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Theorem dvhopN 31603
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 30636 . . . . . 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 31255 . . . . . 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 31602 . . . . 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 31259 . . . . . 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 31257 . . . . . 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 3956 . . . 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 2440 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  U  e.  E ) )  -> 
( U S <. (  _I  |`  B ) ,  (  _I  |`  T )
>. )  =  <. (  _I  |`  B ) ,  U >. )
1918oveq2d 6060 . 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 31276 . . . 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 31601 . . 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 30610 . . . . 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 5637 . . . 4  |-  ( F : B -1-1-onto-> B  ->  F : B
--> B )
30 fcoi1 5580 . . . 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 31277 . . . 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 3956 . 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 2445 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 3781    e. cmpt 4230    _I cid 4457    X. cxp 4839    |` cres 4843    o. ccom 4845   -->wf 5413   -1-1-onto->wf1o 5416   ` cfv 5417  (class class class)co 6044    e. cmpt2 6046   1stc1st 6310   2ndc2nd 6311   Basecbs 13428   HLchlt 29837   LHypclh 30470   LTrncltrn 30587   TEndoctendo 31238
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 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
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 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-iin 4060  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-undef 6506  df-riota 6512  df-map 6983  df-poset 14362  df-plt 14374  df-lub 14390  df-glb 14391  df-join 14392  df-meet 14393  df-p0 14427  df-p1 14428  df-lat 14434  df-clat 14496  df-oposet 29663  df-ol 29665  df-oml 29666  df-covers 29753  df-ats 29754  df-atl 29785  df-cvlat 29809  df-hlat 29838  df-llines 29984  df-lplanes 29985  df-lvols 29986  df-lines 29987  df-psubsp 29989  df-pmap 29990  df-padd 30282  df-lhyp 30474  df-laut 30475  df-ldil 30590  df-ltrn 30591  df-trl 30645  df-tendo 31241
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