This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem tendopltp

Description: Trace-preserving property of endomorphism sum operation P , based on Theorems trlco . Part of remark in Crawley p. 118, 2nd line, "it is clear from the second part of G (our trlco ) that Delta is a subring of E." (In our development, we will bypass their E and go directly to their Delta, whose base set is our ( TEndoK )W .) (Contributed by NM, 9-Jun-2013)

Ref Expression
Hypotheses tendopl.h 
|- H = ( LHyp ` K )
tendopl.t 
|- T = ( ( LTrn ` K ) ` W )
tendopl.e 
|- E = ( ( TEndo ` K ) ` W )
tendopl.p 
|- P = ( s e. E , t e. E |-> ( f e. T |-> ( ( s ` f ) o. ( t ` f ) ) ) )
tendopltp.l 
|- .<_ = ( le ` K )
tendopltp.r 
|- R = ( ( trL ` K ) ` W )
Assertion tendopltp 
|- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ F e. T ) -> ( R ` ( ( U P V ) ` F ) ) .<_ ( R ` F ) )

Proof

Step Hyp Ref Expression
1 tendopl.h 
 |-  H = ( LHyp ` K )
2 tendopl.t 
 |-  T = ( ( LTrn ` K ) ` W )
3 tendopl.e 
 |-  E = ( ( TEndo ` K ) ` W )
4 tendopl.p 
 |-  P = ( s e. E , t e. E |-> ( f e. T |-> ( ( s ` f ) o. ( t ` f ) ) ) )
5 tendopltp.l 
 |-  .<_ = ( le ` K )
6 tendopltp.r 
 |-  R = ( ( trL ` K ) ` W )
7 eqid 
 |-  ( Base ` K ) = ( Base ` K )
8 simp1l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ F e. T ) -> K e. HL )
9 8 hllatd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ F e. T ) -> K e. Lat )
10 simp1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ F e. T ) -> ( K e. HL /\ W e. H ) )
11 1 2 3 4 tendoplcl2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ F e. T ) -> ( ( U P V ) ` F ) e. T )
12 7 1 2 6 trlcl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( U P V ) ` F ) e. T ) -> ( R ` ( ( U P V ) ` F ) ) e. ( Base ` K ) )
13 10 11 12 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ F e. T ) -> ( R ` ( ( U P V ) ` F ) ) e. ( Base ` K ) )
14 1 2 3 tendocl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ U e. E /\ F e. T ) -> ( U ` F ) e. T )
15 14 3adant2r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ F e. T ) -> ( U ` F ) e. T )
16 7 1 2 6 trlcl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( U ` F ) e. T ) -> ( R ` ( U ` F ) ) e. ( Base ` K ) )
17 10 15 16 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ F e. T ) -> ( R ` ( U ` F ) ) e. ( Base ` K ) )
18 1 2 3 tendocl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ V e. E /\ F e. T ) -> ( V ` F ) e. T )
19 18 3adant2l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ F e. T ) -> ( V ` F ) e. T )
20 7 1 2 6 trlcl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( V ` F ) e. T ) -> ( R ` ( V ` F ) ) e. ( Base ` K ) )
21 10 19 20 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ F e. T ) -> ( R ` ( V ` F ) ) e. ( Base ` K ) )
22 eqid 
 |-  ( join ` K ) = ( join ` K )
23 7 22 latjcl 
 |-  ( ( K e. Lat /\ ( R ` ( U ` F ) ) e. ( Base ` K ) /\ ( R ` ( V ` F ) ) e. ( Base ` K ) ) -> ( ( R ` ( U ` F ) ) ( join ` K ) ( R ` ( V ` F ) ) ) e. ( Base ` K ) )
24 9 17 21 23 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ F e. T ) -> ( ( R ` ( U ` F ) ) ( join ` K ) ( R ` ( V ` F ) ) ) e. ( Base ` K ) )
25 simp3 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ F e. T ) -> F e. T )
26 7 1 2 6 trlcl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( R ` F ) e. ( Base ` K ) )
27 10 25 26 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ F e. T ) -> ( R ` F ) e. ( Base ` K ) )
28 simp2l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ F e. T ) -> U e. E )
29 simp2r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ F e. T ) -> V e. E )
30 4 2 tendopl2 
 |-  ( ( U e. E /\ V e. E /\ F e. T ) -> ( ( U P V ) ` F ) = ( ( U ` F ) o. ( V ` F ) ) )
31 28 29 25 30 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ F e. T ) -> ( ( U P V ) ` F ) = ( ( U ` F ) o. ( V ` F ) ) )
32 31 fveq2d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ F e. T ) -> ( R ` ( ( U P V ) ` F ) ) = ( R ` ( ( U ` F ) o. ( V ` F ) ) ) )
33 5 22 1 2 6 trlco 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( U ` F ) e. T /\ ( V ` F ) e. T ) -> ( R ` ( ( U ` F ) o. ( V ` F ) ) ) .<_ ( ( R ` ( U ` F ) ) ( join ` K ) ( R ` ( V ` F ) ) ) )
34 10 15 19 33 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ F e. T ) -> ( R ` ( ( U ` F ) o. ( V ` F ) ) ) .<_ ( ( R ` ( U ` F ) ) ( join ` K ) ( R ` ( V ` F ) ) ) )
35 32 34 eqbrtrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ F e. T ) -> ( R ` ( ( U P V ) ` F ) ) .<_ ( ( R ` ( U ` F ) ) ( join ` K ) ( R ` ( V ` F ) ) ) )
36 5 1 2 6 3 tendotp 
 |-  ( ( ( K e. HL /\ W e. H ) /\ U e. E /\ F e. T ) -> ( R ` ( U ` F ) ) .<_ ( R ` F ) )
37 36 3adant2r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ F e. T ) -> ( R ` ( U ` F ) ) .<_ ( R ` F ) )
38 5 1 2 6 3 tendotp 
 |-  ( ( ( K e. HL /\ W e. H ) /\ V e. E /\ F e. T ) -> ( R ` ( V ` F ) ) .<_ ( R ` F ) )
39 38 3adant2l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ F e. T ) -> ( R ` ( V ` F ) ) .<_ ( R ` F ) )
40 7 5 22 latjle12 
 |-  ( ( K e. Lat /\ ( ( R ` ( U ` F ) ) e. ( Base ` K ) /\ ( R ` ( V ` F ) ) e. ( Base ` K ) /\ ( R ` F ) e. ( Base ` K ) ) ) -> ( ( ( R ` ( U ` F ) ) .<_ ( R ` F ) /\ ( R ` ( V ` F ) ) .<_ ( R ` F ) ) <-> ( ( R ` ( U ` F ) ) ( join ` K ) ( R ` ( V ` F ) ) ) .<_ ( R ` F ) ) )
41 9 17 21 27 40 syl13anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ F e. T ) -> ( ( ( R ` ( U ` F ) ) .<_ ( R ` F ) /\ ( R ` ( V ` F ) ) .<_ ( R ` F ) ) <-> ( ( R ` ( U ` F ) ) ( join ` K ) ( R ` ( V ` F ) ) ) .<_ ( R ` F ) ) )
42 37 39 41 mpbi2and 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ F e. T ) -> ( ( R ` ( U ` F ) ) ( join ` K ) ( R ` ( V ` F ) ) ) .<_ ( R ` F ) )
43 7 5 9 13 24 27 35 42 lattrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ F e. T ) -> ( R ` ( ( U P V ) ` F ) ) .<_ ( R ` F ) )