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Theorem adjvalval 22517
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 22512 . . 3  |-  ( ( T  e.  dom  adjh  /\  A  e.  ~H )  ->  ( ( adjh `  T
) `  A )  e.  ~H )
2 eqcom 2285 . . . . . . . . 9  |-  ( ( ( T `  x
)  .ih  A )  =  ( x  .ih  w )  <->  ( x  .ih  w )  =  ( ( T `  x
)  .ih  A )
)
3 adj2 22514 . . . . . . . . . . . 12  |-  ( ( T  e.  dom  adjh  /\  x  e.  ~H  /\  A  e.  ~H )  ->  ( ( T `  x )  .ih  A
)  =  ( x 
.ih  ( ( adjh `  T ) `  A
) ) )
433com23 1157 . . . . . . . . . . 11  |-  ( ( T  e.  dom  adjh  /\  A  e.  ~H  /\  x  e.  ~H )  ->  ( ( T `  x )  .ih  A
)  =  ( x 
.ih  ( ( adjh `  T ) `  A
) ) )
543expa 1151 . . . . . . . . . 10  |-  ( ( ( T  e.  dom  adjh  /\  A  e.  ~H )  /\  x  e.  ~H )  ->  ( ( T `
 x )  .ih  A )  =  ( x 
.ih  ( ( adjh `  T ) `  A
) ) )
65eqeq2d 2294 . . . . . . . . 9  |-  ( ( ( 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 248 . . . . . . . 8  |-  ( ( ( 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 2559 . . . . . . 7  |-  ( ( 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 451 . . . . . 6  |-  ( ( ( 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 447 . . . . . . 7  |-  ( ( ( T  e.  dom  adjh  /\  A  e.  ~H )  /\  w  e.  ~H )  ->  w  e.  ~H )
111adantr 451 . . . . . . 7  |-  ( ( ( T  e.  dom  adjh  /\  A  e.  ~H )  /\  w  e.  ~H )  ->  ( ( adjh `  T ) `  A
)  e.  ~H )
12 hial2eq2 21686 . . . . . . 7  |-  ( ( 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 642 . . . . . 6  |-  ( ( ( 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 244 . . . . 5  |-  ( ( ( 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 )
) )
15143adant2 974 . . . 4  |-  ( ( ( T  e.  dom  adjh  /\  A  e.  ~H )  /\  ( ( adjh `  T ) `  A
)  e.  ~H  /\  w  e.  ~H )  ->  ( A. x  e. 
~H  ( ( T `
 x )  .ih  A )  =  ( x 
.ih  w )  <->  w  =  ( ( adjh `  T
) `  A )
) )
1615riota5OLD 6331 . . 3  |-  ( ( ( T  e.  dom  adjh  /\  A  e.  ~H )  /\  ( ( adjh `  T ) `  A
)  e.  ~H )  ->  ( iota_ w  e.  ~H A. x  e.  ~H  (
( T `  x
)  .ih  A )  =  ( x  .ih  w ) )  =  ( ( adjh `  T
) `  A )
)
171, 16mpdan 649 . 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 )
)
1817eqcomd 2288 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 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   dom cdm 4689   ` cfv 5255  (class class class)co 5858   iota_crio 6297   ~Hchil 21499    .ih csp 21502   adjhcado 21535
This theorem is referenced by:  nmopadjlei  22668
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-hilex 21579  ax-hfvadd 21580  ax-hvcom 21581  ax-hvass 21582  ax-hv0cl 21583  ax-hvaddid 21584  ax-hfvmul 21585  ax-hvmulid 21586  ax-hvdistr2 21589  ax-hvmul0 21590  ax-hfi 21658  ax-his1 21661  ax-his2 21662  ax-his3 21663  ax-his4 21664
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-2 9804  df-cj 11584  df-re 11585  df-im 11586  df-hvsub 21551  df-adjh 22429
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