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Mirrors > Home > MPE Home > Th. List > nrgabv | Structured version Visualization version GIF version |
Description: The norm of a normed ring is an absolute value. (Contributed by Mario Carneiro, 4-Oct-2015.) |
Ref | Expression |
---|---|
isnrg.1 | ⊢ 𝑁 = (norm‘𝑅) |
isnrg.2 | ⊢ 𝐴 = (AbsVal‘𝑅) |
Ref | Expression |
---|---|
nrgabv | ⊢ (𝑅 ∈ NrmRing → 𝑁 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isnrg.1 | . . 3 ⊢ 𝑁 = (norm‘𝑅) | |
2 | isnrg.2 | . . 3 ⊢ 𝐴 = (AbsVal‘𝑅) | |
3 | 1, 2 | isnrg 24697 | . 2 ⊢ (𝑅 ∈ NrmRing ↔ (𝑅 ∈ NrmGrp ∧ 𝑁 ∈ 𝐴)) |
4 | 3 | simprbi 496 | 1 ⊢ (𝑅 ∈ NrmRing → 𝑁 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ‘cfv 6563 AbsValcabv 20826 normcnm 24605 NrmGrpcngp 24606 NrmRingcnrg 24608 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-iota 6516 df-fv 6571 df-nrg 24614 |
This theorem is referenced by: nrgring 24700 nmmul 24701 nm1 24704 nrgdomn 24708 subrgnrg 24710 sranlm 24721 |
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