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 ) >. ) ) )