Theorem dvh0g 36400

Description: The zero vector of vector space H has the zero translation as its first member and the zero trace-preserving endomorphism as the second. (Contributed by NM, 9-Mar-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)

Hypotheses

Ref	Expression
dvh0g.b	|- B = ( Base ` K )
dvh0g.h	|- H = ( LHyp ` K )
dvh0g.t	|- T = ( ( LTrn ` K ) ` W )
dvh0g.u	|- U = ( ( DVecH ` K ) ` W )
dvh0g.z	|- .0. = ( 0g ` U )
dvh0g.o	|- O = ( f e. T |-> ( _I |` B ) )

Assertion

Ref	Expression
dvh0g	|- ( ( K e. HL /\ W e. H ) -> .0. = <. ( _I |` B ) , O >. )

Distinct variable groups:   B, f   f, H   f, K   T, f   f, W

Allowed substitution hints:   U( f)   O( f)   .0. ( f)

Proof of Theorem dvh0g

Dummy variables s t are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 22	. . . 4 |- ( ( K e. HL /\ W e. H ) -> ( K e. HL /\ W e. H ) )
2		dvh0g.b	. . . . 5 |- B = ( Base ` K )
3		dvh0g.h	. . . . 5 |- H = ( LHyp ` K )
4		dvh0g.t	. . . . 5 |- T = ( ( LTrn ` K ) ` W )
5	2, 3, 4	idltrn 35436	. . . 4 |- ( ( K e. HL /\ W e. H ) -> ( _I |` B ) e. T )
6		eqid 2622	. . . . 5 |- ( ( TEndo ` K ) ` W ) = ( ( TEndo ` K ) ` W )
7		dvh0g.o	. . . . 5 |- O = ( f e. T |-> ( _I |` B ) )
8	2, 3, 4, 6, 7	tendo0cl 36078	. . . 4 |- ( ( K e. HL /\ W e. H ) -> O e. ( ( TEndo ` K ) ` W ) )
9		dvh0g.u	. . . . 5 |- U = ( ( DVecH ` K ) ` W )
10		eqid 2622	. . . . 5 |- (Scalar ` U ) = (Scalar ` U )
11		eqid 2622	. . . . 5 |- ( +g ` U ) = ( +g ` U )
12		eqid 2622	. . . . 5 |- ( +g ` (Scalar ` U ) ) = ( +g ` (Scalar ` U ) )
13	3, 4, 6, 9, 10, 11, 12	dvhopvadd 36382	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( _I |` B ) e. T /\ O e. ( ( TEndo ` K ) ` W ) ) /\ ( ( _I |` B ) e. T /\ O e. ( ( TEndo ` K ) ` W ) ) ) -> ( <. ( _I |` B ) , O >. ( +g ` U ) <. ( _I |` B ) , O >. ) = <. ( ( _I |` B ) o. ( _I |` B ) ) , ( O ( +g ` (Scalar ` U ) ) O ) >. )
14	1, 5, 8, 5, 8, 13	syl122anc 1335	. . 3 |- ( ( K e. HL /\ W e. H ) -> ( <. ( _I |` B ) , O >. ( +g ` U ) <. ( _I |` B ) , O >. ) = <. ( ( _I |` B ) o. ( _I |` B ) ) , ( O ( +g ` (Scalar ` U ) ) O ) >. )
15		f1oi 6174	. . . . . 6 |- ( _I |` B ) : B -1-1-onto-> B
16		f1of 6137	. . . . . 6 |- ( ( _I |` B ) : B -1-1-onto-> B -> ( _I |` B ) : B --> B )
17		fcoi2 6079	. . . . . 6 |- ( ( _I |` B ) : B --> B -> ( ( _I |` B ) o. ( _I |` B ) ) = ( _I |` B ) )
18	15, 16, 17	mp2b 10	. . . . 5 |- ( ( _I |` B ) o. ( _I |` B ) ) = ( _I |` B )
19	18	a1i 11	. . . 4 |- ( ( K e. HL /\ W e. H ) -> ( ( _I |` B ) o. ( _I |` B ) ) = ( _I |` B ) )
20		eqid 2622	. . . . . . 7 |- ( s e. ( ( TEndo ` K ) ` W ) , t e. ( ( TEndo ` K ) ` W ) |-> ( f e. T |-> ( ( s ` f ) o. ( t ` f ) ) ) ) = ( s e. ( ( TEndo ` K ) ` W ) , t e. ( ( TEndo ` K ) ` W ) |-> ( f e. T |-> ( ( s ` f ) o. ( t ` f ) ) ) )
21	3, 4, 6, 9, 10, 20, 12	dvhfplusr 36373	. . . . . 6 |- ( ( K e. HL /\ W e. H ) -> ( +g ` (Scalar ` U ) ) = ( s e. ( ( TEndo ` K ) ` W ) , t e. ( ( TEndo ` K ) ` W ) |-> ( f e. T |-> ( ( s ` f ) o. ( t ` f ) ) ) ) )
