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Theorem strlem2 22847
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 5541 . . . . 5  |-  ( x  =  C  ->  ( proj  h `  x )  =  ( proj  h `  C ) )
21fveq1d 5543 . . . 4  |-  ( x  =  C  ->  (
( proj  h `  x
) `  u )  =  ( ( proj 
h `  C ) `  u ) )
32fveq2d 5545 . . 3  |-  ( x  =  C  ->  ( normh `  ( ( proj 
h `  x ) `  u ) )  =  ( normh `  ( ( proj  h `  C ) `
 u ) ) )
43oveq1d 5889 . 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 5899 . 2  |-  ( (
normh `  ( ( proj 
h `  C ) `  u ) ) ^
2 )  e.  _V
74, 5, 6fvmpt 5618 1  |-  ( C  e.  CH  ->  ( S `  C )  =  ( ( normh `  ( ( proj  h `  C ) `  u
) ) ^ 2 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   2c2 9811   ^cexp 11120   normhcno 21519   CHcch 21525   proj 
hcpjh 21533
This theorem is referenced by:  strlem3a  22848  strlem4  22850  strlem5  22851  jplem2  22865
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-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
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-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  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-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877
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