![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > isnrg | Structured version Visualization version GIF version |
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 | β’ π = (normβπ ) |
isnrg.2 | β’ π΄ = (AbsValβπ ) |
Ref | Expression |
---|---|
isnrg | β’ (π β NrmRing β (π β NrmGrp β§ π β π΄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6882 | . . . 4 β’ (π = π β (normβπ) = (normβπ )) | |
2 | isnrg.1 | . . . 4 β’ π = (normβπ ) | |
3 | 1, 2 | eqtr4di 2782 | . . 3 β’ (π = π β (normβπ) = π) |
4 | fveq2 6882 | . . . 4 β’ (π = π β (AbsValβπ) = (AbsValβπ )) | |
5 | isnrg.2 | . . . 4 β’ π΄ = (AbsValβπ ) | |
6 | 4, 5 | eqtr4di 2782 | . . 3 β’ (π = π β (AbsValβπ) = π΄) |
7 | 3, 6 | eleq12d 2819 | . 2 β’ (π = π β ((normβπ) β (AbsValβπ) β π β π΄)) |
8 | df-nrg 24438 | . 2 β’ NrmRing = {π β NrmGrp β£ (normβπ) β (AbsValβπ)} | |
9 | 7, 8 | elrab2 3679 | 1 β’ (π β NrmRing β (π β NrmGrp β§ π β π΄)) |
Colors of variables: wff setvar class |
Syntax hints: β wb 205 β§ wa 395 = wceq 1533 β wcel 2098 βcfv 6534 AbsValcabv 20655 normcnm 24429 NrmGrpcngp 24430 NrmRingcnrg 24432 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-iota 6486 df-fv 6542 df-nrg 24438 |
This theorem is referenced by: nrgabv 24522 nrgngp 24523 subrgnrg 24534 tngnrg 24535 cnnrg 24641 zhmnrg 33466 |
Copyright terms: Public domain | W3C validator |