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Theorem cdleme10 34927
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 6288 . 2  |-  ( S 
.\/  D )  =  ( S  .\/  (
( R  .\/  S
)  ./\  W )
)
3 simp1l 1015 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  K  e.  HL )
4 simp3l 1019 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  S  e.  A )
5 simp2 992 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  R  e.  A )
6 eqid 2462 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
7 cdleme10.j . . . . . 6  |-  .\/  =  ( join `  K )
8 cdleme10.a . . . . . 6  |-  A  =  ( Atoms `  K )
96, 7, 8hlatjcl 34040 . . . . 5  |-  ( ( K  e.  HL  /\  R  e.  A  /\  S  e.  A )  ->  ( R  .\/  S
)  e.  ( Base `  K ) )
103, 5, 4, 9syl3anc 1223 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  ( R  .\/  S )  e.  (
Base `  K )
)
11 simp1r 1016 . . . . 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 34671 . . . . 5  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
1411, 13syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  W  e.  ( Base `  K )
)
15 hllat 34037 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
163, 15syl 16 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  K  e.  Lat )
176, 8atbase 33963 . . . . . 6  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
18173ad2ant2 1013 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  R  e.  ( Base `  K )
)
196, 8atbase 33963 . . . . . 6  |-  ( S  e.  A  ->  S  e.  ( Base `  K
) )
204, 19syl 16 . . . . 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 15539 . . . . 5  |-  ( ( K  e.  Lat  /\  R  e.  ( Base `  K )  /\  S  e.  ( Base `  K
) )  ->  S  .<_  ( R  .\/  S
) )
2316, 18, 20, 22syl3anc 1223 . . . 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 34537 . . . 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 1236 . . 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 15537 . . . . 5  |-  ( ( K  e.  Lat  /\  R  e.  ( Base `  K )  /\  S  e.  ( Base `  K
) )  ->  ( R  .\/  S )  =  ( S  .\/  R
) )
2816, 18, 20, 27syl3anc 1223 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  ( R  .\/  S )  =  ( S  .\/  R ) )
29 eqid 2462 . . . . . 6  |-  ( 1.
`  K )  =  ( 1. `  K
)
3021, 7, 29, 8, 12lhpjat2 34694 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  -> 
( S  .\/  W
)  =  ( 1.
`  K ) )
31303adant2 1010 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  ( S  .\/  W )  =  ( 1. `  K ) )
3228, 31oveq12d 6295 . . 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 34035 . . . . 5  |-  ( K  e.  HL  ->  K  e.  OL )
343, 33syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  K  e.  OL )
356, 7latjcl 15529 . . . . 5  |-  ( ( K  e.  Lat  /\  S  e.  ( Base `  K )  /\  R  e.  ( Base `  K
) )  ->  ( S  .\/  R )  e.  ( Base `  K
) )
3616, 20, 18, 35syl3anc 1223 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  ( S  .\/  R )  e.  (
Base `  K )
)
376, 24, 29olm11 33901 . . . 4  |-  ( ( K  e.  OL  /\  ( S  .\/  R )  e.  ( Base `  K
) )  ->  (
( S  .\/  R
)  ./\  ( 1. `  K ) )  =  ( S  .\/  R
) )
3834, 36, 37syl2anc 661 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  ( ( S  .\/  R )  ./\  ( 1. `  K ) )  =  ( S 
.\/  R ) )
3926, 32, 383eqtrd 2507 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  ( S  .\/  ( ( R  .\/  S )  ./\  W )
)  =  ( S 
.\/  R ) )
402, 39syl5eq 2515 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  ( S  .\/  D )  =  ( S  .\/  R ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   class class class wbr 4442   ` cfv 5581  (class class class)co 6277   Basecbs 14481   lecple 14553   joincjn 15422   meetcmee 15423   1.cp1 15516   Latclat 15523   OLcol 33848   Atomscatm 33937   HLchlt 34024   LHypclh 34657
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-iin 4323  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6776  df-2nd 6777  df-poset 15424  df-plt 15436  df-lub 15452  df-glb 15453  df-join 15454  df-meet 15455  df-p0 15517  df-p1 15518  df-lat 15524  df-clat 15586  df-oposet 33850  df-ol 33852  df-oml 33853  df-covers 33940  df-ats 33941  df-atl 33972  df-cvlat 33996  df-hlat 34025  df-psubsp 34176  df-pmap 34177  df-padd 34469  df-lhyp 34661
This theorem is referenced by:  cdleme10tN  34931  cdleme20aN  34982  cdleme20g  34988
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