Theorem tendovalco 36053

Description: Value of composition of translations in a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013.)

Hypotheses

Ref	Expression
tendof.h	|- H = ( LHyp ` K )
tendof.t	|- T = ( ( LTrn ` K ) ` W )
tendof.e	|- E = ( ( TEndo ` K ) ` W )

Assertion

Ref	Expression
tendovalco	|- ( ( ( K e. V /\ W e. H /\ S e. E ) /\ ( F e. T /\ G e. T ) ) -> ( S ` ( F o. G ) ) = ( ( S ` F ) o. ( S ` G ) ) )

Proof of Theorem tendovalco

Dummy variables f g are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . . . 5 |- ( le ` K ) = ( le ` K )
2		tendof.h	. . . . 5 |- H = ( LHyp ` K )
3		tendof.t	. . . . 5 |- T = ( ( LTrn ` K ) ` W )
4		eqid 2622	. . . . 5 |- ( ( trL ` K ) ` W ) = ( ( trL ` K ) ` W )
5		tendof.e	. . . . 5 |- E = ( ( TEndo ` K ) ` W )
6	1, 2, 3, 4, 5	istendo 36048	. . . 4 |- ( ( 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 ( ( ( trL ` K ) ` W ) ` ( S ` f ) ) ( le ` K ) ( ( ( trL ` K ) ` W ) ` f ) ) ) )
7		coeq1 5279	. . . . . . . . 9 |- ( f = F -> ( f o. g ) = ( F o. g ) )
8	7	fveq2d 6195	. . . . . . . 8 |- ( f = F -> ( S ` ( f o. g ) ) = ( S ` ( F o. g ) ) )
9		fveq2 6191	. . . . . . . . 9 |- ( f = F -> ( S ` f ) = ( S ` F ) )
10	9	coeq1d 5283	. . . . . . . 8 |- ( f = F -> ( ( S ` f ) o. ( S ` g ) ) = ( ( S ` F ) o. ( S ` g ) ) )
11	8, 10	eqeq12d 2637	. . . . . . 7 |- ( f = F -> ( ( S ` ( f o. g ) ) = ( ( S ` f ) o. ( S ` g ) ) <-> ( S ` ( F o. g ) ) = ( ( S ` F ) o. ( S ` g ) ) ) )
12		coeq2 5280	. . . . . . . . 9 |- ( g = G -> ( F o. g ) = ( F o. G ) )
13	12	fveq2d 6195	. . . . . . . 8 |- ( g = G -> ( S ` ( F o. g ) ) = ( S ` ( F o. G ) ) )
14		fveq2 6191	. . . . . . . . 9 |- ( g = G -> ( S ` g ) = ( S ` G ) )
15	14	coeq2d 5284	. . . . . . . 8 |- ( g = G -> ( ( S ` F ) o. ( S ` g ) ) = ( ( S ` F ) o. ( S ` G ) ) )
16	13, 15	eqeq12d 2637	. . . . . . 7 |- ( g = G -> ( ( S ` ( F o. g ) ) = ( ( S ` F ) o. ( S ` g ) ) <-> ( S ` ( F o. G ) ) = ( ( S ` F ) o. ( S ` G ) ) ) )
17	11, 16	rspc2v 3322	. . . . . 6 |- ( ( F e. T /\ G e. T ) -> ( A. f e. T A. g e. T ( S ` ( f o. g ) ) = ( ( S ` f ) o. ( S ` g ) ) -> ( S ` ( F o. G ) ) = ( ( S ` F ) o. ( S ` G ) ) ) )
18	17	com12 32	. . . . 5 |- ( A. f e. T A. g e. T ( S ` ( f o. g ) ) = ( ( S ` f ) o. ( S ` g ) ) -> ( ( F e. T /\ G e. T ) -> ( S ` ( F o. G ) ) = ( ( S ` F ) o. ( S ` G ) ) ) )
19	18	3ad2ant2 1083	. . . 4 |- ( ( S : T --> T /\ A. f e. T A. g e. T ( S ` ( f o. g ) ) = ( ( S ` f ) o. ( S ` g ) ) /\ A. f e. T ( ( ( trL ` K ) ` W ) ` ( S ` f ) ) ( le ` K ) ( ( ( trL ` K ) ` W ) ` f ) ) -> ( ( F e. T /\ G e. T ) -> ( S ` ( F o. G ) ) = ( ( S ` F ) o. ( S ` G ) ) ) )
20	6, 19	syl6bi 243	. . 3 |- ( ( K e. V /\ W e. H ) -> ( S e. E -> ( ( F e. T /\ G e. T ) -> ( S ` ( F o. G ) ) = ( ( S ` F ) o. ( S ` G ) ) ) ) )
21	20	3impia 1261	. 2 |- ( ( K e. V /\ W e. H /\ S e. E ) -> ( ( F e. T /\ G e. T ) -> ( S ` ( F o. G ) ) = ( ( S ` F ) o. ( S ` G ) ) ) )
22	21	imp 445	1 |- ( ( ( K e. V /\ W e. H /\ S e. E ) /\ ( F e. T /\ G e. T ) ) -> ( S ` ( F o. G ) ) = ( ( S ` F ) o. ( S ` G ) ) )

Syntax hints:   -> wi 4   /\ 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:  tendoco2  36056  tendococl  36060  tendodi1  36072  tendoicl  36084  cdlemi2  36107  tendospdi1  36309