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Metamath Proof Explorer

Theorem dvhopN

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.)

Ref Expression
Hypotheses 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 dvhopN 
|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ U e. E ) ) -> <. F , U >. = ( <. F , O >. A ( U S <. ( _I |` B ) , ( _I |` T ) >. ) ) )

Proof

Step Hyp Ref Expression
1 dvhop.b 
 |-  B = ( Base ` K )
2 dvhop.h 
 |-  H = ( LHyp ` K )
3 dvhop.t 
 |-  T = ( ( LTrn ` K ) ` W )
4 dvhop.e 
 |-  E = ( ( TEndo ` K ) ` W )
5 dvhop.p 
 |-  P = ( a e. E , b e. E |-> ( c e. T |-> ( ( a ` c ) o. ( b ` c ) ) ) )
6 dvhop.a 
 |-  A = ( f e. ( T X. E ) , g e. ( T X. E ) |-> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( ( 2nd ` f ) P ( 2nd ` g ) ) >. )
7 dvhop.s 
 |-  S = ( s e. E , f e. ( T X. E ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. )
8 dvhop.o 
 |-  O = ( c e. T |-> ( _I |` B ) )
9 simprr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ U e. E ) ) -> U e. E )
10 1 2 3 idltrn 
 |-  ( ( K e. HL /\ W e. H ) -> ( _I |` B ) e. T )
11 10 adantr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ U e. E ) ) -> ( _I |` B ) e. T )
12 2 3 4 tendoidcl 
 |-  ( ( K e. HL /\ W e. H ) -> ( _I |` T ) e. E )
13 12 adantr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ U e. E ) ) -> ( _I |` T ) e. E )
14 7 dvhopspN 
 |-  ( ( U e. E /\ ( ( _I |` B ) e. T /\ ( _I |` T ) e. E ) ) -> ( U S <. ( _I |` B ) , ( _I |` T ) >. ) = <. ( U ` ( _I |` B ) ) , ( U o. ( _I |` T ) ) >. )
15 9 11 13 14 syl12anc 
 |-  ( ( ( 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 ) ) >. )
16 1 2 4 tendoid 
 |-  ( ( ( K e. HL /\ W e. H ) /\ U e. E ) -> ( U ` ( _I |` B ) ) = ( _I |` B ) )
17 16 adantrl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ U e. E ) ) -> ( U ` ( _I |` B ) ) = ( _I |` B ) )
18 2 3 4 tendo1mulr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ U e. E ) -> ( U o. ( _I |` T ) ) = U )
19 18 adantrl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ U e. E ) ) -> ( U o. ( _I |` T ) ) = U )
20 17 19 opeq12d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ U e. E ) ) -> <. ( U ` ( _I |` B ) ) , ( U o. ( _I |` T ) ) >. = <. ( _I |` B ) , U >. )
21 15 20 eqtrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ U e. E ) ) -> ( U S <. ( _I |` B ) , ( _I |` T ) >. ) = <. ( _I |` B ) , U >. )
22 21 oveq2d 
 |-  ( ( ( 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 >. ) )
23 simprl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ U e. E ) ) -> F e. T )
24 1 2 3 4 8 tendo0cl 
 |-  ( ( K e. HL /\ W e. H ) -> O e. E )
25 24 adantr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ U e. E ) ) -> O e. E )
26 6 dvhopaddN 
 |-  ( ( ( 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 ) >. )
27 23 25 11 9 26 syl22anc 
 |-  ( ( ( 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 ) >. )
28 1 2 3 ltrn1o 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> F : B -1-1-onto-> B )
29 28 adantrr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ U e. E ) ) -> F : B -1-1-onto-> B )
30 f1of 
 |-  ( F : B -1-1-onto-> B -> F : B --> B )
31 fcoi1 
 |-  ( F : B --> B -> ( F o. ( _I |` B ) ) = F )
32 29 30 31 3syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ U e. E ) ) -> ( F o. ( _I |` B ) ) = F )
33 1 2 3 4 8 5 tendo0pl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ U e. E ) -> ( O P U ) = U )
34 33 adantrl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ U e. E ) ) -> ( O P U ) = U )
35 32 34 opeq12d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ U e. E ) ) -> <. ( F o. ( _I |` B ) ) , ( O P U ) >. = <. F , U >. )
36 22 27 35 3eqtrrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ U e. E ) ) -> <. F , U >. = ( <. F , O >. A ( U S <. ( _I |` B ) , ( _I |` T ) >. ) ) )