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Theorem dia2dimlem4 31704
Description: Lemma for dia2dim 31714. 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 31355 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  D  e.  T  /\  G  e.  T
)  ->  ( D  o.  G )  e.  T
)
71, 2, 3, 6syl3anc 1184 . 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 446 . . . 4  |-  ( ph  ->  P  e.  A )
11 dia2dimlem4.l . . . . 5  |-  .<_  =  ( le `  K )
12 dia2dimlem4.a . . . . 5  |-  A  =  ( Atoms `  K )
1311, 12, 4, 5ltrncoval 30781 . . . 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 1189 . . 3  |-  ( ph  ->  ( ( D  o.  G ) `  P
)  =  ( D `
 ( G `  P ) ) )
15 dia2dimlem4.gv . . . 4  |-  ( ph  ->  ( G `  P
)  =  Q )
1615fveq2d 5723 . . 3  |-  ( ph  ->  ( D `  ( G `  P )
)  =  ( D `
 Q ) )
17 dia2dimlem4.dv . . 3  |-  ( ph  ->  ( D `  Q
)  =  ( F `
 P ) )
1814, 16, 173eqtrd 2471 . 2  |-  ( ph  ->  ( ( D  o.  G ) `  P
)  =  ( F `
 P ) )
1911, 12, 4, 5cdlemd 30843 . 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 1198 1  |-  ( ph  ->  ( D  o.  G
)  =  F )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   class class class wbr 4204    o. ccom 4873   ` cfv 5445   lecple 13524   Atomscatm 29900   HLchlt 29987   LHypclh 30620   LTrncltrn 30737
This theorem is referenced by:  dia2dimlem5  31705
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-1st 6340  df-2nd 6341  df-undef 6534  df-riota 6540  df-map 7011  df-poset 14391  df-plt 14403  df-lub 14419  df-glb 14420  df-join 14421  df-meet 14422  df-p0 14456  df-p1 14457  df-lat 14463  df-clat 14525  df-oposet 29813  df-ol 29815  df-oml 29816  df-covers 29903  df-ats 29904  df-atl 29935  df-cvlat 29959  df-hlat 29988  df-llines 30134  df-lplanes 30135  df-lvols 30136  df-lines 30137  df-psubsp 30139  df-pmap 30140  df-padd 30432  df-lhyp 30624  df-laut 30625  df-ldil 30740  df-ltrn 30741  df-trl 30795
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