Theorem dvhfvadd 35888

Description: The vector sum operation for the constructed full vector space H. (Contributed by NM, 26-Oct-2013.) (Revised by Mario Carneiro, 23-Jun-2014.)

Hypotheses

Ref	Expression
dvhfvadd.h	|- H = ( LHyp ` K )
dvhfvadd.t	|- T = ( ( LTrn ` K ) ` W )
dvhfvadd.e	|- E = ( ( TEndo ` K ) ` W )
dvhfvadd.u	|- U = ( ( DVecH ` K ) ` W )
dvhfvadd.f	|- D = (Scalar ` U )
dvhfvadd.p	|- .+^ = ( +g ` D )
dvhfvadd.a	|- .+b = ( f e. ( T X. E ) , g e. ( T X. E ) |-> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( ( 2nd ` f ) .+^ ( 2nd ` g ) ) >. )
dvhfvadd.s	|- .+ = ( +g ` U )

Assertion

Ref	Expression
dvhfvadd	|- ( ( K e. HL /\ W e. H ) -> .+ = .+b )

Distinct variable groups:   f, g, E   f, H, g   f, K, g   T, f, g   f, W, g

Allowed substitution hints:   D( f, g)   .+ ( f, g)   .+^ ( f, g)   .+b ( f, g)   U( f, g)

