Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdlemn2 Structured version   Unicode version

Theorem cdlemn2 34195
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 997 . . . . 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 1030 . . . . . 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 1031 . . . . . 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 33578 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  F  e.  T )
101, 2, 3, 9syl3anc 1230 . . . . 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 2402 . . . . . 6  |-  ( meet `  K )  =  (
meet `  K )
13 cdlemn2.r . . . . . 6  |-  R  =  ( ( trL `  K
) `  W )
144, 11, 12, 5, 6, 7, 13trlval2 33161 . . . . 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 1230 . . . 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 33580 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  ( F `  Q )  =  S )
171, 2, 3, 16syl3anc 1230 . . . . . 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 6293 . . . . 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 6292 . . . 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 2443 . . 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 1021 . . . . . . 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 32361 . . . . . . 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 1114 . . . . . . 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 32287 . . . . . . 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 1118 . . . . . 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 16012 . . . . . 6  |-  ( ( K  e.  Lat  /\  Q  e.  B  /\  X  e.  B )  ->  Q  .<_  ( Q  .\/  X ) )
3023, 27, 28, 29syl3anc 1230 . . . . 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 999 . . . . 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 1116 . . . . . . 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 32287 . . . . . . 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 16003 . . . . . . 7  |-  ( ( K  e.  Lat  /\  Q  e.  B  /\  X  e.  B )  ->  ( Q  .\/  X
)  e.  B )
3623, 27, 28, 35syl3anc 1230 . . . . . 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 16014 . . . . . 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 1232 . . . . 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 922 . . . 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 32364 . . . . . 6  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  S  e.  A )  ->  ( Q  .\/  S
)  e.  B )
4121, 24, 32, 40syl3anc 1230 . . . . 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 1022 . . . . . 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 32995 . . . . . 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 16033 . . . . 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 1232 . . . 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 15 . . 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 4414 . 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 1032 . . 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 33039 . . 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 1230 . 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 4418 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   class class class wbr 4394   ` cfv 5568   iota_crio 6238  (class class class)co 6277   Basecbs 14839   lecple 14914   joincjn 15895   meetcmee 15896   Latclat 15997   Atomscatm 32261   HLchlt 32348   LHypclh 32981   LTrncltrn 33098   trLctrl 33156
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-riotaBAD 31957
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-iin 4273  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6783  df-2nd 6784  df-undef 7004  df-map 7458  df-preset 15879  df-poset 15897  df-plt 15910  df-lub 15926  df-glb 15927  df-join 15928  df-meet 15929  df-p0 15991  df-p1 15992  df-lat 15998  df-clat 16060  df-oposet 32174  df-ol 32176  df-oml 32177  df-covers 32264  df-ats 32265  df-atl 32296  df-cvlat 32320  df-hlat 32349  df-llines 32495  df-lplanes 32496  df-lvols 32497  df-lines 32498  df-psubsp 32500  df-pmap 32501  df-padd 32793  df-lhyp 32985  df-laut 32986  df-ldil 33101  df-ltrn 33102  df-trl 33157
This theorem is referenced by:  cdlemn2a  34196
  Copyright terms: Public domain W3C validator