Theorem 2sqlem1 27581

Description: Lemma for 2sq 27594. (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 6881	. . . . 5 ⊢ (𝑤 = 𝑥 → (abs‘𝑤) = (abs‘𝑥))
4	3	oveq1d 7425	. . . 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 7443	. . 3 ⊢ ((abs‘𝑥)↑2) ∈ V
7	5, 6	elrnmpti 5952	. 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))

Syntax hints:   ↔ wb 209   = wceq 1570   ∈ wcel 2143  ∃wrex 3089   ↦ cmpt 5192  ran crn 5662  ‘cfv 6536  (class class class)co 7410  2c2 12290  ↑cexp 14093  abscabs 15281  ℤ[i]cgz 16984

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-cnv 5669  df-dm 5671  df-rn 5672  df-iota 6492  df-fv 6544  df-ov 7413

This theorem is referenced by:  2sqlem2  27582  mul2sq  27583  2sqlem3  27584  2sqlem9  27591  2sqlem10  27592