Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dihjatcclem3 Unicode version

Theorem dihjatcclem3 30983
Description: Lemma for dihjatcc 30985. (Contributed by NM, 28-Sep-2014.)
Hypotheses
Ref Expression
dihjatcclem.b  |-  B  =  ( Base `  K
)
dihjatcclem.l  |-  .<_  =  ( le `  K )
dihjatcclem.h  |-  H  =  ( LHyp `  K
)
dihjatcclem.j  |-  .\/  =  ( join `  K )
dihjatcclem.m  |-  ./\  =  ( meet `  K )
dihjatcclem.a  |-  A  =  ( Atoms `  K )
dihjatcclem.u  |-  U  =  ( ( DVecH `  K
) `  W )
dihjatcclem.s  |-  .(+)  =  (
LSSum `  U )
dihjatcclem.i  |-  I  =  ( ( DIsoH `  K
) `  W )
dihjatcclem.v  |-  V  =  ( ( P  .\/  Q )  ./\  W )
dihjatcclem.k  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
dihjatcclem.p  |-  ( ph  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
dihjatcclem.q  |-  ( ph  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
dihjatcc.w  |-  C  =  ( ( oc `  K ) `  W
)
dihjatcc.t  |-  T  =  ( ( LTrn `  K
) `  W )
dihjatcc.r  |-  R  =  ( ( trL `  K
) `  W )
dihjatcc.e  |-  E  =  ( ( TEndo `  K
) `  W )
dihjatcc.g  |-  G  =  ( iota_ d  e.  T
( d `  C
)  =  P )
dihjatcc.dd  |-  D  =  ( iota_ d  e.  T
( d `  C
)  =  Q )
Assertion
Ref Expression
dihjatcclem3  |-  ( ph  ->  ( R `  ( G  o.  `' D
) )  =  V )
Distinct variable groups:    .<_ , d    A, d    B, d    C, d    H, d    P, d    K, d    Q, d    T, d    W, d
Allowed substitution hints:    ph( d)    D( d)   
.(+) ( d)    R( d)    U( d)    E( d)    G( d)    I( d)    .\/ ( d)    ./\ ( d)    V( d)

Proof of Theorem dihjatcclem3
StepHypRef Expression
1 dihjatcclem.k . . 3  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
2 dihjatcclem.l . . . . . . 7  |-  .<_  =  ( le `  K )
3 dihjatcclem.a . . . . . . 7  |-  A  =  ( Atoms `  K )
4 dihjatcclem.h . . . . . . 7  |-  H  =  ( LHyp `  K
)
5 dihjatcc.w . . . . . . 7  |-  C  =  ( ( oc `  K ) `  W
)
62, 3, 4, 5lhpocnel2 29581 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( C  e.  A  /\  -.  C  .<_  W ) )
71, 6syl 15 . . . . 5  |-  ( ph  ->  ( C  e.  A  /\  -.  C  .<_  W ) )
8 dihjatcclem.p . . . . 5  |-  ( ph  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
9 dihjatcc.t . . . . . 6  |-  T  =  ( ( LTrn `  K
) `  W )
10 dihjatcc.g . . . . . 6  |-  G  =  ( iota_ d  e.  T
( d `  C
)  =  P )
112, 3, 4, 9, 10ltrniotacl 30141 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( C  e.  A  /\  -.  C  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  G  e.  T )
121, 7, 8, 11syl3anc 1182 . . . 4  |-  ( ph  ->  G  e.  T )
13 dihjatcclem.q . . . . . 6  |-  ( ph  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
14 dihjatcc.dd . . . . . . 7  |-  D  =  ( iota_ d  e.  T
( d `  C
)  =  Q )
152, 3, 4, 9, 14ltrniotacl 30141 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( C  e.  A  /\  -.  C  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  D  e.  T )
161, 7, 13, 15syl3anc 1182 . . . . 5  |-  ( ph  ->  D  e.  T )
174, 9ltrncnv 29708 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  D  e.  T
)  ->  `' D  e.  T )
181, 16, 17syl2anc 642 . . . 4  |-  ( ph  ->  `' D  e.  T
)
194, 9ltrnco 30281 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  `' D  e.  T
)  ->  ( G  o.  `' D )  e.  T
)
201, 12, 18, 19syl3anc 1182 . . 3  |-  ( ph  ->  ( G  o.  `' D )  e.  T
)
21 dihjatcclem.j . . . 4  |-  .\/  =  ( join `  K )
22 dihjatcclem.m . . . 4  |-  ./\  =  ( meet `  K )
