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Theorem strlem2 23595
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 5661 . . . . 5  |-  ( x  =  C  ->  ( proj  h `  x )  =  ( proj  h `  C ) )
21fveq1d 5663 . . . 4  |-  ( x  =  C  ->  (
( proj  h `  x
) `  u )  =  ( ( proj 
h `  C ) `  u ) )
32fveq2d 5665 . . 3  |-  ( x  =  C  ->  ( normh `  ( ( proj 
h `  x ) `  u ) )  =  ( normh `  ( ( proj  h `  C ) `
 u ) ) )
43oveq1d 6028 . 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 6038 . 2  |-  ( (
normh `  ( ( proj 
h `  C ) `  u ) ) ^
2 )  e.  _V
74, 5, 6fvmpt 5738 1  |-  ( C  e.  CH  ->  ( S `  C )  =  ( ( normh `  ( ( proj  h `  C ) `  u
) ) ^ 2 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717    e. cmpt 4200   ` cfv 5387  (class class class)co 6013   2c2 9974   ^cexp 11302   normhcno 22267   CHcch 22273   proj 
hcpjh 22281
This theorem is referenced by:  strlem3a  23596  strlem4  23598  strlem5  23599  jplem2  23613
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-iota 5351  df-fun 5389  df-fv 5395  df-ov 6016
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