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Theorem adjadj 22508
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 22506 . . . . 5  |-  ( ( T  e.  dom  adjh  /\  x  e.  ~H  /\  y  e.  ~H )  ->  ( ( T `  x )  .ih  y
)  =  ( x 
.ih  ( ( adjh `  T ) `  y
) ) )
2 dmadjrn 22467 . . . . . 6  |-  ( T  e.  dom  adjh  ->  (
adjh `  T )  e.  dom  adjh )
3 adj1 22505 . . . . . 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 2317 . . . 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 2635 . 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 22467 . . . 4  |-  ( (
adjh `  T )  e.  dom  adjh  ->  ( adjh `  ( adjh `  T
) )  e.  dom  adjh )
9 dmadjop 22460 . . . 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 22460 . . 3  |-  ( T  e.  dom  adjh  ->  T : ~H --> ~H )
12 hoeq1 22402 . . 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 1628    e. wcel 1688   A.wral 2544   dom cdm 4688   -->wf 5217   ` cfv 5221  (class class class)co 5819   ~Hchil 21491    .ih csp 21494   adjhcado 21527
This theorem is referenced by:  adjbd1o  22657  adjsslnop  22659  nmopadji  22662  adjeq0  22663  nmopcoadji  22673  nmopcoadj2i  22674
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-13 1690  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-resscn 8789  ax-1cn 8790  ax-icn 8791  ax-addcl 8792  ax-addrcl 8793  ax-mulcl 8794  ax-mulrcl 8795  ax-mulcom 8796  ax-addass 8797  ax-mulass 8798  ax-distr 8799  ax-i2m1 8800  ax-1ne0 8801  ax-1rid 8802  ax-rnegex 8803  ax-rrecex 8804  ax-cnre 8805  ax-pre-lttri 8806  ax-pre-lttrn 8807  ax-pre-ltadd 8808  ax-pre-mulgt0 8809  ax-hilex 21571  ax-hfvadd 21572  ax-hvcom 21573  ax-hvass 21574  ax-hv0cl 21575  ax-hvaddid 21576  ax-hfvmul 21577  ax-hvmulid 21578  ax-hvdistr2 21581  ax-hvmul0 21582  ax-hfi 21650  ax-his1 21653  ax-his2 21654  ax-his3 21655  ax-his4 21656
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 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-po 4313  df-so 4314  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-iota 6252  df-riota 6299  df-er 6655  df-map 6769  df-en 6859  df-dom 6860  df-sdom 6861  df-pnf 8864  df-mnf 8865  df-xr 8866  df-ltxr 8867  df-le 8868  df-sub 9034  df-neg 9035  df-div 9419  df-2 9799  df-cj 11578  df-re 11579  df-im 11580  df-hvsub 21543  df-adjh 22421
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