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Theorem homulid2 25357
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 9452 . . . . 5  |-  1  e.  CC
2 homval 25298 . . . . 5  |-  ( ( 1  e.  CC  /\  T : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( 1  .op 
T ) `  x
)  =  ( 1  .h  ( T `  x ) ) )
31, 2mp3an1 1302 . . . 4  |-  ( ( T : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( 1  .op 
T ) `  x
)  =  ( 1  .h  ( T `  x ) ) )
4 ffvelrn 5951 . . . . 5  |-  ( ( T : ~H --> ~H  /\  x  e.  ~H )  ->  ( T `  x
)  e.  ~H )
5 ax-hvmulid 24561 . . . . 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 2495 . . 3  |-  ( ( T : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( 1  .op 
T ) `  x
)  =  ( T `
 x ) )
87ralrimiva 2830 . 2  |-  ( T : ~H --> ~H  ->  A. x  e.  ~H  (
( 1  .op  T
) `  x )  =  ( T `  x ) )
9 homulcl 25316 . . . 4  |-  ( ( 1  e.  CC  /\  T : ~H --> ~H )  ->  ( 1  .op  T
) : ~H --> ~H )
101, 9mpan 670 . . 3  |-  ( T : ~H --> ~H  ->  ( 1  .op  T ) : ~H --> ~H )
11 hoeq 25317 . . 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 669 . 2  |-  ( T : ~H --> ~H  ->  ( A. x  e.  ~H  ( ( 1  .op 
T ) `  x
)  =  ( T `
 x )  <->  ( 1 
.op  T )  =  T ) )
138, 12mpbid 210 1  |-  ( T : ~H --> ~H  ->  ( 1  .op  T )  =  T )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2799   -->wf 5523   ` cfv 5527  (class class class)co 6201   CCcc 9392   1c1 9395   ~Hchil 24474    .h csm 24476    .op chot 24494
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-1cn 9452  ax-hilex 24554  ax-hfvmul 24560  ax-hvmulid 24561
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-map 7327  df-homul 25288
This theorem is referenced by:  honegneg  25363  ho2times  25376  leopmul  25691  nmopleid  25696  opsqrlem1  25697  opsqrlem6  25702
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