Theorem dvhvaddcomN 36385

Description: Commutativity of vector sum. (Contributed by NM, 26-Oct-2013.) (Revised by Mario Carneiro, 23-Jun-2014.) (New usage is discouraged.)

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
dvhvaddcomN	|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. ( T X. E ) /\ G e. ( T X. E ) ) ) -> ( F .+ G ) = ( G .+ F ) )

Proof of Theorem dvhvaddcomN

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

Step	Hyp	Ref	Expression
1		simpl 473	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. ( T X. E ) /\ G e. ( T X. E ) ) ) -> ( K e. HL /\ W e. H ) )
2		xp1st 7198	. . . . 5 |- ( F e. ( T X. E ) -> ( 1st ` F ) e. T )
3	2	ad2antrl 764	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. ( T X. E ) /\ G e. ( T X. E ) ) ) -> ( 1st ` F ) e. T )
4		xp1st 7198	. . . . 5 |- ( G e. ( T X. E ) -> ( 1st ` G ) e. T )
5	4	ad2antll 765	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. ( T X. E ) /\ G e. ( T X. E ) ) ) -> ( 1st ` G ) e. T )
6		dvhvaddcl.h	. . . . 5 |- H = ( LHyp ` K )
7		dvhvaddcl.t	. . . . 5 |- T = ( ( LTrn ` K ) ` W )
8	6, 7	ltrncom 36026	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( 1st ` F ) e. T /\ ( 1st ` G ) e. T ) -> ( ( 1st ` F ) o. ( 1st ` G ) ) = ( ( 1st ` G ) o. ( 1st ` F ) ) )
9	1, 3, 5, 8	syl3anc 1326	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. ( T X. E ) /\ G e. ( T X. E ) ) ) -> ( ( 1st ` F ) o. ( 1st ` G ) ) = ( ( 1st ` G ) o. ( 1st ` F ) ) )
10		xp2nd 7199	. . . . . 6 |- ( F e. ( T X. E ) -> ( 2nd ` F ) e. E )
11		xp2nd 7199	. . . . . 6 |- ( G e. ( T X. E ) -> ( 2nd ` G ) e. E )
12	10, 11	anim12i 590	. . . . 5 |- ( ( F e. ( T X. E ) /\ G e. ( T X. E ) ) -> ( ( 2nd ` F ) e. E /\ ( 2nd ` G ) e. E ) )
13		dvhvaddcl.e	. . . . . . 7 |- E = ( ( TEndo ` K ) ` W )
14		eqid 2622	. . . . . . 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 ) ) ) )
15	6, 7, 13, 14	tendoplcom 36070	. . . . . 6 |- ( ( ( 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 ) ) = ( ( 2nd ` G ) ( a e. E , b e. E |-> ( c e. T |-> ( ( a ` c ) o. ( b ` c ) ) ) ) ( 2nd ` F ) ) )
16	15	3expb 1266	. . . . 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 ) ) = ( ( 2nd ` G ) ( a e. E , b e. E |-> ( c e. T |-> ( ( a ` c ) o. ( b ` c ) ) ) ) ( 2nd ` F ) ) )
17	12, 16	sylan2 491	. . . 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 ) ) = ( ( 2nd ` G ) ( a e. E , b e. E |-> ( c e. T |-> ( ( a ` c ) o. ( b ` c ) ) ) ) ( 2nd ` F ) ) )
18		dvhvaddcl.u	. . . . . . 7 |- U = ( ( DVecH ` K ) ` W )
19		dvhvaddcl.d	. . . . . . 7 |- D = (Scalar ` U )
20		dvhvaddcl.p	. . . . . . 7 |- .+^ = ( +g ` D )
21	6, 7, 13, 18, 19, 14, 20	dvhfplusr 36373	. . . . . 6 |- ( ( K e. HL /\ W e. H ) -> .+^ = ( a e. E , b e. E |-> ( c e. T |-> ( ( a ` c ) o. ( b ` c ) ) ) ) )
22	21	adantr 481	. . . . 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 ) ) ) ) )
23	22	oveqd 6667	. . . 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 ) ) )
24	22	oveqd 6667	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. ( T X. E ) /\ G e. ( T X. E ) ) ) -> ( ( 2nd ` G ) .+^ ( 2nd ` F ) ) = ( ( 2nd ` G ) ( a e. E , b e. E |-> ( c e. T |-> ( ( a ` c ) o. ( b ` c ) ) ) ) ( 2nd ` F ) ) )
25	17, 23, 24	3eqtr4d 2666	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. ( T X. E ) /\ G e. ( T X. E ) ) ) -> ( ( 2nd ` F ) .+^ ( 2nd ` G ) ) = ( ( 2nd ` G ) .+^ ( 2nd ` F ) ) )
26	9, 25	opeq12d 4410	. 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 ) ) >. = <. ( ( 1st ` G ) o. ( 1st ` F ) ) , ( ( 2nd ` G ) .+^ ( 2nd ` F ) ) >. )
27		dvhvaddcl.a	. . 3 |- .+ = ( +g ` U )
28	6, 7, 13, 18, 19, 27, 20	dvhvadd 36381	. 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 ) ) >. )
29	6, 7, 13, 18, 19, 27, 20	dvhvadd 36381	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( G e. ( T X. E ) /\ F e. ( T X. E ) ) ) -> ( G .+ F ) = <. ( ( 1st ` G ) o. ( 1st ` F ) ) , ( ( 2nd ` G ) .+^ ( 2nd ` F ) ) >. )
30	29	ancom2s 844	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. ( T X. E ) /\ G e. ( T X. E ) ) ) -> ( G .+ F ) = <. ( ( 1st ` G ) o. ( 1st ` F ) ) , ( ( 2nd ` G ) .+^ ( 2nd ` F ) ) >. )
31	26, 28, 30	3eqtr4d 2666	1 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. ( T X. E ) /\ G e. ( T X. E ) ) ) -> ( F .+ G ) = ( G .+ F ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   <.cop 4183   |-> cmpt 4729   X. cxp 5112   o. ccom 5118   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1stc1st 7166   2ndc2nd 7167   +g cplusg 15941  Scalarcsca 15944   HLchlt 34637   LHypclh 35270   LTrncltrn 35387   TEndoctendo 36040   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  ax-riotaBAD 34239

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-rmo 2920  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-iin 4523  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-undef 7399  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  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-preset 16928  df-poset 16946  df-plt 16958  df-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-p0 17039  df-p1 17040  df-lat 17046  df-clat 17108  df-oposet 34463  df-ol 34465  df-oml 34466  df-covers 34553  df-ats 34554  df-atl 34585  df-cvlat 34609  df-hlat 34638  df-llines 34784  df-lplanes 34785  df-lvols 34786  df-lines 34787  df-psubsp 34789  df-pmap 34790  df-padd 35082  df-lhyp 35274  df-laut 35275  df-ldil 35390  df-ltrn 35391  df-trl 35446  df-tendo 36043  df-edring 36045  df-dvech 36368

This theorem is referenced by: (None)