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Theorem 2sqlem1 21178
Description: Lemma for 2sq 21191. (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 2506 . 2  |-  ( A  e.  S  <->  A  e.  ran  ( w  e.  ZZ [ _i ]  |->  ( ( abs `  w ) ^ 2 ) ) )
3 fveq2 5757 . . . . 5  |-  ( w  =  x  ->  ( abs `  w )  =  ( abs `  x
) )
43oveq1d 6125 . . . 4  |-  ( w  =  x  ->  (
( abs `  w
) ^ 2 )  =  ( ( abs `  x ) ^ 2 ) )
54cbvmptv 4325 . . 3  |-  ( w  e.  ZZ [ _i ]  |->  ( ( abs `  w ) ^ 2 ) )  =  ( x  e.  ZZ [
_i ]  |->  ( ( abs `  x ) ^ 2 ) )
6 ovex 6135 . . 3  |-  ( ( abs `  x ) ^ 2 )  e. 
_V
75, 6elrnmpti 5150 . 2  |-  ( A  e.  ran  ( w  e.  ZZ [ _i ]  |->  ( ( abs `  w ) ^ 2 ) )  <->  E. x  e.  ZZ [ _i ]  A  =  ( ( abs `  x ) ^
2 ) )
82, 7bitri 242 1  |-  ( A  e.  S  <->  E. x  e.  ZZ [ _i ]  A  =  ( ( abs `  x ) ^
2 ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    = wceq 1653    e. wcel 1727   E.wrex 2712    e. cmpt 4291   ran crn 4908   ` cfv 5483  (class class class)co 6110   2c2 10080   ^cexp 11413   abscabs 12070   ZZ [ _i ]cgz 13328
This theorem is referenced by:  2sqlem2  21179  mul2sq  21180  2sqlem3  21181  2sqlem9  21188  2sqlem10  21189
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pr 4432
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-sbc 3168  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-br 4238  df-opab 4292  df-mpt 4293  df-cnv 4915  df-dm 4917  df-rn 4918  df-iota 5447  df-fv 5491  df-ov 6113
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