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Mirrors > Home > MPE Home > Th. List > unitnmn0 | Structured version Visualization version GIF version |
Description: The norm of a unit is nonzero in a nonzero normed ring. (Contributed by Mario Carneiro, 5-Oct-2015.) |
Ref | Expression |
---|---|
nminvr.n | β’ π = (normβπ ) |
nminvr.u | β’ π = (Unitβπ ) |
Ref | Expression |
---|---|
unitnmn0 | β’ ((π β NrmRing β§ π β NzRing β§ π΄ β π) β (πβπ΄) β 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nrgngp 24400 | . . 3 β’ (π β NrmRing β π β NrmGrp) | |
2 | 1 | 3ad2ant1 1132 | . 2 β’ ((π β NrmRing β§ π β NzRing β§ π΄ β π) β π β NrmGrp) |
3 | eqid 2731 | . . . 4 β’ (Baseβπ ) = (Baseβπ ) | |
4 | nminvr.u | . . . 4 β’ π = (Unitβπ ) | |
5 | 3, 4 | unitcl 20267 | . . 3 β’ (π΄ β π β π΄ β (Baseβπ )) |
6 | 5 | 3ad2ant3 1134 | . 2 β’ ((π β NrmRing β§ π β NzRing β§ π΄ β π) β π΄ β (Baseβπ )) |
7 | eqid 2731 | . . . 4 β’ (0gβπ ) = (0gβπ ) | |
8 | 4, 7 | nzrunit 20414 | . . 3 β’ ((π β NzRing β§ π΄ β π) β π΄ β (0gβπ )) |
9 | 8 | 3adant1 1129 | . 2 β’ ((π β NrmRing β§ π β NzRing β§ π΄ β π) β π΄ β (0gβπ )) |
10 | nminvr.n | . . 3 β’ π = (normβπ ) | |
11 | 3, 10, 7 | nmne0 24349 | . 2 β’ ((π β NrmGrp β§ π΄ β (Baseβπ ) β§ π΄ β (0gβπ )) β (πβπ΄) β 0) |
12 | 2, 6, 9, 11 | syl3anc 1370 | 1 β’ ((π β NrmRing β§ π β NzRing β§ π΄ β π) β (πβπ΄) β 0) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1086 = wceq 1540 β wcel 2105 β wne 2939 βcfv 6543 0cc0 11114 Basecbs 17149 0gc0g 17390 Unitcui 20247 NzRingcnzr 20404 normcnm 24306 NrmGrpcngp 24307 NrmRingcnrg 24309 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 ax-pre-sup 11192 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-tpos 8215 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-er 8707 df-map 8826 df-en 8944 df-dom 8945 df-sdom 8946 df-sup 9441 df-inf 9442 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-div 11877 df-nn 12218 df-2 12280 df-3 12281 df-n0 12478 df-z 12564 df-uz 12828 df-q 12938 df-rp 12980 df-xneg 13097 df-xadd 13098 df-xmul 13099 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-0g 17392 df-topgen 17394 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-grp 18859 df-minusg 18860 df-cmn 19692 df-abl 19693 df-mgp 20030 df-rng 20048 df-ur 20077 df-ring 20130 df-oppr 20226 df-dvdsr 20249 df-unit 20250 df-invr 20280 df-nzr 20405 df-psmet 21137 df-xmet 21138 df-bl 21140 df-mopn 21141 df-top 22617 df-topon 22634 df-topsp 22656 df-bases 22670 df-xms 24047 df-ms 24048 df-nm 24312 df-ngp 24313 df-nrg 24315 |
This theorem is referenced by: nminvr 24407 nmdvr 24408 |
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