23 dihjatcc.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
242, 21, 22, 3, 4, 9, 23trlval2 29725 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( G  o.  `' D )  e.  T  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( R `  ( G  o.  `' D ) )  =  ( ( Q  .\/  ( ( G  o.  `' D ) `  Q
) )  ./\  W
) )
251, 20, 13, 24syl3anc 1182 . 2  |-  ( ph  ->  ( R `  ( G  o.  `' D
) )  =  ( ( Q  .\/  (
( G  o.  `' D ) `  Q
) )  ./\  W
) )
2613simpld 445 . . . . . . . 8  |-  ( ph  ->  Q  e.  A )
272, 3, 4, 9ltrncoval 29707 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( G  e.  T  /\  `' D  e.  T )  /\  Q  e.  A )  ->  (
( G  o.  `' D ) `  Q
)  =  ( G `
 ( `' D `  Q ) ) )
281, 12, 18, 26, 27syl121anc 1187 . . . . . . 7  |-  ( ph  ->  ( ( G  o.  `' D ) `  Q
)  =  ( G `
 ( `' D `  Q ) ) )
292, 3, 4, 9, 14ltrniotacnvval 30144 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( C  e.  A  /\  -.  C  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( `' D `  Q )  =  C )
301, 7, 13, 29syl3anc 1182 . . . . . . . . 9  |-  ( ph  ->  ( `' D `  Q )  =  C )
3130fveq2d 5529 . . . . . . . 8  |-  ( ph  ->  ( G `  ( `' D `  Q ) )  =  ( G `
 C ) )
322, 3, 4, 9, 10ltrniotaval 30143 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( C  e.  A  /\  -.  C  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( G `  C )  =  P )
331, 7, 8, 32syl3anc 1182 . . . . . . . 8  |-  ( ph  ->  ( G `  C
)  =  P )
3431, 33eqtrd 2315 . . . . . . 7  |-  ( ph  ->  ( G `  ( `' D `  Q ) )  =  P )
3528, 34eqtrd 2315 . . . . . 6  |-  ( ph  ->  ( ( G  o.  `' D ) `  Q
)  =  P )
3635oveq2d 5874 . . . . 5  |-  ( ph  ->  ( Q  .\/  (
( G  o.  `' D ) `  Q
) )  =  ( Q  .\/  P ) )
371simpld 445 . . . . . 6  |-  ( ph  ->  K  e.  HL )
388simpld 445 . . . . . 6  |-  ( ph  ->  P  e.  A )
3921, 3hlatjcom 28930 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  =  ( Q 
.\/  P ) )
4037, 38, 26, 39syl3anc 1182 . . . . 5  |-  ( ph  ->  ( P  .\/  Q
)  =  ( Q 
.\/  P ) )
4136, 40eqtr4d 2318 . . . 4  |-  ( ph  ->  ( Q  .\/  (
( G  o.  `' D ) `  Q
) )  =  ( P  .\/  Q ) )
4241oveq1d 5873 . . 3  |-  ( ph  ->  ( ( Q  .\/  ( ( G  o.  `' D ) `  Q
) )  ./\  W
)  =  ( ( P  .\/  Q ) 
./\  W ) )
43 dihjatcclem.v . . 3  |-  V  =  ( ( P  .\/  Q )  ./\  W )
4442, 43syl6eqr 2333 . 2  |-  ( ph  ->  ( ( Q  .\/  ( ( G  o.  `' D ) `  Q
) )  ./\  W
)  =  V )
4525, 44eqtrd 2315 1  |-  ( ph  ->  ( R `  ( G  o.  `' D
) )  =  V )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   class class class wbr 4023   `'ccnv 4688    o. ccom 4693   ` cfv 5255  (class class class)co 5858   iota_crio 6297   Basecbs 13148   lecple 13215   occoc 13216   joincjn 14078   meetcmee 14079   LSSumclsm 14945   Atomscatm 28826   HLchlt 28913   LHypclh 29546   LTrncltrn 29663   trLctrl 29720   TEndoctendo 30314   DVecHcdvh 30641   DIsoHcdih 30791
This theorem is referenced by:  dihjatcclem4  30984
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-map 6774  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-oposet 28739  df-ol 28741  df-oml 28742  df-covers 28829  df-ats 28830  df-atl 28861  df-cvlat 28885  df-hlat 28914  df-llines 29060  df-lplanes 29061  df-lvols 29062  df-lines 29063  df-psubsp 29065  df-pmap 29066  df-padd 29358  df-lhyp 29550  df-laut 29551  df-ldil 29666  df-ltrn 29667  df-trl 29721
  Copyright terms: Public domain W3C validator