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Theorem tendopltp 30969
Description: Trace-preserving property of endomorphism sum operation  P, based on theorem trlco 30916. Part of remark in [Crawley] p. 118, 2nd line, "it is clear from the second part of G (our trlco 30916) 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
StepHypRef Expression
1 eqid 2283 . 2  |-  ( Base `  K )  =  (
Base `  K )
2 tendopltp.l . 2  |-  .<_  =  ( le `  K )
3 simp1l 979 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  F  e.  T )  ->  K  e.  HL )
4 hllat 29553 . . 3  |-  ( K  e.  HL  ->  K  e.  Lat )
53, 4syl 15 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  F  e.  T )  ->  K  e.  Lat )
6 simp1 955 . . 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 )
) ) )
117, 8, 9, 10tendoplcl2 30967 . . 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 )
131, 7, 8, 12trlcl 30353 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( U P V ) `  F )  e.  T
)  ->  ( R `  ( ( U P V ) `  F
) )  e.  (
Base `  K )
)
146, 11, 13syl2anc 642 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  F  e.  T )  ->  ( R `  ( ( U P V ) `  F ) )  e.  ( Base `  K
) )
157, 8, 9tendocl 30956 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  F  e.  T
)  ->  ( U `  F )  e.  T
)
16153adant2r 1177 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  F  e.  T )  ->  ( U `  F )  e.  T )
171, 7, 8, 12trlcl 30353 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U `  F )  e.  T
)  ->  ( R `  ( U `  F
) )  e.  (
Base `  K )
)
186, 16, 17syl2anc 642 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  F  e.  T )  ->  ( R `  ( U `  F ) )  e.  ( Base `  K
) )
197, 8, 9tendocl 30956 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  V  e.  E  /\  F  e.  T
)  ->  ( V `  F )  e.  T
)
20193adant2l 1176 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  F  e.  T )  ->  ( V `  F )  e.  T )
211, 7, 8, 12trlcl 30353 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( V `  F )  e.  T
)  ->  ( R `  ( V `  F
) )  e.  (
Base `  K )
)
226, 20, 21syl2anc 642 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  F  e.  T )  ->  ( R `  ( V `  F ) )  e.  ( Base `  K
) )
23 eqid 2283 . . . 4  |-  ( join `  K )  =  (
join `  K )
241, 23latjcl 14156 . . 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 )
)
255, 18, 22, 24syl3anc 1182 . 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 957 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  F  e.  T )  ->  F  e.  T )
271, 7, 8, 12trlcl 30353 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( R `  F )  e.  (
Base `  K )
)
286, 26, 27syl2anc 642 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  F  e.  T )  ->  ( R `  F )  e.  ( Base `  K
) )
29 simp2l 981 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  F  e.  T )  ->  U  e.  E )
30 simp2r 982 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  F  e.  T )  ->  V  e.  E )
3110, 8tendopl2 30966 . . . . 5  |-  ( ( U  e.  E  /\  V  e.  E  /\  F  e.  T )  ->  ( ( U P V ) `  F
)  =  ( ( U `  F )  o.  ( V `  F ) ) )
3229, 30, 26, 31syl3anc 1182 . . . 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
) ) )
3332fveq2d 5529 . . 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 ) ) ) )
342, 23, 7, 8, 12trlco 30916 . . . 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 ) ) ) )
356, 16, 20, 34syl3anc 1182 . . 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 ) ) ) )
3633, 35eqbrtrd 4043 . 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 ) ) ) )
372, 7, 8, 12, 9tendotp 30950 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  F  e.  T
)  ->  ( R `  ( U `  F
) )  .<_  ( R `
 F ) )
38373adant2r 1177 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  F  e.  T )  ->  ( R `  ( U `  F ) )  .<_  ( R `  F ) )
392, 7, 8, 12, 9tendotp 30950 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  V  e.  E  /\  F  e.  T
)  ->  ( R `  ( V `  F
) )  .<_  ( R `
 F ) )
40393adant2l 1176 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  F  e.  T )  ->  ( R `  ( V `  F ) )  .<_  ( R `  F ) )
411, 2, 23latjle12 14168 . . . 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 ) ) )
425, 18, 22, 28, 41syl13anc 1184 . . 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 ) ) )
4338, 40, 42mpbi2and 887 . 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 ) )
441, 2, 5, 14, 25, 28, 36, 43lattrd 14164 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  F  e.  T )  ->  ( R `  ( ( U P V ) `  F ) )  .<_  ( R `  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   class class class wbr 4023    e. cmpt 4077    o. ccom 4693   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   Basecbs 13148   lecple 13215   joincjn 14078   Latclat 14151   HLchlt 29540   LHypclh 30173   LTrncltrn 30290   trLctrl 30347   TEndoctendo 30941
This theorem is referenced by:  tendoplcl  30970
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-map 6774  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-llines 29687  df-lplanes 29688  df-lvols 29689  df-lines 29690  df-psubsp 29692  df-pmap 29693  df-padd 29985  df-lhyp 30177  df-laut 30178  df-ldil 30293  df-ltrn 30294  df-trl 30348  df-tendo 30944
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