Theorem tendo0tp 36077

Description: Trace-preserving property of endomorphism additive identity. (Contributed by NM, 11-Jun-2013.)

Hypotheses

Ref	Expression
tendo0.b	|- B = ( Base ` K )
tendo0.h	|- H = ( LHyp ` K )
tendo0.t	|- T = ( ( LTrn ` K ) ` W )
tendo0.e	|- E = ( ( TEndo ` K ) ` W )
tendo0.o	|- O = ( f e. T |-> ( _I |` B ) )
tendo0tp.l	|- .<_ = ( le ` K )
tendo0tp.r	|- R = ( ( trL ` K ) ` W )

Assertion

Ref	Expression
tendo0tp	|- ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( R ` ( O ` F ) ) .<_ ( R ` F ) )

Distinct variable groups:   B, f   T, f

Allowed substitution hints:   R( f)   E( f)   F( f)   H( f)   K( f)   .<_ ( f)   O( f)   W( f)

Proof of Theorem tendo0tp

Step	Hyp	Ref	Expression
1		tendo0.o	. . . . . 6 |- O = ( f e. T |-> ( _I |` B ) )
2		tendo0.b	. . . . . 6 |- B = ( Base ` K )
3	1, 2	tendo02 36075	. . . . 5 |- ( F e. T -> ( O ` F ) = ( _I |` B ) )
4	3	adantl 482	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( O ` F ) = ( _I |` B ) )
5	4	fveq2d 6195	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( R ` ( O ` F ) ) = ( R ` ( _I |` B ) ) )
6		eqid 2622	. . . . 5 |- ( 0. ` K ) = ( 0. ` K )
7		tendo0.h	. . . . 5 |- H = ( LHyp ` K )
8		tendo0tp.r	. . . . 5 |- R = ( ( trL ` K ) ` W )
9	2, 6, 7, 8	trlid0 35463	. . . 4 |- ( ( K e. HL /\ W e. H ) -> ( R ` ( _I |` B ) ) = ( 0. ` K ) )
10	9	adantr 481	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( R ` ( _I |` B ) ) = ( 0. ` K ) )
11	5, 10	eqtrd 2656	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( R ` ( O ` F ) ) = ( 0. ` K ) )
12		hlop 34649	. . . 4 |- ( K e. HL -> K e. OP )
13	12	ad2antrr 762	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> K e. OP )
14		tendo0.t	. . . 4 |- T = ( ( LTrn ` K ) ` W )
15	2, 7, 14, 8	trlcl 35451	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( R ` F ) e. B )
16		tendo0tp.l	. . . 4 |- .<_ = ( le ` K )
17	2, 16, 6	op0le 34473	. . 3 |- ( ( K e. OP /\ ( R ` F ) e. B ) -> ( 0. ` K ) .<_ ( R ` F ) )
18	13, 15, 17	syl2anc 693	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( 0. ` K ) .<_ ( R ` F ) )
19	11, 18	eqbrtrd 4675	1 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( R ` ( O ` F ) ) .<_ ( R ` F ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   class class class wbr 4653   |-> cmpt 4729   _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

This theorem is referenced by:  tendo0cl  36078