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Mirrors > Home > MPE Home > Th. List > cnnvnm | Structured version Visualization version GIF version |
Description: The norm operation of the normed complex vector space of complex numbers. (Contributed by NM, 12-Jan-2008.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cnnvnm.6 | ⊢ 𝑈 = 〈〈 + , · 〉, abs〉 |
Ref | Expression |
---|---|
cnnvnm | ⊢ abs = (normCV‘𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2739 | . . 3 ⊢ (normCV‘𝑈) = (normCV‘𝑈) | |
2 | 1 | nmcvfval 28554 | . 2 ⊢ (normCV‘𝑈) = (2nd ‘𝑈) |
3 | cnnvnm.6 | . . 3 ⊢ 𝑈 = 〈〈 + , · 〉, abs〉 | |
4 | 3 | fveq2i 6689 | . 2 ⊢ (2nd ‘𝑈) = (2nd ‘〈〈 + , · 〉, abs〉) |
5 | opex 5332 | . . 3 ⊢ 〈 + , · 〉 ∈ V | |
6 | absf 14799 | . . . 4 ⊢ abs:ℂ⟶ℝ | |
7 | cnex 10708 | . . . 4 ⊢ ℂ ∈ V | |
8 | fex 7011 | . . . 4 ⊢ ((abs:ℂ⟶ℝ ∧ ℂ ∈ V) → abs ∈ V) | |
9 | 6, 7, 8 | mp2an 692 | . . 3 ⊢ abs ∈ V |
10 | 5, 9 | op2nd 7735 | . 2 ⊢ (2nd ‘〈〈 + , · 〉, abs〉) = abs |
11 | 2, 4, 10 | 3eqtrri 2767 | 1 ⊢ abs = (normCV‘𝑈) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2114 Vcvv 3400 〈cop 4532 ⟶wf 6345 ‘cfv 6349 2nd c2nd 7725 ℂcc 10625 ℝcr 10626 + caddc 10630 · cmul 10632 abscabs 14695 normCVcnmcv 28537 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7491 ax-cnex 10683 ax-resscn 10684 ax-1cn 10685 ax-icn 10686 ax-addcl 10687 ax-addrcl 10688 ax-mulcl 10689 ax-mulrcl 10690 ax-mulcom 10691 ax-addass 10692 ax-mulass 10693 ax-distr 10694 ax-i2m1 10695 ax-1ne0 10696 ax-1rid 10697 ax-rnegex 10698 ax-rrecex 10699 ax-cnre 10700 ax-pre-lttri 10701 ax-pre-lttrn 10702 ax-pre-ltadd 10703 ax-pre-mulgt0 10704 ax-pre-sup 10705 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-iun 4893 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6185 df-on 6186 df-lim 6187 df-suc 6188 df-iota 6307 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7139 df-ov 7185 df-oprab 7186 df-mpo 7187 df-om 7612 df-2nd 7727 df-wrecs 7988 df-recs 8049 df-rdg 8087 df-er 8332 df-en 8568 df-dom 8569 df-sdom 8570 df-sup 8991 df-pnf 10767 df-mnf 10768 df-xr 10769 df-ltxr 10770 df-le 10771 df-sub 10962 df-neg 10963 df-div 11388 df-nn 11729 df-2 11791 df-3 11792 df-n0 11989 df-z 12075 df-uz 12337 df-rp 12485 df-seq 13473 df-exp 13534 df-cj 14560 df-re 14561 df-im 14562 df-sqrt 14696 df-abs 14697 df-nmcv 28547 |
This theorem is referenced by: cnims 28640 ipblnfi 28802 htthlem 28864 |
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