Theorem dvhvaddcl 34079

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 34076	. 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 455	. . . 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 6766	. . . . 5 |- ( F e. ( T X. E ) -> ( 1st ` F ) e. T )
11	10	ad2antrl 726	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. ( T X. E ) /\ G e. ( T X. E ) ) ) -> ( 1st ` F ) e. T )
12		xp1st 6766	. . . . 5 |- ( G e. ( T X. E ) -> ( 1st ` G ) e. T )
13	12	ad2antll 727	. . . 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 33702	. . . 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 1228	. . 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 2400	. . . . . . 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 34068	. . . . . 6 |- ( ( K e. HL /\ W e. H ) -> .+^ = ( a e. E , b e. E |-> ( c e. T |-> ( ( a ` c ) o. ( b ` c ) ) ) ) )
18	17	adantr 463	. . . . 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 6249	. . . 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 6767	. . . . . 6 |- ( F e. ( T X. E ) -> ( 2nd ` F ) e. E )
21	20	ad2antrl 726	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. ( T X. E ) /\ G e. ( T X. E ) ) ) -> ( 2nd ` F ) e. E )
22		xp2nd 6767	. . . . . 6 |- ( G e. ( T X. E ) -> ( 2nd ` G ) e. E )
23	22	ad2antll 727	. . . . 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 33764	. . . . 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 1228	. . . 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 2488	. . 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 4972	. . 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 659	. 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 2488	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 367   = wceq 1403   e. wcel 1840   <.cop 3975   |-> cmpt 4450   X. cxp 4938   o. ccom 4944   ` cfv 5523  (class class class)co 6232   |-> cmpt2 6234   1stc1st 6734   2ndc2nd 6735   +g cplusg 14799  Scalarcsca 14802   HLchlt 32332   LHypclh 32965   LTrncltrn 33082   TEndoctendo 33735   DVecHcdvh 34062

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528  ax-cnex 9496  ax-resscn 9497  ax-1cn 9498  ax-icn 9499  ax-addcl 9500  ax-addrcl 9501  ax-mulcl 9502  ax-mulrcl 9503  ax-mulcom 9504  ax-addass 9505  ax-mulass 9506  ax-distr 9507  ax-i2m1 9508  ax-1ne0 9509  ax-1rid 9510  ax-rnegex 9511  ax-rrecex 9512  ax-cnre 9513  ax-pre-lttri 9514  ax-pre-lttrn 9515  ax-pre-ltadd 9516  ax-pre-mulgt0 9517  ax-riotaBAD 31941

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-nel 2599  df-ral 2756  df-rex 2757  df-reu 2758  df-rmo 2759  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-pss 3427  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-tp 3974  df-op 3976  df-uni 4189  df-int 4225  df-iun 4270  df-iin 4271  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4487  df-eprel 4731  df-id 4735  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-riota 6194  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-om 6637  df-1st 6736  df-2nd 6737  df-undef 6957  df-recs 6997  df-rdg 7031  df-1o 7085  df-oadd 7089  df-er 7266  df-map 7377  df-en 7473  df-dom 7474  df-sdom 7475  df-fin 7476  df-pnf 9578  df-mnf 9579  df-xr 9580  df-ltxr 9581  df-le 9582  df-sub 9761  df-neg 9762  df-nn 10495  df-2 10553  df-3 10554  df-4 10555  df-5 10556  df-6 10557  df-n0 10755  df-z 10824  df-uz 11044  df-fz 11642  df-struct 14733  df-ndx 14734  df-slot 14735  df-base 14736  df-plusg 14812  df-mulr 14813  df-sca 14815  df-vsca 14816  df-preset 15771  df-poset 15789  df-plt 15802  df-lub 15818  df-glb 15819  df-join 15820  df-meet 15821  df-p0 15883  df-p1 15884  df-lat 15890  df-clat 15952  df-oposet 32158  df-ol 32160  df-oml 32161  df-covers 32248  df-ats 32249  df-atl 32280  df-cvlat 32304  df-hlat 32333  df-llines 32479  df-lplanes 32480  df-lvols 32481  df-lines 32482  df-psubsp 32484  df-pmap 32485  df-padd 32777  df-lhyp 32969  df-laut 32970  df-ldil 33085  df-ltrn 33086  df-trl 33141  df-tendo 33738  df-edring 33740  df-dvech 34063

This theorem is referenced by:  dvhvaddass  34081  dvhgrp  34091  dvhlveclem  34092