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Theorem dvhopN 31377
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 733 . . . . 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 30410 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  B )  e.  T )
65adantr 451 . . . . 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 31029 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  T )  e.  E )
98adantr 451 . . . . 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 31376 . . . . 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 1181 . . . 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 31033 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E
)  ->  ( U `  (  _I  |`  B ) )  =  (  _I  |`  B ) )
1413adantrl 696 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  U  e.  E ) )  -> 
( U `  (  _I  |`  B ) )  =  (  _I  |`  B ) )
153, 4, 7tendo1mulr 31031 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E
)  ->  ( U  o.  (  _I  |`  T ) )  =  U )
1615adantrl 696 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  U  e.  E ) )  -> 
( U  o.  (  _I  |`  T ) )  =  U )
1714, 16opeq12d 3906 . . . 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 2398 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  U  e.  E ) )  -> 
( U S <. (  _I  |`  B ) ,  (  _I  |`  T )
>. )  =  <. (  _I  |`  B ) ,  U >. )
1918oveq2d 5997 . 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 732 . . 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 31050 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  O  e.  E )
2322adantr 451 . . 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 31375 . . 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 1184 . 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 30384 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  F : B
-1-1-onto-> B )
2827adantrr 697 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  U  e.  E ) )  ->  F : B -1-1-onto-> B )
29 f1of 5578 . . . 4  |-  ( F : B -1-1-onto-> B  ->  F : B
--> B )
30 fcoi1 5521 . . . 4  |-  ( F : B --> B  -> 
( F  o.  (  _I  |`  B ) )  =  F )
3128, 29, 303syl 18 . . 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 31051 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E
)  ->  ( O P U )  =  U )
3433adantrl 696 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  U  e.  E ) )  -> 
( O P U )  =  U )
3531, 34opeq12d 3906 . 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 2403 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 358    = wceq 1647    e. wcel 1715   <.cop 3732    e. cmpt 4179    _I cid 4407    X. cxp 4790    |` cres 4794    o. ccom 4796   -->wf 5354   -1-1-onto->wf1o 5357   ` cfv 5358  (class class class)co 5981    e. cmpt2 5983   1stc1st 6247   2ndc2nd 6248   Basecbs 13356   HLchlt 29611   LHypclh 30244   LTrncltrn 30361   TEndoctendo 31012
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-iin 4010  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-undef 6440  df-riota 6446  df-map 6917  df-poset 14290  df-plt 14302  df-lub 14318  df-glb 14319  df-join 14320  df-meet 14321  df-p0 14355  df-p1 14356  df-lat 14362  df-clat 14424  df-oposet 29437  df-ol 29439  df-oml 29440  df-covers 29527  df-ats 29528  df-atl 29559  df-cvlat 29583  df-hlat 29612  df-llines 29758  df-lplanes 29759  df-lvols 29760  df-lines 29761  df-psubsp 29763  df-pmap 29764  df-padd 30056  df-lhyp 30248  df-laut 30249  df-ldil 30364  df-ltrn 30365  df-trl 30419  df-tendo 31015
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