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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 10479 | . . . . 5 ⊢ 0 ∈ ℂ | |
2 | eqid 2795 | . . . . . 6 ⊢ (abs ∘ − ) = (abs ∘ − ) | |
3 | 2 | cnmetdval 23062 | . . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 0 ∈ ℂ) → (𝑥(abs ∘ − )0) = (abs‘(𝑥 − 0))) |
4 | 1, 3 | mpan2 687 | . . . 4 ⊢ (𝑥 ∈ ℂ → (𝑥(abs ∘ − )0) = (abs‘(𝑥 − 0))) |
5 | subid1 10754 | . . . . 5 ⊢ (𝑥 ∈ ℂ → (𝑥 − 0) = 𝑥) | |
6 | 5 | fveq2d 6542 | . . . 4 ⊢ (𝑥 ∈ ℂ → (abs‘(𝑥 − 0)) = (abs‘𝑥)) |
7 | 4, 6 | eqtrd 2831 | . . 3 ⊢ (𝑥 ∈ ℂ → (𝑥(abs ∘ − )0) = (abs‘𝑥)) |
8 | 7 | mpteq2ia 5051 | . 2 ⊢ (𝑥 ∈ ℂ ↦ (𝑥(abs ∘ − )0)) = (𝑥 ∈ ℂ ↦ (abs‘𝑥)) |
9 | eqid 2795 | . . 3 ⊢ (norm‘ℂfld) = (norm‘ℂfld) | |
10 | cnfldbas 20231 | . . 3 ⊢ ℂ = (Base‘ℂfld) | |
11 | cnfld0 20251 | . . 3 ⊢ 0 = (0g‘ℂfld) | |
12 | cnfldds 20237 | . . 3 ⊢ (abs ∘ − ) = (dist‘ℂfld) | |
13 | 9, 10, 11, 12 | nmfval 22881 | . 2 ⊢ (norm‘ℂfld) = (𝑥 ∈ ℂ ↦ (𝑥(abs ∘ − )0)) |
14 | absf 14531 | . . . . 5 ⊢ abs:ℂ⟶ℝ | |
15 | 14 | a1i 11 | . . . 4 ⊢ (⊤ → abs:ℂ⟶ℝ) |
16 | 15 | feqmptd 6601 | . . 3 ⊢ (⊤ → abs = (𝑥 ∈ ℂ ↦ (abs‘𝑥))) |
17 | 16 | mptru 1529 | . 2 ⊢ abs = (𝑥 ∈ ℂ ↦ (abs‘𝑥)) |
18 | 8, 13, 17 | 3eqtr4ri 2830 | 1 ⊢ abs = (norm‘ℂfld) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1522 ⊤wtru 1523 ∈ wcel 2081 ↦ cmpt 5041 ∘ ccom 5447 ⟶wf 6221 ‘cfv 6225 (class class class)co 7016 ℂcc 10381 ℝcr 10382 0cc0 10383 − cmin 10717 abscabs 14427 ℂfldccnfld 20227 normcnm 22869 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 ax-pre-sup 10461 ax-addf 10462 ax-mulf 10463 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-int 4783 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-om 7437 df-1st 7545 df-2nd 7546 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-1o 7953 df-oadd 7957 df-er 8139 df-en 8358 df-dom 8359 df-sdom 8360 df-fin 8361 df-sup 8752 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-div 11146 df-nn 11487 df-2 11548 df-3 11549 df-4 11550 df-5 11551 df-6 11552 df-7 11553 df-8 11554 df-9 11555 df-n0 11746 df-z 11830 df-dec 11948 df-uz 12094 df-rp 12240 df-fz 12743 df-seq 13220 df-exp 13280 df-cj 14292 df-re 14293 df-im 14294 df-sqrt 14428 df-abs 14429 df-struct 16314 df-ndx 16315 df-slot 16316 df-base 16318 df-sets 16319 df-plusg 16407 df-mulr 16408 df-starv 16409 df-tset 16413 df-ple 16414 df-ds 16416 df-unif 16417 df-0g 16544 df-mgm 17681 df-sgrp 17723 df-mnd 17734 df-grp 17864 df-cmn 18635 df-mgp 18930 df-ring 18989 df-cring 18990 df-cnfld 20228 df-nm 22875 |
This theorem is referenced by: cnngp 23071 cnnrg 23072 abscn 23137 clmabs 23370 isncvsngp 23436 cnnm 23447 cnncvsabsnegdemo 23452 tcphcph 23523 zringnm 30818 cnzh 30828 rezh 30829 |
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