Theorem tendotr 36118

Description: The trace of the value of a nonzero trace-preserving endomorphism equals the trace of the argument. (Contributed by NM, 11-Aug-2013.)

Hypotheses

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

Assertion

Ref	Expression
tendotr	|- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ U =/= O ) /\ F e. T ) -> ( R ` ( U ` F ) ) = ( R ` F ) )

Distinct variable groups:   B, f   T, f

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

Proof of Theorem tendotr

Step	Hyp	Ref	Expression
1		simpl1 1064	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ U =/= O ) /\ F e. T ) /\ F = ( _I |` B ) ) -> ( K e. HL /\ W e. H ) )
2		simpl2l 1114	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ U =/= O ) /\ F e. T ) /\ F = ( _I |` B ) ) -> U e. E )
3		tendotr.b	. . . . . 6 |- B = ( Base ` K )
4		tendotr.h	. . . . . 6 |- H = ( LHyp ` K )
5		tendotr.e	. . . . . 6 |- E = ( ( TEndo ` K ) ` W )
6	3, 4, 5	tendoid 36061	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ U e. E ) -> ( U ` ( _I |` B ) ) = ( _I |` B ) )
7	1, 2, 6	syl2anc 693	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ U =/= O ) /\ F e. T ) /\ F = ( _I |` B ) ) -> ( U ` ( _I |` B ) ) = ( _I |` B ) )
8		simpr 477	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ U =/= O ) /\ F e. T ) /\ F = ( _I |` B ) ) -> F = ( _I |` B ) )
9	8	fveq2d 6195	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ U =/= O ) /\ F e. T ) /\ F = ( _I |` B ) ) -> ( U ` F ) = ( U ` ( _I |` B ) ) )
10	7, 9, 8	3eqtr4d 2666	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ U =/= O ) /\ F e. T ) /\ F = ( _I |` B ) ) -> ( U ` F ) = F )
11	10	fveq2d 6195	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ U =/= O ) /\ F e. T ) /\ F = ( _I |` B ) ) -> ( R ` ( U ` F ) ) = ( R ` F ) )
12		simpl1 1064	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ U =/= O ) /\ F e. T ) /\ F =/= ( _I |` B ) ) -> ( K e. HL /\ W e. H ) )
13		simpl2l 1114	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ U =/= O ) /\ F e. T ) /\ F =/= ( _I |` B ) ) -> U e. E )
14		simpl3 1066	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ U =/= O ) /\ F e. T ) /\ F =/= ( _I |` B ) ) -> F e. T )
15		eqid 2622	. . . . 5 |- ( le ` K ) = ( le ` K )
16		tendotr.t	. . . . 5 |- T = ( ( LTrn ` K ) ` W )
17		tendotr.r	. . . . 5 |- R = ( ( trL ` K ) ` W )
18	15, 4, 16, 17, 5	tendotp 36049	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ U e. E /\ F e. T ) -> ( R ` ( U ` F ) ) ( le ` K ) ( R ` F ) )
19	12, 13, 14, 18	syl3anc 1326	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ U =/= O ) /\ F e. T ) /\ F =/= ( _I |` B ) ) -> ( R ` ( U ` F ) ) ( le ` K ) ( R ` F ) )
20		simpl1l 1112	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ U =/= O ) /\ F e. T ) /\ F =/= ( _I |` B ) ) -> K e. HL )
21		hlatl 34647	. . . . 5 |- ( K e. HL -> K e. AtLat )
22	20, 21	syl 17	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ U =/= O ) /\ F e. T ) /\ F =/= ( _I |` B ) ) -> K e. AtLat )
23	4, 16, 5	tendocl 36055	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ U e. E /\ F e. T ) -> ( U ` F ) e. T )
24	12, 13, 14, 23	syl3anc 1326	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ U =/= O ) /\ F e. T ) /\ F =/= ( _I |` B ) ) -> ( U ` F ) e. T )
25		simpl2r 1115	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ U =/= O ) /\ F e. T ) /\ F =/= ( _I |` B ) ) -> U =/= O )
26		simpr 477	. . . . . . . 8 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ U =/= O ) /\ F e. T ) /\ F =/= ( _I |` B ) ) -> F =/= ( _I |` B ) )
27		tendotr.o	. . . . . . . . 9 |- O = ( f e. T |-> ( _I |` B ) )
28	3, 4, 16, 5, 27	tendoid0 36113	. . . . . . . 8 |- ( ( ( K e. HL /\ W e. H ) /\ U e. E /\ ( F e. T /\ F =/= ( _I |` B ) ) ) -> ( ( U ` F ) = ( _I |` B ) <-> U = O ) )
29	12, 13, 14, 26, 28	syl112anc 1330	. . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ U =/= O ) /\ F e. T ) /\ F =/= ( _I |` B ) ) -> ( ( U ` F ) = ( _I |` B ) <-> U = O ) )
30	29	necon3bid 2838	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ U =/= O ) /\ F e. T ) /\ F =/= ( _I |` B ) ) -> ( ( U ` F ) =/= ( _I |` B ) <-> U =/= O ) )
31	25, 30	mpbird 247	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ U =/= O ) /\ F e. T ) /\ F =/= ( _I |` B ) ) -> ( U ` F ) =/= ( _I |` B ) )
32		eqid 2622	. . . . . 6 |- ( Atoms ` K ) = ( Atoms ` K )
33	3, 32, 4, 16, 17	trlnidat 35460	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( U ` F ) e. T /\ ( U ` F ) =/= ( _I |` B ) ) -> ( R ` ( U ` F ) ) e. ( Atoms ` K ) )
34	12, 24, 31, 33	syl3anc 1326	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ U =/= O ) /\ F e. T ) /\ F =/= ( _I |` B ) ) -> ( R ` ( U ` F ) ) e. ( Atoms ` K ) )
35	3, 32, 4, 16, 17	trlnidat 35460	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ F =/= ( _I |` B ) ) -> ( R ` F ) e. ( Atoms ` K ) )
36	12, 14, 26, 35	syl3anc 1326	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ U =/= O ) /\ F e. T ) /\ F =/= ( _I |` B ) ) -> ( R ` F ) e. ( Atoms ` K ) )
37	15, 32	atcmp 34598	. . . 4 |- ( ( K e. AtLat /\ ( R ` ( U ` F ) ) e. ( Atoms ` K ) /\ ( R ` F ) e. ( Atoms ` K ) ) -> ( ( R ` ( U ` F ) ) ( le ` K ) ( R ` F ) <-> ( R ` ( U ` F ) ) = ( R ` F ) ) )
38	22, 34, 36, 37	syl3anc 1326	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ U =/= O ) /\ F e. T ) /\ F =/= ( _I |` B ) ) -> ( ( R ` ( U ` F ) ) ( le ` K ) ( R ` F ) <-> ( R ` ( U ` F ) ) = ( R ` F ) ) )
39	19, 38	mpbid 222	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ U =/= O ) /\ F e. T ) /\ F =/= ( _I |` B ) ) -> ( R ` ( U ` F ) ) = ( R ` F ) )
40	11, 39	pm2.61dane 2881	1 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ U =/= O ) /\ F e. T ) -> ( R ` ( U ` F ) ) = ( R ` F ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   |-> cmpt 4729   _I cid 5023   |` cres 5116   ` cfv 5888   Basecbs 15857   lecple 15948   Atomscatm 34550   AtLatcal 34551   HLchlt 34637   LHypclh 35270   LTrncltrn 35387   trLctrl 35445   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:  cdleml6  36269