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Theorem dia2dimlem4 30387
Description: Lemma for dia2dim 30397. 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 30038 . . 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 29464 . . . 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 5427 . . 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 29526 . 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 3963    o. ccom 4630   ` cfv 4638   lecple 13142   Atomscatm 28583   HLchlt 28670   LHypclh 29303   LTrncltrn 29420
This theorem is referenced by:  dia2dimlem5  30388
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 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449
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 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-iun 3848  df-iin 3849  df-br 3964  df-opab 4018  df-mpt 4019  df-id 4246  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-1st 6021  df-2nd 6022  df-iota 6190  df-undef 6229  df-riota 6237  df-map 6707  df-poset 14007  df-plt 14019  df-lub 14035  df-glb 14036  df-join 14037  df-meet 14038  df-p0 14072  df-p1 14073  df-lat 14079  df-clat 14141  df-oposet 28496  df-ol 28498  df-oml 28499  df-covers 28586  df-ats 28587  df-atl 28618  df-cvlat 28642  df-hlat 28671  df-llines 28817  df-lplanes 28818  df-lvols 28819  df-lines 28820  df-psubsp 28822  df-pmap 28823  df-padd 29115  df-lhyp 29307  df-laut 29308  df-ldil 29423  df-ltrn 29424  df-trl 29478
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