Proof of Theorem dvhfvadd

Dummy variables h s are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvhfvadd.h	. . . . 5 |- H = ( LHyp ` K )
2		dvhfvadd.t	. . . . 5 |- T = ( ( LTrn ` K ) ` W )
3		dvhfvadd.e	. . . . 5 |- E = ( ( TEndo ` K ) ` W )
4		eqid 2467	. . . . 5 |- ( ( EDRing ` K ) ` W ) = ( ( EDRing ` K ) ` W )
5		dvhfvadd.u	. . . . 5 |- U = ( ( DVecH ` K ) ` W )
6	1, 2, 3, 4, 5	dvhset 35878	. . . 4 |- ( ( K e. HL /\ W e. H ) -> U = ( { <. ( Base ` ndx ) , ( T X. E ) >. , <. ( +g ` ndx ) , ( f e. ( T X. E ) , g e. ( T X. E ) |-> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( h e. T |-> ( ( ( 2nd ` f ) ` h ) o. ( ( 2nd ` g ) ` h ) ) ) >. ) >. , <. (Scalar ` ndx ) , ( ( EDRing ` K ) ` W ) >. } u. { <. ( .s ` ndx ) , ( s e. E , f e. ( T X. E ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. ) >. } ) )
7	6	fveq2d 5868	. . 3 |- ( ( K e. HL /\ W e. H ) -> ( +g ` U ) = ( +g ` ( { <. ( Base ` ndx ) , ( T X. E ) >. , <. ( +g ` ndx ) , ( f e. ( T X. E ) , g e. ( T X. E ) |-> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( h e. T |-> ( ( ( 2nd ` f ) ` h ) o. ( ( 2nd ` g ) ` h ) ) ) >. ) >. , <. (Scalar ` ndx ) , ( ( EDRing ` K ) ` W ) >. } u. { <. ( .s ` ndx ) , ( s e. E , f e. ( T X. E ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. ) >. } ) ) )
8		dvhfvadd.p	. . . . . . . . . 10 |- .+^ = ( +g ` D )
9		dvhfvadd.f	. . . . . . . . . . . 12 |- D = (Scalar ` U )
10	1, 4, 5, 9	dvhsca 35879	. . . . . . . . . . 11 |- ( ( K e. HL /\ W e. H ) -> D = ( ( EDRing ` K ) ` W ) )
11	10	fveq2d 5868	. . . . . . . . . 10 |- ( ( K e. HL /\ W e. H ) -> ( +g ` D ) = ( +g ` ( ( EDRing ` K ) ` W ) ) )
12	8, 11	syl5eq 2520	. . . . . . . . 9 |- ( ( K e. HL /\ W e. H ) -> .+^ = ( +g ` ( ( EDRing ` K ) ` W ) ) )
13	12	oveqd 6299	. . . . . . . 8 |- ( ( K e. HL /\ W e. H ) -> ( ( 2nd ` f ) .+^ ( 2nd ` g ) ) = ( ( 2nd ` f ) ( +g ` ( ( EDRing ` K ) ` W ) ) ( 2nd ` g ) ) )
14	13	3ad2ant1 1017	. . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ f e. ( T X. E ) /\ g e. ( T X. E ) ) -> ( ( 2nd ` f ) .+^ ( 2nd ` g ) ) = ( ( 2nd ` f ) ( +g ` ( ( EDRing ` K ) ` W ) ) ( 2nd ` g ) ) )
15		xp2nd 6812	. . . . . . . . . 10 |- ( f e. ( T X. E ) -> ( 2nd ` f ) e. E )
16		xp2nd 6812	. . . . . . . . . 10 |- ( g e. ( T X. E ) -> ( 2nd ` g ) e. E )
17	15, 16	anim12i 566	. . . . . . . . 9 |- ( ( f e. ( T X. E ) /\ g e. ( T X. E ) ) -> ( ( 2nd ` f ) e. E /\ ( 2nd ` g ) e. E ) )
18		eqid 2467	. . . . . . . . . 10 |- ( +g ` ( ( EDRing ` K ) ` W ) ) = ( +g ` ( ( EDRing ` K ) ` W ) )
19	1, 2, 3, 4, 18	erngplus 35599	. . . . . . . . 9 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( 2nd ` f ) e. E /\ ( 2nd ` g ) e. E ) ) -> ( ( 2nd ` f ) ( +g ` ( ( EDRing ` K ) ` W ) ) ( 2nd ` g ) ) = ( h e. T |-> ( ( ( 2nd ` f ) ` h ) o. ( ( 2nd ` g ) ` h ) ) ) )
20	17, 19	sylan2 474	. . . . . . . 8 |- ( ( ( K e. HL /\ W e. H ) /\ ( f e. ( T X. E ) /\ g e. ( T X. E ) ) ) -> ( ( 2nd ` f ) ( +g ` ( ( EDRing ` K ) ` W ) ) ( 2nd ` g ) ) = ( h e. T |-> ( ( ( 2nd ` f ) ` h ) o. ( ( 2nd ` g ) ` h ) ) ) )
21	20	3impb 1192	. . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ f e. ( T X. E ) /\ g e. ( T X. E ) ) -> ( ( 2nd ` f ) ( +g ` ( ( EDRing ` K ) ` W ) ) ( 2nd ` g ) ) = ( h e. T |-> ( ( ( 2nd ` f ) ` h ) o. ( ( 2nd ` g ) ` h ) ) ) )
22	14, 21	eqtrd 2508	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ f e. ( T X. E ) /\ g e. ( T X. E ) ) -> ( ( 2nd ` f ) .+^ ( 2nd ` g ) ) = ( h e. T |-> ( ( ( 2nd ` f ) ` h ) o. ( ( 2nd ` g ) ` h ) ) ) )
23	22	opeq2d 4220	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ f e. ( T X. E ) /\ g e. ( T X. E ) ) -> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( ( 2nd ` f ) .+^ ( 2nd ` g ) ) >. = <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( h e. T |-> ( ( ( 2nd ` f ) ` h ) o. ( ( 2nd ` g ) ` h ) ) ) >. )
24	23	mpt2eq3dva 6343	. . . 4 |- ( ( K e. HL /\ W e. H ) -> ( f e. ( T X. E ) , g e. ( T X. E ) |-> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( ( 2nd ` f ) .+^ ( 2nd ` g ) ) >. ) = ( f e. ( T X. E ) , g e. ( T X. E ) |-> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( h e. T |-> ( ( ( 2nd ` f ) ` h ) o. ( ( 2nd ` g ) ` h ) ) ) >. ) )
25		fvex 5874	. . . . . . . 8 |- ( ( LTrn ` K ) ` W ) e. _V
26	2, 25	eqeltri 2551	. . . . . . 7 |- T e. _V
