Theorem dvhopN 36405

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

Step	Hyp	Ref	Expression
1		simprr 796	. . . . 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 )
5	2, 3, 4	idltrn 35436	. . . . . 6 |- ( ( K e. HL /\ W e. H ) -> ( _I |` B ) e. T )
6	5	adantr 481	. . . . 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 )
8	3, 4, 7	tendoidcl 36057	. . . . . 6 |- ( ( K e. HL /\ W e. H ) -> ( _I |` T ) e. E )
9	8	adantr 481	. . . . 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 ) ) >. )
11	10	dvhopspN 36404	. . . . 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 ) ) >. )
12	1, 6, 9, 11	syl12anc 1324	. . . 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 ) ) >. )
13	2, 3, 7	tendoid 36061	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ U e. E ) -> ( U ` ( _I |` B ) ) = ( _I |` B ) )
14	13	adantrl 752	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ U e. E ) ) -> ( U ` ( _I |` B ) ) = ( _I |` B ) )
15	3, 4, 7	tendo1mulr 36059	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ U e. E ) -> ( U o. ( _I |` T ) ) = U )
16	15	adantrl 752	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ U e. E ) ) -> ( U o. ( _I |` T ) ) = U )
17	14, 16	opeq12d 4410	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ U e. E ) ) -> <. ( U ` ( _I |` B ) ) , ( U o. ( _I |` T ) ) >. = <. ( _I |` B ) , U >. )
18	12, 17	eqtrd 2656	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ U e. E ) ) -> ( U S <. ( _I |` B ) , ( _I |` T ) >. ) = <. ( _I |` B ) , U >. )
19	18	oveq2d 6666	. 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 794	. . 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 ) )
22	2, 3, 4, 7, 21	tendo0cl 36078	. . . 4 |- ( ( K e. HL /\ W e. H ) -> O e. E )
23	22	adantr 481	. . 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 ) ) >. )
25	24	dvhopaddN 36403	. . 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 ) >. )
26	20, 23, 6, 1, 25	syl22anc 1327	. 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 ) >. )
27	2, 3, 4	ltrn1o 35410	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> F : B -1-1-onto-> B )
28	27	adantrr 753	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ U e. E ) ) -> F : B -1-1-onto-> B )
29		f1of 6137	. . . 4 |- ( F : B -1-1-onto-> B -> F : B --> B )
30		fcoi1 6078	. . . 4 |- ( F : B --> B -> ( F o. ( _I |` B ) ) = F )
31	28, 29, 30	3syl 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 ) ) ) )
33	2, 3, 4, 7, 21, 32	tendo0pl 36079	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ U e. E ) -> ( O P U ) = U )
34	33	adantrl 752	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ U e. E ) ) -> ( O P U ) = U )
35	31, 34	opeq12d 4410	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ U e. E ) ) -> <. ( F o. ( _I |` B ) ) , ( O P U ) >. = <. F , U >. )
36	19, 26, 35	3eqtrrd 2661	1 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ U e. E ) ) -> <. F , U >. = ( <. F , O >. A ( U S <. ( _I |` B ) , ( _I |` T ) >. ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   <.cop 4183   |-> cmpt 4729   _I cid 5023   X. cxp 5112   |` cres 5116   o. ccom 5118   -->wf 5884   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1stc1st 7166   2ndc2nd 7167   Basecbs 15857   HLchlt 34637   LHypclh 35270   LTrncltrn 35387   TEndoctendo 36040

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-riotaBAD 34239

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-undef 7399  df-map 7859  df-preset 16928  df-poset 16946  df-plt 16958  df-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-p0 17039  df-p1 17040  df-lat 17046  df-clat 17108  df-oposet 34463  df-ol 34465  df-oml 34466  df-covers 34553  df-ats 34554  df-atl 34585  df-cvlat 34609  df-hlat 34638  df-llines 34784  df-lplanes 34785  df-lvols 34786  df-lines 34787  df-psubsp 34789  df-pmap 34790  df-padd 35082  df-lhyp 35274  df-laut 35275  df-ldil 35390  df-ltrn 35391  df-trl 35446  df-tendo 36043

This theorem is referenced by: (None)