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Theorem adjvalval 23478
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 23473 . . 3  |-  ( ( T  e.  dom  adjh  /\  A  e.  ~H )  ->  ( ( adjh `  T
) `  A )  e.  ~H )
2 eqcom 2445 . . . . . . 7  |-  ( ( ( T `  x
)  .ih  A )  =  ( x  .ih  w )  <->  ( x  .ih  w )  =  ( ( T `  x
)  .ih  A )
)
3 adj2 23475 . . . . . . . . . 10  |-  ( ( T  e.  dom  adjh  /\  x  e.  ~H  /\  A  e.  ~H )  ->  ( ( T `  x )  .ih  A
)  =  ( x 
.ih  ( ( adjh `  T ) `  A
) ) )
433com23 1160 . . . . . . . . 9  |-  ( ( T  e.  dom  adjh  /\  A  e.  ~H  /\  x  e.  ~H )  ->  ( ( T `  x )  .ih  A
)  =  ( x 
.ih  ( ( adjh `  T ) `  A
) ) )
543expa 1154 . . . . . . . 8  |-  ( ( ( T  e.  dom  adjh  /\  A  e.  ~H )  /\  x  e.  ~H )  ->  ( ( T `
 x )  .ih  A )  =  ( x 
.ih  ( ( adjh `  T ) `  A
) ) )
65eqeq2d 2454 . . . . . . 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 250 . . . . . 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 2728 . . . . 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 453 . . . 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 449 . . . . 5  |-  ( ( ( T  e.  dom  adjh  /\  A  e.  ~H )  /\  w  e.  ~H )  ->  w  e.  ~H )
111adantr 453 . . . . 5  |-  ( ( ( T  e.  dom  adjh  /\  A  e.  ~H )  /\  w  e.  ~H )  ->  ( ( adjh `  T ) `  A
)  e.  ~H )
12 hial2eq2 22647 . . . . 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 644 . . . 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 246 . . 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 6611 . 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 2448 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 178    /\ wa 360    = wceq 1654    e. wcel 1728   A.wral 2712   dom cdm 4913   ` cfv 5489  (class class class)co 6117   iota_crio 6578   ~Hchil 22460    .ih csp 22463   adjhcado 22496
This theorem is referenced by:  nmopadjlei  23629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-13 1730  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-sep 4361  ax-nul 4369  ax-pow 4412  ax-pr 4438  ax-un 4736  ax-resscn 9085  ax-1cn 9086  ax-icn 9087  ax-addcl 9088  ax-addrcl 9089  ax-mulcl 9090  ax-mulrcl 9091  ax-mulcom 9092  ax-addass 9093  ax-mulass 9094  ax-distr 9095  ax-i2m1 9096  ax-1ne0 9097  ax-1rid 9098  ax-rnegex 9099  ax-rrecex 9100  ax-cnre 9101  ax-pre-lttri 9102  ax-pre-lttrn 9103  ax-pre-ltadd 9104  ax-pre-mulgt0 9105  ax-hilex 22540  ax-hfvadd 22541  ax-hvcom 22542  ax-hvass 22543  ax-hv0cl 22544  ax-hvaddid 22545  ax-hfvmul 22546  ax-hvmulid 22547  ax-hvdistr2 22550  ax-hvmul0 22551  ax-hfi 22619  ax-his1 22622  ax-his2 22623  ax-his3 22624  ax-his4 22625
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2717  df-rex 2718  df-reu 2719  df-rmo 2720  df-rab 2721  df-v 2967  df-sbc 3171  df-csb 3271  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-nul 3617  df-if 3768  df-pw 3830  df-sn 3849  df-pr 3850  df-op 3852  df-uni 4045  df-iun 4124  df-br 4244  df-opab 4298  df-mpt 4299  df-id 4533  df-po 4538  df-so 4539  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5453  df-fun 5491  df-fn 5492  df-f 5493  df-f1 5494  df-fo 5495  df-f1o 5496  df-fv 5497  df-ov 6120  df-oprab 6121  df-mpt2 6122  df-riota 6585  df-er 6941  df-map 7056  df-en 7146  df-dom 7147  df-sdom 7148  df-pnf 9160  df-mnf 9161  df-xr 9162  df-ltxr 9163  df-le 9164  df-sub 9331  df-neg 9332  df-div 9716  df-2 10096  df-cj 11942  df-re 11943  df-im 11944  df-hvsub 22512  df-adjh 23390
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