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Mirrors > Home > MPE Home > Th. List > cncms | Structured version Visualization version GIF version |
Description: The field of complex numbers is a complete metric space. (Contributed by Mario Carneiro, 15-Oct-2015.) |
Ref | Expression |
---|---|
cncms | ⊢ ℂfld ∈ CMetSp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnfldms 24810 | . 2 ⊢ ℂfld ∈ MetSp | |
2 | eqid 2734 | . . 3 ⊢ (abs ∘ − ) = (abs ∘ − ) | |
3 | 2 | cncmet 25368 | . 2 ⊢ (abs ∘ − ) ∈ (CMet‘ℂ) |
4 | cnfldbas 21386 | . . 3 ⊢ ℂ = (Base‘ℂfld) | |
5 | cnmet 24806 | . . . . . 6 ⊢ (abs ∘ − ) ∈ (Met‘ℂ) | |
6 | metf 24354 | . . . . . 6 ⊢ ((abs ∘ − ) ∈ (Met‘ℂ) → (abs ∘ − ):(ℂ × ℂ)⟶ℝ) | |
7 | 5, 6 | ax-mp 5 | . . . . 5 ⊢ (abs ∘ − ):(ℂ × ℂ)⟶ℝ |
8 | ffn 6746 | . . . . 5 ⊢ ((abs ∘ − ):(ℂ × ℂ)⟶ℝ → (abs ∘ − ) Fn (ℂ × ℂ)) | |
9 | fnresdm 6698 | . . . . 5 ⊢ ((abs ∘ − ) Fn (ℂ × ℂ) → ((abs ∘ − ) ↾ (ℂ × ℂ)) = (abs ∘ − )) | |
10 | 7, 8, 9 | mp2b 10 | . . . 4 ⊢ ((abs ∘ − ) ↾ (ℂ × ℂ)) = (abs ∘ − ) |
11 | cnfldds 21394 | . . . . 5 ⊢ (abs ∘ − ) = (dist‘ℂfld) | |
12 | 11 | reseq1i 6004 | . . . 4 ⊢ ((abs ∘ − ) ↾ (ℂ × ℂ)) = ((dist‘ℂfld) ↾ (ℂ × ℂ)) |
13 | 10, 12 | eqtr3i 2764 | . . 3 ⊢ (abs ∘ − ) = ((dist‘ℂfld) ↾ (ℂ × ℂ)) |
14 | 4, 13 | iscms 25391 | . 2 ⊢ (ℂfld ∈ CMetSp ↔ (ℂfld ∈ MetSp ∧ (abs ∘ − ) ∈ (CMet‘ℂ))) |
15 | 1, 3, 14 | mpbir2an 710 | 1 ⊢ ℂfld ∈ CMetSp |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2103 × cxp 5697 ↾ cres 5701 ∘ ccom 5703 Fn wfn 6567 ⟶wf 6568 ‘cfv 6572 ℂcc 11178 ℝcr 11179 − cmin 11516 abscabs 15279 distcds 17315 Metcmet 21368 ℂfldccnfld 21382 MetSpcms 24342 CMetccmet 25300 CMetSpccms 25378 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-rep 5306 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 ax-cnex 11236 ax-resscn 11237 ax-1cn 11238 ax-icn 11239 ax-addcl 11240 ax-addrcl 11241 ax-mulcl 11242 ax-mulrcl 11243 ax-mulcom 11244 ax-addass 11245 ax-mulass 11246 ax-distr 11247 ax-i2m1 11248 ax-1ne0 11249 ax-1rid 11250 ax-rnegex 11251 ax-rrecex 11252 ax-cnre 11253 ax-pre-lttri 11254 ax-pre-lttrn 11255 ax-pre-ltadd 11256 ax-pre-mulgt0 11257 ax-pre-sup 11258 ax-addf 11259 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3383 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-pss 3990 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4973 df-iun 5021 df-iin 5022 df-br 5170 df-opab 5232 df-mpt 5253 df-tr 5287 df-id 5597 df-eprel 5603 df-po 5611 df-so 5612 df-fr 5654 df-se 5655 df-we 5656 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-pred 6331 df-ord 6397 df-on 6398 df-lim 6399 df-suc 6400 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-isom 6581 df-riota 7401 df-ov 7448 df-oprab 7449 df-mpo 7450 df-of 7710 df-om 7900 df-1st 8026 df-2nd 8027 df-supp 8198 df-frecs 8318 df-wrecs 8349 df-recs 8423 df-rdg 8462 df-1o 8518 df-2o 8519 df-er 8759 df-map 8882 df-ixp 8952 df-en 9000 df-dom 9001 df-sdom 9002 df-fin 9003 df-fsupp 9428 df-fi 9476 df-sup 9507 df-inf 9508 df-oi 9575 df-card 10004 df-pnf 11322 df-mnf 11323 df-xr 11324 df-ltxr 11325 df-le 11326 df-sub 11518 df-neg 11519 df-div 11944 df-nn 12290 df-2 12352 df-3 12353 df-4 12354 df-5 12355 df-6 12356 df-7 12357 df-8 12358 df-9 12359 df-n0 12550 df-z 12636 df-dec 12755 df-uz 12900 df-q 13010 df-rp 13054 df-xneg 13171 df-xadd 13172 df-xmul 13173 df-ioo 13407 df-ico 13409 df-icc 13410 df-fz 13564 df-fzo 13708 df-seq 14049 df-exp 14109 df-hash 14376 df-cj 15144 df-re 15145 df-im 15146 df-sqrt 15280 df-abs 15281 df-struct 17189 df-sets 17206 df-slot 17224 df-ndx 17236 df-base 17254 df-ress 17283 df-plusg 17319 df-mulr 17320 df-starv 17321 df-sca 17322 df-vsca 17323 df-ip 17324 df-tset 17325 df-ple 17326 df-ds 17328 df-unif 17329 df-hom 17330 df-cco 17331 df-rest 17477 df-topn 17478 df-0g 17496 df-gsum 17497 df-topgen 17498 df-pt 17499 df-prds 17502 df-xrs 17557 df-qtop 17562 df-imas 17563 df-xps 17565 df-mre 17639 df-mrc 17640 df-acs 17642 df-mgm 18673 df-sgrp 18752 df-mnd 18768 df-submnd 18814 df-mulg 19103 df-cntz 19352 df-cmn 19819 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-fbas 21379 df-fg 21380 df-cnfld 21383 df-top 22914 df-topon 22931 df-topsp 22953 df-bases 22967 df-cld 23041 df-ntr 23042 df-cls 23043 df-nei 23120 df-cn 23249 df-cnp 23250 df-haus 23337 df-cmp 23409 df-tx 23584 df-hmeo 23777 df-fil 23868 df-flim 23961 df-fcls 23963 df-xms 24344 df-ms 24345 df-tms 24346 df-cncf 24916 df-cfil 25301 df-cmet 25303 df-cms 25381 |
This theorem is referenced by: cnfldcusp 25403 resscdrg 25404 ishl2 25416 csschl 25422 recms 25426 |
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