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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 30509 | . . . 4 ⊢ + ∈ AbelOp | |
2 | ablogrpo 30475 | . . . 4 ⊢ ( + ∈ AbelOp → + ∈ GrpOp) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ + ∈ GrpOp |
4 | ax-addf 11226 | . . . 4 ⊢ + :(ℂ × ℂ)⟶ℂ | |
5 | 4 | fdmi 6729 | . . 3 ⊢ dom + = (ℂ × ℂ) |
6 | 3, 5 | grporn 30449 | . 2 ⊢ ℂ = ran + |
7 | cnidOLD 30510 | . 2 ⊢ 0 = (GId‘ + ) | |
8 | cncvcOLD 30511 | . 2 ⊢ 〈 + , · 〉 ∈ CVecOLD | |
9 | absf 15335 | . 2 ⊢ abs:ℂ⟶ℝ | |
10 | abs00 15287 | . . 3 ⊢ (𝑥 ∈ ℂ → ((abs‘𝑥) = 0 ↔ 𝑥 = 0)) | |
11 | 10 | biimpa 475 | . 2 ⊢ ((𝑥 ∈ ℂ ∧ (abs‘𝑥) = 0) → 𝑥 = 0) |
12 | absmul 15292 | . 2 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (abs‘(𝑦 · 𝑥)) = ((abs‘𝑦) · (abs‘𝑥))) | |
13 | abstri 15328 | . 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 30541 | 1 ⊢ 𝑈 ∈ NrmCVec |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∈ wcel 2099 〈cop 4630 × cxp 5671 ‘cfv 6544 ℂcc 11145 0cc0 11147 + caddc 11150 · cmul 11152 abscabs 15232 GrpOpcgr 30417 AbelOpcablo 30472 NrmCVeccnv 30512 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7736 ax-cnex 11203 ax-resscn 11204 ax-1cn 11205 ax-icn 11206 ax-addcl 11207 ax-addrcl 11208 ax-mulcl 11209 ax-mulrcl 11210 ax-mulcom 11211 ax-addass 11212 ax-mulass 11213 ax-distr 11214 ax-i2m1 11215 ax-1ne0 11216 ax-1rid 11217 ax-rnegex 11218 ax-rrecex 11219 ax-cnre 11220 ax-pre-lttri 11221 ax-pre-lttrn 11222 ax-pre-ltadd 11223 ax-pre-mulgt0 11224 ax-pre-sup 11225 ax-addf 11226 ax-mulf 11227 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3365 df-reu 3366 df-rab 3421 df-v 3465 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4324 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4907 df-iun 4996 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6303 df-ord 6369 df-on 6370 df-lim 6371 df-suc 6372 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7867 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-er 8724 df-en 8965 df-dom 8966 df-sdom 8967 df-sup 9476 df-pnf 11289 df-mnf 11290 df-xr 11291 df-ltxr 11292 df-le 11293 df-sub 11485 df-neg 11486 df-div 11911 df-nn 12257 df-2 12319 df-3 12320 df-n0 12517 df-z 12603 df-uz 12867 df-rp 13021 df-seq 14014 df-exp 14074 df-cj 15097 df-re 15098 df-im 15099 df-sqrt 15233 df-abs 15234 df-grpo 30421 df-gid 30422 df-ablo 30473 df-vc 30487 df-nv 30520 |
This theorem is referenced by: cnnvm 30610 elimnvu 30612 cnims 30621 cncph 30747 ipblnfi 30783 cnbn 30797 htthlem 30845 |
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