Theorem dvhvaddcl 35098

Description: Closure of 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
dvhvaddcl.h	|- H = ( LHyp ` K )
dvhvaddcl.t	|- T = ( ( LTrn ` K ) ` W )
dvhvaddcl.e	|- E = ( ( TEndo ` K ) ` W )
dvhvaddcl.u	|- U = ( ( DVecH ` K ) ` W )
dvhvaddcl.d	|- D = (Scalar ` U )
dvhvaddcl.p	|- .+^ = ( +g ` D )
dvhvaddcl.a	|- .+ = ( +g ` U )

Assertion

Ref	Expression
dvhvaddcl	|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. ( T X. E ) /\ G e. ( T X. E ) ) ) -> ( F .+ G ) e. ( T X. E ) )

Proof of Theorem dvhvaddcl

Dummy variables a b c are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvhvaddcl.h	. . 3 |- H = ( LHyp ` K )
2		dvhvaddcl.t	. . 3 |- T = ( ( LTrn ` K ) ` W )
3		dvhvaddcl.e	. . 3 |- E = ( ( TEndo ` K ) ` W )
4		dvhvaddcl.u	. . 3 |- U = ( ( DVecH ` K ) ` W )
5		dvhvaddcl.d	. . 3 |- D = (Scalar ` U )
6		dvhvaddcl.a	. . 3 |- .+ = ( +g ` U )
7		dvhvaddcl.p	. . 3 |- .+^ = ( +g ` D )
8	1, 2, 3, 4, 5, 6, 7	dvhvadd 35095	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. ( T X. E ) /\ G e. ( T X. E ) ) ) -> ( F .+ G ) = <. ( ( 1st ` F ) o. ( 1st ` G ) ) , ( ( 2nd ` F ) .+^ ( 2nd ` G ) ) >. )
9		simpl 457	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. ( T X. E ) /\ G e. ( T X. E ) ) ) -> ( K e. HL /\ W e. H ) )
10		xp1st 6719	. . . . 5 |- ( F e. ( T X. E ) -> ( 1st ` F ) e. T )
11	10	ad2antrl 727	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. ( T X. E ) /\ G e. ( T X. E ) ) ) -> ( 1st ` F ) e. T )
12		xp1st 6719	. . . . 5 |- ( G e. ( T X. E ) -> ( 1st ` G ) e. T )
13	12	ad2antll 728	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. ( T X. E ) /\ G e. ( T X. E ) ) ) -> ( 1st ` G ) e. T )
14	1, 2	ltrnco 34721	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( 1st ` F ) e. T /\ ( 1st ` G ) e. T ) -> ( ( 1st ` F ) o. ( 1st ` G ) ) e. T )
15	9, 11, 13, 14	syl3anc 1219	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. ( T X. E ) /\ G e. ( T X. E ) ) ) -> ( ( 1st ` F ) o. ( 1st ` G ) ) e. T )
16		eqid 2454	. . . . . . 7 |- ( a e. E , b e. E |-> ( c e. T |-> ( ( a ` c ) o. ( b ` c ) ) ) ) = ( a e. E , b e. E |-> ( c e. T |-> ( ( a ` c ) o. ( b ` c ) ) ) )
17	1, 2, 3, 4, 5, 16, 7	dvhfplusr 35087	. . . . . 6 |- ( ( K e. HL /\ W e. H ) -> .+^ = ( a e. E , b e. E |-> ( c e. T |-> ( ( a ` c ) o. ( b ` c ) ) ) ) )
18	17	adantr 465	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. ( T X. E ) /\ G e. ( T X. E ) ) ) -> .+^ = ( a e. E , b e. E |-> ( c e. T |-> ( ( a ` c ) o. ( b ` c ) ) ) ) )
19	18	oveqd 6220	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. ( T X. E ) /\ G e. ( T X. E ) ) ) -> ( ( 2nd ` F ) .+^ ( 2nd ` G ) ) = ( ( 2nd ` F ) ( a e. E , b e. E |-> ( c e. T |-> ( ( a ` c ) o. ( b ` c ) ) ) ) ( 2nd ` G ) ) )
20		xp2nd 6720	. . . . . 6 |- ( F e. ( T X. E ) -> ( 2nd ` F ) e. E )
21	20	ad2antrl 727	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. ( T X. E ) /\ G e. ( T X. E ) ) ) -> ( 2nd ` F ) e. E )
