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Theorem lnophmlem1 23519
Description: Lemma for lnophmi 23521. (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 20 . . . . 5  |-  ( x  =  A  ->  x  =  A )
4 fveq2 5728 . . . . 5  |-  ( x  =  A  ->  ( T `  x )  =  ( T `  A ) )
53, 4oveq12d 6099 . . . 4  |-  ( x  =  A  ->  (
x  .ih  ( T `  x ) )  =  ( A  .ih  ( T `  A )
) )
65eleq1d 2502 . . 3  |-  ( x  =  A  ->  (
( x  .ih  ( T `  x )
)  e.  RR  <->  ( A  .ih  ( T `  A
) )  e.  RR ) )
76rspcv 3048 . 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 set class
Syntax hints:    = wceq 1652    e. wcel 1725   A.wral 2705   ` cfv 5454  (class class class)co 6081   RRcr 8989   ~Hchil 22422    .ih csp 22425   LinOpclo 22450
This theorem is referenced by:  lnophmlem2  23520
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-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
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-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-iota 5418  df-fv 5462  df-ov 6084
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