27		fvex 5874	. . . . . . . 8 |- ( ( TEndo ` K ) ` W ) e. _V
28	3, 27	eqeltri 2551	. . . . . . 7 |- E e. _V
29	26, 28	xpex 6711	. . . . . 6 |- ( T X. E ) e. _V
30	29, 29	mpt2ex 6857	. . . . 5 |- ( f e. ( T X. E ) , g e. ( T X. E ) |-> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( h e. T |-> ( ( ( 2nd ` f ) ` h ) o. ( ( 2nd ` g ) ` h ) ) ) >. ) e. _V
31		eqid 2467	. . . . . 6 |- ( { <. ( Base ` ndx ) , ( T X. E ) >. , <. ( +g ` ndx ) , ( f e. ( T X. E ) , g e. ( T X. E ) |-> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( h e. T |-> ( ( ( 2nd ` f ) ` h ) o. ( ( 2nd ` g ) ` h ) ) ) >. ) >. , <. (Scalar ` ndx ) , ( ( EDRing ` K ) ` W ) >. } u. { <. ( .s ` ndx ) , ( s e. E , f e. ( T X. E ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. ) >. } ) = ( { <. ( Base ` ndx ) , ( T X. E ) >. , <. ( +g ` ndx ) , ( f e. ( T X. E ) , g e. ( T X. E ) |-> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( h e. T |-> ( ( ( 2nd ` f ) ` h ) o. ( ( 2nd ` g ) ` h ) ) ) >. ) >. , <. (Scalar ` ndx ) , ( ( EDRing ` K ) ` W ) >. } u. { <. ( .s ` ndx ) , ( s e. E , f e. ( T X. E ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. ) >. } )
32	31	lmodplusg 14614	. . . . 5 |- ( ( f e. ( T X. E ) , g e. ( T X. E ) |-> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( h e. T |-> ( ( ( 2nd ` f ) ` h ) o. ( ( 2nd ` g ) ` h ) ) ) >. ) e. _V -> ( f e. ( T X. E ) , g e. ( T X. E ) |-> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( h e. T |-> ( ( ( 2nd ` f ) ` h ) o. ( ( 2nd ` g ) ` h ) ) ) >. ) = ( +g ` ( { <. ( Base ` ndx ) , ( T X. E ) >. , <. ( +g ` ndx ) , ( f e. ( T X. E ) , g e. ( T X. E ) |-> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( h e. T |-> ( ( ( 2nd ` f ) ` h ) o. ( ( 2nd ` g ) ` h ) ) ) >. ) >. , <. (Scalar ` ndx ) , ( ( EDRing ` K ) ` W ) >. } u. { <. ( .s ` ndx ) , ( s e. E , f e. ( T X. E ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. ) >. } ) ) )
33	30, 32	ax-mp 5	. . . 4 |- ( f e. ( T X. E ) , g e. ( T X. E ) |-> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( h e. T |-> ( ( ( 2nd ` f ) ` h ) o. ( ( 2nd ` g ) ` h ) ) ) >. ) = ( +g ` ( { <. ( Base ` ndx ) , ( T X. E ) >. , <. ( +g ` ndx ) , ( f e. ( T X. E ) , g e. ( T X. E ) |-> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( h e. T |-> ( ( ( 2nd ` f ) ` h ) o. ( ( 2nd ` g ) ` h ) ) ) >. ) >. , <. (Scalar ` ndx ) , ( ( EDRing ` K ) ` W ) >. } u. { <. ( .s ` ndx ) , ( s e. E , f e. ( T X. E ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. ) >. } ) )
34	24, 33	syl6req 2525	. . 3 |- ( ( K e. HL /\ W e. H ) -> ( +g ` ( { <. ( Base ` ndx ) , ( T X. E ) >. , <. ( +g ` ndx ) , ( f e. ( T X. E ) , g e. ( T X. E ) |-> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( h e. T |-> ( ( ( 2nd ` f ) ` h ) o. ( ( 2nd ` g ) ` h ) ) ) >. ) >. , <. (Scalar ` ndx ) , ( ( EDRing ` K ) ` W ) >. } u. { <. ( .s ` ndx ) , ( s e. E , f e. ( T X. E ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. ) >. } ) ) = ( f e. ( T X. E ) , g e. ( T X. E ) |-> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( ( 2nd ` f ) .+^ ( 2nd ` g ) ) >. ) )
35	7, 34	eqtrd 2508	. 2 |- ( ( K e. HL /\ W e. H ) -> ( +g ` U ) = ( f e. ( T X. E ) , g e. ( T X. E ) |-> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( ( 2nd ` f ) .+^ ( 2nd ` g ) ) >. ) )
36		dvhfvadd.s	. 2 |- .+ = ( +g ` U )
37		dvhfvadd.a	. 2 |- .+b = ( f e. ( T X. E ) , g e. ( T X. E ) |-> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( ( 2nd ` f ) .+^ ( 2nd ` g ) ) >. )
38	35, 36, 37	3eqtr4g 2533	1 |- ( ( K e. HL /\ W e. H ) -> .+ = .+b )

Syntax hints:   -> wi 4   /\ wa 369   /\ w3a 973   = wceq 1379   e. wcel 1767   _Vcvv 3113   u. cun 3474   {csn 4027   {ctp 4031   <.cop 4033   |-> cmpt 4505   X. cxp 4997   o. ccom 5003   ` cfv 5586  (class class class)co 6282   |-> cmpt2 6284   1stc1st 6779   2ndc2nd 6780   ndxcnx 14480   Basecbs 14483   +g cplusg 14548  Scalarcsca 14551   .scvsca 14552   HLchlt 34147   LHypclh 34780   LTrncltrn 34897   TEndoctendo 35548   EDRingcedring 35549   DVecHcdvh 35875

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-struct 14485  df-ndx 14486  df-slot 14487  df-base 14488  df-plusg 14561  df-mulr 14562  df-sca 14564  df-vsca 14565  df-edring 35553  df-dvech 35876

This theorem is referenced by:  dvhvadd  35889