemg3a  30711
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 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-iin 4038  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-undef 6479  df-riota 6485  df-poset 14330  df-plt 14342  df-lub 14358  df-glb 14359  df-join 14360  df-meet 14361  df-p0 14395  df-p1 14396  df-lat 14402  df-clat 14464  df-oposet 29291  df-ol 29293  df-oml 29294  df-covers 29381  df-ats 29382  df-atl 29413  df-cvlat 29437  df-hlat 29466  df-psubsp 29617  df-pmap 29618  df-padd 29910  df-lhyp 30102
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