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

Theorem 2sqlem1 26768
Description: Lemma for 2sq 26781. (Contributed by Mario Carneiro, 19-Jun-2015.)
Hypothesis
Ref Expression
2sq.1 𝑆 = ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2))
Assertion
Ref Expression
2sqlem1 (𝐴𝑆 ↔ ∃𝑥 ∈ ℤ[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 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2))
21eleq2i 2830 . 2 (𝐴𝑆𝐴 ∈ ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2)))
3 fveq2 6843 . . . . 5 (𝑤 = 𝑥 → (abs‘𝑤) = (abs‘𝑥))
43oveq1d 7373 . . . 4 (𝑤 = 𝑥 → ((abs‘𝑤)↑2) = ((abs‘𝑥)↑2))
54cbvmptv 5219 . . 3 (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2)) = (𝑥 ∈ ℤ[i] ↦ ((abs‘𝑥)↑2))
6 ovex 7391 . . 3 ((abs‘𝑥)↑2) ∈ V
75, 6elrnmpti 5916 . 2 (𝐴 ∈ ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2)) ↔ ∃𝑥 ∈ ℤ[i] 𝐴 = ((abs‘𝑥)↑2))
82, 7bitri 275 1 (𝐴𝑆 ↔ ∃𝑥 ∈ ℤ[i] 𝐴 = ((abs‘𝑥)↑2))
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1542  wcel 2107  wrex 3074  cmpt 5189  ran crn 5635  cfv 6497  (class class class)co 7358  2c2 12209  cexp 13968  abscabs 15120  ℤ[i]cgz 16802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-cnv 5642  df-dm 5644  df-rn 5645  df-iota 6449  df-fv 6505  df-ov 7361
This theorem is referenced by:  2sqlem2  26769  mul2sq  26770  2sqlem3  26771  2sqlem9  26778  2sqlem10  26779
  Copyright terms: Public domain W3C validator