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Theorem cdleme42a 29349
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 2253 . . . . 5  |-  ( 1.
`  K )  =  ( 1. `  K
)
4 cdleme42.a . . . . 5  |-  A  =  ( Atoms `  K )
5 cdleme42.h . . . . 5  |-  H  =  ( LHyp `  K
)
61, 2, 3, 4, 5lhpjat2 28899 . . . 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 5726 . 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 5721 . . 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 28245 . . . . 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 28876 . . . . 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 28253 . . . . 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 28742 . . . 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 2298 . 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 28240 . . . 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 28106 . . 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 2294 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 3920   ` cfv 4592  (class class class)co 5710   Basecbs 13022   lecple 13089   joincjn 13922   meetcmee 13923   1.cp1 13988   OLcol 28053   Atomscatm 28142   HLchlt 28229   LHypclh 28862
This theorem is referenced by:  cdleme42d  29351  cdleme42f  29358  cdleme42g  29359  cdleme42keg  29364  cdleme43cN  29369
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 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-iin 3806  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-undef 6182  df-riota 6190  df-poset 13924  df-plt 13936  df-lub 13952  df-glb 13953  df-join 13954  df-meet 13955  df-p0 13989  df-p1 13990  df-lat 13996  df-clat 14058  df-oposet 28055  df-ol 28057  df-oml 28058  df-covers 28145  df-ats 28146  df-atl 28177  df-cvlat 28201  df-hlat 28230  df-psubsp 28381  df-pmap 28382  df-padd 28674  df-lhyp 28866
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