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Mirrors > Home > MPE Home > Th. List > 2sqlem1 | Structured version Visualization version GIF version |
Description: Lemma for 2sq 27376. (Contributed by Mario Carneiro, 19-Jun-2015.) |
Ref | Expression |
---|---|
2sq.1 | ⊢ 𝑆 = ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2)) |
Ref | Expression |
---|---|
2sqlem1 | ⊢ (𝐴 ∈ 𝑆 ↔ ∃𝑥 ∈ ℤ[i] 𝐴 = ((abs‘𝑥)↑2)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2sq.1 | . . 3 ⊢ 𝑆 = ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2)) | |
2 | 1 | eleq2i 2821 | . 2 ⊢ (𝐴 ∈ 𝑆 ↔ 𝐴 ∈ ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2))) |
3 | fveq2 6897 | . . . . 5 ⊢ (𝑤 = 𝑥 → (abs‘𝑤) = (abs‘𝑥)) | |
4 | 3 | oveq1d 7435 | . . . 4 ⊢ (𝑤 = 𝑥 → ((abs‘𝑤)↑2) = ((abs‘𝑥)↑2)) |
5 | 4 | cbvmptv 5261 | . . 3 ⊢ (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2)) = (𝑥 ∈ ℤ[i] ↦ ((abs‘𝑥)↑2)) |
6 | ovex 7453 | . . 3 ⊢ ((abs‘𝑥)↑2) ∈ V | |
7 | 5, 6 | elrnmpti 5962 | . 2 ⊢ (𝐴 ∈ ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2)) ↔ ∃𝑥 ∈ ℤ[i] 𝐴 = ((abs‘𝑥)↑2)) |
8 | 2, 7 | bitri 275 | 1 ⊢ (𝐴 ∈ 𝑆 ↔ ∃𝑥 ∈ ℤ[i] 𝐴 = ((abs‘𝑥)↑2)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1534 ∈ wcel 2099 ∃wrex 3067 ↦ cmpt 5231 ran crn 5679 ‘cfv 6548 (class class class)co 7420 2c2 12298 ↑cexp 14059 abscabs 15214 ℤ[i]cgz 16898 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-cnv 5686 df-dm 5688 df-rn 5689 df-iota 6500 df-fv 6556 df-ov 7423 |
This theorem is referenced by: 2sqlem2 27364 mul2sq 27365 2sqlem3 27366 2sqlem9 27373 2sqlem10 27374 |
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