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Theorem adjvalval 23401
Description: Value of the value of the adjoint function. (Contributed by NM, 22-Feb-2006.) (Proof shortened by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.)
Assertion
Ref Expression
adjvalval  |-  ( ( T  e.  dom  adjh  /\  A  e.  ~H )  ->  ( ( adjh `  T
) `  A )  =  ( iota_ w  e. 
~H A. x  e.  ~H  ( ( T `  x )  .ih  A
)  =  ( x 
.ih  w ) ) )
Distinct variable groups:    x, w, A    x, T, w

Proof of Theorem adjvalval
StepHypRef Expression
1 adjcl 23396 . . 3  |-  ( ( T  e.  dom  adjh  /\  A  e.  ~H )  ->  ( ( adjh `  T
) `  A )  e.  ~H )
2 eqcom 2414 . . . . . . 7  |-  ( ( ( T `  x
)  .ih  A )  =  ( x  .ih  w )  <->  ( x  .ih  w )  =  ( ( T `  x
)  .ih  A )
)
3 adj2 23398 . . . . . . . . . 10  |-  ( ( T  e.  dom  adjh  /\  x  e.  ~H  /\  A  e.  ~H )  ->  ( ( T `  x )  .ih  A
)  =  ( x 
.ih  ( ( adjh `  T ) `  A
) ) )
433com23 1159 . . . . . . . . 9  |-  ( ( T  e.  dom  adjh  /\  A  e.  ~H  /\  x  e.  ~H )  ->  ( ( T `  x )  .ih  A
)  =  ( x 
.ih  ( ( adjh `  T ) `  A
) ) )
543expa 1153 . . . . . . . 8  |-  ( ( ( T  e.  dom  adjh  /\  A  e.  ~H )  /\  x  e.  ~H )  ->  ( ( T `
 x )  .ih  A )  =  ( x 
.ih  ( ( adjh `  T ) `  A
) ) )
65eqeq2d 2423 . . . . . . 7  |-  ( ( ( T  e.  dom  adjh  /\  A  e.  ~H )  /\  x  e.  ~H )  ->  ( ( x 
.ih  w )  =  ( ( T `  x )  .ih  A
)  <->  ( x  .ih  w )  =  ( x  .ih  ( (
adjh `  T ) `  A ) ) ) )
72, 6syl5bb 249 . . . . . 6  |-  ( ( ( T  e.  dom  adjh  /\  A  e.  ~H )  /\  x  e.  ~H )  ->  ( ( ( T `  x ) 
.ih  A )  =  ( x  .ih  w
)  <->  ( x  .ih  w )  =  ( x  .ih  ( (
adjh `  T ) `  A ) ) ) )
87ralbidva 2690 . . . . 5  |-  ( ( T  e.  dom  adjh  /\  A  e.  ~H )  ->  ( A. x  e. 
~H  ( ( T `
 x )  .ih  A )  =  ( x 
.ih  w )  <->  A. x  e.  ~H  ( x  .ih  w )  =  ( x  .ih  ( (
adjh `  T ) `  A ) ) ) )
98adantr 452 . . . 4  |-  ( ( ( T  e.  dom  adjh  /\  A  e.  ~H )  /\  w  e.  ~H )  ->  ( A. x  e.  ~H  ( ( T `
 x )  .ih  A )  =  ( x 
.ih  w )  <->  A. x  e.  ~H  ( x  .ih  w )  =  ( x  .ih  ( (
adjh `  T ) `  A ) ) ) )
10 simpr 448 . . . . 5  |-  ( ( ( T  e.  dom  adjh  /\  A  e.  ~H )  /\  w  e.  ~H )  ->  w  e.  ~H )
111adantr 452 . . . . 5  |-  ( ( ( T  e.  dom  adjh  /\  A  e.  ~H )  /\  w  e.  ~H )  ->  ( ( adjh `  T ) `  A
)  e.  ~H )
12 hial2eq2 22570 . . . . 5  |-  ( ( w  e.  ~H  /\  ( ( adjh `  T
) `  A )  e.  ~H )  ->  ( A. x  e.  ~H  ( x  .ih  w )  =  ( x  .ih  ( ( adjh `  T
) `  A )
)  <->  w  =  (
( adjh `  T ) `  A ) ) )
1310, 11, 12syl2anc 643 . . . 4  |-  ( ( ( T  e.  dom  adjh  /\  A  e.  ~H )  /\  w  e.  ~H )  ->  ( A. x  e.  ~H  ( x  .ih  w )  =  ( x  .ih  ( (
adjh `  T ) `  A ) )  <->  w  =  ( ( adjh `  T
) `  A )
) )
149, 13bitrd 245 . . 3  |-  ( ( ( T  e.  dom  adjh  /\  A  e.  ~H )  /\  w  e.  ~H )  ->  ( A. x  e.  ~H  ( ( T `
 x )  .ih  A )  =  ( x 
.ih  w )  <->  w  =  ( ( adjh `  T
) `  A )
) )
151, 14riota5 6542 . 2  |-  ( ( T  e.  dom  adjh  /\  A  e.  ~H )  ->  ( iota_ w  e.  ~H A. x  e.  ~H  (
( T `  x
)  .ih  A )  =  ( x  .ih  w ) )  =  ( ( adjh `  T
) `  A )
)
1615eqcomd 2417 1  |-  ( ( T  e.  dom  adjh  /\  A  e.  ~H )  ->  ( ( adjh `  T
) `  A )  =  ( iota_ w  e. 
~H A. x  e.  ~H  ( ( T `  x )  .ih  A
)  =  ( x 
.ih  w ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2674   dom cdm 4845   ` cfv 5421  (class class class)co 6048   iota_crio 6509   ~Hchil 22383    .ih csp 22386   adjhcado 22419
This theorem is referenced by:  nmopadjlei  23552
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031  ax-hilex 22463  ax-hfvadd 22464  ax-hvcom 22465  ax-hvass 22466  ax-hv0cl 22467  ax-hvaddid 22468  ax-hfvmul 22469  ax-hvmulid 22470  ax-hvdistr2 22473  ax-hvmul0 22474  ax-hfi 22542  ax-his1 22545  ax-his2 22546  ax-his3 22547  ax-his4 22548
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-po 4471  df-so 4472  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-riota 6516  df-er 6872  df-map 6987  df-en 7077  df-dom 7078  df-sdom 7079  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642  df-2 10022  df-cj 11867  df-re 11868  df-im 11869  df-hvsub 22435  df-adjh 23313
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