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Theorem dia2dimlem4 30524
Description: Lemma for dia2dim 30534. 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 30175 . . 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 29601 . . . 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 5489 . . 3  |-  ( ph  ->  ( D `  ( G `  P )
)  =  ( D `
 Q ) )
17 dia2dimlem4.dv . . 3  |-  ( ph  ->  ( D `  Q
)  =  ( F `
 P ) )
1814, 16, 173eqtrd 2320 . 2  |-  ( ph  ->  ( ( D  o.  G ) `  P
)  =  ( F `
 P ) )
1911, 12, 4, 5cdlemd 29663 . 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 1628    e. wcel 1688   class class class wbr 4024    o. ccom 4692   ` cfv 5221   lecple 13209   Atomscatm 28720   HLchlt 28807   LHypclh 29440   LTrncltrn 29557
This theorem is referenced by:  dia2dimlem5  30525
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-13 1690  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
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 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-iin 3909  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-iota 6252  df-undef 6291  df-riota 6299  df-map 6769  df-poset 14074  df-plt 14086  df-lub 14102  df-glb 14103  df-join 14104  df-meet 14105  df-p0 14139  df-p1 14140  df-lat 14146  df-clat 14208  df-oposet 28633  df-ol 28635  df-oml 28636  df-covers 28723  df-ats 28724  df-atl 28755  df-cvlat 28779  df-hlat 28808  df-llines 28954  df-lplanes 28955  df-lvols 28956  df-lines 28957  df-psubsp 28959  df-pmap 28960  df-padd 29252  df-lhyp 29444  df-laut 29445  df-ldil 29560  df-ltrn 29561  df-trl 29615
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