Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 23271 | . 2 ⊢ (𝑅 ∈ NrmRing ↔ (𝑅 ∈ NrmGrp ∧ 𝑁 ∈ 𝐴)) |
4 | 3 | simprbi 499 | 1 ⊢ (𝑅 ∈ NrmRing → 𝑁 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ‘cfv 6357 AbsValcabv 19589 normcnm 23188 NrmGrpcngp 23189 NrmRingcnrg 23191 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-iota 6316 df-fv 6365 df-nrg 23197 |
This theorem is referenced by: nrgring 23274 nmmul 23275 nm1 23278 nrgdomn 23282 subrgnrg 23284 sranlm 23295 |
Copyright terms: Public domain | W3C validator |