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Theorem 2sqlem1 24151
Description: Lemma for 2sq 24164. (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 2498 . 2  |-  ( A  e.  S  <->  A  e.  ran  ( w  e.  ZZ[_i]  |->  ( ( abs `  w
) ^ 2 ) ) )
3 fveq2 5872 . . . . 5  |-  ( w  =  x  ->  ( abs `  w )  =  ( abs `  x
) )
43oveq1d 6311 . . . 4  |-  ( w  =  x  ->  (
( abs `  w
) ^ 2 )  =  ( ( abs `  x ) ^ 2 ) )
54cbvmptv 4509 . . 3  |-  ( w  e.  ZZ[_i]  |->  ( ( abs `  w ) ^ 2 ) )  =  ( x  e.  ZZ[_i]  |->  ( ( abs `  x ) ^ 2 ) )
6 ovex 6324 . . 3  |-  ( ( abs `  x ) ^ 2 )  e. 
_V
75, 6elrnmpti 5096 . 2  |-  ( A  e.  ran  ( w  e.  ZZ[_i]  |->  ( ( abs `  w ) ^ 2 ) )  <->  E. x  e.  ZZ[_i]  A  =  ( ( abs `  x ) ^
2 ) )
82, 7bitri 252 1  |-  ( A  e.  S  <->  E. x  e.  ZZ[_i]  A  =  ( ( abs `  x ) ^
2 ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    = wceq 1437    e. wcel 1867   E.wrex 2774    |-> cmpt 4475   ran crn 4846   ` cfv 5592  (class class class)co 6296   2c2 10648   ^cexp 12258   abscabs 13265   ZZ[_i]cgz 14825
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pr 4652
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-opab 4476  df-mpt 4477  df-cnv 4853  df-dm 4855  df-rn 4856  df-iota 5556  df-fv 5600  df-ov 6299
This theorem is referenced by:  2sqlem2  24152  mul2sq  24153  2sqlem3  24154  2sqlem9  24161  2sqlem10  24162
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