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Mirrors > Home > MPE Home > Th. List > ncvsm1 | Structured version Visualization version GIF version |
Description: The norm of the opposite of a vector. (Contributed by NM, 28-Nov-2006.) (Revised by AV, 8-Oct-2021.) |
Ref | Expression |
---|---|
ncvsprp.v | ⊢ 𝑉 = (Base‘𝑊) |
ncvsprp.n | ⊢ 𝑁 = (norm‘𝑊) |
ncvsprp.s | ⊢ · = ( ·𝑠 ‘𝑊) |
Ref | Expression |
---|---|
ncvsm1 | ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴 ∈ 𝑉) → (𝑁‘(-1 · 𝐴)) = (𝑁‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 481 | . . 3 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴 ∈ 𝑉) → 𝑊 ∈ (NrmVec ∩ ℂVec)) | |
2 | elin 3960 | . . . . 5 ⊢ (𝑊 ∈ (NrmVec ∩ ℂVec) ↔ (𝑊 ∈ NrmVec ∧ 𝑊 ∈ ℂVec)) | |
3 | id 22 | . . . . . . 7 ⊢ (𝑊 ∈ ℂVec → 𝑊 ∈ ℂVec) | |
4 | 3 | cvsclm 25114 | . . . . . 6 ⊢ (𝑊 ∈ ℂVec → 𝑊 ∈ ℂMod) |
5 | eqid 2725 | . . . . . . 7 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
6 | eqid 2725 | . . . . . . 7 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
7 | 5, 6 | clmneg1 25070 | . . . . . 6 ⊢ (𝑊 ∈ ℂMod → -1 ∈ (Base‘(Scalar‘𝑊))) |
8 | 4, 7 | syl 17 | . . . . 5 ⊢ (𝑊 ∈ ℂVec → -1 ∈ (Base‘(Scalar‘𝑊))) |
9 | 2, 8 | simplbiim 503 | . . . 4 ⊢ (𝑊 ∈ (NrmVec ∩ ℂVec) → -1 ∈ (Base‘(Scalar‘𝑊))) |
10 | 9 | adantr 479 | . . 3 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴 ∈ 𝑉) → -1 ∈ (Base‘(Scalar‘𝑊))) |
11 | simpr 483 | . . 3 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴 ∈ 𝑉) → 𝐴 ∈ 𝑉) | |
12 | ncvsprp.v | . . . 4 ⊢ 𝑉 = (Base‘𝑊) | |
13 | ncvsprp.n | . . . 4 ⊢ 𝑁 = (norm‘𝑊) | |
14 | ncvsprp.s | . . . 4 ⊢ · = ( ·𝑠 ‘𝑊) | |
15 | 12, 13, 14, 5, 6 | ncvsprp 25141 | . . 3 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ -1 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝐴 ∈ 𝑉) → (𝑁‘(-1 · 𝐴)) = ((abs‘-1) · (𝑁‘𝐴))) |
16 | 1, 10, 11, 15 | syl3anc 1368 | . 2 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴 ∈ 𝑉) → (𝑁‘(-1 · 𝐴)) = ((abs‘-1) · (𝑁‘𝐴))) |
17 | ax-1cn 11203 | . . . . . 6 ⊢ 1 ∈ ℂ | |
18 | 17 | absnegi 15391 | . . . . 5 ⊢ (abs‘-1) = (abs‘1) |
19 | abs1 15288 | . . . . 5 ⊢ (abs‘1) = 1 | |
20 | 18, 19 | eqtri 2753 | . . . 4 ⊢ (abs‘-1) = 1 |
21 | 20 | oveq1i 7429 | . . 3 ⊢ ((abs‘-1) · (𝑁‘𝐴)) = (1 · (𝑁‘𝐴)) |
22 | nvcnlm 24674 | . . . . . . . . 9 ⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ NrmMod) | |
23 | nlmngp 24655 | . . . . . . . . 9 ⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ NrmGrp) | |
24 | 22, 23 | syl 17 | . . . . . . . 8 ⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ NrmGrp) |
25 | 24 | adantr 479 | . . . . . . 7 ⊢ ((𝑊 ∈ NrmVec ∧ 𝑊 ∈ ℂVec) → 𝑊 ∈ NrmGrp) |
