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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 6446 | . . . 4 ⊢ (𝑟 = 𝑅 → (norm‘𝑟) = (norm‘𝑅)) | |
2 | isnrg.1 | . . . 4 ⊢ 𝑁 = (norm‘𝑅) | |
3 | 1, 2 | syl6eqr 2832 | . . 3 ⊢ (𝑟 = 𝑅 → (norm‘𝑟) = 𝑁) |
4 | fveq2 6446 | . . . 4 ⊢ (𝑟 = 𝑅 → (AbsVal‘𝑟) = (AbsVal‘𝑅)) | |
5 | isnrg.2 | . . . 4 ⊢ 𝐴 = (AbsVal‘𝑅) | |
6 | 4, 5 | syl6eqr 2832 | . . 3 ⊢ (𝑟 = 𝑅 → (AbsVal‘𝑟) = 𝐴) |
7 | 3, 6 | eleq12d 2853 | . 2 ⊢ (𝑟 = 𝑅 → ((norm‘𝑟) ∈ (AbsVal‘𝑟) ↔ 𝑁 ∈ 𝐴)) |
8 | df-nrg 22798 | . 2 ⊢ NrmRing = {𝑟 ∈ NrmGrp ∣ (norm‘𝑟) ∈ (AbsVal‘𝑟)} | |
9 | 7, 8 | elrab2 3576 | 1 ⊢ (𝑅 ∈ NrmRing ↔ (𝑅 ∈ NrmGrp ∧ 𝑁 ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ‘cfv 6135 AbsValcabv 19208 normcnm 22789 NrmGrpcngp 22790 NrmRingcnrg 22792 |
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 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 |
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-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-rex 3096 df-rab 3099 df-v 3400 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-br 4887 df-iota 6099 df-fv 6143 df-nrg 22798 |
This theorem is referenced by: nrgabv 22873 nrgngp 22874 subrgnrg 22885 tngnrg 22886 cnnrg 22992 zhmnrg 30609 |
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