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Theorem dia2dimlem4 30408
Description: Lemma for dia2dim 30418. Show that the composition (sum) of translations (vectors)  G and  D equals  F. Part of proof of Lemma M in [Crawley] p. 121 line 5. (Contributed by NM, 8-Sep-2014.)
Hypotheses
Ref Expression
dia2dimlem4.l  |-  .<_  =  ( le `  K )
dia2dimlem4.a  |-  A  =  ( Atoms `  K )
dia2dimlem4.h  |-  H  =  ( LHyp `  K
)
dia2dimlem4.t  |-  T  =  ( ( LTrn `  K
) `  W )
dia2dimlem4.k  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
dia2dimlem4.p  |-  ( ph  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
dia2dimlem4.f  |-  ( ph  ->  F  e.  T )
dia2dimlem4.g  |-  ( ph  ->  G  e.  T )
dia2dimlem4.gv  |-  ( ph  ->  ( G `  P
)  =  Q )
dia2dimlem4.d  |-  ( ph  ->  D  e.  T )
dia2dimlem4.dv  |-  ( ph  ->  ( D `  Q
)  =  ( F `
 P ) )
Assertion
Ref Expression
dia2dimlem4  |-  ( ph  ->  ( D  o.  G
)  =  F )

Proof of Theorem dia2dimlem4
StepHypRef Expression
1 dia2dimlem4.k . 2  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
2 dia2dimlem4.d . . 3  |-  ( ph  ->  D  e.  T )
3 dia2dimlem4.g . . 3  |-  ( ph  ->  G  e.  T )
4 dia2dimlem4.h . . . 4  |-  H  =  ( LHyp `  K
)
5 dia2dimlem4.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
64, 5ltrnco 30059 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  D  e.  T  /\  G  e.  T
)  ->  ( D  o.  G )  e.  T
)
71, 2, 3, 6syl3anc 1187 . 2  |-  ( ph  ->  ( D  o.  G
)  e.  T )
8 dia2dimlem4.f . 2  |-  ( ph  ->  F  e.  T )
9 dia2dimlem4.p . 2  |-  ( ph  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
109simpld 447 . . . 4  |-  ( ph  ->  P  e.  A )
11 dia2dimlem4.l . . . . 5  |-  .<_  =  ( le `  K )
12 dia2dimlem4.a . . . . 5  |-  A  =  ( Atoms `  K )
1311, 12, 4, 5ltrncoval 29485 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( D  e.  T  /\  G  e.  T )  /\  P  e.  A )  ->  (
( D  o.  G
) `  P )  =  ( D `  ( G `  P ) ) )
141, 2, 3, 10, 13syl121anc 1192 . . 3  |-  ( ph  ->  ( ( D  o.  G ) `  P
)  =  ( D `
 ( G `  P ) ) )
15 dia2dimlem4.gv . . . 4  |-  ( ph  ->  ( G `  P
)  =  Q )
1615fveq2d 5448 . . 3  |-  ( ph  ->  ( D `  ( G `  P )
)  =  ( D `
 Q ) )
17 dia2dimlem4.dv . . 3  |-  ( ph  ->  ( D `  Q
)  =  ( F `
 P ) )
1814, 16, 173eqtrd 2292 . 2  |-  ( ph  ->  ( ( D  o.  G ) `  P
)  =  ( F `
 P ) )
1911, 12, 4, 5cdlemd 29547 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( D  o.  G )  e.  T  /\  F  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( ( D  o.  G ) `  P )  =  ( F `  P ) )  ->  ( D  o.  G )  =  F )
201, 7, 8, 9, 18, 19syl311anc 1201 1  |-  ( ph  ->  ( D  o.  G
)  =  F )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621   class class class wbr 3983    o. ccom 4651   ` cfv 4659   lecple 13163   Atomscatm 28604   HLchlt 28691   LHypclh 29324   LTrncltrn 29441
This theorem is referenced by:  dia2dimlem5  30409
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2521  df-rex 2522  df-reu 2523  df-rmo 2524  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-op 3609  df-uni 3788  df-iun 3867  df-iin 3868  df-br 3984  df-opab 4038  df-mpt 4039  df-id 4267  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-1st 6042  df-2nd 6043  df-iota 6211  df-undef 6250  df-riota 6258  df-map 6728  df-poset 14028  df-plt 14040  df-lub 14056  df-glb 14057  df-join 14058  df-meet 14059  df-p0 14093  df-p1 14094  df-lat 14100  df-clat 14162  df-oposet 28517  df-ol 28519  df-oml 28520  df-covers 28607  df-ats 28608  df-atl 28639  df-cvlat 28663  df-hlat 28692  df-llines 28838  df-lplanes 28839  df-lvols 28840  df-lines 28841  df-psubsp 28843  df-pmap 28844  df-padd 29136  df-lhyp 29328  df-laut 29329  df-ldil 29444  df-ltrn 29445  df-trl 29499
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