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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 11202 | . . . . 5 ⊢ 0 ∈ ℂ | |
2 | eqid 2732 | . . . . . 6 ⊢ (abs ∘ − ) = (abs ∘ − ) | |
3 | 2 | cnmetdval 24278 | . . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 0 ∈ ℂ) → (𝑥(abs ∘ − )0) = (abs‘(𝑥 − 0))) |
4 | 1, 3 | mpan2 689 | . . . 4 ⊢ (𝑥 ∈ ℂ → (𝑥(abs ∘ − )0) = (abs‘(𝑥 − 0))) |
5 | subid1 11476 | . . . . 5 ⊢ (𝑥 ∈ ℂ → (𝑥 − 0) = 𝑥) | |
6 | 5 | fveq2d 6892 | . . . 4 ⊢ (𝑥 ∈ ℂ → (abs‘(𝑥 − 0)) = (abs‘𝑥)) |
7 | 4, 6 | eqtrd 2772 | . . 3 ⊢ (𝑥 ∈ ℂ → (𝑥(abs ∘ − )0) = (abs‘𝑥)) |
8 | 7 | mpteq2ia 5250 | . 2 ⊢ (𝑥 ∈ ℂ ↦ (𝑥(abs ∘ − )0)) = (𝑥 ∈ ℂ ↦ (abs‘𝑥)) |
9 | eqid 2732 | . . 3 ⊢ (norm‘ℂfld) = (norm‘ℂfld) | |
10 | cnfldbas 20940 | . . 3 ⊢ ℂ = (Base‘ℂfld) | |
11 | cnfld0 20961 | . . 3 ⊢ 0 = (0g‘ℂfld) | |
12 | cnfldds 20946 | . . 3 ⊢ (abs ∘ − ) = (dist‘ℂfld) | |
13 | 9, 10, 11, 12 | nmfval 24088 | . 2 ⊢ (norm‘ℂfld) = (𝑥 ∈ ℂ ↦ (𝑥(abs ∘ − )0)) |
14 | absf 15280 | . . . . 5 ⊢ abs:ℂ⟶ℝ | |
15 | 14 | a1i 11 | . . . 4 ⊢ (⊤ → abs:ℂ⟶ℝ) |
16 | 15 | feqmptd 6957 | . . 3 ⊢ (⊤ → abs = (𝑥 ∈ ℂ ↦ (abs‘𝑥))) |
17 | 16 | mptru 1548 | . 2 ⊢ abs = (𝑥 ∈ ℂ ↦ (abs‘𝑥)) |
18 | 8, 13, 17 | 3eqtr4ri 2771 | 1 ⊢ abs = (norm‘ℂfld) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ⊤wtru 1542 ∈ wcel 2106 ↦ cmpt 5230 ∘ ccom 5679 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 ℂcc 11104 ℝcr 11105 0cc0 11106 − cmin 11440 abscabs 15177 ℂfldccnfld 20936 normcnm 24076 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-rp 12971 df-fz 13481 df-seq 13963 df-exp 14024 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-plusg 17206 df-mulr 17207 df-starv 17208 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-0g 17383 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-grp 18818 df-cmn 19644 df-mgp 19982 df-ring 20051 df-cring 20052 df-cnfld 20937 df-nm 24082 |
This theorem is referenced by: cnngp 24287 cnnrg 24288 abscn 24353 clmabs 24590 isncvsngp 24657 cnnm 24668 cnncvsabsnegdemo 24673 tcphcph 24745 zringnm 32926 cnzh 32938 rezh 32939 |
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