Theorem istendod 36050

Description: Deduce the predicate "is a trace-preserving endomorphism". (Contributed by NM, 9-Jun-2013.)

Hypotheses

Ref	Expression
tendoset.l	|- .<_ = ( le ` K )
tendoset.h	|- H = ( LHyp ` K )
tendoset.t	|- T = ( ( LTrn ` K ) ` W )
tendoset.r	|- R = ( ( trL ` K ) ` W )
tendoset.e	|- E = ( ( TEndo ` K ) ` W )
istendod.1	|- ( ph -> ( K e. V /\ W e. H ) )
istendod.2	|- ( ph -> S : T --> T )
istendod.3	|- ( ( ph /\ f e. T /\ g e. T ) -> ( S ` ( f o. g ) ) = ( ( S ` f ) o. ( S ` g ) ) )
istendod.4	|- ( ( ph /\ f e. T ) -> ( R ` ( S ` f ) ) .<_ ( R ` f ) )

Assertion

Ref	Expression
istendod	|- ( ph -> S e. E )

Distinct variable groups:   f, g, K   T, f, g   f, W, g   S, f, g   .<_ , f   R, f   ph, f, g

Allowed substitution hints:   R( g)   E( f, g)   H( f, g)   .<_ ( g)   V( f, g)

Proof of Theorem istendod

Step	Hyp	Ref	Expression
1		istendod.2	. 2 |- ( ph -> S : T --> T )
2		istendod.3	. . . 4 |- ( ( ph /\ f e. T /\ g e. T ) -> ( S ` ( f o. g ) ) = ( ( S ` f ) o. ( S ` g ) ) )
3	2	3expb 1266	. . 3 |- ( ( ph /\ ( f e. T /\ g e. T ) ) -> ( S ` ( f o. g ) ) = ( ( S ` f ) o. ( S ` g ) ) )
4	3	ralrimivva 2971	. 2 |- ( ph -> A. f e. T A. g e. T ( S ` ( f o. g ) ) = ( ( S ` f ) o. ( S ` g ) ) )
5		istendod.4	. . 3 |- ( ( ph /\ f e. T ) -> ( R ` ( S ` f ) ) .<_ ( R ` f ) )
6	5	ralrimiva 2966	. 2 |- ( ph -> A. f e. T ( R ` ( S ` f ) ) .<_ ( R ` f ) )
7		istendod.1	. . 3 |- ( ph -> ( K e. V /\ W e. H ) )
8		tendoset.l	. . . 4 |- .<_ = ( le ` K )
9		tendoset.h	. . . 4 |- H = ( LHyp ` K )
10		tendoset.t	. . . 4 |- T = ( ( LTrn ` K ) ` W )
11		tendoset.r	. . . 4 |- R = ( ( trL ` K ) ` W )
12		tendoset.e	. . . 4 |- E = ( ( TEndo ` K ) ` W )
13	8, 9, 10, 11, 12	istendo 36048	. . 3 |- ( ( K e. V /\ W e. H ) -> ( S e. E <-> ( S : T --> T /\ A. f e. T A. g e. T ( S ` ( f o. g ) ) = ( ( S ` f ) o. ( S ` g ) ) /\ A. f e. T ( R ` ( S ` f ) ) .<_ ( R ` f ) ) ) )
14	7, 13	syl 17	. 2 |- ( ph -> ( S e. E <-> ( S : T --> T /\ A. f e. T A. g e. T ( S ` ( f o. g ) ) = ( ( S ` f ) o. ( S ` g ) ) /\ A. f e. T ( R ` ( S ` f ) ) .<_ ( R ` f ) ) ) )
15	1, 4, 6, 14	mpbir3and 1245	1 |- ( ph -> S e. E )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   class class class wbr 4653   o. ccom 5118   -->wf 5884   ` cfv 5888   lecple 15948   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-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-tendo 36043

This theorem is referenced by:  tendoidcl  36057  tendococl  36060  tendoplcl  36069  tendo0cl  36078  tendoicl  36084  cdlemk56  36259