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Theorem 2sqlem1 20604
Description: Lemma for 2sq 20617. (Contributed by Mario Carneiro, 19-Jun-2015.)
Hypothesis
Ref Expression
2sq.1  |-  S  =  ran  ( w  e.  ZZ [ _i ]  |->  ( ( abs `  w
) ^ 2 ) )
Assertion
Ref Expression
2sqlem1  |-  ( A  e.  S  <->  E. x  e.  ZZ [ _i ]  A  =  ( ( abs `  x ) ^
2 ) )
Distinct variable groups:    x, w    x, A    x, S
Allowed substitution hints:    A( w)    S( w)

Proof of Theorem 2sqlem1
StepHypRef Expression
1 2sq.1 . . 3  |-  S  =  ran  ( w  e.  ZZ [ _i ]  |->  ( ( abs `  w
) ^ 2 ) )
21eleq2i 2349 . 2  |-  ( A  e.  S  <->  A  e.  ran  ( w  e.  ZZ [ _i ]  |->  ( ( abs `  w ) ^ 2 ) ) )
3 fveq2 5527 . . . . 5  |-  ( w  =  x  ->  ( abs `  w )  =  ( abs `  x
) )
43oveq1d 5875 . . . 4  |-  ( w  =  x  ->  (
( abs `  w
) ^ 2 )  =  ( ( abs `  x ) ^ 2 ) )
54cbvmptv 4113 . . 3  |-  ( w  e.  ZZ [ _i ]  |->  ( ( abs `  w ) ^ 2 ) )  =  ( x  e.  ZZ [
_i ]  |->  ( ( abs `  x ) ^ 2 ) )
6 ovex 5885 . . 3  |-  ( ( abs `  x ) ^ 2 )  e. 
_V
75, 6elrnmpti 4932 . 2  |-  ( A  e.  ran  ( w  e.  ZZ [ _i ]  |->  ( ( abs `  w ) ^ 2 ) )  <->  E. x  e.  ZZ [ _i ]  A  =  ( ( abs `  x ) ^
2 ) )
82, 7bitri 240 1  |-  ( A  e.  S  <->  E. x  e.  ZZ [ _i ]  A  =  ( ( abs `  x ) ^
2 ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1625    e. wcel 1686   E.wrex 2546    e. cmpt 4079   ran crn 4692   ` cfv 5257  (class class class)co 5860   2c2 9797   ^cexp 11106   abscabs 11721   ZZ [ _i ]cgz 12978
This theorem is referenced by:  2sqlem2  20605  mul2sq  20606  2sqlem3  20607  2sqlem9  20614  2sqlem10  20615
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4216
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-mpt 4081  df-cnv 4699  df-dm 4701  df-rn 4702  df-iota 5221  df-fv 5265  df-ov 5863
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