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Mirrors > Home > MPE Home > Th. List > cnngp | Structured version Visualization version GIF version |
Description: The complex numbers form a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.) |
Ref | Expression |
---|---|
cnngp | ⊢ ℂfld ∈ NrmGrp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnring 20417 | . . 3 ⊢ ℂfld ∈ Ring | |
2 | ringgrp 19599 | . . 3 ⊢ (ℂfld ∈ Ring → ℂfld ∈ Grp) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ ℂfld ∈ Grp |
4 | cnfldms 23704 | . 2 ⊢ ℂfld ∈ MetSp | |
5 | ssid 3939 | . 2 ⊢ (abs ∘ − ) ⊆ (abs ∘ − ) | |
6 | cnfldnm 23707 | . . 3 ⊢ abs = (norm‘ℂfld) | |
7 | cnfldsub 20423 | . . 3 ⊢ − = (-g‘ℂfld) | |
8 | cnfldds 20405 | . . 3 ⊢ (abs ∘ − ) = (dist‘ℂfld) | |
9 | 6, 7, 8 | isngp 23525 | . 2 ⊢ (ℂfld ∈ NrmGrp ↔ (ℂfld ∈ Grp ∧ ℂfld ∈ MetSp ∧ (abs ∘ − ) ⊆ (abs ∘ − ))) |
10 | 3, 4, 5, 9 | mpbir3an 1343 | 1 ⊢ ℂfld ∈ NrmGrp |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2112 ⊆ wss 3883 ∘ ccom 5572 − cmin 11091 abscabs 14829 Grpcgrp 18397 Ringcrg 19594 ℂfldccnfld 20395 MetSpcms 23247 NrmGrpcngp 23506 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2710 ax-rep 5195 ax-sep 5208 ax-nul 5215 ax-pow 5274 ax-pr 5338 ax-un 7544 ax-cnex 10814 ax-resscn 10815 ax-1cn 10816 ax-icn 10817 ax-addcl 10818 ax-addrcl 10819 ax-mulcl 10820 ax-mulrcl 10821 ax-mulcom 10822 ax-addass 10823 ax-mulass 10824 ax-distr 10825 ax-i2m1 10826 ax-1ne0 10827 ax-1rid 10828 ax-rnegex 10829 ax-rrecex 10830 ax-cnre 10831 ax-pre-lttri 10832 ax-pre-lttrn 10833 ax-pre-ltadd 10834 ax-pre-mulgt0 10835 ax-pre-sup 10836 ax-addf 10837 ax-mulf 10838 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2818 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rmo 3072 df-rab 3073 df-v 3425 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4456 df-pw 4531 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4836 df-iun 4922 df-br 5070 df-opab 5132 df-mpt 5152 df-tr 5178 df-id 5471 df-eprel 5477 df-po 5485 df-so 5486 df-fr 5526 df-we 5528 df-xp 5574 df-rel 5575 df-cnv 5576 df-co 5577 df-dm 5578 df-rn 5579 df-res 5580 df-ima 5581 df-pred 6178 df-ord 6236 df-on 6237 df-lim 6238 df-suc 6239 df-iota 6358 df-fun 6402 df-fn 6403 df-f 6404 df-f1 6405 df-fo 6406 df-f1o 6407 df-fv 6408 df-riota 7191 df-ov 7237 df-oprab 7238 df-mpo 7239 df-om 7666 df-1st 7782 df-2nd 7783 df-wrecs 8070 df-recs 8131 df-rdg 8169 df-1o 8225 df-er 8414 df-map 8533 df-en 8650 df-dom 8651 df-sdom 8652 df-fin 8653 df-sup 9087 df-inf 9088 df-pnf 10898 df-mnf 10899 df-xr 10900 df-ltxr 10901 df-le 10902 df-sub 11093 df-neg 11094 df-div 11519 df-nn 11860 df-2 11922 df-3 11923 df-4 11924 df-5 11925 df-6 11926 df-7 11927 df-8 11928 df-9 11929 df-n0 12120 df-z 12206 df-dec 12323 df-uz 12468 df-q 12574 df-rp 12616 df-xneg 12733 df-xadd 12734 df-xmul 12735 df-fz 13125 df-seq 13606 df-exp 13667 df-cj 14694 df-re 14695 df-im 14696 df-sqrt 14830 df-abs 14831 df-struct 16732 df-sets 16749 df-slot 16767 df-ndx 16777 df-base 16793 df-plusg 16847 df-mulr 16848 df-starv 16849 df-tset 16853 df-ple 16854 df-ds 16856 df-unif 16857 df-rest 16959 df-topn 16960 df-0g 16978 df-topgen 16980 df-mgm 18146 df-sgrp 18195 df-mnd 18206 df-grp 18400 df-minusg 18401 df-sbg 18402 df-cmn 19204 df-mgp 19537 df-ring 19596 df-cring 19597 df-psmet 20387 df-xmet 20388 df-met 20389 df-bl 20390 df-mopn 20391 df-cnfld 20396 df-top 21822 df-topon 21839 df-topsp 21861 df-bases 21874 df-xms 23249 df-ms 23250 df-nm 23511 df-ngp 23512 |
This theorem is referenced by: cnnrg 23709 abscn 23774 cnncvsabsnegdemo 24093 |
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