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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 6842 | . . . 4 ⊢ (𝑟 = 𝑅 → (norm‘𝑟) = (norm‘𝑅)) | |
2 | isnrg.1 | . . . 4 ⊢ 𝑁 = (norm‘𝑅) | |
3 | 1, 2 | eqtr4di 2794 | . . 3 ⊢ (𝑟 = 𝑅 → (norm‘𝑟) = 𝑁) |
4 | fveq2 6842 | . . . 4 ⊢ (𝑟 = 𝑅 → (AbsVal‘𝑟) = (AbsVal‘𝑅)) | |
5 | isnrg.2 | . . . 4 ⊢ 𝐴 = (AbsVal‘𝑅) | |
6 | 4, 5 | eqtr4di 2794 | . . 3 ⊢ (𝑟 = 𝑅 → (AbsVal‘𝑟) = 𝐴) |
7 | 3, 6 | eleq12d 2832 | . 2 ⊢ (𝑟 = 𝑅 → ((norm‘𝑟) ∈ (AbsVal‘𝑟) ↔ 𝑁 ∈ 𝐴)) |
8 | df-nrg 23941 | . 2 ⊢ NrmRing = {𝑟 ∈ NrmGrp ∣ (norm‘𝑟) ∈ (AbsVal‘𝑟)} | |
9 | 7, 8 | elrab2 3648 | 1 ⊢ (𝑅 ∈ NrmRing ↔ (𝑅 ∈ NrmGrp ∧ 𝑁 ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ‘cfv 6496 AbsValcabv 20275 normcnm 23932 NrmGrpcngp 23933 NrmRingcnrg 23935 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-rab 3408 df-v 3447 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-br 5106 df-iota 6448 df-fv 6504 df-nrg 23941 |
This theorem is referenced by: nrgabv 24025 nrgngp 24026 subrgnrg 24037 tngnrg 24038 cnnrg 24144 zhmnrg 32548 |
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