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Theorem homulid2 22396
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 8811 . . . . 5  |-  1  e.  CC
2 homval 22337 . . . . 5  |-  ( ( 1  e.  CC  /\  T : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( 1  .op 
T ) `  x
)  =  ( 1  .h  ( T `  x ) ) )
31, 2mp3an1 1264 . . . 4  |-  ( ( T : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( 1  .op 
T ) `  x
)  =  ( 1  .h  ( T `  x ) ) )
4 ffvelrn 5679 . . . . 5  |-  ( ( T : ~H --> ~H  /\  x  e.  ~H )  ->  ( T `  x
)  e.  ~H )
5 ax-hvmulid 21602 . . . . 5  |-  ( ( T `  x )  e.  ~H  ->  (
1  .h  ( T `
 x ) )  =  ( T `  x ) )
64, 5syl 15 . . . 4  |-  ( ( T : ~H --> ~H  /\  x  e.  ~H )  ->  ( 1  .h  ( T `  x )
)  =  ( T `
 x ) )
73, 6eqtrd 2328 . . 3  |-  ( ( T : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( 1  .op 
T ) `  x
)  =  ( T `
 x ) )
87ralrimiva 2639 . 2  |-  ( T : ~H --> ~H  ->  A. x  e.  ~H  (
( 1  .op  T
) `  x )  =  ( T `  x ) )
9 homulcl 22355 . . . 4  |-  ( ( 1  e.  CC  /\  T : ~H --> ~H )  ->  ( 1  .op  T
) : ~H --> ~H )
101, 9mpan 651 . . 3  |-  ( T : ~H --> ~H  ->  ( 1  .op  T ) : ~H --> ~H )
11 hoeq 22356 . . 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 650 . 2  |-  ( T : ~H --> ~H  ->  ( A. x  e.  ~H  ( ( 1  .op 
T ) `  x
)  =  ( T `
 x )  <->  ( 1 
.op  T )  =  T ) )
138, 12mpbid 201 1  |-  ( T : ~H --> ~H  ->  ( 1  .op  T )  =  T )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   -->wf 5267   ` cfv 5271  (class class class)co 5874   CCcc 8751   1c1 8754   ~Hchil 21515    .h csm 21517    .op chot 21535
This theorem is referenced by:  honegneg  22402  ho2times  22415  leopmul  22730  nmopleid  22735  opsqrlem1  22736  opsqrlem6  22741
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-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-1cn 8811  ax-hilex 21595  ax-hfvmul 21601  ax-hvmulid 21602
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-ral 2561  df-rex 2562  df-reu 2563  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-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-map 6790  df-homul 22327
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