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Theorem tendoid 34726
Description: The identity value of a trace-preserving endomorphism. (Contributed by NM, 21-Jun-2013.)
Hypotheses
Ref Expression
tendoid.b  |-  B  =  ( Base `  K
)
tendoid.h  |-  H  =  ( LHyp `  K
)
tendoid.e  |-  E  =  ( ( TEndo `  K
) `  W )
Assertion
Ref Expression
tendoid  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E
)  ->  ( S `  (  _I  |`  B ) )  =  (  _I  |`  B ) )

Proof of Theorem tendoid
StepHypRef Expression
1 tendoid.b . . . . . . 7  |-  B  =  ( Base `  K
)
2 tendoid.h . . . . . . 7  |-  H  =  ( LHyp `  K
)
3 eqid 2451 . . . . . . 7  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
41, 2, 3idltrn 34103 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  B )  e.  ( ( LTrn `  K ) `  W
) )
54adantr 465 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E
)  ->  (  _I  |`  B )  e.  ( ( LTrn `  K
) `  W )
)
6 eqid 2451 . . . . . 6  |-  ( le
`  K )  =  ( le `  K
)
7 eqid 2451 . . . . . 6  |-  ( ( trL `  K ) `
 W )  =  ( ( trL `  K
) `  W )
8 tendoid.e . . . . . 6  |-  E  =  ( ( TEndo `  K
) `  W )
96, 2, 3, 7, 8tendotp 34714 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E  /\  (  _I  |`  B )  e.  ( ( LTrn `  K ) `  W
) )  ->  (
( ( trL `  K
) `  W ) `  ( S `  (  _I  |`  B ) ) ) ( le `  K ) ( ( ( trL `  K
) `  W ) `  (  _I  |`  B ) ) )
105, 9mpd3an3 1316 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E
)  ->  ( (
( trL `  K
) `  W ) `  ( S `  (  _I  |`  B ) ) ) ( le `  K ) ( ( ( trL `  K
) `  W ) `  (  _I  |`  B ) ) )
11 eqid 2451 . . . . . 6  |-  ( 0.
`  K )  =  ( 0. `  K
)
121, 11, 2, 7trlid0 34129 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( ( trL `  K ) `  W
) `  (  _I  |`  B ) )  =  ( 0. `  K
) )
1312adantr 465 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E
)  ->  ( (
( trL `  K
) `  W ) `  (  _I  |`  B ) )  =  ( 0.
`  K ) )
1410, 13breqtrd 4417 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E
)  ->  ( (
( trL `  K
) `  W ) `  ( S `  (  _I  |`  B ) ) ) ( le `  K ) ( 0.
`  K ) )
15 hlop 33316 . . . . 5  |-  ( K  e.  HL  ->  K  e.  OP )
1615ad2antrr 725 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E
)  ->  K  e.  OP )
172, 3, 8tendocl 34720 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E  /\  (  _I  |`  B )  e.  ( ( LTrn `  K ) `  W
) )  ->  ( S `  (  _I  |`  B ) )  e.  ( ( LTrn `  K
) `  W )
)
185, 17mpd3an3 1316 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E
)  ->  ( S `  (  _I  |`  B ) )  e.  ( (
LTrn `  K ) `  W ) )
191, 2, 3, 7trlcl 34117 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S `  (  _I  |`  B ) )  e.  ( (
LTrn `  K ) `  W ) )  -> 
( ( ( trL `  K ) `  W
) `  ( S `  (  _I  |`  B ) ) )  e.  B
)
2018, 19syldan 470 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E
)  ->  ( (
( trL `  K
) `  W ) `  ( S `  (  _I  |`  B ) ) )  e.  B )
211, 6, 11ople0 33141 . . . 4  |-  ( ( K  e.  OP  /\  ( ( ( trL `  K ) `  W
) `  ( S `  (  _I  |`  B ) ) )  e.  B
)  ->  ( (
( ( trL `  K
) `  W ) `  ( S `  (  _I  |`  B ) ) ) ( le `  K ) ( 0.
`  K )  <->  ( (
( trL `  K
) `  W ) `  ( S `  (  _I  |`  B ) ) )  =  ( 0.
`  K ) ) )
2216, 20, 21syl2anc 661 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E
)  ->  ( (
( ( trL `  K
) `  W ) `  ( S `  (  _I  |`  B ) ) ) ( le `  K ) ( 0.
`  K )  <->  ( (
( trL `  K
) `  W ) `  ( S `  (  _I  |`  B ) ) )  =  ( 0.
`  K ) ) )
2314, 22mpbid 210 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E
)  ->  ( (
( trL `  K
) `  W ) `  ( S `  (  _I  |`  B ) ) )  =  ( 0.
`  K ) )
241, 11, 2, 3, 7trlid0b 34131 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S `  (  _I  |`  B ) )  e.  ( (
LTrn `  K ) `  W ) )  -> 
( ( S `  (  _I  |`  B ) )  =  (  _I  |`  B )  <->  ( (
( trL `  K
) `  W ) `  ( S `  (  _I  |`  B ) ) )  =  ( 0.
`  K ) ) )
2518, 24syldan 470 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E
)  ->  ( ( S `  (  _I  |`  B ) )  =  (  _I  |`  B )  <-> 
( ( ( trL `  K ) `  W
) `  ( S `  (  _I  |`  B ) ) )  =  ( 0. `  K ) ) )
2623, 25mpbird 232 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E
)  ->  ( S `  (  _I  |`  B ) )  =  (  _I  |`  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   class class class wbr 4393    _I cid 4732    |` cres 4943   ` cfv 5519   Basecbs 14285   lecple 14356   0.cp0 15318   OPcops 33126   HLchlt 33304   LHypclh 33937   LTrncltrn 34054   trLctrl 34111   TEndoctendo 34705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-map 7319  df-poset 15227  df-plt 15239  df-lub 15255  df-glb 15256  df-join 15257  df-meet 15258  df-p0 15320  df-p1 15321  df-lat 15327  df-clat 15389  df-oposet 33130  df-ol 33132  df-oml 33133  df-covers 33220  df-ats 33221  df-atl 33252  df-cvlat 33276  df-hlat 33305  df-lhyp 33941  df-laut 33942  df-ldil 34057  df-ltrn 34058  df-trl 34112  df-tendo 34708
This theorem is referenced by:  tendoeq2  34727  tendo0mulr  34780  tendotr  34783  tendocnv  34975  dvhopN  35070  dihpN  35290
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