Theorem tendo0mulr 36115

Description: Additive identity multiplied by a trace-preserving endomorphism. (Contributed by NM, 13-Feb-2014.)

Hypotheses

Ref	Expression
tendoid0.b	|- B = ( Base ` K )
tendoid0.h	|- H = ( LHyp ` K )
tendoid0.t	|- T = ( ( LTrn ` K ) ` W )
tendoid0.e	|- E = ( ( TEndo ` K ) ` W )
tendoid0.o	|- O = ( f e. T |-> ( _I |` B ) )

Assertion

Ref	Expression
tendo0mulr	|- ( ( ( K e. HL /\ W e. H ) /\ U e. E ) -> ( U o. O ) = O )

Distinct variable groups:   B, f   T, f

Allowed substitution hints:   U( f)   E( f)   H( f)   K( f)   O( f)   W( f)

Proof of Theorem tendo0mulr

Dummy variable g is distinct from all other variables.

Step	Hyp	Ref	Expression
1		tendoid0.b	. . . 4 |- B = ( Base ` K )
2		tendoid0.h	. . . 4 |- H = ( LHyp ` K )
3		tendoid0.t	. . . 4 |- T = ( ( LTrn ` K ) ` W )
4	1, 2, 3	cdlemftr0 35856	. . 3 |- ( ( K e. HL /\ W e. H ) -> E. g e. T g =/= ( _I |` B ) )
5	4	adantr 481	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ U e. E ) -> E. g e. T g =/= ( _I |` B ) )
6		simpll 790	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ U e. E ) /\ ( g e. T /\ g =/= ( _I |` B ) ) ) -> ( K e. HL /\ W e. H ) )
7		simplr 792	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ U e. E ) /\ ( g e. T /\ g =/= ( _I |` B ) ) ) -> U e. E )
8		tendoid0.e	. . . . . 6 |- E = ( ( TEndo ` K ) ` W )
9		tendoid0.o	. . . . . 6 |- O = ( f e. T |-> ( _I |` B ) )
10	1, 2, 3, 8, 9	tendo0cl 36078	. . . . 5 |- ( ( K e. HL /\ W e. H ) -> O e. E )
11	10	ad2antrr 762	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ U e. E ) /\ ( g e. T /\ g =/= ( _I |` B ) ) ) -> O e. E )
12	2, 8	tendococl 36060	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ U e. E /\ O e. E ) -> ( U o. O ) e. E )
13	6, 7, 11, 12	syl3anc 1326	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ U e. E ) /\ ( g e. T /\ g =/= ( _I |` B ) ) ) -> ( U o. O ) e. E )
14	9, 1	tendo02 36075	. . . . . . 7 |- ( g e. T -> ( O ` g ) = ( _I |` B ) )
15	14	ad2antrl 764	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ U e. E ) /\ ( g e. T /\ g =/= ( _I |` B ) ) ) -> ( O ` g ) = ( _I |` B ) )
16	15	fveq2d 6195	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ U e. E ) /\ ( g e. T /\ g =/= ( _I |` B ) ) ) -> ( U ` ( O ` g ) ) = ( U ` ( _I |` B ) ) )
17	1, 2, 8	tendoid 36061	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ U e. E ) -> ( U ` ( _I |` B ) ) = ( _I |` B ) )
18	17	adantr 481	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ U e. E ) /\ ( g e. T /\ g =/= ( _I |` B ) ) ) -> ( U ` ( _I |` B ) ) = ( _I |` B ) )
19	16, 18	eqtrd 2656	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ U e. E ) /\ ( g e. T /\ g =/= ( _I |` B ) ) ) -> ( U ` ( O ` g ) ) = ( _I |` B ) )
20		simprl 794	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ U e. E ) /\ ( g e. T /\ g =/= ( _I |` B ) ) ) -> g e. T )
21	2, 3, 8	tendocoval 36054	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ O e. E ) /\ g e. T ) -> ( ( U o. O ) ` g ) = ( U ` ( O ` g ) ) )
22	6, 7, 11, 20, 21	syl121anc 1331	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ U e. E ) /\ ( g e. T /\ g =/= ( _I |` B ) ) ) -> ( ( U o. O ) ` g ) = ( U ` ( O ` g ) ) )
23	19, 22, 15	3eqtr4d 2666	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ U e. E ) /\ ( g e. T /\ g =/= ( _I |` B ) ) ) -> ( ( U o. O ) ` g ) = ( O ` g ) )
24		simpr 477	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ U e. E ) /\ ( g e. T /\ g =/= ( _I |` B ) ) ) -> ( g e. T /\ g =/= ( _I |` B ) ) )
25	1, 2, 3, 8	tendocan 36112	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( U o. O ) e. E /\ O e. E /\ ( ( U o. O ) ` g ) = ( O ` g ) ) /\ ( g e. T /\ g =/= ( _I |` B ) ) ) -> ( U o. O ) = O )
26	6, 13, 11, 23, 24, 25	syl131anc 1339	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ U e. E ) /\ ( g e. T /\ g =/= ( _I |` B ) ) ) -> ( U o. O ) = O )
27	5, 26	rexlimddv 3035	1 |- ( ( ( K e. HL /\ W e. H ) /\ U e. E ) -> ( U o. O ) = O )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   |-> cmpt 4729   _I cid 5023   |` cres 5116   o. ccom 5118   ` cfv 5888   Basecbs 15857   HLchlt 34637   LHypclh 35270   LTrncltrn 35387   TEndoctendo 36040

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-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-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  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-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-1st 7168  df-2nd 7169  df-undef 7399  df-map 7859  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

This theorem is referenced by:  dib1dim2  36457  diblss  36459