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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 6645 | . . . 4 ⊢ (𝑟 = 𝑅 → (norm‘𝑟) = (norm‘𝑅)) | |
2 | isnrg.1 | . . . 4 ⊢ 𝑁 = (norm‘𝑅) | |
3 | 1, 2 | eqtr4di 2851 | . . 3 ⊢ (𝑟 = 𝑅 → (norm‘𝑟) = 𝑁) |
4 | fveq2 6645 | . . . 4 ⊢ (𝑟 = 𝑅 → (AbsVal‘𝑟) = (AbsVal‘𝑅)) | |
5 | isnrg.2 | . . . 4 ⊢ 𝐴 = (AbsVal‘𝑅) | |
6 | 4, 5 | eqtr4di 2851 | . . 3 ⊢ (𝑟 = 𝑅 → (AbsVal‘𝑟) = 𝐴) |
7 | 3, 6 | eleq12d 2884 | . 2 ⊢ (𝑟 = 𝑅 → ((norm‘𝑟) ∈ (AbsVal‘𝑟) ↔ 𝑁 ∈ 𝐴)) |
8 | df-nrg 23192 | . 2 ⊢ NrmRing = {𝑟 ∈ NrmGrp ∣ (norm‘𝑟) ∈ (AbsVal‘𝑟)} | |
9 | 7, 8 | elrab2 3631 | 1 ⊢ (𝑅 ∈ NrmRing ↔ (𝑅 ∈ NrmGrp ∧ 𝑁 ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ‘cfv 6324 AbsValcabv 19580 normcnm 23183 NrmGrpcngp 23184 NrmRingcnrg 23186 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-rab 3115 df-v 3443 df-un 3886 df-in 3888 df-ss 3898 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-iota 6283 df-fv 6332 df-nrg 23192 |
This theorem is referenced by: nrgabv 23267 nrgngp 23268 subrgnrg 23279 tngnrg 23280 cnnrg 23386 zhmnrg 31318 |
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