26 | 2, 25 | sylbi 216 | . . . . . 6 ⊢ (𝑊 ∈ (NrmVec ∩ ℂVec) → 𝑊 ∈ NrmGrp) |
27 | 12, 13 | nmcl 24586 | . . . . . 6 ⊢ ((𝑊 ∈ NrmGrp ∧ 𝐴 ∈ 𝑉) → (𝑁‘𝐴) ∈ ℝ) |
28 | 26, 27 | sylan 578 | . . . . 5 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴 ∈ 𝑉) → (𝑁‘𝐴) ∈ ℝ) |
29 | 28 | recnd 11279 | . . . 4 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴 ∈ 𝑉) → (𝑁‘𝐴) ∈ ℂ) |
30 | 29 | mullidd 11269 | . . 3 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴 ∈ 𝑉) → (1 · (𝑁‘𝐴)) = (𝑁‘𝐴)) |
31 | 21, 30 | eqtrid 2777 | . 2 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴 ∈ 𝑉) → ((abs‘-1) · (𝑁‘𝐴)) = (𝑁‘𝐴)) |
32 | 16, 31 | eqtrd 2765 | 1 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴 ∈ 𝑉) → (𝑁‘(-1 · 𝐴)) = (𝑁‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∩ cin 3943 ‘cfv 6549 (class class class)co 7419 ℝcr 11144 1c1 11146 · cmul 11150 -cneg 11482 abscabs 15225 Basecbs 17199 Scalarcsca 17255 ·𝑠 cvsca 17256 normcnm 24546 NrmGrpcngp 24547 NrmModcnlm 24550 NrmVeccnvc 24551 ℂModcclm 25050 ℂVecccvs 25111 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11201 ax-resscn 11202 ax-1cn 11203 ax-icn 11204 ax-addcl 11205 ax-addrcl 11206 ax-mulcl 11207 ax-mulrcl 11208 ax-mulcom 11209 ax-addass 11210 ax-mulass 11211 ax-distr 11212 ax-i2m1 11213 ax-1ne0 11214 ax-1rid 11215 ax-rnegex 11216 ax-rrecex 11217 ax-cnre 11218 ax-pre-lttri 11219 ax-pre-lttrn 11220 ax-pre-ltadd 11221 ax-pre-mulgt0 11222 ax-pre-sup 11223 ax-addf 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9472 df-inf 9473 df-pnf 11287 df-mnf 11288 df-xr 11289 df-ltxr 11290 df-le 11291 df-sub 11483 df-neg 11484 df-div 11909 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-q 12971 df-rp 13015 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-fz 13525 df-seq 14008 df-exp 14068 df-cj 15090 df-re 15091 df-im 15092 df-sqrt 15226 df-abs 15227 df-struct 17135 df-sets 17152 df-slot 17170 df-ndx 17182 df-base 17200 df-ress 17229 df-plusg 17265 df-mulr 17266 df-starv 17267 df-tset 17271 df-ple 17272 df-ds 17274 df-unif 17275 df-0g 17442 df-topgen 17444 df-mgm 18619 df-sgrp 18698 df-mnd 18714 df-grp 18917 df-minusg 18918 df-mulg 19048 df-subg 19103 df-cmn 19766 df-mgp 20104 df-ur 20151 df-ring 20204 df-cring 20205 df-subrg 20537 df-psmet 21305 df-xmet 21306 df-met 21307 df-bl 21308 df-mopn 21309 df-cnfld 21314 df-top 22857 df-topon 22874 df-topsp 22896 df-bases 22910 df-xms 24287 df-ms 24288 df-nm 24552 df-ngp 24553 df-nlm 24556 df-nvc 24557 df-clm 25051 df-cvs 25112 |
This theorem is referenced by: (None) |
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