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Theorem dvhopN 31599
Description: Decompose a  DVecH vector expressed as an ordered pair into the sum of two components, the first from the translation group vector base of  DVecA and the other from the one-dimensional vector subspace  E. Part of Lemma M of [Crawley] p. 121, line 18. We represent their e, sigma, f by 
<. (  _I  |`  B ) ,  (  _I  |`  T )
>.,  U,  <. F ,  O >.. We swapped the order of vector sum (their juxtaposition i.e. composition) to show  <. F ,  O >. first. Note that  O and  (  _I  |`  T ) are the zero and one of the division ring  E, and  (  _I  |`  B ) is the zero of the translation group.  S is the scalar product. (Contributed by NM, 21-Nov-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
dvhop.b  |-  B  =  ( Base `  K
)
dvhop.h  |-  H  =  ( LHyp `  K
)
dvhop.t  |-  T  =  ( ( LTrn `  K
) `  W )
dvhop.e  |-  E  =  ( ( TEndo `  K
) `  W )
dvhop.p  |-  P  =  ( a  e.  E ,  b  e.  E  |->  ( c  e.  T  |->  ( ( a `  c )  o.  (
b `  c )
) ) )
dvhop.a  |-  A  =  ( f  e.  ( T  X.  E ) ,  g  e.  ( T  X.  E ) 
|->  <. ( ( 1st `  f )  o.  ( 1st `  g ) ) ,  ( ( 2nd `  f ) P ( 2nd `  g ) ) >. )
dvhop.s  |-  S  =  ( s  e.  E ,  f  e.  ( T  X.  E )  |->  <.
( s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f
) ) >. )
dvhop.o  |-  O  =  ( c  e.  T  |->  (  _I  |`  B ) )
Assertion
Ref Expression
dvhopN  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  U  e.  E ) )  ->  <. F ,  U >.  =  ( <. F ,  O >. A ( U S
<. (  _I  |`  B ) ,  (  _I  |`  T )
>. ) ) )
Distinct variable groups:    B, c    a, b, f, g, s, E    H, c    K, c    P, f, g    a, c, T, b, f, g, s    W, a, b, c
Allowed substitution hints:    A( f, g, s, a, b, c)    B( f, g, s, a, b)    P( s, a, b, c)    S( f, g, s, a, b, c)    U( f, g, s, a, b, c)    E( c)    F( f, g, s, a, b, c)    H( f, g, s, a, b)    K( f, g, s, a, b)    O( f, g, s, a, b, c)    W( f, g, s)

Proof of Theorem dvhopN
StepHypRef Expression
1 simprr 734 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  U  e.  E ) )  ->  U  e.  E )
2 dvhop.b . . . . . . 7  |-  B  =  ( Base `  K
)
3 dvhop.h . . . . . . 7  |-  H  =  ( LHyp `  K
)
4 dvhop.t . . . . . . 7  |-  T  =  ( ( LTrn `  K
) `  W )
52, 3, 4idltrn 30632 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  B )  e.  T )
65adantr 452 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  U  e.  E ) )  -> 
(  _I  |`  B )  e.  T )
7 dvhop.e . . . . . . 7  |-  E  =  ( ( TEndo `  K
) `  W )
83, 4, 7tendoidcl 31251 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  T )  e.  E )
98adantr 452 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  U  e.  E ) )  -> 
(  _I  |`  T )  e.  E )
10 dvhop.s . . . . . 6  |-  S  =  ( s  e.  E ,  f  e.  ( T  X.  E )  |->  <.
