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Theorem dihjatcclem3 35088
Description: Lemma for dihjatcc 35090. (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 33686 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( C  e.  A  /\  -.  C  .<_  W ) )
71, 6syl 16 . . . . 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 34246 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( C  e.  A  /\  -.  C  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  G  e.  T )
121, 7, 8, 11syl3anc 1218 . . . 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 34246 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( C  e.  A  /\  -.  C  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  D  e.  T )
161, 7, 13, 15syl3anc 1218 . . . . 5  |-  ( ph  ->  D  e.  T )
174, 9ltrncnv 33813 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  D  e.  T
)  ->  `' D  e.  T )
181, 16, 17syl2anc 661 . . . 4  |-  ( ph  ->  `' D  e.  T
)
194, 9ltrnco 34386 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  `' D  e.  T
)  ->  ( G  o.  `' D )  e.  T
)
201, 12, 18, 19syl3anc 1218 . . 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 33830 . . 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 1218 . 2  |-  ( ph  ->  ( R `  ( G  o.  `' D
) )  =  ( ( Q  .\/  (
( G  o.  `' D ) `  Q
) )  ./\  W
) )
2613simpld 459 . . . . . . . 8  |-  ( ph  ->  Q  e.  A )
272, 3, 4, 9ltrncoval 33812 . . . . . . . 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 1223 . . . . . . 7  |-  ( ph  ->  ( ( G  o.  `' D ) `  Q
)  =  ( G `
 ( `' D `  Q ) ) )
292, 3, 4, 9, 14ltrniotacnvval 34249 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( C  e.  A  /\  -.  C  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( `' D `  Q )  =  C )
301, 7, 13, 29syl3anc 1218 . . . . . . . . 9  |-  ( ph  ->  ( `' D `  Q )  =  C )
3130fveq2d 5714 . . . . . . . 8  |-  ( ph  ->  ( G `  ( `' D `  Q ) )  =  ( G `
 C ) )
322, 3, 4, 9, 10ltrniotaval 34248 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( C  e.  A  /\  -.  C  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( G `  C )  =  P )
331, 7, 8, 32syl3anc 1218 . . . . . . . 8  |-  ( ph  ->  ( G `  C
)  =  P )
3431, 33eqtrd 2475 . . . . . . 7  |-  ( ph  ->  ( G `  ( `' D `  Q ) )  =  P )
3528, 34eqtrd 2475 . . . . . 6  |-  ( ph  ->  ( ( G  o.  `' D ) `  Q
)  =  P )
3635oveq2d 6126 . . . . 5  |-  ( ph  ->  ( Q  .\/  (
( G  o.  `' D ) `  Q
) )  =  ( Q  .\/  P ) )
371simpld 459 . . . . . 6  |-  ( ph  ->  K  e.  HL )
388simpld 459 . . . . . 6  |-  ( ph  ->  P  e.  A )
3921, 3hlatjcom 33035 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  =  ( Q 
.\/  P ) )
4037, 38, 26, 39syl3anc 1218 . . . . 5  |-  ( ph  ->  ( P  .\/  Q
)  =  ( Q 
.\/  P ) )
4136, 40eqtr4d 2478 . . . 4  |-  ( ph  ->  ( Q  .\/  (
( G  o.  `' D ) `  Q
) )  =  ( P  .\/  Q ) )
4241oveq1d 6125 . . 3  |-  ( ph  ->  ( ( Q  .\/  ( ( G  o.  `' D ) `  Q
) )  ./\  W
)  =  ( ( P  .\/  Q ) 
./\  W ) )
43 dihjatcclem.v . . 3  |-  V  =  ( ( P  .\/  Q )  ./\  W )
4442, 43syl6eqr 2493 . 2  |-  ( ph  ->  ( ( Q  .\/  ( ( G  o.  `' D ) `  Q
) )  ./\  W
)  =  V )
4525, 44eqtrd 2475 1  |-  ( ph  ->  ( R `  ( G  o.  `' D
) )  =  V )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   class class class wbr 4311   `'ccnv 4858    o. ccom 4863   ` cfv 5437   iota_crio 6070  (class class class)co 6110   Basecbs 14193   lecple 14264   occoc 14265   joincjn 15133   meetcmee 15134   LSSumclsm 16152   Atomscatm 32931   HLchlt 33018   LHypclh 33651   LTrncltrn 33768   trLctrl 33825   TEndoctendo 34419   DVecHcdvh 34746   DIsoHcdih 34896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4422  ax-sep 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550  ax-un 6391  ax-riotaBAD 32627
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2739  df-rex 2740  df-reu 2741  df-rmo 2742  df-rab 2743  df-v 2993  df-sbc 3206  df-csb 3308  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-nul 3657  df-if 3811  df-pw 3881  df-sn 3897  df-pr 3899  df-op 3903  df-uni 4111  df-iun 4192  df-iin 4193  df-br 4312  df-opab 4370  df-mpt 4371  df-id 4655  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-f1 5442  df-fo 5443  df-f1o 5444  df-fv 5445  df-riota 6071  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6596  df-2nd 6597  df-undef 6811  df-map 7235  df-poset 15135  df-plt 15147  df-lub 15163  df-glb 15164  df-join 15165  df-meet 15166  df-p0 15228  df-p1 15229  df-lat 15235  df-clat 15297  df-oposet 32844  df-ol 32846  df-oml 32847  df-covers 32934  df-ats 32935  df-atl 32966  df-cvlat 32990  df-hlat 33019  df-llines 33165  df-lplanes 33166  df-lvols 33167  df-lines 33168  df-psubsp 33170  df-pmap 33171  df-padd 33463  df-lhyp 33655  df-laut 33656  df-ldil 33771  df-ltrn 33772  df-trl 33826
This theorem is referenced by:  dihjatcclem4  35089
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