MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  2sqlem1 Unicode version

Theorem 2sqlem1 21135
Description: Lemma for 2sq 21148. (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 2499 . 2  |-  ( A  e.  S  <->  A  e.  ran  ( w  e.  ZZ [ _i ]  |->  ( ( abs `  w ) ^ 2 ) ) )
3 fveq2 5719 . . . . 5  |-  ( w  =  x  ->  ( abs `  w )  =  ( abs `  x
) )
43oveq1d 6087 . . . 4  |-  ( w  =  x  ->  (
( abs `  w
) ^ 2 )  =  ( ( abs `  x ) ^ 2 ) )
54cbvmptv 4292 . . 3  |-  ( w  e.  ZZ [ _i ]  |->  ( ( abs `  w ) ^ 2 ) )  =  ( x  e.  ZZ [
_i ]  |->  ( ( abs `  x ) ^ 2 ) )
6 ovex 6097 . . 3  |-  ( ( abs `  x ) ^ 2 )  e. 
_V
75, 6elrnmpti 5112 . 2  |-  ( A  e.  ran  ( w  e.  ZZ [ _i ]  |->  ( ( abs `  w ) ^ 2 ) )  <->  E. x  e.  ZZ [ _i ]  A  =  ( ( abs `  x ) ^
2 ) )
82, 7bitri 241 1  |-  ( A  e.  S  <->  E. x  e.  ZZ [ _i ]  A  =  ( ( abs `  x ) ^
2 ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    = wceq 1652    e. wcel 1725   E.wrex 2698    e. cmpt 4258   ran crn 4870   ` cfv 5445  (class class class)co 6072   2c2 10038   ^cexp 11370   abscabs 12027   ZZ [ _i ]cgz 13285
This theorem is referenced by:  2sqlem2  21136  mul2sq  21137  2sqlem3  21138  2sqlem9  21145  2sqlem10  21146
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-cnv 4877  df-dm 4879  df-rn 4880  df-iota 5409  df-fv 5453  df-ov 6075
  Copyright terms: Public domain W3C validator