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| 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 6906 | . . . 4 ⊢ (𝑟 = 𝑅 → (norm‘𝑟) = (norm‘𝑅)) | |
| 2 | isnrg.1 | . . . 4 ⊢ 𝑁 = (norm‘𝑅) | |
| 3 | 1, 2 | eqtr4di 2795 | . . 3 ⊢ (𝑟 = 𝑅 → (norm‘𝑟) = 𝑁) | 
| 4 | fveq2 6906 | . . . 4 ⊢ (𝑟 = 𝑅 → (AbsVal‘𝑟) = (AbsVal‘𝑅)) | |
| 5 | isnrg.2 | . . . 4 ⊢ 𝐴 = (AbsVal‘𝑅) | |
| 6 | 4, 5 | eqtr4di 2795 | . . 3 ⊢ (𝑟 = 𝑅 → (AbsVal‘𝑟) = 𝐴) | 
| 7 | 3, 6 | eleq12d 2835 | . 2 ⊢ (𝑟 = 𝑅 → ((norm‘𝑟) ∈ (AbsVal‘𝑟) ↔ 𝑁 ∈ 𝐴)) | 
| 8 | df-nrg 24598 | . 2 ⊢ NrmRing = {𝑟 ∈ NrmGrp ∣ (norm‘𝑟) ∈ (AbsVal‘𝑟)} | |
| 9 | 7, 8 | elrab2 3695 | 1 ⊢ (𝑅 ∈ NrmRing ↔ (𝑅 ∈ NrmGrp ∧ 𝑁 ∈ 𝐴)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 AbsValcabv 20809 normcnm 24589 NrmGrpcngp 24590 NrmRingcnrg 24592 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-nrg 24598 | 
| This theorem is referenced by: nrgabv 24682 nrgngp 24683 subrgnrg 24694 tngnrg 24695 cnnrg 24801 zhmnrg 33966 | 
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