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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 6897 | . . . 4 β’ (π = π β (normβπ) = (normβπ )) | |
2 | isnrg.1 | . . . 4 β’ π = (normβπ ) | |
3 | 1, 2 | eqtr4di 2786 | . . 3 β’ (π = π β (normβπ) = π) |
4 | fveq2 6897 | . . . 4 β’ (π = π β (AbsValβπ) = (AbsValβπ )) | |
5 | isnrg.2 | . . . 4 β’ π΄ = (AbsValβπ ) | |
6 | 4, 5 | eqtr4di 2786 | . . 3 β’ (π = π β (AbsValβπ) = π΄) |
7 | 3, 6 | eleq12d 2823 | . 2 β’ (π = π β ((normβπ) β (AbsValβπ) β π β π΄)) |
8 | df-nrg 24493 | . 2 β’ NrmRing = {π β NrmGrp β£ (normβπ) β (AbsValβπ)} | |
9 | 7, 8 | elrab2 3685 | 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 1534 β wcel 2099 βcfv 6548 AbsValcabv 20695 normcnm 24484 NrmGrpcngp 24485 NrmRingcnrg 24487 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-iota 6500 df-fv 6556 df-nrg 24493 |
This theorem is referenced by: nrgabv 24577 nrgngp 24578 subrgnrg 24589 tngnrg 24590 cnnrg 24696 zhmnrg 33568 |
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