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Theorem homulid2 22380
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 8795 . . . . 5  |-  1  e.  CC
2 homval 22321 . . . . 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 5663 . . . . 5  |-  ( ( T : ~H --> ~H  /\  x  e.  ~H )  ->  ( T `  x
)  e.  ~H )
5 ax-hvmulid 21586 . . . . 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 2315 . . 3  |-  ( ( T : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( 1  .op 
T ) `  x
)  =  ( T `
 x ) )
87ralrimiva 2626 . 2  |-  ( T : ~H --> ~H  ->  A. x  e.  ~H  (
( 1  .op  T
) `  x )  =  ( T `  x ) )
9 homulcl 22339 . . . 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 22340 . . 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 1623    e. wcel 1684   A.wral 2543   -->wf 5251   ` cfv 5255  (class class class)co 5858   CCcc 8735   1c1 8738   ~Hchil 21499    .h csm 21501    .op chot 21519
This theorem is referenced by:  honegneg  22386  ho2times  22399  leopmul  22714  nmopleid  22719  opsqrlem1  22720  opsqrlem6  22725
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-1cn 8795  ax-hilex 21579  ax-hfvmul 21585  ax-hvmulid 21586
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-homul 22311
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