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Theorem cdleme42a 30033
Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 3-Mar-2013.)
Hypotheses
Ref Expression
cdleme42.b  |-  B  =  ( Base `  K
)
cdleme42.l  |-  .<_  =  ( le `  K )
cdleme42.j  |-  .\/  =  ( join `  K )
cdleme42.m  |-  ./\  =  ( meet `  K )
cdleme42.a  |-  A  =  ( Atoms `  K )
cdleme42.h  |-  H  =  ( LHyp `  K
)
cdleme42.v  |-  V  =  ( ( R  .\/  S )  ./\  W )
Assertion
Ref Expression
cdleme42a  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  ( R  .\/  S )  =  ( R  .\/  V ) )

Proof of Theorem cdleme42a
StepHypRef Expression
1 cdleme42.l . . . . 5  |-  .<_  =  ( le `  K )
2 cdleme42.j . . . . 5  |-  .\/  =  ( join `  K )
3 eqid 2283 . . . . 5  |-  ( 1.
`  K )  =  ( 1. `  K
)
4 cdleme42.a . . . . 5  |-  A  =  ( Atoms `  K )
5 cdleme42.h . . . . 5  |-  H  =  ( LHyp `  K
)
61, 2, 3, 4, 5lhpjat2 29583 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  -> 
( R  .\/  W
)  =  ( 1.
`  K ) )
763adant3 975 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  ( R  .\/  W )  =  ( 1. `  K ) )
87oveq2d 5874 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  ( ( R  .\/  S )  ./\  ( R  .\/  W ) )  =  ( ( R  .\/  S ) 
./\  ( 1. `  K ) ) )
9 cdleme42.v . . . 4  |-  V  =  ( ( R  .\/  S )  ./\  W )
109oveq2i 5869 . . 3  |-  ( R 
.\/  V )  =  ( R  .\/  (
( R  .\/  S
)  ./\  W )
)
11 simp1l 979 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  K  e.  HL )
12 simp2l 981 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  R  e.  A )
13 simp3l 983 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  S  e.  A )
14 cdleme42.b . . . . . 6  |-  B  =  ( Base `  K
)
1514, 2, 4hlatjcl 28929 . . . . 5  |-  ( ( K  e.  HL  /\  R  e.  A  /\  S  e.  A )  ->  ( R  .\/  S
)  e.  B )
1611, 12, 13, 15syl3anc 1182 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  ( R  .\/  S )  e.  B
)
17 simp1r 980 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  W  e.  H )
1814, 5lhpbase 29560 . . . . 5  |-  ( W  e.  H  ->  W  e.  B )
1917, 18syl 15 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  W  e.  B )
201, 2, 4hlatlej1 28937 . . . . 5  |-  ( ( K  e.  HL  /\  R  e.  A  /\  S  e.  A )  ->  R  .<_  ( R  .\/  S ) )
2111, 12, 13, 20syl3anc 1182 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  R  .<_  ( R  .\/  S ) )
22 cdleme42.m . . . . 5  |-  ./\  =  ( meet `  K )
2314, 1, 2, 22, 4atmod3i1 29426 . . . 4  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  ( R  .\/  S
)  e.  B  /\  W  e.  B )  /\  R  .<_  ( R 
.\/  S ) )  ->  ( R  .\/  ( ( R  .\/  S )  ./\  W )
)  =  ( ( R  .\/  S ) 
./\  ( R  .\/  W ) ) )
2411, 12, 16, 19, 21, 23syl131anc 1195 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  ( R  .\/  ( ( R  .\/  S )  ./\  W )
)  =  ( ( R  .\/  S ) 
./\  ( R  .\/  W ) ) )
2510, 24syl5req 2328 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  ( ( R  .\/  S )  ./\  ( R  .\/  W ) )  =  ( R 
.\/  V ) )
26 hlol 28924 . . . 4  |-  ( K  e.  HL  ->  K  e.  OL )
2711, 26syl 15 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  K  e.  OL )
2814, 22, 3olm11 28790 . . 3  |-  ( ( K  e.  OL  /\  ( R  .\/  S )  e.  B )  -> 
( ( R  .\/  S )  ./\  ( 1. `  K ) )  =  ( R  .\/  S
) )
2927, 16, 28syl2anc 642 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  ( ( R  .\/  S )  ./\  ( 1. `  K ) )  =  ( R 
.\/  S ) )
308, 25, 293eqtr3rd 2324 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  ( R  .\/  S )  =  ( R  .\/  V ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   joincjn 14078   meetcmee 14079   1.cp1 14144   OLcol 28737   Atomscatm 28826   HLchlt 28913   LHypclh 29546
This theorem is referenced by:  cdleme42d  30035  cdleme42f  30042  cdleme42g  30043  cdleme42keg  30048  cdleme43cN  30053
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  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-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-oposet 28739  df-ol 28741  df-oml 28742  df-covers 28829  df-ats 28830  df-atl 28861  df-cvlat 28885  df-hlat 28914  df-psubsp 29065  df-pmap 29066  df-padd 29358  df-lhyp 29550
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