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Theorem dvhopN 31379
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 30412 . . . . . 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 31031 . . . . . 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 31378 . . . . 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 1180 . . . 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 31035 . . . . . 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 31033 . . . . . 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 3806 . . . 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 2317 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  U  e.  E ) )  -> 
( U S <. (  _I  |`  B ) ,  (  _I  |`  T )
>. )  =  <. (  _I  |`  B ) ,  U >. )
1918oveq2d 5876 . 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 31052 . . . 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 31377 . . 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 1183 . 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 30386 . . . . 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 5474 . . . 4  |-  ( F : B -1-1-onto-> B  ->  F : B
--> B )
30 fcoi1 5417 . . . 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 31053 . . . 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 3806 . 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 2322 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 1625    e. wcel 1686   <.cop 3645    e. cmpt 4079    _I cid 4306    X. cxp 4689    |` cres 4693    o. ccom 4695   -->wf 5253   -1-1-onto->wf1o 5256   ` cfv 5257  (class class class)co 5860    e. cmpt2 5862   1stc1st 6122   2ndc2nd 6123   Basecbs 13150   HLchlt 29613   LHypclh 30246   LTrncltrn 30363   TEndoctendo 31014
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-iin 3910  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-1st 6124  df-2nd 6125  df-undef 6300  df-riota 6306  df-map 6776  df-poset 14082  df-plt 14094  df-lub 14110  df-glb 14111  df-join 14112  df-meet 14113  df-p0 14147  df-p1 14148  df-lat 14154  df-clat 14216  df-oposet 29439  df-ol 29441  df-oml 29442  df-covers 29529  df-ats 29530  df-atl 29561  df-cvlat 29585  df-hlat 29614  df-llines 29760  df-lplanes 29761  df-lvols 29762  df-lines 29763  df-psubsp 29765  df-pmap 29766  df-padd 30058  df-lhyp 30250  df-laut 30251  df-ldil 30366  df-ltrn 30367  df-trl 30421  df-tendo 31017
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