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

Theorem 2sqlem1 25999
Description: Lemma for 2sq 26012. (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 2905 . 2 (𝐴𝑆𝐴 ∈ ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2)))
3 fveq2 6652 . . . . 5 (𝑤 = 𝑥 → (abs‘𝑤) = (abs‘𝑥))
43oveq1d 7155 . . . 4 (𝑤 = 𝑥 → ((abs‘𝑤)↑2) = ((abs‘𝑥)↑2))
54cbvmptv 5145 . . 3 (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2)) = (𝑥 ∈ ℤ[i] ↦ ((abs‘𝑥)↑2))
6 ovex 7173 . . 3 ((abs‘𝑥)↑2) ∈ V
75, 6elrnmpti 5809 . 2 (𝐴 ∈ ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2)) ↔ ∃𝑥 ∈ ℤ[i] 𝐴 = ((abs‘𝑥)↑2))
82, 7bitri 278 1 (𝐴𝑆 ↔ ∃𝑥 ∈ ℤ[i] 𝐴 = ((abs‘𝑥)↑2))
Colors of variables: wff setvar class
Syntax hints:  wb 209   = wceq 1538  wcel 2114  wrex 3131  cmpt 5122  ran crn 5533  cfv 6334  (class class class)co 7140  2c2 11680  cexp 13425  abscabs 14584  ℤ[i]cgz 16254
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-mpt 5123  df-cnv 5540  df-dm 5542  df-rn 5543  df-iota 6293  df-fv 6342  df-ov 7143
This theorem is referenced by:  2sqlem2  26000  mul2sq  26001  2sqlem3  26002  2sqlem9  26009  2sqlem10  26010
  Copyright terms: Public domain W3C validator