Theorem dvhfvadd 36380

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 2622	. . . . 5 |- ( ( EDRing ` K ) ` W ) = ( ( EDRing ` K ) ` W )
5		dvhfvadd.u	. . . . 5 |- U = ( ( DVecH ` K ) ` W )
6	1, 2, 3, 4, 5	dvhset 36370	. . . 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 6195	. . 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 36371	. . . . . . . . . . 11 |- ( ( K e. HL /\ W e. H ) -> D = ( ( EDRing ` K ) ` W ) )
11	10	fveq2d 6195	. . . . . . . . . 10 |- ( ( K e. HL /\ W e. H ) -> ( +g ` D ) = ( +g ` ( ( EDRing ` K ) ` W ) ) )
12	8, 11	syl5eq 2668	. . . . . . . . 9 |- ( ( K e. HL /\ W e. H ) -> .+^ = ( +g ` ( ( EDRing ` K ) ` W ) ) )
13	12	oveqd 6667	. . . . . . . 8 |- ( ( K e. HL /\ W e. H ) -> ( ( 2nd ` f ) .+^ ( 2nd ` g ) ) = ( ( 2nd ` f ) ( +g ` ( ( EDRing ` K ) ` W ) ) ( 2nd ` g ) ) )
14	13	3ad2ant1 1082	. . . . . . 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 7199	. . . . . . . . . 10 |- ( f e. ( T X. E ) -> ( 2nd ` f ) e. E )
16		xp2nd 7199	. . . . . . . . . 10 |- ( g e. ( T X. E ) -> ( 2nd ` g ) e. E )
17	15, 16	anim12i 590	. . . . . . . . 9 |- ( ( f e. ( T X. E ) /\ g e. ( T X. E ) ) -> ( ( 2nd ` f ) e. E /\ ( 2nd ` g ) e. E ) )
18		eqid 2622	. . . . . . . . . 10 |- ( +g ` ( ( EDRing ` K ) ` W ) ) = ( +g ` ( ( EDRing ` K ) ` W ) )
19	1, 2, 3, 4, 18	erngplus 36091	. . . . . . . . 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 491	. . . . . . . 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 1260	. . . . . . 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 2656	. . . . . 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 4409	. . . . 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 6719	. . . 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 6201	. . . . . . . 8 |- ( ( LTrn ` K ) ` W ) e. _V
26	2, 25	eqeltri 2697	. . . . . . 7 |- T e. _V
27		fvex 6201	. . . . . . . 8 |- ( ( TEndo ` K ) ` W ) e. _V
28	3, 27	eqeltri 2697	. . . . . . 7 |- E e. _V
29	26, 28	xpex 6962	. . . . . 6 |- ( T X. E ) e. _V
30	29, 29	mpt2ex 7247	. . . . 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 2622	. . . . . 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 16019	. . . . 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 2673	. . 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 2656	. 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 2681	1 |- ( ( K e. HL /\ W e. H ) -> .+ = .+b )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   u. cun 3572   {csn 4177   {ctp 4181   <.cop 4183   |-> cmpt 4729   X. cxp 5112   o. ccom 5118   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1stc1st 7166   2ndc2nd 7167   ndxcnx 15854   Basecbs 15857   +g cplusg 15941  Scalarcsca 15944   .scvsca 15945   HLchlt 34637   LHypclh 35270   LTrncltrn 35387   TEndoctendo 36040   EDRingcedring 36041   DVecHcdvh 36367

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-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

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-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-edring 36045  df-dvech 36368

This theorem is referenced by:  dvhvadd  36381