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Mirrors > Home > MPE Home > Th. List > 2sqlem1 | Structured version Visualization version GIF version |
Description: Lemma for 2sq 26006. (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 2904 | . 2 ⊢ (𝐴 ∈ 𝑆 ↔ 𝐴 ∈ ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2))) |
3 | fveq2 6670 | . . . . 5 ⊢ (𝑤 = 𝑥 → (abs‘𝑤) = (abs‘𝑥)) | |
4 | 3 | oveq1d 7171 | . . . 4 ⊢ (𝑤 = 𝑥 → ((abs‘𝑤)↑2) = ((abs‘𝑥)↑2)) |
5 | 4 | cbvmptv 5169 | . . 3 ⊢ (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2)) = (𝑥 ∈ ℤ[i] ↦ ((abs‘𝑥)↑2)) |
6 | ovex 7189 | . . 3 ⊢ ((abs‘𝑥)↑2) ∈ V | |
7 | 5, 6 | elrnmpti 5832 | . 2 ⊢ (𝐴 ∈ ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2)) ↔ ∃𝑥 ∈ ℤ[i] 𝐴 = ((abs‘𝑥)↑2)) |
8 | 2, 7 | bitri 277 | 1 ⊢ (𝐴 ∈ 𝑆 ↔ ∃𝑥 ∈ ℤ[i] 𝐴 = ((abs‘𝑥)↑2)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1537 ∈ wcel 2114 ∃wrex 3139 ↦ cmpt 5146 ran crn 5556 ‘cfv 6355 (class class class)co 7156 2c2 11693 ↑cexp 13430 abscabs 14593 ℤ[i]cgz 16265 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-cnv 5563 df-dm 5565 df-rn 5566 df-iota 6314 df-fv 6363 df-ov 7159 |
This theorem is referenced by: 2sqlem2 25994 mul2sq 25995 2sqlem3 25996 2sqlem9 26003 2sqlem10 26004 |
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