Theorem 2sqlem1 27756

Description: Lemma for 2sq 27769. (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 2855	. 2 ⊢ (𝐴 ∈ 𝑆 ↔ 𝐴 ∈ ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2)))
3		fveq2 6893	. . . . 5 ⊢ (𝑤 = 𝑥 → (abs‘𝑤) = (abs‘𝑥))
4	3	oveq1d 7442	. . . 4 ⊢ (𝑤 = 𝑥 → ((abs‘𝑤)↑2) = ((abs‘𝑥)↑2))
5	4	cbvmptv 5211	. . 3 ⊢ (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2)) = (𝑥 ∈ ℤ[i] ↦ ((abs‘𝑥)↑2))
6		ovex 7460	. . 3 ⊢ ((abs‘𝑥)↑2) ∈ V
7	5, 6	elrnmpti 5948	. 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 1569   ∈ wcel 2146  ∃wrex 3089   ↦ cmpt 5188  ran crn 5655  ‘cfv 6546  (class class class)co 7427  2c2 12407  ↑cexp 14214  abscabs 15413  ℤ[i]cgz 17119

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5251  ax-nul 5262  ax-pr 5394

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3416  df-v 3455  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-br 5106  df-opab 5170  df-mpt 5189  df-cnv 5662  df-dm 5664  df-rn 5665  df-iota 6500  df-fv 6554  df-ov 7430

This theorem is used by:  2sqlem2  27757  mul2sq  27758  2sqlem3  27759  2sqlem9  27766  2sqlem10  27767