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Theorem lnophmlem1 26708
Description: Lemma for lnophmi 26710. (Contributed by NM, 24-Jan-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
lnophmlem.1  |-  A  e. 
~H
lnophmlem.2  |-  B  e. 
~H
lnophmlem.3  |-  T  e. 
LinOp
lnophmlem.4  |-  A. x  e.  ~H  ( x  .ih  ( T `  x ) )  e.  RR
Assertion
Ref Expression
lnophmlem1  |-  ( A 
.ih  ( T `  A ) )  e.  RR
Distinct variable groups:    x, A    x, B    x, T

Proof of Theorem lnophmlem1
StepHypRef Expression
1 lnophmlem.1 . 2  |-  A  e. 
~H
2 lnophmlem.4 . 2  |-  A. x  e.  ~H  ( x  .ih  ( T `  x ) )  e.  RR
3 id 22 . . . . 5  |-  ( x  =  A  ->  x  =  A )
4 fveq2 5866 . . . . 5  |-  ( x  =  A  ->  ( T `  x )  =  ( T `  A ) )
53, 4oveq12d 6303 . . . 4  |-  ( x  =  A  ->  (
x  .ih  ( T `  x ) )  =  ( A  .ih  ( T `  A )
) )
65eleq1d 2536 . . 3  |-  ( x  =  A  ->  (
( x  .ih  ( T `  x )
)  e.  RR  <->  ( A  .ih  ( T `  A
) )  e.  RR ) )
76rspcv 3210 . 2  |-  ( A  e.  ~H  ->  ( A. x  e.  ~H  ( x  .ih  ( T `
 x ) )  e.  RR  ->  ( A  .ih  ( T `  A ) )  e.  RR ) )
81, 2, 7mp2 9 1  |-  ( A 
.ih  ( T `  A ) )  e.  RR
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379    e. wcel 1767   A.wral 2814   ` cfv 5588  (class class class)co 6285   RRcr 9492   ~Hchil 25609    .ih csp 25612   LinOpclo 25637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-iota 5551  df-fv 5596  df-ov 6288
This theorem is referenced by:  lnophmlem2  26709
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