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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 22464 | . 2 ⊢ (𝑅 ∈ NrmRing ↔ (𝑅 ∈ NrmGrp ∧ 𝑁 ∈ 𝐴)) |
4 | 3 | simprbi 480 | 1 ⊢ (𝑅 ∈ NrmRing → 𝑁 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ‘cfv 5888 AbsValcabv 18816 normcnm 22381 NrmGrpcngp 22382 NrmRingcnrg 22384 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-iota 5851 df-fv 5896 df-nrg 22390 |
This theorem is referenced by: nrgring 22467 nmmul 22468 nm1 22471 nrgdomn 22475 subrgnrg 22477 sranlm 22488 |
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