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Theorem lnophmlem1 22596
Description: Lemma for lnophmi 22598. (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 19 . . . . 5  |-  ( x  =  A  ->  x  =  A )
4 fveq2 5525 . . . . 5  |-  ( x  =  A  ->  ( T `  x )  =  ( T `  A ) )
53, 4oveq12d 5876 . . . 4  |-  ( x  =  A  ->  (
x  .ih  ( T `  x ) )  =  ( A  .ih  ( T `  A )
) )
65eleq1d 2349 . . 3  |-  ( x  =  A  ->  (
( x  .ih  ( T `  x )
)  e.  RR  <->  ( A  .ih  ( T `  A
) )  e.  RR ) )
76rspcv 2880 . 2  |-  ( A  e.  ~H  ->  ( A. x  e.  ~H  ( x  .ih  ( T `
 x ) )  e.  RR  ->  ( A  .ih  ( T `  A ) )  e.  RR ) )
81, 2, 7mp2 17 1  |-  ( A 
.ih  ( T `  A ) )  e.  RR
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. wcel 1684   A.wral 2543   ` cfv 5255  (class class class)co 5858   RRcr 8736   ~Hchil 21499    .ih csp 21502   LinOpclo 21527
This theorem is referenced by:  lnophmlem2  22597
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
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-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861
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