Theorem 2sqlem1 27801

Description: Lemma for 2sq 27814. (Contributed by Mario Carneiro, 19-Jun-2015.)

Hypothesis

Ref	Expression
2sq.1	⊢ 𝑆 = ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2))

Assertion

Ref	Expression
2sqlem1	⊢ (𝐴 ∈ 𝑆 ↔ ∃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

Step	Hyp	Ref	Expression
1		2sq.1	. . 3 ⊢ 𝑆 = ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2))
2	1	eleq2i 2856	. 2 ⊢ (𝐴 ∈ 𝑆 ↔ 𝐴 ∈ ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2)))
3		fveq2 6916	. . . . 5 ⊢ (𝑤 = 𝑥 → (abs‘𝑤) = (abs‘𝑥))
4	3	oveq1d 7466	. . . 4 ⊢ (𝑤 = 𝑥 → ((abs‘𝑤)↑2) = ((abs‘𝑥)↑2))
5	4	cbvmptv 5215	. . 3 ⊢ (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2)) = (𝑥 ∈ ℤ[i] ↦ ((abs‘𝑥)↑2))
6		ovex 7484	. . 3 ⊢ ((abs‘𝑥)↑2) ∈ V
7	5, 6	elrnmpti 5956	. 2 ⊢ (𝐴 ∈ ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2)) ↔ ∃𝑥 ∈ ℤ[i] 𝐴 = ((abs‘𝑥)↑2))
8	2, 7	bitri 278	1 ⊢ (𝐴 ∈ 𝑆 ↔ ∃𝑥 ∈ ℤ[i] 𝐴 = ((abs‘𝑥)↑2))

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570   ∈ wcel 2147  ∃wrex 3090   ↦ cmpt 5192  ran crn 5659  ‘cfv 6566  (class class class)co 7451  2c2 12437  ↑cexp 14244  abscabs 15444  ℤ[i]cgz 17151

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-cnv 5666  df-dm 5668  df-rn 5669  df-iota 6520  df-fv 6574  df-ov 7454

This theorem is used by:  2sqlem2  27802  mul2sq  27803  2sqlem3  27804  2sqlem9  27811  2sqlem10  27812