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Mirrors > Home > MPE Home > Th. List > nvm1 | Structured version Visualization version GIF version |
Description: The norm of the negative of a vector. (Contributed by NM, 28-Nov-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nvs.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
nvs.4 | ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) |
nvs.6 | ⊢ 𝑁 = (normCV‘𝑈) |
Ref | Expression |
---|---|
nvm1 | ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘(-1𝑆𝐴)) = (𝑁‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neg1cn 11749 | . . 3 ⊢ -1 ∈ ℂ | |
2 | nvs.1 | . . . 4 ⊢ 𝑋 = (BaseSet‘𝑈) | |
3 | nvs.4 | . . . 4 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) | |
4 | nvs.6 | . . . 4 ⊢ 𝑁 = (normCV‘𝑈) | |
5 | 2, 3, 4 | nvs 28438 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ -1 ∈ ℂ ∧ 𝐴 ∈ 𝑋) → (𝑁‘(-1𝑆𝐴)) = ((abs‘-1) · (𝑁‘𝐴))) |
6 | 1, 5 | mp3an2 1444 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘(-1𝑆𝐴)) = ((abs‘-1) · (𝑁‘𝐴))) |
7 | ax-1cn 10592 | . . . . . 6 ⊢ 1 ∈ ℂ | |
8 | 7 | absnegi 14756 | . . . . 5 ⊢ (abs‘-1) = (abs‘1) |
9 | abs1 14653 | . . . . 5 ⊢ (abs‘1) = 1 | |
10 | 8, 9 | eqtri 2843 | . . . 4 ⊢ (abs‘-1) = 1 |
11 | 10 | oveq1i 7163 | . . 3 ⊢ ((abs‘-1) · (𝑁‘𝐴)) = (1 · (𝑁‘𝐴)) |
12 | 2, 4 | nvcl 28436 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ ℝ) |
13 | 12 | recnd 10666 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ ℂ) |
14 | 13 | mulid2d 10656 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (1 · (𝑁‘𝐴)) = (𝑁‘𝐴)) |
15 | 11, 14 | syl5eq 2867 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((abs‘-1) · (𝑁‘𝐴)) = (𝑁‘𝐴)) |
16 | 6, 15 | eqtrd 2855 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘(-1𝑆𝐴)) = (𝑁‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ‘cfv 6352 (class class class)co 7153 ℂcc 10532 1c1 10535 · cmul 10539 -cneg 10868 abscabs 14589 NrmCVeccnv 28359 BaseSetcba 28361 ·𝑠OLD cns 28362 normCVcnmcv 28365 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5187 ax-sep 5200 ax-nul 5207 ax-pow 5263 ax-pr 5327 ax-un 7458 ax-cnex 10590 ax-resscn 10591 ax-1cn 10592 ax-icn 10593 ax-addcl 10594 ax-addrcl 10595 ax-mulcl 10596 ax-mulrcl 10597 ax-mulcom 10598 ax-addass 10599 ax-mulass 10600 ax-distr 10601 ax-i2m1 10602 ax-1ne0 10603 ax-1rid 10604 ax-rnegex 10605 ax-rrecex 10606 ax-cnre 10607 ax-pre-lttri 10608 ax-pre-lttrn 10609 ax-pre-ltadd 10610 ax-pre-mulgt0 10611 ax-pre-sup 10612 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3495 df-sbc 3771 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4465 df-pw 4538 df-sn 4565 df-pr 4567 df-tp 4569 df-op 4571 df-uni 4836 df-iun 4918 df-br 5064 df-opab 5126 df-mpt 5144 df-tr 5170 df-id 5457 df-eprel 5462 df-po 5471 df-so 5472 df-fr 5511 df-we 5513 df-xp 5558 df-rel 5559 df-cnv 5560 df-co 5561 df-dm 5562 df-rn 5563 df-res 5564 df-ima 5565 df-pred 6145 df-ord 6191 df-on 6192 df-lim 6193 df-suc 6194 df-iota 6311 df-fun 6354 df-fn 6355 df-f 6356 df-f1 6357 df-fo 6358 df-f1o 6359 df-fv 6360 df-riota 7111 df-ov 7156 df-oprab 7157 df-mpo 7158 df-om 7578 df-1st 7686 df-2nd 7687 df-wrecs 7944 df-recs 8005 df-rdg 8043 df-er 8286 df-en 8507 df-dom 8508 df-sdom 8509 df-sup 8903 df-pnf 10674 df-mnf 10675 df-xr 10676 df-ltxr 10677 df-le 10678 df-sub 10869 df-neg 10870 df-div 11295 df-nn 11636 df-2 11698 df-3 11699 df-n0 11896 df-z 11980 df-uz 12242 df-rp 12388 df-seq 13368 df-exp 13428 df-cj 14454 df-re 14455 df-im 14456 df-sqrt 14590 df-abs 14591 df-vc 28334 df-nv 28367 df-va 28370 df-ba 28371 df-sm 28372 df-0v 28373 df-nmcv 28375 |
This theorem is referenced by: nvdif 28441 nvge0 28448 |
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