( s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f
) ) >. )
1110dvhopspN 31598 . . . . 5  |-  ( ( U  e.  E  /\  ( (  _I  |`  B )  e.  T  /\  (  _I  |`  T )  e.  E ) )  -> 
( U S <. (  _I  |`  B ) ,  (  _I  |`  T )
>. )  =  <. ( U `  (  _I  |`  B ) ) ,  ( U  o.  (  _I  |`  T ) )
>. )
121, 6, 9, 11syl12anc 1182 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  U  e.  E ) )  -> 
( U S <. (  _I  |`  B ) ,  (  _I  |`  T )
>. )  =  <. ( U `  (  _I  |`  B ) ) ,  ( U  o.  (  _I  |`  T ) )
>. )
132, 3, 7tendoid 31255 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E
)  ->  ( U `  (  _I  |`  B ) )  =  (  _I  |`  B ) )
1413adantrl 697 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  U  e.  E ) )  -> 
( U `  (  _I  |`  B ) )  =  (  _I  |`  B ) )
153, 4, 7tendo1mulr 31253 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E
)  ->  ( U  o.  (  _I  |`  T ) )  =  U )
1615adantrl 697 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  U  e.  E ) )  -> 
( U  o.  (  _I  |`  T ) )  =  U )
1714, 16opeq12d 3952 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  U  e.  E ) )  ->  <. ( U `  (  _I  |`  B ) ) ,  ( U  o.  (  _I  |`  T ) ) >.  =  <. (  _I  |`  B ) ,  U >. )
1812, 17eqtrd 2436 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  U  e.  E ) )  -> 
( U S <. (  _I  |`  B ) ,  (  _I  |`  T )
>. )  =  <. (  _I  |`  B ) ,  U >. )
1918oveq2d 6056 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  U  e.  E ) )  -> 
( <. F ,  O >. A ( U S
<. (  _I  |`  B ) ,  (  _I  |`  T )
>. ) )  =  (
<. F ,  O >. A
<. (  _I  |`  B ) ,  U >. )
)
20 simprl 733 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  U  e.  E ) )  ->  F  e.  T )
21 dvhop.o . . . . 5  |-  O  =  ( c  e.  T  |->  (  _I  |`  B ) )
222, 3, 4, 7, 21tendo0cl 31272 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  O  e.  E )
2322adantr 452 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  U  e.  E ) )  ->  O  e.  E )
24 dvhop.a . . . 4  |-  A  =  ( f  e.  ( T  X.  E ) ,  g  e.  ( T  X.  E ) 
|->  <. ( ( 1st `  f )  o.  ( 1st `  g ) ) ,  ( ( 2nd `  f ) P ( 2nd `  g ) ) >. )
2524dvhopaddN 31597 . . 3  |-  ( ( ( F  e.  T  /\  O  e.  E
)  /\  ( (  _I  |`  B )  e.  T  /\  U  e.  E ) )  -> 
( <. F ,  O >. A <. (  _I  |`  B ) ,  U >. )  =  <. ( F  o.  (  _I  |`  B ) ) ,  ( O P U ) >.
)
2620, 23, 6, 1, 25syl22anc 1185 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  U  e.  E ) )  -> 
( <. F ,  O >. A <. (  _I  |`  B ) ,  U >. )  =  <. ( F  o.  (  _I  |`  B ) ) ,  ( O P U ) >.
)
272, 3, 4ltrn1o 30606 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  F : B
-1-1-onto-> B )
2827adantrr 698 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  U  e.  E ) )  ->  F : B -1-1-onto-> B )
29 f1of 5633 . . . 4  |-  ( F : B -1-1-onto-> B  ->  F : B
--> B )
30 fcoi1 5576 . . . 4  |-  ( F : B --> B  -> 
( F  o.  (  _I  |`  B ) )  =  F )
3128, 29, 303syl 19 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  U  e.  E ) )  -> 
( F  o.  (  _I  |`  B ) )  =  F )
32 dvhop.p . . . . 5  |-  P  =  ( a  e.  E ,  b  e.  E  |->  ( c  e.  T  |->  ( ( a `  c )  o.  (
b `  c )
) ) )
332, 3, 4, 7, 21, 32tendo0pl 31273 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E
)  ->  ( O P U )  =  U )
3433adantrl 697 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  U  e.  E ) )  -> 
( O P U )  =  U )
3531, 34opeq12d 3952 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  U  e.  E ) )  ->  <. ( F  o.  (  _I  |`  B ) ) ,  ( O P U ) >.  =  <. F ,  U >. )
3619, 26, 353eqtrrd 2441 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  U  e.  E ) )  ->  <. F ,  U >.  =  ( <. F ,  O >. A ( U S
<. (  _I  |`  B ) ,  (  _I  |`  T )
>. ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   <.cop 3777    e. cmpt 4226    _I cid 4453    X. cxp 4835    |` cres 4839    o. ccom 4841   -->wf 5409   -1-1-onto->wf1o 5412   ` cfv 5413  (class class class)co 6040    e. cmpt2 6042   1stc1st 6306   2ndc2nd 6307   Basecbs 13424   HLchlt 29833   LHypclh 30466   LTrncltrn 30583   TEndoctendo 31234
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-map 6979  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-p1 14424  df-lat 14430  df-clat 14492  df-oposet 29659  df-ol 29661  df-oml 29662  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834  df-llines 29980  df-lplanes 29981  df-lvols 29982  df-lines 29983  df-psubsp 29985  df-pmap 29986  df-padd 30278  df-lhyp 30470  df-laut 30471  df-ldil 30586  df-ltrn 30587  df-trl 30641  df-tendo 31237
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