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Mirrors > Home > MPE Home > Th. List > cnnv | Structured version Visualization version GIF version |
Description: The set of complex numbers is a normed complex vector space. The vector operation is +, the scalar product is ·, and the norm function is abs. (Contributed by Steve Rodriguez, 3-Dec-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cnnv.6 | ⊢ 𝑈 = 〈〈 + , · 〉, abs〉 |
Ref | Expression |
---|---|
cnnv | ⊢ 𝑈 ∈ NrmCVec |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnaddabloOLD 28022 | . . . 4 ⊢ + ∈ AbelOp | |
2 | ablogrpo 27988 | . . . 4 ⊢ ( + ∈ AbelOp → + ∈ GrpOp) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ + ∈ GrpOp |
4 | ax-addf 10351 | . . . 4 ⊢ + :(ℂ × ℂ)⟶ℂ | |
5 | 4 | fdmi 6301 | . . 3 ⊢ dom + = (ℂ × ℂ) |
6 | 3, 5 | grporn 27962 | . 2 ⊢ ℂ = ran + |
7 | cnidOLD 28023 | . 2 ⊢ 0 = (GId‘ + ) | |
8 | cncvcOLD 28024 | . 2 ⊢ 〈 + , · 〉 ∈ CVecOLD | |
9 | absf 14484 | . 2 ⊢ abs:ℂ⟶ℝ | |
10 | abs00 14436 | . . 3 ⊢ (𝑥 ∈ ℂ → ((abs‘𝑥) = 0 ↔ 𝑥 = 0)) | |
11 | 10 | biimpa 470 | . 2 ⊢ ((𝑥 ∈ ℂ ∧ (abs‘𝑥) = 0) → 𝑥 = 0) |
12 | absmul 14441 | . 2 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (abs‘(𝑦 · 𝑥)) = ((abs‘𝑦) · (abs‘𝑥))) | |
13 | abstri 14477 | . 2 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (abs‘(𝑥 + 𝑦)) ≤ ((abs‘𝑥) + (abs‘𝑦))) | |
14 | cnnv.6 | . 2 ⊢ 𝑈 = 〈〈 + , · 〉, abs〉 | |
15 | 6, 7, 8, 9, 11, 12, 13, 14 | isnvi 28054 | 1 ⊢ 𝑈 ∈ NrmCVec |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1601 ∈ wcel 2106 〈cop 4403 × cxp 5353 ‘cfv 6135 ℂcc 10270 0cc0 10272 + caddc 10275 · cmul 10277 abscabs 14381 GrpOpcgr 27930 AbelOpcablo 27985 NrmCVeccnv 28025 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350 ax-addf 10351 ax-mulf 10352 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-sup 8636 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-3 11439 df-n0 11643 df-z 11729 df-uz 11993 df-rp 12138 df-seq 13120 df-exp 13179 df-cj 14246 df-re 14247 df-im 14248 df-sqrt 14382 df-abs 14383 df-grpo 27934 df-gid 27935 df-ablo 27986 df-vc 28000 df-nv 28033 |
This theorem is referenced by: cnnvm 28123 elimnvu 28125 cnims 28134 cncph 28260 ipblnfi 28297 cnbn 28311 htthlem 28360 |
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