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Theorem 2sqlem1 22714
Description: Lemma for 2sq 22727. (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 2507 . 2  |-  ( A  e.  S  <->  A  e.  ran  ( w  e.  ZZ[_i]  |->  ( ( abs `  w
) ^ 2 ) ) )
3 fveq2 5703 . . . . 5  |-  ( w  =  x  ->  ( abs `  w )  =  ( abs `  x
) )
43oveq1d 6118 . . . 4  |-  ( w  =  x  ->  (
( abs `  w
) ^ 2 )  =  ( ( abs `  x ) ^ 2 ) )
54cbvmptv 4395 . . 3  |-  ( w  e.  ZZ[_i]  |->  ( ( abs `  w ) ^ 2 ) )  =  ( x  e.  ZZ[_i]  |->  ( ( abs `  x ) ^ 2 ) )
6 ovex 6128 . . 3  |-  ( ( abs `  x ) ^ 2 )  e. 
_V
75, 6elrnmpti 5102 . 2  |-  ( A  e.  ran  ( w  e.  ZZ[_i]  |->  ( ( abs `  w ) ^ 2 ) )  <->  E. x  e.  ZZ[_i]  A  =  ( ( abs `  x ) ^
2 ) )
82, 7bitri 249 1  |-  ( A  e.  S  <->  E. x  e.  ZZ[_i]  A  =  ( ( abs `  x ) ^
2 ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1369    e. wcel 1756   E.wrex 2728    e. cmpt 4362   ran crn 4853   ` cfv 5430  (class class class)co 6103   2c2 10383   ^cexp 11877   abscabs 12735   ZZ[_i]cgz 14002
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pr 4543
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-rab 2736  df-v 2986  df-sbc 3199  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-br 4305  df-opab 4363  df-mpt 4364  df-cnv 4860  df-dm 4862  df-rn 4863  df-iota 5393  df-fv 5438  df-ov 6106
This theorem is referenced by:  2sqlem2  22715  mul2sq  22716  2sqlem3  22717  2sqlem9  22724  2sqlem10  22725
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