22		xp2nd 6720	. . . . . 6 |- ( G e. ( T X. E ) -> ( 2nd ` G ) e. E )
23	22	ad2antll 728	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. ( T X. E ) /\ G e. ( T X. E ) ) ) -> ( 2nd ` G ) e. E )
24	1, 2, 3, 16	tendoplcl 34783	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( 2nd ` F ) e. E /\ ( 2nd ` G ) e. E ) -> ( ( 2nd ` F ) ( a e. E , b e. E |-> ( c e. T |-> ( ( a ` c ) o. ( b ` c ) ) ) ) ( 2nd ` G ) ) e. E )
25	9, 21, 23, 24	syl3anc 1219	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. ( T X. E ) /\ G e. ( T X. E ) ) ) -> ( ( 2nd ` F ) ( a e. E , b e. E |-> ( c e. T |-> ( ( a ` c ) o. ( b ` c ) ) ) ) ( 2nd ` G ) ) e. E )
26	19, 25	eqeltrd 2542	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. ( T X. E ) /\ G e. ( T X. E ) ) ) -> ( ( 2nd ` F ) .+^ ( 2nd ` G ) ) e. E )
27		opelxpi 4982	. . 3 |- ( ( ( ( 1st ` F ) o. ( 1st ` G ) ) e. T /\ ( ( 2nd ` F ) .+^ ( 2nd ` G ) ) e. E ) -> <. ( ( 1st ` F ) o. ( 1st ` G ) ) , ( ( 2nd ` F ) .+^ ( 2nd ` G ) ) >. e. ( T X. E ) )
28	15, 26, 27	syl2anc 661	. 2 |- ( ( ( 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 ) ) >. e. ( T X. E ) )
29	8, 28	eqeltrd 2542	1 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. ( T X. E ) /\ G e. ( T X. E ) ) ) -> ( F .+ G ) e. ( T X. E ) )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1370   e. wcel 1758   <.cop 3994   |-> cmpt 4461   X. cxp 4949   o. ccom 4955   ` cfv 5529  (class class class)co 6203   |-> cmpt2 6205   1stc1st 6688   2ndc2nd 6689   +g cplusg 14360  Scalarcsca 14363   HLchlt 33353   LHypclh 33986   LTrncltrn 34103   TEndoctendo 34754   DVecHcdvh 35081

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 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473  ax-riotaBAD 32962

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-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-iin 4285  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-undef 6905  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-map 7329  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-nn 10437  df-2 10494  df-3 10495  df-4 10496  df-5 10497  df-6 10498  df-n0 10694  df-z 10761  df-uz 10976  df-fz 11558  df-struct 14297  df-ndx 14298  df-slot 14299  df-base 14300  df-plusg 14373  df-mulr 14374  df-sca 14376  df-vsca 14377  df-poset 15238  df-plt 15250  df-lub 15266  df-glb 15267  df-join 15268  df-meet 15269  df-p0 15331  df-p1 15332  df-lat 15338  df-clat 15400  df-oposet 33179  df-ol 33181  df-oml 33182  df-covers 33269  df-ats 33270  df-atl 33301  df-cvlat 33325  df-hlat 33354  df-llines 33500  df-lplanes 33501  df-lvols 33502  df-lines 33503  df-psubsp 33505  df-pmap 33506  df-padd 33798  df-lhyp 33990  df-laut 33991  df-ldil 34106  df-ltrn 34107  df-trl 34161  df-tendo 34757  df-edring 34759  df-dvech 35082

This theorem is referenced by:  dvhvaddass  35100  dvhgrp  35110  dvhlveclem  35111