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Theorem dia2dimlem4 35020
Description: Lemma for dia2dim 35030. 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 34671 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  D  e.  T  /\  G  e.  T
)  ->  ( D  o.  G )  e.  T
)
71, 2, 3, 6syl3anc 1219 . 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 459 . . . 4  |-  ( ph  ->  P  e.  A )
11 dia2dimlem4.l . . . . 5  |-  .<_  =  ( le `  K )
12 dia2dimlem4.a . . . . 5  |-  A  =  ( Atoms `  K )
1311, 12, 4, 5ltrncoval 34097 . . . 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 1224 . . 3  |-  ( ph  ->  ( ( D  o.  G ) `  P
)  =  ( D `
 ( G `  P ) ) )
15 dia2dimlem4.gv . . . 4  |-  ( ph  ->  ( G `  P
)  =  Q )
1615fveq2d 5795 . . 3  |-  ( ph  ->  ( D `  ( G `  P )
)  =  ( D `
 Q ) )
17 dia2dimlem4.dv . . 3  |-  ( ph  ->  ( D `  Q
)  =  ( F `
 P ) )
1814, 16, 173eqtrd 2496 . 2  |-  ( ph  ->  ( ( D  o.  G ) `  P
)  =  ( F `
 P ) )
1911, 12, 4, 5cdlemd 34159 . 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 1233 1  |-  ( ph  ->  ( D  o.  G
)  =  F )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   class class class wbr 4392    o. ccom 4944   ` cfv 5518   lecple 14349   Atomscatm 33216   HLchlt 33303   LHypclh 33936   LTrncltrn 34053
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-riotaBAD 32912
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-iun 4273  df-iin 4274  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-1st 6679  df-2nd 6680  df-undef 6894  df-map 7318  df-poset 15220  df-plt 15232  df-lub 15248  df-glb 15249  df-join 15250  df-meet 15251  df-p0 15313  df-p1 15314  df-lat 15320  df-clat 15382  df-oposet 33129  df-ol 33131  df-oml 33132  df-covers 33219  df-ats 33220  df-atl 33251  df-cvlat 33275  df-hlat 33304  df-llines 33450  df-lplanes 33451  df-lvols 33452  df-lines 33453  df-psubsp 33455  df-pmap 33456  df-padd 33748  df-lhyp 33940  df-laut 33941  df-ldil 34056  df-ltrn 34057  df-trl 34111
This theorem is referenced by:  dia2dimlem5  35021
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