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

Theorem cdlemn2 31832
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 957 . . . . 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 990 . . . . . 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 991 . . . . . 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 31215 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  F  e.  T )
101, 2, 3, 9syl3anc 1184 . . . . 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 2435 . . . . . 6  |-  ( meet `  K )  =  (
meet `  K )
13 cdlemn2.r . . . . . 6  |-  R  =  ( ( trL `  K
) `  W )
144, 11, 12, 5, 6, 7, 13trlval2 30799 . . . . 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 1184 . . . 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 31217 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  ( F `  Q )  =  S )
171, 2, 3, 16syl3anc 1184 . . . . . 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 6088 . . . . 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 6087 . . . 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 2467 . . 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 981 . . . . . . 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 30000 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  Lat )
2321, 22syl 16 . . . . . 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 1074 . . . . . . 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 29926 . . . . . . 7  |-  ( Q  e.  A  ->  Q  e.  B )
2724, 26syl 16 . . . . . 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 1078 . . . . . 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 14477 . . . . . 6  |-  ( ( K  e.  Lat  /\  Q  e.  B  /\  X  e.  B )  ->  Q  .<_  ( Q  .\/  X ) )
3023, 27, 28, 29syl3anc 1184 . . . . 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 959 . . . . 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 1076 . . . . . . 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 29926 . . . . . . 7  |-  ( S  e.  A  ->  S  e.  B )
3432, 33syl 16 . . . . . 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 14467 . . . . . . 7  |-  ( ( K  e.  Lat  /\  Q  e.  B  /\  X  e.  B )  ->  ( Q  .\/  X
)  e.  B )
3623, 27, 28, 35syl3anc 1184 . . . . . 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 14479 . . . . . 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 1186 . . . . 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 888 . . . 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 30003 . . . . . 6  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  S  e.  A )  ->  ( Q  .\/  S
)  e.  B )
4121, 24, 32, 40syl3anc 1184 . . . . 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 982 . . . . . 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 30634 . . . . . 6  |-  ( W  e.  H  ->  W  e.  B )
4442, 43syl 16 . . . . 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 14498 . . . . 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 1186 . . . 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 4224 . 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 992 . . 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 30678 . . 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 1184 . 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 4228 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 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   class class class wbr 4204   ` cfv 5445  (class class class)co 6072   iota_crio 6533   Basecbs 13457   lecple 13524   joincjn 14389   meetcmee 14390   Latclat 14462   Atomscatm 29900   HLchlt 29987   LHypclh 30620   LTrncltrn 30737   trLctrl 30794
This theorem is referenced by:  cdlemn2a  31833
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-1st 6340  df-2nd 6341  df-undef 6534  df-riota 6540  df-map 7011  df-poset 14391  df-plt 14403  df-lub 14419  df-glb 14420  df-join 14421  df-meet 14422  df-p0 14456  df-p1 14457  df-lat 14463  df-clat 14525  df-oposet 29813  df-ol 29815  df-oml 29816  df-covers 29903  df-ats 29904  df-atl 29935  df-cvlat 29959  df-hlat 29988  df-llines 30134  df-lplanes 30135  df-lvols 30136  df-lines 30137  df-psubsp 30139  df-pmap 30140  df-padd 30432  df-lhyp 30624  df-laut 30625  df-ldil 30740  df-ltrn 30741  df-trl 30795
  Copyright terms: Public domain W3C validator