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Theorem 2sqlem1 25362
Description: Lemma for 2sq 25375. (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 2841 . 2 (𝐴𝑆𝐴 ∈ ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2)))
3 fveq2 6332 . . . . 5 (𝑤 = 𝑥 → (abs‘𝑤) = (abs‘𝑥))
43oveq1d 6807 . . . 4 (𝑤 = 𝑥 → ((abs‘𝑤)↑2) = ((abs‘𝑥)↑2))
54cbvmptv 4882 . . 3 (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2)) = (𝑥 ∈ ℤ[i] ↦ ((abs‘𝑥)↑2))
6 ovex 6822 . . 3 ((abs‘𝑥)↑2) ∈ V
75, 6elrnmpti 5514 . 2 (𝐴 ∈ ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2)) ↔ ∃𝑥 ∈ ℤ[i] 𝐴 = ((abs‘𝑥)↑2))
82, 7bitri 264 1 (𝐴𝑆 ↔ ∃𝑥 ∈ ℤ[i] 𝐴 = ((abs‘𝑥)↑2))
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1630  wcel 2144  wrex 3061  cmpt 4861  ran crn 5250  cfv 6031  (class class class)co 6792  2c2 11271  cexp 13066  abscabs 14181  ℤ[i]cgz 15839
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-cnv 5257  df-dm 5259  df-rn 5260  df-iota 5994  df-fv 6039  df-ov 6795
This theorem is referenced by:  2sqlem2  25363  mul2sq  25364  2sqlem3  25365  2sqlem9  25372  2sqlem10  25373
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