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Theorem adjvalval 22533
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 22528 . . 3  |-  ( ( T  e.  dom  adjh  /\  A  e.  ~H )  ->  ( ( adjh `  T
) `  A )  e.  ~H )
2 eqcom 2298 . . . . . . . . 9  |-  ( ( ( T `  x
)  .ih  A )  =  ( x  .ih  w )  <->  ( x  .ih  w )  =  ( ( T `  x
)  .ih  A )
)
3 adj2 22530 . . . . . . . . . . . 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 2307 . . . . . . . . 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 2572 . . . . . . 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 21702 . . . . . . 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 6347 . . 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 2301 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 1632    e. wcel 1696   A.wral 2556   dom cdm 4705   ` cfv 5271  (class class class)co 5874   iota_crio 6313   ~Hchil 21515    .ih csp 21518   adjhcado 21551
This theorem is referenced by:  nmopadjlei  22684
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-hilex 21595  ax-hfvadd 21596  ax-hvcom 21597  ax-hvass 21598  ax-hv0cl 21599  ax-hvaddid 21600  ax-hfvmul 21601  ax-hvmulid 21602  ax-hvdistr2 21605  ax-hvmul0 21606  ax-hfi 21674  ax-his1 21677  ax-his2 21678  ax-his3 21679  ax-his4 21680
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-2 9820  df-cj 11600  df-re 11601  df-im 11602  df-hvsub 21567  df-adjh 22445
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