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Theorem cdlemn2 30536
Description: Part of proof of Lemma N of [Crawley] p. 121 line 30. (Contributed by NM, 21-Feb-2014.)
Hypotheses
Ref Expression
cdlemn2.b  |-  B  =  ( Base `  K
)
cdlemn2.l  |-  .<_  =  ( le `  K )
cdlemn2.j  |-  .\/  =  ( join `  K )
cdlemn2.a  |-  A  =  ( Atoms `  K )
cdlemn2.h  |-  H  =  ( LHyp `  K
)
cdlemn2.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemn2.r  |-  R  =  ( ( trL `  K
) `  W )
cdlemn2.f  |-  F  =  ( iota_ h  e.  T
( h `  Q
)  =  S )
Assertion
Ref Expression
cdlemn2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  S  .<_  ( Q  .\/  X
) )  ->  ( R `  F )  .<_  X )
Distinct variable groups:    .<_ , h    A, h    h, H    h, K    Q, h    S, h    T, h   
h, W
Allowed substitution hints:    B( h)    R( h)    F( h)    .\/ ( h)    X( h)

Proof of Theorem cdlemn2
StepHypRef Expression
1 simp1 960 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  S  .<_  ( Q  .\/  X
) )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 simp21 993 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  S  .<_  ( Q  .\/  X
) )  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
3 simp22 994 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  S  .<_  ( Q  .\/  X
) )  ->  ( S  e.  A  /\  -.  S  .<_  W ) )
4 cdlemn2.l . . . . . . 7  |-  .<_  =  ( le `  K )
5 cdlemn2.a . . . . . . 7  |-  A  =  ( Atoms `  K )
6 cdlemn2.h . . . . . . 7  |-  H  =  ( LHyp `  K
)
7 cdlemn2.t . . . . . . 7  |-  T  =  ( ( LTrn `  K
) `  W )
8 cdlemn2.f . . . . . . 7  |-  F  =  ( iota_ h  e.  T
( h `  Q
)  =  S )
94, 5, 6, 7, 8ltrniotacl 29919 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  F  e.  T )
101, 2, 3, 9syl3anc 1187 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  S  .<_  ( Q  .\/  X
) )  ->  F  e.  T )
11 cdlemn2.j . . . . . 6  |-  .\/  =  ( join `  K )
12 eqid 2256 . . . . . 6  |-  ( meet `  K )  =  (
meet `  K )
13 cdlemn2.r . . . . . 6  |-  R  =  ( ( trL `  K
) `  W )
144, 11, 12, 5, 6, 7, 13trlval2 29503 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( R `  F )  =  ( ( Q  .\/  ( F `  Q )
) ( meet `  K
) W ) )
151, 10, 2, 14syl3anc 1187 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  S  .<_  ( Q  .\/  X
) )  ->  ( R `  F )  =  ( ( Q 
.\/  ( F `  Q ) ) (
meet `  K ) W ) )
164, 5, 6, 7, 8ltrniotaval 29921 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  ( F `  Q )  =  S )
171, 2, 3, 16syl3anc 1187 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  S  .<_  ( Q  .\/  X
) )  ->  ( F `  Q )  =  S )
1817oveq2d 5794 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  S  .<_  ( Q  .\/  X
) )  ->  ( Q  .\/  ( F `  Q ) )  =  ( Q  .\/  S
) )
1918oveq1d 5793 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  S  .<_  ( Q  .\/  X
) )  ->  (
( Q  .\/  ( F `  Q )
) ( meet `  K
) W )  =  ( ( Q  .\/  S ) ( meet `  K
) W ) )
2015, 19eqtrd 2288 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  S  .<_  ( Q  .\/  X
) )  ->  ( R `  F )  =  ( ( Q 
.\/  S ) (
meet `  K ) W ) )
21 simp1l 984 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  S  .<_  ( Q  .\/  X
) )  ->  K  e.  HL )
22 hllat 28704 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  Lat )
2321, 22syl 17 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  S  .<_  ( Q  .\/  X
) )  ->  K  e.  Lat )
24 simp21l 1077 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  S  .<_  ( Q  .\/  X
) )  ->  Q  e.  A )
25 cdlemn2.b . . . . . . . 8  |-  B  =  ( Base `  K
)
2625, 5atbase 28630 . . . . . . 7  |-  ( Q  e.  A  ->  Q  e.  B )
2724, 26syl 17 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  S  .<_  ( Q  .\/  X
) )  ->  Q  e.  B )
28 simp23l 1081 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  S  .<_  ( Q  .\/  X
) )  ->  X  e.  B )
2925, 4, 11latlej1 14114 . . . . . 6  |-  ( ( K  e.  Lat  /\  Q  e.  B  /\  X  e.  B )  ->  Q  .<_  ( Q  .\/  X ) )
3023, 27, 28, 29syl3anc 1187 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  S  .<_  ( Q  .\/  X
) )  ->  Q  .<_  ( Q  .\/  X
) )
31 simp3 962 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  S  .<_  ( Q  .\/  X
) )  ->  S  .<_  ( Q  .\/  X
) )
32 simp22l 1079 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  S  .<_  ( Q  .\/  X
) )  ->  S  e.  A )
