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Theorem lnophmlem1 22612
Description: Lemma for lnophmi 22614. (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 5541 . . . . 5  |-  ( x  =  A  ->  ( T `  x )  =  ( T `  A ) )
53, 4oveq12d 5892 . . . 4  |-  ( x  =  A  ->  (
x  .ih  ( T `  x ) )  =  ( A  .ih  ( T `  A )
) )
65eleq1d 2362 . . 3  |-  ( x  =  A  ->  (
( x  .ih  ( T `  x )
)  e.  RR  <->  ( A  .ih  ( T `  A
) )  e.  RR ) )
76rspcv 2893 . 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 1632    e. wcel 1696   A.wral 2556   ` cfv 5271  (class class class)co 5874   RRcr 8752   ~Hchil 21515    .ih csp 21518   LinOpclo 21543
This theorem is referenced by:  lnophmlem2  22613
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-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
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-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-ov 5877
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