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

Theorem 2sqlem1 23463
Description: Lemma for 2sq 23476. (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 2545 . 2  |-  ( A  e.  S  <->  A  e.  ran  ( w  e.  ZZ[_i]  |->  ( ( abs `  w
) ^ 2 ) ) )
3 fveq2 5866 . . . . 5  |-  ( w  =  x  ->  ( abs `  w )  =  ( abs `  x
) )
43oveq1d 6300 . . . 4  |-  ( w  =  x  ->  (
( abs `  w
) ^ 2 )  =  ( ( abs `  x ) ^ 2 ) )
54cbvmptv 4538 . . 3  |-  ( w  e.  ZZ[_i]  |->  ( ( abs `  w ) ^ 2 ) )  =  ( x  e.  ZZ[_i]  |->  ( ( abs `  x ) ^ 2 ) )
6 ovex 6310 . . 3  |-  ( ( abs `  x ) ^ 2 )  e. 
_V
75, 6elrnmpti 5253 . 2  |-  ( A  e.  ran  ( w  e.  ZZ[_i]  |->  ( ( abs `  w ) ^ 2 ) )  <->  E. x  e.  ZZ[_i]  A  =  ( ( abs `  x ) ^
2 ) )
82, 7bitri 249 1  |-  ( A  e.  S  <->  E. x  e.  ZZ[_i]  A  =  ( ( abs `  x ) ^
2 ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1379    e. wcel 1767   E.wrex 2815    |-> cmpt 4505   ran crn 5000   ` cfv 5588  (class class class)co 6285   2c2 10586   ^cexp 12135   abscabs 13033   ZZ[_i]cgz 14309
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-cnv 5007  df-dm 5009  df-rn 5010  df-iota 5551  df-fv 5596  df-ov 6288
This theorem is referenced by:  2sqlem2  23464  mul2sq  23465  2sqlem3  23466  2sqlem9  23473  2sqlem10  23474
  Copyright terms: Public domain W3C validator