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Mirrors > Home > MPE Home > Th. List > 2sqlem1 | Structured version Visualization version GIF version |
Description: Lemma for 2sq 26781. (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 2830 | . 2 ⊢ (𝐴 ∈ 𝑆 ↔ 𝐴 ∈ ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2))) |
3 | fveq2 6843 | . . . . 5 ⊢ (𝑤 = 𝑥 → (abs‘𝑤) = (abs‘𝑥)) | |
4 | 3 | oveq1d 7373 | . . . 4 ⊢ (𝑤 = 𝑥 → ((abs‘𝑤)↑2) = ((abs‘𝑥)↑2)) |
5 | 4 | cbvmptv 5219 | . . 3 ⊢ (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2)) = (𝑥 ∈ ℤ[i] ↦ ((abs‘𝑥)↑2)) |
6 | ovex 7391 | . . 3 ⊢ ((abs‘𝑥)↑2) ∈ V | |
7 | 5, 6 | elrnmpti 5916 | . 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 1542 ∈ wcel 2107 ∃wrex 3074 ↦ cmpt 5189 ran crn 5635 ‘cfv 6497 (class class class)co 7358 2c2 12209 ↑cexp 13968 abscabs 15120 ℤ[i]cgz 16802 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-cnv 5642 df-dm 5644 df-rn 5645 df-iota 6449 df-fv 6505 df-ov 7361 |
This theorem is referenced by: 2sqlem2 26769 mul2sq 26770 2sqlem3 26771 2sqlem9 26778 2sqlem10 26779 |
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