Theorem tendopltp 36068

Description: Trace-preserving property of endomorphism sum operation P, based on theorem trlco 36015. Part of remark in [Crawley] p. 118, 2nd line, "it is clear from the second part of G (our trlco 36015) 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 ( TEndo ` K ) ` W.) (Contributed by NM, 9-Jun-2013.)

Hypotheses

Ref	Expression
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

Ref	Expression
tendopltp	|- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ F e. T ) -> ( R ` ( ( U P V ) ` F ) ) .<_ ( R ` F ) )

Distinct variable groups:   t, s, E   f, s, t, T   f, W, s, t

Allowed substitution hints:   P( t, f, s)   R( t, f, s)   U( t, f, s)   E( f)   F( t, f, s)   H( t, f, s)   K( t, f, s)   .<_ ( t, f, s)   V( t, f, s)

Proof of Theorem tendopltp

Step	Hyp	Ref	Expression
1		eqid 2622	. 2 |- ( Base ` K ) = ( Base ` K )
2		tendopltp.l	. 2 |- .<_ = ( le ` K )
3		simp1l 1085	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ F e. T ) -> K e. HL )
4		hllat 34650	. . 3 |- ( K e. HL -> K e. Lat )
5	3, 4	syl 17	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ F e. T ) -> K e. Lat )
6		simp1 1061	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ F e. T ) -> ( K e. HL /\ W e. H ) )
7		tendopl.h	. . . 4 |- H = ( LHyp ` K )
8		tendopl.t	. . . 4 |- T = ( ( LTrn ` K ) ` W )
9		tendopl.e	. . . 4 |- E = ( ( TEndo ` K ) ` W )
10		tendopl.p	. . . 4 |- P = ( s e. E , t e. E |-> ( f e. T |-> ( ( s ` f ) o. ( t ` f ) ) ) )
11	7, 8, 9, 10	tendoplcl2 36066	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ F e. T ) -> ( ( U P V ) ` F ) e. T )
12		tendopltp.r	. . . 4 |- R = ( ( trL ` K ) ` W )
13	1, 7, 8, 12	trlcl 35451	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( U P V ) ` F ) e. T ) -> ( R ` ( ( U P V ) ` F ) ) e. ( Base ` K ) )
14	6, 11, 13	syl2anc 693	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ F e. T ) -> ( R ` ( ( U P V ) ` F ) ) e. ( Base ` K ) )
15	7, 8, 9	tendocl 36055	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ U e. E /\ F e. T ) -> ( U ` F ) e. T )
16	15	3adant2r 1321	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ F e. T ) -> ( U ` F ) e. T )
17	1, 7, 8, 12	trlcl 35451	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( U ` F ) e. T ) -> ( R ` ( U ` F ) ) e. ( Base ` K ) )
18	6, 16, 17	syl2anc 693	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ F e. T ) -> ( R ` ( U ` F ) ) e. ( Base ` K ) )
19	7, 8, 9	tendocl 36055	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ V e. E /\ F e. T ) -> ( V ` F ) e. T )
20	19	3adant2l 1320	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ F e. T ) -> ( V ` F ) e. T )
21	1, 7, 8, 12	trlcl 35451	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( V ` F ) e. T ) -> ( R ` ( V ` F ) ) e. ( Base ` K ) )
22	6, 20, 21	syl2anc 693	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ F e. T ) -> ( R ` ( V ` F ) ) e. ( Base ` K ) )
23		eqid 2622	. . . 4 |- ( join ` K ) = ( join ` K )
24	1, 23	latjcl 17051	. . 3 |- ( ( 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 ) )
25	5, 18, 22, 24	syl3anc 1326	. 2 |- ( ( ( 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 ) )
26		simp3 1063	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ F e. T ) -> F e. T )
27	1, 7, 8, 12	trlcl 35451	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( R ` F ) e. ( Base ` K ) )
28	6, 26, 27	syl2anc 693	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ F e. T ) -> ( R ` F ) e. ( Base ` K ) )
29		simp2l 1087	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ F e. T ) -> U e. E )
30		simp2r 1088	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ F e. T ) -> V e. E )
31	10, 8	tendopl2 36065	. . . . 5 |- ( ( U e. E /\ V e. E /\ F e. T ) -> ( ( U P V ) ` F ) = ( ( U ` F ) o. ( V ` F ) ) )
32	29, 30, 26, 31	syl3anc 1326	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ F e. T ) -> ( ( U P V ) ` F ) = ( ( U ` F ) o. ( V ` F ) ) )
33	32	fveq2d 6195	. . 3 |- ( ( ( 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 ) ) ) )
34	2, 23, 7, 8, 12	trlco 36015	. . . 4 |- ( ( ( 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 ) ) ) )
35	6, 16, 20, 34	syl3anc 1326	. . 3 |- ( ( ( 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 ) ) ) )
36	33, 35	eqbrtrd 4675	. 2 |- ( ( ( 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 ) ) ) )
37	2, 7, 8, 12, 9	tendotp 36049	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ U e. E /\ F e. T ) -> ( R ` ( U ` F ) ) .<_ ( R ` F ) )
38	37	3adant2r 1321	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ F e. T ) -> ( R ` ( U ` F ) ) .<_ ( R ` F ) )
39	2, 7, 8, 12, 9	tendotp 36049	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ V e. E /\ F e. T ) -> ( R ` ( V ` F ) ) .<_ ( R ` F ) )
40	39	3adant2l 1320	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ F e. T ) -> ( R ` ( V ` F ) ) .<_ ( R ` F ) )
41	1, 2, 23	latjle12 17062	. . . 4 |- ( ( 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 ) ) )
42	5, 18, 22, 28, 41	syl13anc 1328	. . 3 |- ( ( ( 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 ) ) )
43	38, 40, 42	mpbi2and 956	. 2 |- ( ( ( 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 ) )
44	1, 2, 5, 14, 25, 28, 36, 43	lattrd 17058	1 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ F e. T ) -> ( R ` ( ( U P V ) ` F ) ) .<_ ( R ` F ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   class class class wbr 4653   |-> cmpt 4729   o. ccom 5118   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   Basecbs 15857   lecple 15948   joincjn 16944   Latclat 17045   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  ax-riotaBAD 34239

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  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-iin 4523  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-1st 7168  df-2nd 7169  df-undef 7399  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-llines 34784  df-lplanes 34785  df-lvols 34786  df-lines 34787  df-psubsp 34789  df-pmap 34790  df-padd 35082  df-lhyp 35274  df-laut 35275  df-ldil 35390  df-ltrn 35391  df-trl 35446  df-tendo 36043

This theorem is referenced by:  tendoplcl  36069