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Theorem dihjatcclem3 32292
Description: Lemma for dihjatcc 32294. (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 30890 . . . . . 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 31450 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( C  e.  A  /\  -.  C  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  G  e.  T )
121, 7, 8, 11syl3anc 1185 . . . 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 31450 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( C  e.  A  /\  -.  C  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  D  e.  T )
161, 7, 13, 15syl3anc 1185 . . . . 5  |-  ( ph  ->  D  e.  T )
174, 9ltrncnv 31017 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  D  e.  T
)  ->  `' D  e.  T )
181, 16, 17syl2anc 644 . . . 4  |-  ( ph  ->  `' D  e.  T
)
194, 9ltrnco 31590 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  `' D  e.  T
)  ->  ( G  o.  `' D )  e.  T
)
201, 12, 18, 19syl3anc 1185 . . 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 31034 . . 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 1185 . 2  |-  ( ph  ->  ( R `  ( G  o.  `' D
) )  =  ( ( Q  .\/  (
( G  o.  `' D ) `  Q
) )  ./\  W
) )
2613simpld 447 . . . . . . . 8  |-  ( ph  ->  Q  e.  A )
272, 3, 4, 9ltrncoval 31016 . . . . . . . 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 1190 . . . . . . 7  |-  ( ph  ->  ( ( G  o.  `' D ) `  Q
)  =  ( G `
 ( `' D `  Q ) ) )
292, 3, 4, 9, 14ltrniotacnvval 31453 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( C  e.  A  /\  -.  C  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( `' D `  Q )  =  C )
301, 7, 13, 29syl3anc 1185 . . . . . . . . 9  |-  ( ph  ->  ( `' D `  Q )  =  C )
3130fveq2d 5735 . . . . . . . 8  |-  ( ph  ->  ( G `  ( `' D `  Q ) )  =  ( G `
 C ) )
322, 3, 4, 9, 10ltrniotaval 31452 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( C  e.  A  /\  -.  C  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( G `  C )  =  P )
331, 7, 8, 32syl3anc 1185 . . . . . . . 8  |-  ( ph  ->  ( G `  C
)  =  P )
3431, 33eqtrd 2470 . . . . . . 7  |-  ( ph  ->  ( G `  ( `' D `  Q ) )  =  P )
3528, 34eqtrd 2470 . . . . . 6  |-  ( ph  ->  ( ( G  o.  `' D ) `  Q
)  =  P )
3635oveq2d 6100 . . . . 5  |-  ( ph  ->  ( Q  .\/  (
( G  o.  `' D ) `  Q
) )  =  ( Q  .\/  P ) )
371simpld 447 . . . . . 6  |-  ( ph  ->  K  e.  HL )
388simpld 447 . . . . . 6  |-  ( ph  ->  P  e.  A )
3921, 3hlatjcom 30239 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  =  ( Q 
.\/  P ) )
4037, 38, 26, 39syl3anc 1185 . . . . 5  |-  ( ph  ->  ( P  .\/  Q
)  =  ( Q 
.\/  P ) )
4136, 40eqtr4d 2473 . . . 4  |-  ( ph  ->  ( Q  .\/  (
( G  o.  `' D ) `  Q
) )  =  ( P  .\/  Q ) )
4241oveq1d 6099 . . 3  |-  ( ph  ->  ( ( Q  .\/  ( ( G  o.  `' D ) `  Q
) )  ./\  W
)  =  ( ( P  .\/  Q ) 
./\  W ) )
43 dihjatcclem.v . . 3  |-  V  =  ( ( P  .\/  Q )  ./\  W )
4442, 43syl6eqr 2488 . 2  |-  ( ph  ->  ( ( Q  .\/  ( ( G  o.  `' D ) `  Q
) )  ./\  W
)  =  V )
4525, 44eqtrd 2470 1  |-  ( ph  ->  ( R `  ( G  o.  `' D
) )  =  V )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   class class class wbr 4215   `'ccnv 4880    o. ccom 4885   ` cfv 5457  (class class class)co 6084   iota_crio 6545   Basecbs 13474   lecple 13541   occoc 13542   joincjn 14406   meetcmee 14407   LSSumclsm 15273   Atomscatm 30135   HLchlt 30222   LHypclh 30855   LTrncltrn 30972   trLctrl 31029   TEndoctendo 31623   DVecHcdvh 31950   DIsoHcdih 32100
This theorem is referenced by:  dihjatcclem4  32293
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-iin 4098  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-undef 6546  df-riota 6552  df-map 7023  df-poset 14408  df-plt 14420  df-lub 14436  df-glb 14437  df-join 14438  df-meet 14439  df-p0 14473  df-p1 14474  df-lat 14480  df-clat 14542  df-oposet 30048  df-ol 30050  df-oml 30051  df-covers 30138  df-ats 30139  df-atl 30170  df-cvlat 30194  df-hlat 30223  df-llines 30369  df-lplanes 30370  df-lvols 30371  df-lines 30372  df-psubsp 30374  df-pmap 30375  df-padd 30667  df-lhyp 30859  df-laut 30860  df-ldil 30975  df-ltrn 30976  df-trl 31030
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