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Mirrors > Home > MPE Home > Th. List > nminvr | Structured version Visualization version GIF version |
Description: The norm of an inverse in a nonzero normed ring. (Contributed by Mario Carneiro, 5-Oct-2015.) |
Ref | Expression |
---|---|
nminvr.n | β’ π = (normβπ ) |
nminvr.u | β’ π = (Unitβπ ) |
nminvr.i | β’ πΌ = (invrβπ ) |
Ref | Expression |
---|---|
nminvr | β’ ((π β NrmRing β§ π β NzRing β§ π΄ β π) β (πβ(πΌβπ΄)) = (1 / (πβπ΄))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nrgngp 24170 | . . . . 5 β’ (π β NrmRing β π β NrmGrp) | |
2 | 1 | 3ad2ant1 1133 | . . . 4 β’ ((π β NrmRing β§ π β NzRing β§ π΄ β π) β π β NrmGrp) |
3 | eqid 2732 | . . . . . 6 β’ (Baseβπ ) = (Baseβπ ) | |
4 | nminvr.u | . . . . . 6 β’ π = (Unitβπ ) | |
5 | 3, 4 | unitcl 20181 | . . . . 5 β’ (π΄ β π β π΄ β (Baseβπ )) |
6 | 5 | 3ad2ant3 1135 | . . . 4 β’ ((π β NrmRing β§ π β NzRing β§ π΄ β π) β π΄ β (Baseβπ )) |
7 | nminvr.n | . . . . 5 β’ π = (normβπ ) | |
8 | 3, 7 | nmcl 24116 | . . . 4 β’ ((π β NrmGrp β§ π΄ β (Baseβπ )) β (πβπ΄) β β) |
9 | 2, 6, 8 | syl2anc 584 | . . 3 β’ ((π β NrmRing β§ π β NzRing β§ π΄ β π) β (πβπ΄) β β) |
10 | 9 | recnd 11238 | . 2 β’ ((π β NrmRing β§ π β NzRing β§ π΄ β π) β (πβπ΄) β β) |
11 | nzrring 20287 | . . . . . 6 β’ (π β NzRing β π β Ring) | |
12 | 11 | 3ad2ant2 1134 | . . . . 5 β’ ((π β NrmRing β§ π β NzRing β§ π΄ β π) β π β Ring) |
13 | simp3 1138 | . . . . 5 β’ ((π β NrmRing β§ π β NzRing β§ π΄ β π) β π΄ β π) | |
14 | nminvr.i | . . . . . 6 β’ πΌ = (invrβπ ) | |
15 | 4, 14, 3 | ringinvcl 20198 | . . . . 5 β’ ((π β Ring β§ π΄ β π) β (πΌβπ΄) β (Baseβπ )) |
16 | 12, 13, 15 | syl2anc 584 | . . . 4 β’ ((π β NrmRing β§ π β NzRing β§ π΄ β π) β (πΌβπ΄) β (Baseβπ )) |
17 | 3, 7 | nmcl 24116 | . . . 4 β’ ((π β NrmGrp β§ (πΌβπ΄) β (Baseβπ )) β (πβ(πΌβπ΄)) β β) |
18 | 2, 16, 17 | syl2anc 584 | . . 3 β’ ((π β NrmRing β§ π β NzRing β§ π΄ β π) β (πβ(πΌβπ΄)) β β) |
19 | 18 | recnd 11238 | . 2 β’ ((π β NrmRing β§ π β NzRing β§ π΄ β π) β (πβ(πΌβπ΄)) β β) |
20 | 7, 4 | unitnmn0 24176 | . 2 β’ ((π β NrmRing β§ π β NzRing β§ π΄ β π) β (πβπ΄) β 0) |
21 | eqid 2732 | . . . . . 6 β’ (.rβπ ) = (.rβπ ) | |
22 | eqid 2732 | . . . . . 6 β’ (1rβπ ) = (1rβπ ) | |
23 | 4, 14, 21, 22 | unitrinv 20200 | . . . . 5 β’ ((π β Ring β§ π΄ β π) β (π΄(.rβπ )(πΌβπ΄)) = (1rβπ )) |
24 | 12, 13, 23 | syl2anc 584 | . . . 4 β’ ((π β NrmRing β§ π β NzRing β§ π΄ β π) β (π΄(.rβπ )(πΌβπ΄)) = (1rβπ )) |
25 | 24 | fveq2d 6892 | . . 3 β’ ((π β NrmRing β§ π β NzRing β§ π΄ β π) β (πβ(π΄(.rβπ )(πΌβπ΄))) = (πβ(1rβπ ))) |
26 | simp1 1136 | . . . 4 β’ ((π β NrmRing β§ π β NzRing β§ π΄ β π) β π β NrmRing) | |
27 | 3, 7, 21 | nmmul 24172 | . . . 4 β’ ((π β NrmRing β§ π΄ β (Baseβπ ) β§ (πΌβπ΄) β (Baseβπ )) β (πβ(π΄(.rβπ )(πΌβπ΄))) = ((πβπ΄) Β· (πβ(πΌβπ΄)))) |
28 | 26, 6, 16, 27 | syl3anc 1371 | . . 3 β’ ((π β NrmRing β§ π β NzRing β§ π΄ β π) β (πβ(π΄(.rβπ )(πΌβπ΄))) = ((πβπ΄) Β· (πβ(πΌβπ΄)))) |
29 | 7, 22 | nm1 24175 | . . . 4 β’ ((π β NrmRing β§ π β NzRing) β (πβ(1rβπ )) = 1) |
30 | 29 | 3adant3 1132 | . . 3 β’ ((π β NrmRing β§ π β NzRing β§ π΄ β π) β (πβ(1rβπ )) = 1) |
31 | 25, 28, 30 | 3eqtr3d 2780 | . 2 β’ ((π β NrmRing β§ π β NzRing β§ π΄ β π) β ((πβπ΄) Β· (πβ(πΌβπ΄))) = 1) |
32 | 10, 19, 20, 31 | mvllmuld 12042 | 1 β’ ((π β NrmRing β§ π β NzRing β§ π΄ β π) β (πβ(πΌβπ΄)) = (1 / (πβπ΄))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1087 = wceq 1541 β wcel 2106 βcfv 6540 (class class class)co 7405 βcr 11105 1c1 11107 Β· cmul 11111 / cdiv 11867 Basecbs 17140 .rcmulr 17194 1rcur 19998 Ringcrg 20049 Unitcui 20161 invrcinvr 20193 NzRingcnzr 20283 normcnm 24076 NrmGrpcngp 24077 NrmRingcnrg 24079 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-tpos 8207 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-sup 9433 df-inf 9434 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-n0 12469 df-z 12555 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ico 13326 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-0g 17383 df-topgen 17385 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-grp 18818 df-minusg 18819 df-mgp 19982 df-ur 19999 df-ring 20051 df-oppr 20142 df-dvdsr 20163 df-unit 20164 df-invr 20194 df-nzr 20284 df-abv 20417 df-psmet 20928 df-xmet 20929 df-met 20930 df-bl 20931 df-mopn 20932 df-top 22387 df-topon 22404 df-topsp 22426 df-bases 22440 df-xms 23817 df-ms 23818 df-nm 24082 df-ngp 24083 df-nrg 24085 |
This theorem is referenced by: nmdvr 24178 |
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