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Theorem homulid2 23151
Description: An operator equals its scalar product with one. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)
Assertion
Ref Expression
homulid2  |-  ( T : ~H --> ~H  ->  ( 1  .op  T )  =  T )

Proof of Theorem homulid2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ax-1cn 8981 . . . . 5  |-  1  e.  CC
2 homval 23092 . . . . 5  |-  ( ( 1  e.  CC  /\  T : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( 1  .op 
T ) `  x
)  =  ( 1  .h  ( T `  x ) ) )
31, 2mp3an1 1266 . . . 4  |-  ( ( T : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( 1  .op 
T ) `  x
)  =  ( 1  .h  ( T `  x ) ) )
4 ffvelrn 5807 . . . . 5  |-  ( ( T : ~H --> ~H  /\  x  e.  ~H )  ->  ( T `  x
)  e.  ~H )
5 ax-hvmulid 22357 . . . . 5  |-  ( ( T `  x )  e.  ~H  ->  (
1  .h  ( T `
 x ) )  =  ( T `  x ) )
64, 5syl 16 . . . 4  |-  ( ( T : ~H --> ~H  /\  x  e.  ~H )  ->  ( 1  .h  ( T `  x )
)  =  ( T `
 x ) )
73, 6eqtrd 2419 . . 3  |-  ( ( T : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( 1  .op 
T ) `  x
)  =  ( T `
 x ) )
87ralrimiva 2732 . 2  |-  ( T : ~H --> ~H  ->  A. x  e.  ~H  (
( 1  .op  T
) `  x )  =  ( T `  x ) )
9 homulcl 23110 . . . 4  |-  ( ( 1  e.  CC  /\  T : ~H --> ~H )  ->  ( 1  .op  T
) : ~H --> ~H )
101, 9mpan 652 . . 3  |-  ( T : ~H --> ~H  ->  ( 1  .op  T ) : ~H --> ~H )
11 hoeq 23111 . . 3  |-  ( ( ( 1  .op  T
) : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( A. x  e. 
~H  ( ( 1 
.op  T ) `  x )  =  ( T `  x )  <-> 
( 1  .op  T
)  =  T ) )
1210, 11mpancom 651 . 2  |-  ( T : ~H --> ~H  ->  ( A. x  e.  ~H  ( ( 1  .op 
T ) `  x
)  =  ( T `
 x )  <->  ( 1 
.op  T )  =  T ) )
138, 12mpbid 202 1  |-  ( T : ~H --> ~H  ->  ( 1  .op  T )  =  T )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2649   -->wf 5390   ` cfv 5394  (class class class)co 6020   CCcc 8921   1c1 8924   ~Hchil 22270    .h csm 22272    .op chot 22290
This theorem is referenced by:  honegneg  23157  ho2times  23170  leopmul  23485  nmopleid  23490  opsqrlem1  23491  opsqrlem6  23496
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-1cn 8981  ax-hilex 22350  ax-hfvmul 22356  ax-hvmulid 22357
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-map 6956  df-homul 23082
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