Theorem tendoid 36061

Description: The identity value of a trace-preserving endomorphism. (Contributed by NM, 21-Jun-2013.)

Hypotheses

Ref	Expression
tendoid.b	|- B = ( Base ` K )
tendoid.h	|- H = ( LHyp ` K )
tendoid.e	|- E = ( ( TEndo ` K ) ` W )

Assertion

Ref	Expression
tendoid	|- ( ( ( K e. HL /\ W e. H ) /\ S e. E ) -> ( S ` ( _I |` B ) ) = ( _I |` B ) )

Proof of Theorem tendoid

Step	Hyp	Ref	Expression
1		tendoid.b	. . . . . . 7 |- B = ( Base ` K )
2		tendoid.h	. . . . . . 7 |- H = ( LHyp ` K )
3		eqid 2622	. . . . . . 7 |- ( ( LTrn ` K ) ` W ) = ( ( LTrn ` K ) ` W )
4	1, 2, 3	idltrn 35436	. . . . . 6 |- ( ( K e. HL /\ W e. H ) -> ( _I |` B ) e. ( ( LTrn ` K ) ` W ) )
5	4	adantr 481	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ S e. E ) -> ( _I |` B ) e. ( ( LTrn ` K ) ` W ) )
6		eqid 2622	. . . . . 6 |- ( le ` K ) = ( le ` K )
7		eqid 2622	. . . . . 6 |- ( ( trL ` K ) ` W ) = ( ( trL ` K ) ` W )
8		tendoid.e	. . . . . 6 |- E = ( ( TEndo ` K ) ` W )
9	6, 2, 3, 7, 8	tendotp 36049	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ S e. E /\ ( _I |` B ) e. ( ( LTrn ` K ) ` W ) ) -> ( ( ( trL ` K ) ` W ) ` ( S ` ( _I |` B ) ) ) ( le ` K ) ( ( ( trL ` K ) ` W ) ` ( _I |` B ) ) )
10	5, 9	mpd3an3 1425	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ S e. E ) -> ( ( ( trL ` K ) ` W ) ` ( S ` ( _I |` B ) ) ) ( le ` K ) ( ( ( trL ` K ) ` W ) ` ( _I |` B ) ) )
11		eqid 2622	. . . . . 6 |- ( 0. ` K ) = ( 0. ` K )
12	1, 11, 2, 7	trlid0 35463	. . . . 5 |- ( ( K e. HL /\ W e. H ) -> ( ( ( trL ` K ) ` W ) ` ( _I |` B ) ) = ( 0. ` K ) )
13	12	adantr 481	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ S e. E ) -> ( ( ( trL ` K ) ` W ) ` ( _I |` B ) ) = ( 0. ` K ) )
14	10, 13	breqtrd 4679	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ S e. E ) -> ( ( ( trL ` K ) ` W ) ` ( S ` ( _I |` B ) ) ) ( le ` K ) ( 0. ` K ) )
15		hlop 34649	. . . . 5 |- ( K e. HL -> K e. OP )
16	15	ad2antrr 762	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ S e. E ) -> K e. OP )
17	2, 3, 8	tendocl 36055	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ S e. E /\ ( _I |` B ) e. ( ( LTrn ` K ) ` W ) ) -> ( S ` ( _I |` B ) ) e. ( ( LTrn ` K ) ` W ) )
18	5, 17	mpd3an3 1425	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ S e. E ) -> ( S ` ( _I |` B ) ) e. ( ( LTrn ` K ) ` W ) )
19	1, 2, 3, 7	trlcl 35451	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( S ` ( _I |` B ) ) e. ( ( LTrn ` K ) ` W ) ) -> ( ( ( trL ` K ) ` W ) ` ( S ` ( _I |` B ) ) ) e. B )
20	18, 19	syldan 487	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ S e. E ) -> ( ( ( trL ` K ) ` W ) ` ( S ` ( _I |` B ) ) ) e. B )
21	1, 6, 11	ople0 34474	. . . 4 |- ( ( K e. OP /\ ( ( ( trL ` K ) ` W ) ` ( S ` ( _I |` B ) ) ) e. B ) -> ( ( ( ( trL ` K ) ` W ) ` ( S ` ( _I |` B ) ) ) ( le ` K ) ( 0. ` K ) <-> ( ( ( trL ` K ) ` W ) ` ( S ` ( _I |` B ) ) ) = ( 0. ` K ) ) )
22	16, 20, 21	syl2anc 693	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ S e. E ) -> ( ( ( ( trL ` K ) ` W ) ` ( S ` ( _I |` B ) ) ) ( le ` K ) ( 0. ` K ) <-> ( ( ( trL ` K ) ` W ) ` ( S ` ( _I |` B ) ) ) = ( 0. ` K ) ) )
23	14, 22	mpbid 222	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ S e. E ) -> ( ( ( trL ` K ) ` W ) ` ( S ` ( _I |` B ) ) ) = ( 0. ` K ) )
24	1, 11, 2, 3, 7	trlid0b 35465	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( S ` ( _I |` B ) ) e. ( ( LTrn ` K ) ` W ) ) -> ( ( S ` ( _I |` B ) ) = ( _I |` B ) <-> ( ( ( trL ` K ) ` W ) ` ( S ` ( _I |` B ) ) ) = ( 0. ` K ) ) )
25	18, 24	syldan 487	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ S e. E ) -> ( ( S ` ( _I |` B ) ) = ( _I |` B ) <-> ( ( ( trL ` K ) ` W ) ` ( S ` ( _I |` B ) ) ) = ( 0. ` K ) ) )
26	23, 25	mpbird 247	1 |- ( ( ( K e. HL /\ W e. H ) /\ S e. E ) -> ( S ` ( _I |` B ) ) = ( _I |` B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   class class class wbr 4653   _I cid 5023   |` cres 5116   ` cfv 5888   Basecbs 15857   lecple 15948   0.cp0 17037   OPcops 34459   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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-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-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  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-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-lhyp 35274  df-laut 35275  df-ldil 35390  df-ltrn 35391  df-trl 35446  df-tendo 36043

This theorem is referenced by:  tendoeq2  36062  tendo0mulr  36115  tendotr  36118  tendocnv  36310  dvhopN  36405  dihpN  36625