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Theorem 2sqlem1 20602
Description: Lemma for 2sq 20615. (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 2347 . 2  |-  ( A  e.  S  <->  A  e.  ran  ( w  e.  ZZ [ _i ]  |->  ( ( abs `  w ) ^ 2 ) ) )
3 fveq2 5525 . . . . 5  |-  ( w  =  x  ->  ( abs `  w )  =  ( abs `  x
) )
43oveq1d 5873 . . . 4  |-  ( w  =  x  ->  (
( abs `  w
) ^ 2 )  =  ( ( abs `  x ) ^ 2 ) )
54cbvmptv 4111 . . 3  |-  ( w  e.  ZZ [ _i ]  |->  ( ( abs `  w ) ^ 2 ) )  =  ( x  e.  ZZ [
_i ]  |->  ( ( abs `  x ) ^ 2 ) )
6 ovex 5883 . . 3  |-  ( ( abs `  x ) ^ 2 )  e. 
_V
75, 6elrnmpti 4930 . 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 1623    e. wcel 1684   E.wrex 2544    e. cmpt 4077   ran crn 4690   ` cfv 5255  (class class class)co 5858   2c2 9795   ^cexp 11104   abscabs 11719   ZZ [ _i ]cgz 12976
This theorem is referenced by:  2sqlem2  20603  mul2sq  20604  2sqlem3  20605  2sqlem9  20612  2sqlem10  20613
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-cnv 4697  df-dm 4699  df-rn 4700  df-iota 5219  df-fv 5263  df-ov 5861
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