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Mirrors > Home > MPE Home > Th. List > nmvs | Structured version Visualization version Unicode version |
Description: Defining property of a normed module. (Contributed by Mario Carneiro, 4-Oct-2015.) |
Ref | Expression |
---|---|
isnlm.v | |
isnlm.n | |
isnlm.s | |
isnlm.f | Scalar |
isnlm.k | |
isnlm.a |
Ref | Expression |
---|---|
nmvs | NrmMod |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isnlm.v | . . . . 5 | |
2 | isnlm.n | . . . . 5 | |
3 | isnlm.s | . . . . 5 | |
4 | isnlm.f | . . . . 5 Scalar | |
5 | isnlm.k | . . . . 5 | |
6 | isnlm.a | . . . . 5 | |
7 | 1, 2, 3, 4, 5, 6 | isnlm 22479 | . . . 4 NrmMod NrmGrp NrmRing |
8 | 7 | simprbi 480 | . . 3 NrmMod |
9 | oveq1 6657 | . . . . . 6 | |
10 | 9 | fveq2d 6195 | . . . . 5 |
11 | fveq2 6191 | . . . . . 6 | |
12 | 11 | oveq1d 6665 | . . . . 5 |
13 | 10, 12 | eqeq12d 2637 | . . . 4 |
14 | oveq2 6658 | . . . . . 6 | |
15 | 14 | fveq2d 6195 | . . . . 5 |
16 | fveq2 6191 | . . . . . 6 | |
17 | 16 | oveq2d 6666 | . . . . 5 |
18 | 15, 17 | eqeq12d 2637 | . . . 4 |
19 | 13, 18 | rspc2v 3322 | . . 3 |
20 | 8, 19 | syl5com 31 | . 2 NrmMod |
21 | 20 | 3impib 1262 | 1 NrmMod |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 w3a 1037 wceq 1483 wcel 1990 wral 2912 cfv 5888 (class class class)co 6650 cmul 9941 cbs 15857 Scalarcsca 15944 cvsca 15945 clmod 18863 cnm 22381 NrmGrpcngp 22382 NrmRingcnrg 22384 NrmModcnlm 22385 |
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 ax-nul 4789 |
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-eu 2474 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 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-ov 6653 df-nlm 22391 |
This theorem is referenced by: nlmdsdi 22485 nlmdsdir 22486 nlmmul0or 22487 lssnlm 22505 nmoleub2lem3 22915 nmoleub3 22919 ncvsprp 22952 cphnmvs 22990 nmmulg 30012 |
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