3325, 5atbase 28630 . . . . . . 7  |-  ( S  e.  A  ->  S  e.  B )
3432, 33syl 17 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  S  .<_  ( Q  .\/  X
) )  ->  S  e.  B )
3525, 11latjcl 14104 . . . . . . 7  |-  ( ( K  e.  Lat  /\  Q  e.  B  /\  X  e.  B )  ->  ( Q  .\/  X
)  e.  B )
3623, 27, 28, 35syl3anc 1187 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  S  .<_  ( Q  .\/  X
) )  ->  ( Q  .\/  X )  e.  B )
3725, 4, 11latjle12 14116 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( Q  e.  B  /\  S  e.  B  /\  ( Q  .\/  X
)  e.  B ) )  ->  ( ( Q  .<_  ( Q  .\/  X )  /\  S  .<_  ( Q  .\/  X ) )  <->  ( Q  .\/  S )  .<_  ( Q  .\/  X ) ) )
3823, 27, 34, 36, 37syl13anc 1189 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  S  .<_  ( Q  .\/  X
) )  ->  (
( Q  .<_  ( Q 
.\/  X )  /\  S  .<_  ( Q  .\/  X ) )  <->  ( Q  .\/  S )  .<_  ( Q 
.\/  X ) ) )
3930, 31, 38mpbi2and 892 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  S  .<_  ( Q  .\/  X
) )  ->  ( Q  .\/  S )  .<_  ( Q  .\/  X ) )
4025, 11, 5hlatjcl 28707 . . . . . 6  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  S  e.  A )  ->  ( Q  .\/  S
)  e.  B )
4121, 24, 32, 40syl3anc 1187 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  S  .<_  ( Q  .\/  X
) )  ->  ( Q  .\/  S )  e.  B )
42 simp1r 985 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  S  .<_  ( Q  .\/  X
) )  ->  W  e.  H )
4325, 6lhpbase 29338 . . . . . 6  |-  ( W  e.  H  ->  W  e.  B )
4442, 43syl 17 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  S  .<_  ( Q  .\/  X
) )  ->  W  e.  B )
4525, 4, 12latmlem1 14135 . . . . 5  |-  ( ( K  e.  Lat  /\  ( ( Q  .\/  S )  e.  B  /\  ( Q  .\/  X )  e.  B  /\  W  e.  B ) )  -> 
( ( Q  .\/  S )  .<_  ( Q  .\/  X )  ->  (
( Q  .\/  S
) ( meet `  K
) W )  .<_  ( ( Q  .\/  X ) ( meet `  K
) W ) ) )
4623, 41, 36, 44, 45syl13anc 1189 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  S  .<_  ( Q  .\/  X
) )  ->  (
( Q  .\/  S
)  .<_  ( Q  .\/  X )  ->  ( ( Q  .\/  S ) (
meet `  K ) W )  .<_  ( ( Q  .\/  X ) ( meet `  K
) W ) ) )
4739, 46mpd 16 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  S  .<_  ( Q  .\/  X
) )  ->  (
( Q  .\/  S
) ( meet `  K
) W )  .<_  ( ( Q  .\/  X ) ( meet `  K
) W ) )
4820, 47eqbrtrd 4003 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  S  .<_  ( Q  .\/  X
) )  ->  ( R `  F )  .<_  ( ( Q  .\/  X ) ( meet `  K
) W ) )
49 simp23 995 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  S  .<_  ( Q  .\/  X
) )  ->  ( X  e.  B  /\  X  .<_  W ) )
5025, 4, 11, 12, 5, 6lhple 29382 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  ( ( Q 
.\/  X ) (
meet `  K ) W )  =  X )
511, 2, 49, 50syl3anc 1187 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  S  .<_  ( Q  .\/  X
) )  ->  (
( Q  .\/  X
) ( meet `  K
) W )  =  X )
5248, 51breqtrd 4007 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  S  .<_  ( Q  .\/  X
) )  ->  ( R `  F )  .<_  X )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621   class class class wbr 3983   ` cfv 4659  (class class class)co 5778   iota_crio 6249   Basecbs 13096   lecple 13163   joincjn 14026   meetcmee 14027   Latclat 14099   Atomscatm 28604   HLchlt 28691   LHypclh 29324   LTrncltrn 29441   trLctrl 29498
This theorem is referenced by:  cdlemn2a  30537
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 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  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 2521  df-rex 2522  df-reu 2523  df-rmo 2524  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-op 3609  df-uni 3788  df-iun 3867  df-iin 3868  df-br 3984  df-opab 4038  df-mpt 4039  df-id 4267  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-1st 6042  df-2nd 6043  df-iota 6211  df-undef 6250  df-riota 6258  df-map 6728  df-poset 14028  df-plt 14040  df-lub 14056  df-glb 14057  df-join 14058  df-meet 14059  df-p0 14093  df-p1 14094  df-lat 14100  df-clat 14162  df-oposet 28517  df-ol 28519  df-oml 28520  df-covers 28607  df-ats 28608  df-atl 28639  df-cvlat 28663  df-hlat 28692  df-llines 28838  df-lplanes 28839  df-lvols 28840  df-lines 28841  df-psubsp 28843  df-pmap 28844  df-padd 29136  df-lhyp 29328  df-laut 29329  df-ldil 29444  df-ltrn 29445  df-trl 29499
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