HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  strlem2 Structured version   Unicode version

Theorem strlem2 23746
Description: Lemma for strong state theorem. (Contributed by NM, 28-Oct-1999.) (New usage is discouraged.)
Hypothesis
Ref Expression
strlem2.1  |-  S  =  ( x  e.  CH  |->  ( ( normh `  (
( proj  h `  x
) `  u )
) ^ 2 ) )
Assertion
Ref Expression
strlem2  |-  ( C  e.  CH  ->  ( S `  C )  =  ( ( normh `  ( ( proj  h `  C ) `  u
) ) ^ 2 ) )
Distinct variable groups:    x, C    x, u
Allowed substitution hints:    C( u)    S( x, u)

Proof of Theorem strlem2
StepHypRef Expression
1 fveq2 5720 . . . . 5  |-  ( x  =  C  ->  ( proj  h `  x )  =  ( proj  h `  C ) )
21fveq1d 5722 . . . 4  |-  ( x  =  C  ->  (
( proj  h `  x
) `  u )  =  ( ( proj 
h `  C ) `  u ) )
32fveq2d 5724 . . 3  |-  ( x  =  C  ->  ( normh `  ( ( proj 
h `  x ) `  u ) )  =  ( normh `  ( ( proj  h `  C ) `
 u ) ) )
43oveq1d 6088 . 2  |-  ( x  =  C  ->  (
( normh `  ( ( proj  h `  x ) `
 u ) ) ^ 2 )  =  ( ( normh `  (
( proj  h `  C
) `  u )
) ^ 2 ) )
5 strlem2.1 . 2  |-  S  =  ( x  e.  CH  |->  ( ( normh `  (
( proj  h `  x
) `  u )
) ^ 2 ) )
6 ovex 6098 . 2  |-  ( (
normh `  ( ( proj 
h `  C ) `  u ) ) ^
2 )  e.  _V
74, 5, 6fvmpt 5798 1  |-  ( C  e.  CH  ->  ( S `  C )  =  ( ( normh `  ( ( proj  h `  C ) `  u
) ) ^ 2 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725    e. cmpt 4258   ` cfv 5446  (class class class)co 6073   2c2 10041   ^cexp 11374   normhcno 22418   CHcch 22424   proj 
hcpjh 22432
This theorem is referenced by:  strlem3a  23747  strlem4  23749  strlem5  23750  jplem2  23764
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-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
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-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076
  Copyright terms: Public domain W3C validator