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Theorem adjadj 22462
Description: Double adjoint. Theorem 3.11(iv) of [Beran] p. 106. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)
Assertion
Ref Expression
adjadj  |-  ( T  e.  dom  adjh  ->  (
adjh `  ( adjh `  T ) )  =  T )

Proof of Theorem adjadj
StepHypRef Expression
1 adj2 22460 . . . . 5  |-  ( ( T  e.  dom  adjh  /\  x  e.  ~H  /\  y  e.  ~H )  ->  ( ( T `  x )  .ih  y
)  =  ( x 
.ih  ( ( adjh `  T ) `  y
) ) )
2 dmadjrn 22421 . . . . . 6  |-  ( T  e.  dom  adjh  ->  (
adjh `  T )  e.  dom  adjh )
3 adj1 22459 . . . . . 6  |-  ( ( ( adjh `  T
)  e.  dom  adjh  /\  x  e.  ~H  /\  y  e.  ~H )  ->  ( x  .ih  (
( adjh `  T ) `  y ) )  =  ( ( ( adjh `  ( adjh `  T
) ) `  x
)  .ih  y )
)
42, 3syl3an1 1220 . . . . 5  |-  ( ( T  e.  dom  adjh  /\  x  e.  ~H  /\  y  e.  ~H )  ->  ( x  .ih  (
( adjh `  T ) `  y ) )  =  ( ( ( adjh `  ( adjh `  T
) ) `  x
)  .ih  y )
)
51, 4eqtr2d 2289 . . . 4  |-  ( ( T  e.  dom  adjh  /\  x  e.  ~H  /\  y  e.  ~H )  ->  ( ( ( adjh `  ( adjh `  T
) ) `  x
)  .ih  y )  =  ( ( T `
 x )  .ih  y ) )
653expib 1159 . . 3  |-  ( T  e.  dom  adjh  ->  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( ( ( adjh `  ( adjh `  T
) ) `  x
)  .ih  y )  =  ( ( T `
 x )  .ih  y ) ) )
76ralrimivv 2607 . 2  |-  ( T  e.  dom  adjh  ->  A. x  e.  ~H  A. y  e.  ~H  (
( ( adjh `  ( adjh `  T ) ) `
 x )  .ih  y )  =  ( ( T `  x
)  .ih  y )
)
8 dmadjrn 22421 . . . 4  |-  ( (
adjh `  T )  e.  dom  adjh  ->  ( adjh `  ( adjh `  T
) )  e.  dom  adjh )
9 dmadjop 22414 . . . 4  |-  ( (
adjh `  ( adjh `  T ) )  e. 
dom  adjh  ->  ( adjh `  ( adjh `  T
) ) : ~H --> ~H )
102, 8, 93syl 20 . . 3  |-  ( T  e.  dom  adjh  ->  (
adjh `  ( adjh `  T ) ) : ~H --> ~H )
11 dmadjop 22414 . . 3  |-  ( T  e.  dom  adjh  ->  T : ~H --> ~H )
12 hoeq1 22356 . . 3  |-  ( ( ( adjh `  ( adjh `  T ) ) : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( A. x  e. 
~H  A. y  e.  ~H  ( ( ( adjh `  ( adjh `  T
) ) `  x
)  .ih  y )  =  ( ( T `
 x )  .ih  y )  <->  ( adjh `  ( adjh `  T
) )  =  T ) )
1310, 11, 12syl2anc 645 . 2  |-  ( T  e.  dom  adjh  ->  ( A. x  e.  ~H  A. y  e.  ~H  (
( ( adjh `  ( adjh `  T ) ) `
 x )  .ih  y )  =  ( ( T `  x
)  .ih  y )  <->  (
adjh `  ( adjh `  T ) )  =  T ) )
147, 13mpbid 203 1  |-  ( T  e.  dom  adjh  ->  (
adjh `  ( adjh `  T ) )  =  T )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ w3a 939    = wceq 1619    e. wcel 1621   A.wral 2516   dom cdm 4647   -->wf 4655   ` cfv 4659  (class class class)co 5778   ~Hchil 21445    .ih csp 21448   adjhcado 21481
This theorem is referenced by:  adjbd1o  22611  adjsslnop  22613  nmopadji  22616  adjeq0  22617  nmopcoadji  22627  nmopcoadj2i  22628
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 1927  ax-ext 2237  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470  ax-resscn 8748  ax-1cn 8749  ax-icn 8750  ax-addcl 8751  ax-addrcl 8752  ax-mulcl 8753  ax-mulrcl 8754  ax-mulcom 8755  ax-addass 8756  ax-mulass 8757  ax-distr 8758  ax-i2m1 8759  ax-1ne0 8760  ax-1rid 8761  ax-rnegex 8762  ax-rrecex 8763  ax-cnre 8764  ax-pre-lttri 8765  ax-pre-lttrn 8766  ax-pre-ltadd 8767  ax-pre-mulgt0 8768  ax-hilex 21525  ax-hfvadd 21526  ax-hvcom 21527  ax-hvass 21528  ax-hv0cl 21529  ax-hvaddid 21530  ax-hfvmul 21531  ax-hvmulid 21532  ax-hvdistr2 21535  ax-hvmul0 21536  ax-hfi 21604  ax-his1 21607  ax-his2 21608  ax-his3 21609  ax-his4 21610
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 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2521  df-rex 2522  df-reu 2523  df-rmo 2524  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-op 3609  df-uni 3788  df-iun 3867  df-br 3984  df-opab 4038  df-mpt 4039  df-id 4267  df-po 4272  df-so 4273  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-iota 6211  df-riota 6258  df-er 6614  df-map 6728  df-en 6818  df-dom 6819  df-sdom 6820  df-pnf 8823  df-mnf 8824  df-xr 8825  df-ltxr 8826  df-le 8827  df-sub 8993  df-neg 8994  df-div 9378  df-2 9758  df-cj 11535  df-re 11536  df-im 11537  df-hvsub 21497  df-adjh 22375
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