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Theorem cdleme10 29711
Description: Part of proof of Lemma E in [Crawley] p. 113, 2nd paragraph on p. 114.  D represents s2. In their notation, we prove s  \/ s2 = s  \/ r. (Contributed by NM, 9-Jun-2012.)
Hypotheses
Ref Expression
cdleme10.l  |-  .<_  =  ( le `  K )
cdleme10.j  |-  .\/  =  ( join `  K )
cdleme10.m  |-  ./\  =  ( meet `  K )
cdleme10.a  |-  A  =  ( Atoms `  K )
cdleme10.h  |-  H  =  ( LHyp `  K
)
cdleme10.d  |-  D  =  ( ( R  .\/  S )  ./\  W )
Assertion
Ref Expression
cdleme10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  ( S  .\/  D )  =  ( S  .\/  R ) )

Proof of Theorem cdleme10
StepHypRef Expression
1 cdleme10.d . . 3  |-  D  =  ( ( R  .\/  S )  ./\  W )
21oveq2i 5831 . 2  |-  ( S 
.\/  D )  =  ( S  .\/  (
( R  .\/  S
)  ./\  W )
)
3 simp1l 981 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  K  e.  HL )
4 simp3l 985 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  S  e.  A )
5 simp2 958 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  R  e.  A )
6 eqid 2285 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
7 cdleme10.j . . . . . 6  |-  .\/  =  ( join `  K )
8 cdleme10.a . . . . . 6  |-  A  =  ( Atoms `  K )
96, 7, 8hlatjcl 28824 . . . . 5  |-  ( ( K  e.  HL  /\  R  e.  A  /\  S  e.  A )  ->  ( R  .\/  S
)  e.  ( Base `  K ) )
103, 5, 4, 9syl3anc 1184 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  ( R  .\/  S )  e.  (
Base `  K )
)
11 simp1r 982 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  W  e.  H )
12 cdleme10.h . . . . . 6  |-  H  =  ( LHyp `  K
)
136, 12lhpbase 29455 . . . . 5  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
1411, 13syl 17 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  W  e.  ( Base `  K )
)
15 hllat 28821 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
163, 15syl 17 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  K  e.  Lat )
176, 8atbase 28747 . . . . . 6  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
18173ad2ant2 979 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  R  e.  ( Base `  K )
)
196, 8atbase 28747 . . . . . 6  |-  ( S  e.  A  ->  S  e.  ( Base `  K
) )
204, 19syl 17 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  S  e.  ( Base `  K )
)
21 cdleme10.l . . . . . 6  |-  .<_  =  ( le `  K )
226, 21, 7latlej2 14162 . . . . 5  |-  ( ( K  e.  Lat  /\  R  e.  ( Base `  K )  /\  S  e.  ( Base `  K
) )  ->  S  .<_  ( R  .\/  S
) )
2316, 18, 20, 22syl3anc 1184 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  S  .<_  ( R  .\/  S ) )
24 cdleme10.m . . . . 5  |-  ./\  =  ( meet `  K )
256, 21, 7, 24, 8atmod3i1 29321 . . . 4  |-  ( ( K  e.  HL  /\  ( S  e.  A  /\  ( R  .\/  S
)  e.  ( Base `  K )  /\  W  e.  ( Base `  K
) )  /\  S  .<_  ( R  .\/  S
) )  ->  ( S  .\/  ( ( R 
.\/  S )  ./\  W ) )  =  ( ( R  .\/  S
)  ./\  ( S  .\/  W ) ) )
263, 4, 10, 14, 23, 25syl131anc 1197 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  ( S  .\/  ( ( R  .\/  S )  ./\  W )
)  =  ( ( R  .\/  S ) 
./\  ( S  .\/  W ) ) )
276, 7latjcom 14160 . . . . 5  |-  ( ( K  e.  Lat  /\  R  e.  ( Base `  K )  /\  S  e.  ( Base `  K
) )  ->  ( R  .\/  S )  =  ( S  .\/  R
) )
2816, 18, 20, 27syl3anc 1184 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  ( R  .\/  S )  =  ( S  .\/  R ) )
29 eqid 2285 . . . . . 6  |-  ( 1.
`  K )  =  ( 1. `  K
)
3021, 7, 29, 8, 12lhpjat2 29478 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  -> 
( S  .\/  W
)  =  ( 1.
`  K ) )
31303adant2 976 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  ( S  .\/  W )  =  ( 1. `  K ) )
3228, 31oveq12d 5838 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  ( ( R  .\/  S )  ./\  ( S  .\/  W ) )  =  ( ( S  .\/  R ) 
./\  ( 1. `  K ) ) )
33 hlol 28819 . . . . 5  |-  ( K  e.  HL  ->  K  e.  OL )
343, 33syl 17 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  K  e.  OL )
356, 7latjcl 14151 . . . . 5  |-  ( ( K  e.  Lat  /\  S  e.  ( Base `  K )  /\  R  e.  ( Base `  K
) )  ->  ( S  .\/  R )  e.  ( Base `  K
) )
3616, 20, 18, 35syl3anc 1184 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  ( S  .\/  R )  e.  (
Base `  K )
)
376, 24, 29olm11 28685 . . . 4  |-  ( ( K  e.  OL  /\  ( S  .\/  R )  e.  ( Base `  K
) )  ->  (
( S  .\/  R
)  ./\  ( 1. `  K ) )  =  ( S  .\/  R
) )
3834, 36, 37syl2anc 644 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  ( ( S  .\/  R )  ./\  ( 1. `  K ) )  =  ( S 
.\/  R ) )
3926, 32, 383eqtrd 2321 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  ( S  .\/  ( ( R  .\/  S )  ./\  W )
)  =  ( S 
.\/  R ) )
402, 39syl5eq 2329 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  ( S  .\/  D )  =  ( S  .\/  R ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    /\ w3a 936    = wceq 1624    e. wcel 1685   class class class wbr 4025   ` cfv 5222  (class class class)co 5820   Basecbs 13143   lecple 13210   joincjn 14073   meetcmee 14074   1.cp1 14139   Latclat 14146   OLcol 28632   Atomscatm 28721   HLchlt 28808   LHypclh 29441
This theorem is referenced by:  cdleme10tN  29715  cdleme20aN  29766  cdleme20g  29772
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-iin 3910  df-br 4026  df-opab 4080  df-mpt 4081  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-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-iota 6253  df-undef 6292  df-riota 6300  df-poset 14075  df-plt 14087  df-lub 14103  df-glb 14104  df-join 14105  df-meet 14106  df-p0 14140  df-p1 14141  df-lat 14147  df-clat 14209  df-oposet 28634  df-ol 28636  df-oml 28637  df-covers 28724  df-ats 28725  df-atl 28756  df-cvlat 28780  df-hlat 28809  df-psubsp 28960  df-pmap 28961  df-padd 29253  df-lhyp 29445
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