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Mirrors > Home > MPE Home > Th. List > 2sqlem1 | Structured version Visualization version GIF version |
Description: Lemma for 2sq 25607. (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 2850 | . 2 ⊢ (𝐴 ∈ 𝑆 ↔ 𝐴 ∈ ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2))) |
3 | fveq2 6446 | . . . . 5 ⊢ (𝑤 = 𝑥 → (abs‘𝑤) = (abs‘𝑥)) | |
4 | 3 | oveq1d 6937 | . . . 4 ⊢ (𝑤 = 𝑥 → ((abs‘𝑤)↑2) = ((abs‘𝑥)↑2)) |
5 | 4 | cbvmptv 4985 | . . 3 ⊢ (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2)) = (𝑥 ∈ ℤ[i] ↦ ((abs‘𝑥)↑2)) |
6 | ovex 6954 | . . 3 ⊢ ((abs‘𝑥)↑2) ∈ V | |
7 | 5, 6 | elrnmpti 5622 | . 2 ⊢ (𝐴 ∈ ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2)) ↔ ∃𝑥 ∈ ℤ[i] 𝐴 = ((abs‘𝑥)↑2)) |
8 | 2, 7 | bitri 267 | 1 ⊢ (𝐴 ∈ 𝑆 ↔ ∃𝑥 ∈ ℤ[i] 𝐴 = ((abs‘𝑥)↑2)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 = wceq 1601 ∈ wcel 2106 ∃wrex 3090 ↦ cmpt 4965 ran crn 5356 ‘cfv 6135 (class class class)co 6922 2c2 11430 ↑cexp 13178 abscabs 14381 ℤ[i]cgz 16037 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pr 5138 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3399 df-sbc 3652 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-br 4887 df-opab 4949 df-mpt 4966 df-cnv 5363 df-dm 5365 df-rn 5366 df-iota 6099 df-fv 6143 df-ov 6925 |
This theorem is referenced by: 2sqlem2 25595 mul2sq 25596 2sqlem3 25597 2sqlem9 25604 2sqlem10 25605 |
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