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Theorem 2sqlem1 20565
Description: Lemma for 2sq 20578. (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 2322 . 2  |-  ( A  e.  S  <->  A  e.  ran  (  w  e.  ZZ [ _i ]  |->  ( ( abs `  w
) ^ 2 ) ) )
3 fveq2 5458 . . . . 5  |-  ( w  =  x  ->  ( abs `  w )  =  ( abs `  x
) )
43oveq1d 5807 . . . 4  |-  ( w  =  x  ->  (
( abs `  w
) ^ 2 )  =  ( ( abs `  x ) ^ 2 ) )
54cbvmptv 4085 . . 3  |-  ( w  e.  ZZ [ _i ]  |->  ( ( abs `  w ) ^ 2 ) )  =  ( x  e.  ZZ [
_i ]  |->  ( ( abs `  x ) ^ 2 ) )
6 ovex 5817 . . 3  |-  ( ( abs `  x ) ^ 2 )  e. 
_V
75, 6elrnmpti 4918 . 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 1619    e. wcel 1621   E.wrex 2519    e. cmpt 4051   ran crn 4662   ` cfv 4673  (class class class)co 5792   2c2 9763   ^cexp 11071   abscabs 11685   ZZ [ _i ]cgz 12939
This theorem is referenced by:  2sqlem2  20566  mul2sq  20567  2sqlem3  20568  2sqlem9  20575  2sqlem10  20576
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-br 3998  df-opab 4052  df-mpt 4053  df-xp 4675  df-cnv 4677  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fv 4689  df-ov 5795
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