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Theorem tendopltp 29658
Description: Trace-preserving property of endomorphism sum operation  P, based on theorem trlco 29605. Part of remark in [Crawley] p. 118, 2nd line, "it is clear from the second part of G (our trlco 29605) 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 2253 . 2  |-  ( Base `  K )  =  (
Base `  K )
2 tendopltp.l . 2  |-  .<_  =  ( le `  K )
3 simp1l 984 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  F  e.  T )  ->  K  e.  HL )
4 hllat 28242 . . 3  |-  ( K  e.  HL  ->  K  e.  Lat )
53, 4syl 17 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  F  e.  T )  ->  K  e.  Lat )
6 simp1 960 . . 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 29656 . . 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 29042 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( U P V ) `  F )  e.  T
)  ->  ( R `  ( ( U P V ) `  F
) )  e.  (
Base `  K )
)
146, 11, 13syl2anc 645 . 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 29645 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  F  e.  T
)  ->  ( U `  F )  e.  T
)
16153adant2r 1182 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  F  e.  T )  ->  ( U `  F )  e.  T )
171, 7, 8, 12trlcl 29042 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U `  F )  e.  T
)  ->  ( R `  ( U `  F
) )  e.  (
Base `  K )
)
186, 16, 17syl2anc 645 . . 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 29645 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  V  e.  E  /\  F  e.  T
)  ->  ( V `  F )  e.  T
)
20193adant2l 1181 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  F  e.  T )  ->  ( V `  F )  e.  T )
211, 7, 8, 12trlcl 29042 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( V `  F )  e.  T
)  ->  ( R `  ( V `  F
) )  e.  (
Base `  K )
)
226, 20, 21syl2anc 645 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  F  e.  T )  ->  ( R `  ( V `  F ) )  e.  ( Base `  K
) )
23 eqid 2253 . . . 4  |-  ( join `  K )  =  (
join `  K )
241, 23latjcl 14000 . . 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 1187 . 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 962 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  F  e.  T )  ->  F  e.  T )
271, 7, 8, 12trlcl 29042 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( R `  F )  e.  (
Base `  K )
)
286, 26, 27syl2anc 645 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  F  e.  T )  ->  ( R `  F )  e.  ( Base `  K
) )
29 simp2l 986 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  F  e.  T )  ->  U  e.  E )
30 simp2r 987 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  F  e.  T )  ->  V  e.  E )
3110, 8tendopl2 29655 . . . . 5  |-  ( ( U  e.  E  /\  V  e.  E  /\  F  e.  T )  ->  ( ( U P V ) `  F
)  =  ( ( U `  F )  o.  ( V `  F ) ) )
3229, 30, 26, 31syl3anc 1187 . . . 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 5381 . . 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 29605 . . . 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 1187 . . 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 3940 . 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 29639 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  F  e.  T
)  ->  ( R `  ( U `  F
) )  .<_  ( R `
 F ) )
38373adant2r 1182 . . 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 29639 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  V  e.  E  /\  F  e.  T
)  ->  ( R `  ( V `  F
) )  .<_  ( R `
 F ) )
40393adant2l 1181 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  F  e.  T )  ->  ( R `  ( V `  F ) )  .<_  ( R `  F ) )
411, 2, 23latjle12 14012 . . . 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 1189 . . 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 892 . 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 14008 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 6    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621   class class class wbr 3920    e. cmpt 3974    o. ccom 4584   ` cfv 4592  (class class class)co 5710    e. cmpt2 5712   Basecbs 13022   lecple 13089   joincjn 13922   Latclat 13995   HLchlt 28229   LHypclh 28862   LTrncltrn 28979   trLctrl 29036   TEndoctendo 29630
This theorem is referenced by:  tendoplcl  29659
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-iin 3806  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-undef 6182  df-riota 6190  df-map 6660  df-poset 13924  df-plt 13936  df-lub 13952  df-glb 13953  df-join 13954  df-meet 13955  df-p0 13989  df-p1 13990  df-lat 13996  df-clat 14058  df-oposet 28055  df-ol 28057  df-oml 28058  df-covers 28145  df-ats 28146  df-atl 28177  df-cvlat 28201  df-hlat 28230  df-llines 28376  df-lplanes 28377  df-lvols 28378  df-lines 28379  df-psubsp 28381  df-pmap 28382  df-padd 28674  df-lhyp 28866  df-laut 28867  df-ldil 28982  df-ltrn 28983  df-trl 29037  df-tendo 29633
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