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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 23196 | . 2 ⊢ (𝑅 ∈ NrmRing ↔ (𝑅 ∈ NrmGrp ∧ 𝑁 ∈ 𝐴)) |
4 | 3 | simprbi 497 | 1 ⊢ (𝑅 ∈ NrmRing → 𝑁 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ‘cfv 6348 AbsValcabv 19516 normcnm 23113 NrmGrpcngp 23114 NrmRingcnrg 23116 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-iota 6307 df-fv 6356 df-nrg 23122 |
This theorem is referenced by: nrgring 23199 nmmul 23200 nm1 23203 nrgdomn 23207 subrgnrg 23209 sranlm 23220 |
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