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Theorem strlem2 22831
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 5525 . . . . 5  |-  ( x  =  C  ->  ( proj  h `  x )  =  ( proj  h `  C ) )
21fveq1d 5527 . . . 4  |-  ( x  =  C  ->  (
( proj  h `  x
) `  u )  =  ( ( proj 
h `  C ) `  u ) )
32fveq2d 5529 . . 3  |-  ( x  =  C  ->  ( normh `  ( ( proj 
h `  x ) `  u ) )  =  ( normh `  ( ( proj  h `  C ) `
 u ) ) )
43oveq1d 5873 . 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 5883 . 2  |-  ( (
normh `  ( ( proj 
h `  C ) `  u ) ) ^
2 )  e.  _V
74, 5, 6fvmpt 5602 1  |-  ( C  e.  CH  ->  ( S `  C )  =  ( ( normh `  ( ( proj  h `  C ) `  u
) ) ^ 2 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   2c2 9795   ^cexp 11104   normhcno 21503   CHcch 21509   proj 
hcpjh 21517
This theorem is referenced by:  strlem3a  22832  strlem4  22834  strlem5  22835  jplem2  22849
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
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-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  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-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861
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