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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 24592 | . . . . 5 β’ (π β NrmRing β π β NrmGrp) | |
2 | 1 | 3ad2ant1 1131 | . . . 4 β’ ((π β NrmRing β§ π β NzRing β§ π΄ β π) β π β NrmGrp) |
3 | eqid 2728 | . . . . . 6 β’ (Baseβπ ) = (Baseβπ ) | |
4 | nminvr.u | . . . . . 6 β’ π = (Unitβπ ) | |
5 | 3, 4 | unitcl 20314 | . . . . 5 β’ (π΄ β π β π΄ β (Baseβπ )) |
6 | 5 | 3ad2ant3 1133 | . . . 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 24538 | . . . 4 β’ ((π β NrmGrp β§ π΄ β (Baseβπ )) β (πβπ΄) β β) |
9 | 2, 6, 8 | syl2anc 583 | . . 3 β’ ((π β NrmRing β§ π β NzRing β§ π΄ β π) β (πβπ΄) β β) |
10 | 9 | recnd 11273 | . 2 β’ ((π β NrmRing β§ π β NzRing β§ π΄ β π) β (πβπ΄) β β) |
11 | nzrring 20455 | . . . . . 6 β’ (π β NzRing β π β Ring) | |
12 | 11 | 3ad2ant2 1132 | . . . . 5 β’ ((π β NrmRing β§ π β NzRing β§ π΄ β π) β π β Ring) |
13 | simp3 1136 | . . . . 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 20331 | . . . . 5 β’ ((π β Ring β§ π΄ β π) β (πΌβπ΄) β (Baseβπ )) |
16 | 12, 13, 15 | syl2anc 583 | . . . 4 β’ ((π β NrmRing β§ π β NzRing β§ π΄ β π) β (πΌβπ΄) β (Baseβπ )) |
17 | 3, 7 | nmcl 24538 | . . . 4 β’ ((π β NrmGrp β§ (πΌβπ΄) β (Baseβπ )) β (πβ(πΌβπ΄)) β β) |
18 | 2, 16, 17 | syl2anc 583 | . . 3 β’ ((π β NrmRing β§ π β NzRing β§ π΄ β π) β (πβ(πΌβπ΄)) β β) |
19 | 18 | recnd 11273 | . 2 β’ ((π β NrmRing β§ π β NzRing β§ π΄ β π) β (πβ(πΌβπ΄)) β β) |
20 | 7, 4 | unitnmn0 24598 | . 2 β’ ((π β NrmRing β§ π β NzRing β§ π΄ β π) β (πβπ΄) β 0) |
21 | eqid 2728 | . . . . . 6 β’ (.rβπ ) = (.rβπ ) | |
22 | eqid 2728 | . . . . . 6 β’ (1rβπ ) = (1rβπ ) | |
23 | 4, 14, 21, 22 | unitrinv 20333 | . . . . 5 β’ ((π β Ring β§ π΄ β π) β (π΄(.rβπ )(πΌβπ΄)) = (1rβπ )) |
24 | 12, 13, 23 | syl2anc 583 | . . . 4 β’ ((π β NrmRing β§ π β NzRing β§ π΄ β π) β (π΄(.rβπ )(πΌβπ΄)) = (1rβπ )) |
25 | 24 | fveq2d 6901 | . . 3 β’ ((π β NrmRing β§ π β NzRing β§ π΄ β π) β (πβ(π΄(.rβπ )(πΌβπ΄))) = (πβ(1rβπ ))) |
26 | simp1 1134 | . . . 4 β’ ((π β NrmRing β§ π β NzRing β§ π΄ β π) β π β NrmRing) | |
27 | 3, 7, 21 | nmmul 24594 | . . . 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 1369 | . . 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 24597 | . . . 4 β’ ((π β NrmRing β§ π β NzRing) β (πβ(1rβπ )) = 1) |
30 | 29 | 3adant3 1130 | . . 3 β’ ((π β NrmRing β§ π β NzRing β§ π΄ β π) β (πβ(1rβπ )) = 1) |
31 | 25, 28, 30 | 3eqtr3d 2776 | . 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 12077 | 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 1085 = wceq 1534 β wcel 2099 βcfv 6548 (class class class)co 7420 βcr 11138 1c1 11140 Β· cmul 11144 / cdiv 11902 Basecbs 17180 .rcmulr 17234 1rcur 20121 Ringcrg 20173 Unitcui 20294 invrcinvr 20326 NzRingcnzr 20451 normcnm 24498 NrmGrpcngp 24499 NrmRingcnrg 24501 |
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-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-pre-sup 11217 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-tpos 8232 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-sup 9466 df-inf 9467 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-nn 12244 df-2 12306 df-3 12307 df-n0 12504 df-z 12590 df-uz 12854 df-q 12964 df-rp 13008 df-xneg 13125 df-xadd 13126 df-xmul 13127 df-ico 13363 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-ress 17210 df-plusg 17246 df-mulr 17247 df-0g 17423 df-topgen 17425 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-grp 18893 df-minusg 18894 df-cmn 19737 df-abl 19738 df-mgp 20075 df-rng 20093 df-ur 20122 df-ring 20175 df-oppr 20273 df-dvdsr 20296 df-unit 20297 df-invr 20327 df-nzr 20452 df-abv 20697 df-psmet 21271 df-xmet 21272 df-met 21273 df-bl 21274 df-mopn 21275 df-top 22809 df-topon 22826 df-topsp 22848 df-bases 22862 df-xms 24239 df-ms 24240 df-nm 24504 df-ngp 24505 df-nrg 24507 |
This theorem is referenced by: nmdvr 24600 |
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