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Mirrors > Home > MPE Home > Th. List > cnnrg | Structured version Visualization version GIF version |
Description: The complex numbers form a normed ring. (Contributed by Mario Carneiro, 4-Oct-2015.) |
Ref | Expression |
---|---|
cnnrg | ⊢ ℂfld ∈ NrmRing |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnngp 24757 | . 2 ⊢ ℂfld ∈ NrmGrp | |
2 | absabv 21391 | . 2 ⊢ abs ∈ (AbsVal‘ℂfld) | |
3 | cnfldnm 24756 | . . 3 ⊢ abs = (norm‘ℂfld) | |
4 | eqid 2725 | . . 3 ⊢ (AbsVal‘ℂfld) = (AbsVal‘ℂfld) | |
5 | 3, 4 | isnrg 24638 | . 2 ⊢ (ℂfld ∈ NrmRing ↔ (ℂfld ∈ NrmGrp ∧ abs ∈ (AbsVal‘ℂfld))) |
6 | 1, 2, 5 | mpbir2an 709 | 1 ⊢ ℂfld ∈ NrmRing |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2098 ‘cfv 6549 abscabs 15225 AbsValcabv 20725 ℂfldccnfld 21313 NrmGrpcngp 24547 NrmRingcnrg 24549 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11201 ax-resscn 11202 ax-1cn 11203 ax-icn 11204 ax-addcl 11205 ax-addrcl 11206 ax-mulcl 11207 ax-mulrcl 11208 ax-mulcom 11209 ax-addass 11210 ax-mulass 11211 ax-distr 11212 ax-i2m1 11213 ax-1ne0 11214 ax-1rid 11215 ax-rnegex 11216 ax-rrecex 11217 ax-cnre 11218 ax-pre-lttri 11219 ax-pre-lttrn 11220 ax-pre-ltadd 11221 ax-pre-mulgt0 11222 ax-pre-sup 11223 ax-addf 11224 ax-mulf 11225 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 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 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9472 df-inf 9473 df-pnf 11287 df-mnf 11288 df-xr 11289 df-ltxr 11290 df-le 11291 df-sub 11483 df-neg 11484 df-div 11909 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-q 12971 df-rp 13015 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-ico 13370 df-fz 13525 df-seq 14008 df-exp 14068 df-cj 15090 df-re 15091 df-im 15092 df-sqrt 15226 df-abs 15227 df-struct 17135 df-sets 17152 df-slot 17170 df-ndx 17182 df-base 17200 df-plusg 17265 df-mulr 17266 df-starv 17267 df-tset 17271 df-ple 17272 df-ds 17274 df-unif 17275 df-rest 17423 df-topn 17424 df-0g 17442 df-topgen 17444 df-mgm 18619 df-sgrp 18698 df-mnd 18714 df-grp 18917 df-minusg 18918 df-sbg 18919 df-cmn 19766 df-abl 19767 df-mgp 20104 df-rng 20122 df-ur 20151 df-ring 20204 df-cring 20205 df-abv 20726 df-psmet 21305 df-xmet 21306 df-met 21307 df-bl 21308 df-mopn 21309 df-cnfld 21314 df-top 22857 df-topon 22874 df-topsp 22896 df-bases 22910 df-xms 24287 df-ms 24288 df-nm 24552 df-ngp 24553 df-nrg 24555 |
This theorem is referenced by: zringnrg 24765 isncvsngp 25138 cnrnvc 25147 tcphcph 25226 iistmd 33654 cnzh 33722 rezh 33723 rerrext 33761 cnrrext 33762 |
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