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Theorem dihjatcclem3 31586
Description: Lemma for dihjatcc 31588. (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 30184 . . . . . 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 30744 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( C  e.  A  /\  -.  C  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  G  e.  T )
121, 7, 8, 11syl3anc 1184 . . . 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 30744 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( C  e.  A  /\  -.  C  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  D  e.  T )
161, 7, 13, 15syl3anc 1184 . . . . 5  |-  ( ph  ->  D  e.  T )
174, 9ltrncnv 30311 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  D  e.  T
)  ->  `' D  e.  T )
181, 16, 17syl2anc 643 . . . 4  |-  ( ph  ->  `' D  e.  T
)
194, 9ltrnco 30884 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  `' D  e.  T
)  ->  ( G  o.  `' D )  e.  T
)
201, 12, 18, 19syl3anc 1184 . . 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 30328 . . 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 1184 . 2  |-  ( ph  ->  ( R `  ( G  o.  `' D
) )  =  ( ( Q  .\/  (
( G  o.  `' D ) `  Q
) )  ./\  W
) )
2613simpld 446 . . . . . . . 8  |-  ( ph  ->  Q  e.  A )
272, 3, 4, 9ltrncoval 30310 . . . . . . . 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 1189 . . . . . . 7  |-  ( ph  ->  ( ( G  o.  `' D ) `  Q
)  =  ( G `
 ( `' D `  Q ) ) )
292, 3, 4, 9, 14ltrniotacnvval 30747 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( C  e.  A  /\  -.  C  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( `' D `  Q )  =  C )
301, 7, 13, 29syl3anc 1184 . . . . . . . . 9  |-  ( ph  ->  ( `' D `  Q )  =  C )
3130fveq2d 5665 . . . . . . . 8  |-  ( ph  ->  ( G `  ( `' D `  Q ) )  =  ( G `
 C ) )
322, 3, 4, 9, 10ltrniotaval 30746 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( C  e.  A  /\  -.  C  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( G `  C )  =  P )
331, 7, 8, 32syl3anc 1184 . . . . . . . 8  |-  ( ph  ->  ( G `  C
)  =  P )
3431, 33eqtrd 2412 . . . . . . 7  |-  ( ph  ->  ( G `  ( `' D `  Q ) )  =  P )
3528, 34eqtrd 2412 . . . . . 6  |-  ( ph  ->  ( ( G  o.  `' D ) `  Q
)  =  P )
3635oveq2d 6029 . . . . 5  |-  ( ph  ->  ( Q  .\/  (
( G  o.  `' D ) `  Q
) )  =  ( Q  .\/  P ) )
371simpld 446 . . . . . 6  |-  ( ph  ->  K  e.  HL )
388simpld 446 . . . . . 6  |-  ( ph  ->  P  e.  A )
3921, 3hlatjcom 29533 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  =  ( Q 
.\/  P ) )
4037, 38, 26, 39syl3anc 1184 . . . . 5  |-  ( ph  ->  ( P  .\/  Q
)  =  ( Q 
.\/  P ) )
4136, 40eqtr4d 2415 . . . 4  |-  ( ph  ->  ( Q  .\/  (
( G  o.  `' D ) `  Q
) )  =  ( P  .\/  Q ) )
4241oveq1d 6028 . . 3  |-  ( ph  ->  ( ( Q  .\/  ( ( G  o.  `' D ) `  Q
) )  ./\  W
)  =  ( ( P  .\/  Q ) 
./\  W ) )
43 dihjatcclem.v . . 3  |-  V  =  ( ( P  .\/  Q )  ./\  W )
4442, 43syl6eqr 2430 . 2  |-  ( ph  ->  ( ( Q  .\/  ( ( G  o.  `' D ) `  Q
) )  ./\  W
)  =  V )
4525, 44eqtrd 2412 1  |-  ( ph  ->  ( R `  ( G  o.  `' D
) )  =  V )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   class class class wbr 4146   `'ccnv 4810    o. ccom 4815   ` cfv 5387  (class class class)co 6013   iota_crio 6471   Basecbs 13389   lecple 13456   occoc 13457   joincjn 14321   meetcmee 14322   LSSumclsm 15188   Atomscatm 29429   HLchlt 29516   LHypclh 30149   LTrncltrn 30266   trLctrl 30323   TEndoctendo 30917   DVecHcdvh 31244   DIsoHcdih 31394
This theorem is referenced by:  dihjatcclem4  31587
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-iin 4031  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-undef 6472  df-riota 6478  df-map 6949  df-poset 14323  df-plt 14335  df-lub 14351  df-glb 14352  df-join 14353  df-meet 14354  df-p0 14388  df-p1 14389  df-lat 14395  df-clat 14457  df-oposet 29342  df-ol 29344  df-oml 29345  df-covers 29432  df-ats 29433  df-atl 29464  df-cvlat 29488  df-hlat 29517  df-llines 29663  df-lplanes 29664  df-lvols 29665  df-lines 29666  df-psubsp 29668  df-pmap 29669  df-padd 29961  df-lhyp 30153  df-laut 30154  df-ldil 30269  df-ltrn 30270  df-trl 30324
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