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Theorem cdleme42a 29790
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 2256 . . . . 5  |-  ( 1.
`  K )  =  ( 1. `  K
)
4 cdleme42.a . . . . 5  |-  A  =  ( Atoms `  K )
5 cdleme42.h . . . . 5  |-  H  =  ( LHyp `  K
)
61, 2, 3, 4, 5lhpjat2 29340 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  -> 
( R  .\/  W
)  =  ( 1.
`  K ) )
763adant3 980 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  ( R  .\/  W )  =  ( 1. `  K ) )
87oveq2d 5773 . 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 5768 . . 3  |-  ( R 
.\/  V )  =  ( R  .\/  (
( R  .\/  S
)  ./\  W )
)
11 simp1l 984 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  K  e.  HL )
12 simp2l 986 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  R  e.  A )
13 simp3l 988 . . . . 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 28686 . . . . 5  |-  ( ( K  e.  HL  /\  R  e.  A  /\  S  e.  A )  ->  ( R  .\/  S
)  e.  B )
1611, 12, 13, 15syl3anc 1187 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  ( R  .\/  S )  e.  B
)
17 simp1r 985 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  W  e.  H )
1814, 5lhpbase 29317 . . . . 5  |-  ( W  e.  H  ->  W  e.  B )
1917, 18syl 17 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  W  e.  B )
201, 2, 4hlatlej1 28694 . . . . 5  |-  ( ( K  e.  HL  /\  R  e.  A  /\  S  e.  A )  ->  R  .<_  ( R  .\/  S ) )
2111, 12, 13, 20syl3anc 1187 . . . 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 29183 . . . 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 1200 . . 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 2301 . 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 28681 . . . 4  |-  ( K  e.  HL  ->  K  e.  OL )
2711, 26syl 17 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  K  e.  OL )
2814, 22, 3olm11 28547 . . 3  |-  ( ( K  e.  OL  /\  ( R  .\/  S )  e.  B )  -> 
( ( R  .\/  S )  ./\  ( 1. `  K ) )  =  ( R  .\/  S
) )
2927, 16, 28syl2anc 645 . 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 2297 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 5    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621   class class class wbr 3963   ` cfv 4638  (class class class)co 5757   Basecbs 13075   lecple 13142   joincjn 14005   meetcmee 14006   1.cp1 14071   OLcol 28494   Atomscatm 28583   HLchlt 28670   LHypclh 29303
This theorem is referenced by:  cdleme42d  29792  cdleme42f  29799  cdleme42g  29800  cdleme42keg  29805  cdleme43cN  29810
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-iun 3848  df-iin 3849  df-br 3964  df-opab 4018  df-mpt 4019  df-id 4246  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-1st 6021  df-2nd 6022  df-iota 6190  df-undef 6229  df-riota 6237  df-poset 14007  df-plt 14019  df-lub 14035  df-glb 14036  df-join 14037  df-meet 14038  df-p0 14072  df-p1 14073  df-lat 14079  df-clat 14141  df-oposet 28496  df-ol 28498  df-oml 28499  df-covers 28586  df-ats 28587  df-atl 28618  df-cvlat 28642  df-hlat 28671  df-psubsp 28822  df-pmap 28823  df-padd 29115  df-lhyp 29307
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