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Mirrors > Home > MPE Home > Th. List > isnrg | Structured version Visualization version GIF version |
Description: A normed ring is a ring with a norm that makes it into a normed group, and such that the norm is an absolute value on the ring. (Contributed by Mario Carneiro, 4-Oct-2015.) |
Ref | Expression |
---|---|
isnrg.1 | ⊢ 𝑁 = (norm‘𝑅) |
isnrg.2 | ⊢ 𝐴 = (AbsVal‘𝑅) |
Ref | Expression |
---|---|
isnrg | ⊢ (𝑅 ∈ NrmRing ↔ (𝑅 ∈ NrmGrp ∧ 𝑁 ∈ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6103 | . . . 4 ⊢ (𝑟 = 𝑅 → (norm‘𝑟) = (norm‘𝑅)) | |
2 | isnrg.1 | . . . 4 ⊢ 𝑁 = (norm‘𝑅) | |
3 | 1, 2 | syl6eqr 2662 | . . 3 ⊢ (𝑟 = 𝑅 → (norm‘𝑟) = 𝑁) |
4 | fveq2 6103 | . . . 4 ⊢ (𝑟 = 𝑅 → (AbsVal‘𝑟) = (AbsVal‘𝑅)) | |
5 | isnrg.2 | . . . 4 ⊢ 𝐴 = (AbsVal‘𝑅) | |
6 | 4, 5 | syl6eqr 2662 | . . 3 ⊢ (𝑟 = 𝑅 → (AbsVal‘𝑟) = 𝐴) |
7 | 3, 6 | eleq12d 2682 | . 2 ⊢ (𝑟 = 𝑅 → ((norm‘𝑟) ∈ (AbsVal‘𝑟) ↔ 𝑁 ∈ 𝐴)) |
8 | df-nrg 22200 | . 2 ⊢ NrmRing = {𝑟 ∈ NrmGrp ∣ (norm‘𝑟) ∈ (AbsVal‘𝑟)} | |
9 | 7, 8 | elrab2 3333 | 1 ⊢ (𝑅 ∈ NrmRing ↔ (𝑅 ∈ NrmGrp ∧ 𝑁 ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ‘cfv 5804 AbsValcabv 18639 normcnm 22191 NrmGrpcngp 22192 NrmRingcnrg 22194 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 df-nrg 22200 |
This theorem is referenced by: nrgabv 22275 nrgngp 22276 subrgnrg 22287 tngnrg 22288 cnnrg 22394 zhmnrg 29339 |
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