Theorem dvhfvadd 35075

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 2454	. . . . 5 |- ( ( EDRing ` K ) ` W ) = ( ( EDRing ` K ) ` W )
5		dvhfvadd.u	. . . . 5 |- U = ( ( DVecH ` K ) ` W )
6	1, 2, 3, 4, 5	dvhset 35065	. . . 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 5804	. . 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 35066	. . . . . . . . . . 11 |- ( ( K e. HL /\ W e. H ) -> D = ( ( EDRing ` K ) ` W ) )
11	10	fveq2d 5804	. . . . . . . . . 10 |- ( ( K e. HL /\ W e. H ) -> ( +g ` D ) = ( +g ` ( ( EDRing ` K ) ` W ) ) )
12	8, 11	syl5eq 2507	. . . . . . . . 9 |- ( ( K e. HL /\ W e. H ) -> .+^ = ( +g ` ( ( EDRing ` K ) ` W ) ) )
13	12	oveqd 6218	. . . . . . . 8 |- ( ( K e. HL /\ W e. H ) -> ( ( 2nd ` f ) .+^ ( 2nd ` g ) ) = ( ( 2nd ` f ) ( +g ` ( ( EDRing ` K ) ` W ) ) ( 2nd ` g ) ) )
14	13	3ad2ant1 1009	. . . . . . 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 6718	. . . . . . . . . 10 |- ( f e. ( T X. E ) -> ( 2nd ` f ) e. E )
16		xp2nd 6718	. . . . . . . . . 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 2454	. . . . . . . . . 10 |- ( +g ` ( ( EDRing ` K ) ` W ) ) = ( +g ` ( ( EDRing ` K ) ` W ) )
19	1, 2, 3, 4, 18	erngplus 34786	. . . . . . . . 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 1184	. . . . . . 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 2495	. . . . . 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 4175	. . . . 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 6260	. . . 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 5810	. . . . . . . 8 |- ( ( LTrn ` K ) ` W ) e. _V
26	2, 25	eqeltri 2538	. . . . . . 7 |- T e. _V
27		fvex 5810	. . . . . . . 8 |- ( ( TEndo ` K ) ` W ) e. _V
28	3, 27	eqeltri 2538	. . . . . . 7 |- E e. _V
29	26, 28	xpex 6619	. . . . . 6 |- ( T X. E ) e. _V
30	29, 29	mpt2ex 6761	. . . . 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 2454	. . . . . 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 14424	. . . . 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 2512	. . 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 2495	. 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 2520	1 |- ( ( K e. HL /\ W e. H ) -> .+ = .+b )

Syntax hints:   -> wi 4   /\ wa 369   /\ w3a 965   = wceq 1370   e. wcel 1758   _Vcvv 3078   u. cun 3435   {csn 3986   {ctp 3990   <.cop 3992   |-> cmpt 4459   X. cxp 4947   o. ccom 4953   ` cfv 5527  (class class class)co 6201   |-> cmpt2 6203   1stc1st 6686   2ndc2nd 6687   ndxcnx 14290   Basecbs 14293   +g cplusg 14358  Scalarcsca 14361   .scvsca 14362   HLchlt 33334   LHypclh 33967   LTrncltrn 34084   TEndoctendo 34735   EDRingcedring 34736   DVecHcdvh 35062

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-1st 6688  df-2nd 6689  df-recs 6943  df-rdg 6977  df-1o 7031  df-oadd 7035  df-er 7212  df-en 7422  df-dom 7423  df-sdom 7424  df-fin 7425  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-nn 10435  df-2 10492  df-3 10493  df-4 10494  df-5 10495  df-6 10496  df-n0 10692  df-z 10759  df-uz 10974  df-fz 11556  df-struct 14295  df-ndx 14296  df-slot 14297  df-base 14298  df-plusg 14371  df-mulr 14372  df-sca 14374  df-vsca 14375  df-edring 34740  df-dvech 35063

This theorem is referenced by:  dvhvadd  35076