22	21	oveqd 6667	. . . . 5 |- ( ( K e. HL /\ W e. H ) -> ( O ( +g ` (Scalar ` U ) ) O ) = ( O ( s e. ( ( TEndo ` K ) ` W ) , t e. ( ( TEndo ` K ) ` W ) |-> ( f e. T |-> ( ( s ` f ) o. ( t ` f ) ) ) ) O ) )
23	2, 3, 4, 6, 7, 20	tendo0pl 36079	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ O e. ( ( TEndo ` K ) ` W ) ) -> ( O ( s e. ( ( TEndo ` K ) ` W ) , t e. ( ( TEndo ` K ) ` W ) |-> ( f e. T |-> ( ( s ` f ) o. ( t ` f ) ) ) ) O ) = O )
24	8, 23	mpdan 702	. . . . 5 |- ( ( K e. HL /\ W e. H ) -> ( O ( s e. ( ( TEndo ` K ) ` W ) , t e. ( ( TEndo ` K ) ` W ) |-> ( f e. T |-> ( ( s ` f ) o. ( t ` f ) ) ) ) O ) = O )
25	22, 24	eqtrd 2656	. . . 4 |- ( ( K e. HL /\ W e. H ) -> ( O ( +g ` (Scalar ` U ) ) O ) = O )
26	19, 25	opeq12d 4410	. . 3 |- ( ( K e. HL /\ W e. H ) -> <. ( ( _I |` B ) o. ( _I |` B ) ) , ( O ( +g ` (Scalar ` U ) ) O ) >. = <. ( _I |` B ) , O >. )
27	14, 26	eqtrd 2656	. 2 |- ( ( K e. HL /\ W e. H ) -> ( <. ( _I |` B ) , O >. ( +g ` U ) <. ( _I |` B ) , O >. ) = <. ( _I |` B ) , O >. )
28	3, 9, 1	dvhlmod 36399	. . 3 |- ( ( K e. HL /\ W e. H ) -> U e. LMod )
29		eqid 2622	. . . . 5 |- ( Base ` U ) = ( Base ` U )
30	3, 4, 6, 9, 29	dvhelvbasei 36377	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( _I |` B ) e. T /\ O e. ( ( TEndo ` K ) ` W ) ) ) -> <. ( _I |` B ) , O >. e. ( Base ` U ) )
31	1, 5, 8, 30	syl12anc 1324	. . 3 |- ( ( K e. HL /\ W e. H ) -> <. ( _I |` B ) , O >. e. ( Base ` U ) )
32		dvh0g.z	. . . 4 |- .0. = ( 0g ` U )
33	29, 11, 32	lmod0vid 18895	. . 3 |- ( ( U e. LMod /\ <. ( _I |` B ) , O >. e. ( Base ` U ) ) -> ( ( <. ( _I |` B ) , O >. ( +g ` U ) <. ( _I |` B ) , O >. ) = <. ( _I |` B ) , O >. <-> .0. = <. ( _I |` B ) , O >. ) )
34	28, 31, 33	syl2anc 693	. 2 |- ( ( K e. HL /\ W e. H ) -> ( ( <. ( _I |` B ) , O >. ( +g ` U ) <. ( _I |` B ) , O >. ) = <. ( _I |` B ) , O >. <-> .0. = <. ( _I |` B ) , O >. ) )
35	27, 34	mpbid 222	1 |- ( ( K e. HL /\ W e. H ) -> .0. = <. ( _I |` B ) , O >. )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   <.cop 4183   |-> cmpt 4729   _I cid 5023   |` cres 5116   o. ccom 5118   -->wf 5884   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   Basecbs 15857   +g cplusg 15941  Scalarcsca 15944   0gc0g 16100   LModclmod 18863   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-fal 1489  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-tpos 7352  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-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-0g 16102  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-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-mgp 18490  df-ur 18502  df-ring 18549  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-dvr 18683  df-drng 18749  df-lmod 18865  df-lvec 19103  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:  dvheveccl  36401  dib0  36453  dihmeetlem4preN  36595  dihmeetlem13N  36608  dihatlat  36623  dihpN  36625