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Theorem homulid2 23295
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 9040 . . . . 5  |-  1  e.  CC
2 homval 23236 . . . . 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 5860 . . . . 5  |-  ( ( T : ~H --> ~H  /\  x  e.  ~H )  ->  ( T `  x
)  e.  ~H )
5 ax-hvmulid 22501 . . . . 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 2467 . . 3  |-  ( ( T : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( 1  .op 
T ) `  x
)  =  ( T `
 x ) )
87ralrimiva 2781 . 2  |-  ( T : ~H --> ~H  ->  A. x  e.  ~H  (
( 1  .op  T
) `  x )  =  ( T `  x ) )
9 homulcl 23254 . . . 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 23255 . . 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 1652    e. wcel 1725   A.wral 2697   -->wf 5442   ` cfv 5446  (class class class)co 6073   CCcc 8980   1c1 8983   ~Hchil 22414    .h csm 22416    .op chot 22434
This theorem is referenced by:  honegneg  23301  ho2times  23314  leopmul  23629  nmopleid  23634  opsqrlem1  23635  opsqrlem6  23640
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-1cn 9040  ax-hilex 22494  ax-hfvmul 22500  ax-hvmulid 22501
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-map 7012  df-homul 23226
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