![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > cnfldnm | Structured version Visualization version GIF version |
Description: The norm of the field of complex numbers. (Contributed by Mario Carneiro, 4-Oct-2015.) |
Ref | Expression |
---|---|
cnfldnm | ⊢ abs = (norm‘ℂfld) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0cn 10320 | . . . . 5 ⊢ 0 ∈ ℂ | |
2 | eqid 2799 | . . . . . 6 ⊢ (abs ∘ − ) = (abs ∘ − ) | |
3 | 2 | cnmetdval 22902 | . . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 0 ∈ ℂ) → (𝑥(abs ∘ − )0) = (abs‘(𝑥 − 0))) |
4 | 1, 3 | mpan2 683 | . . . 4 ⊢ (𝑥 ∈ ℂ → (𝑥(abs ∘ − )0) = (abs‘(𝑥 − 0))) |
5 | subid1 10593 | . . . . 5 ⊢ (𝑥 ∈ ℂ → (𝑥 − 0) = 𝑥) | |
6 | 5 | fveq2d 6415 | . . . 4 ⊢ (𝑥 ∈ ℂ → (abs‘(𝑥 − 0)) = (abs‘𝑥)) |
7 | 4, 6 | eqtrd 2833 | . . 3 ⊢ (𝑥 ∈ ℂ → (𝑥(abs ∘ − )0) = (abs‘𝑥)) |
8 | 7 | mpteq2ia 4933 | . 2 ⊢ (𝑥 ∈ ℂ ↦ (𝑥(abs ∘ − )0)) = (𝑥 ∈ ℂ ↦ (abs‘𝑥)) |
9 | eqid 2799 | . . 3 ⊢ (norm‘ℂfld) = (norm‘ℂfld) | |
10 | cnfldbas 20072 | . . 3 ⊢ ℂ = (Base‘ℂfld) | |
11 | cnfld0 20092 | . . 3 ⊢ 0 = (0g‘ℂfld) | |
12 | cnfldds 20078 | . . 3 ⊢ (abs ∘ − ) = (dist‘ℂfld) | |
13 | 9, 10, 11, 12 | nmfval 22721 | . 2 ⊢ (norm‘ℂfld) = (𝑥 ∈ ℂ ↦ (𝑥(abs ∘ − )0)) |
14 | absf 14418 | . . . . 5 ⊢ abs:ℂ⟶ℝ | |
15 | 14 | a1i 11 | . . . 4 ⊢ (⊤ → abs:ℂ⟶ℝ) |
16 | 15 | feqmptd 6474 | . . 3 ⊢ (⊤ → abs = (𝑥 ∈ ℂ ↦ (abs‘𝑥))) |
17 | 16 | mptru 1661 | . 2 ⊢ abs = (𝑥 ∈ ℂ ↦ (abs‘𝑥)) |
18 | 8, 13, 17 | 3eqtr4ri 2832 | 1 ⊢ abs = (norm‘ℂfld) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1653 ⊤wtru 1654 ∈ wcel 2157 ↦ cmpt 4922 ∘ ccom 5316 ⟶wf 6097 ‘cfv 6101 (class class class)co 6878 ℂcc 10222 ℝcr 10223 0cc0 10224 − cmin 10556 abscabs 14315 ℂfldccnfld 20068 normcnm 22709 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 ax-pre-sup 10302 ax-addf 10303 ax-mulf 10304 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-int 4668 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-om 7300 df-1st 7401 df-2nd 7402 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-1o 7799 df-oadd 7803 df-er 7982 df-en 8196 df-dom 8197 df-sdom 8198 df-fin 8199 df-sup 8590 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-div 10977 df-nn 11313 df-2 11376 df-3 11377 df-4 11378 df-5 11379 df-6 11380 df-7 11381 df-8 11382 df-9 11383 df-n0 11581 df-z 11667 df-dec 11784 df-uz 11931 df-rp 12075 df-fz 12581 df-seq 13056 df-exp 13115 df-cj 14180 df-re 14181 df-im 14182 df-sqrt 14316 df-abs 14317 df-struct 16186 df-ndx 16187 df-slot 16188 df-base 16190 df-sets 16191 df-plusg 16280 df-mulr 16281 df-starv 16282 df-tset 16286 df-ple 16287 df-ds 16289 df-unif 16290 df-0g 16417 df-mgm 17557 df-sgrp 17599 df-mnd 17610 df-grp 17741 df-cmn 18510 df-mgp 18806 df-ring 18865 df-cring 18866 df-cnfld 20069 df-nm 22715 |
This theorem is referenced by: cnngp 22911 cnnrg 22912 abscn 22977 clmabs 23210 isncvsngp 23276 cnnm 23287 cnncvsabsnegdemo 23292 tcphcph 23363 zringnm 30520 cnzh 30530 rezh 30531 |
Copyright terms: Public domain | W3C validator |