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Theorem dvhopN 31232
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 30265 . . . . . 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 30884 . . . . . 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 31231 . . . . 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 30888 . . . . . 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 30886 . . . . . 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 3935 . . . 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 2420 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  U  e.  E ) )  -> 
( U S <. (  _I  |`  B ) ,  (  _I  |`  T )
>. )  =  <. (  _I  |`  B ) ,  U >. )
1918oveq2d 6037 . 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 30905 . . . 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 31230 . . 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 30239 . . . . 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 5615 . . . 4  |-  ( F : B -1-1-onto-> B  ->  F : B
--> B )
30 fcoi1 5558 . . . 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 30906 . . . 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 3935 . 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 2425 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 1717   <.cop 3761    e. cmpt 4208    _I cid 4435    X. cxp 4817    |` cres 4821    o. ccom 4823   -->wf 5391   -1-1-onto->wf1o 5394   ` cfv 5395  (class class class)co 6021    e. cmpt2 6023   1stc1st 6287   2ndc2nd 6288   Basecbs 13397   HLchlt 29466   LHypclh 30099   LTrncltrn 30216   TEndoctendo 30867
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-iin 4039  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-undef 6480  df-riota 6486  df-map 6957  df-poset 14331  df-plt 14343  df-lub 14359  df-glb 14360  df-join 14361  df-meet 14362  df-p0 14396  df-p1 14397  df-lat 14403  df-clat 14465  df-oposet 29292  df-ol 29294  df-oml 29295  df-covers 29382  df-ats 29383  df-atl 29414  df-cvlat 29438  df-hlat 29467  df-llines 29613  df-lplanes 29614  df-lvols 29615  df-lines 29616  df-psubsp 29618  df-pmap 29619  df-padd 29911  df-lhyp 30103  df-laut 30104  df-ldil 30219  df-ltrn 30220  df-trl 30274  df-tendo 30870
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