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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 485 | . . 3 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴 ∈ 𝑉) → 𝑊 ∈ (NrmVec ∩ ℂVec)) | |
2 | elin 4168 | . . . . 5 ⊢ (𝑊 ∈ (NrmVec ∩ ℂVec) ↔ (𝑊 ∈ NrmVec ∧ 𝑊 ∈ ℂVec)) | |
3 | id 22 | . . . . . . 7 ⊢ (𝑊 ∈ ℂVec → 𝑊 ∈ ℂVec) | |
4 | 3 | cvsclm 23724 | . . . . . 6 ⊢ (𝑊 ∈ ℂVec → 𝑊 ∈ ℂMod) |
5 | eqid 2821 | . . . . . . 7 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
6 | eqid 2821 | . . . . . . 7 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
7 | 5, 6 | clmneg1 23680 | . . . . . 6 ⊢ (𝑊 ∈ ℂMod → -1 ∈ (Base‘(Scalar‘𝑊))) |
8 | 4, 7 | syl 17 | . . . . 5 ⊢ (𝑊 ∈ ℂVec → -1 ∈ (Base‘(Scalar‘𝑊))) |
9 | 2, 8 | simplbiim 507 | . . . 4 ⊢ (𝑊 ∈ (NrmVec ∩ ℂVec) → -1 ∈ (Base‘(Scalar‘𝑊))) |
10 | 9 | adantr 483 | . . 3 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴 ∈ 𝑉) → -1 ∈ (Base‘(Scalar‘𝑊))) |
11 | simpr 487 | . . 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 23750 | . . 3 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ -1 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝐴 ∈ 𝑉) → (𝑁‘(-1 · 𝐴)) = ((abs‘-1) · (𝑁‘𝐴))) |
16 | 1, 10, 11, 15 | syl3anc 1367 | . 2 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴 ∈ 𝑉) → (𝑁‘(-1 · 𝐴)) = ((abs‘-1) · (𝑁‘𝐴))) |
17 | ax-1cn 10589 | . . . . . 6 ⊢ 1 ∈ ℂ | |
18 | 17 | absnegi 14754 | . . . . 5 ⊢ (abs‘-1) = (abs‘1) |
19 | abs1 14651 | . . . . 5 ⊢ (abs‘1) = 1 | |
20 | 18, 19 | eqtri 2844 | . . . 4 ⊢ (abs‘-1) = 1 |
21 | 20 | oveq1i 7160 | . . 3 ⊢ ((abs‘-1) · (𝑁‘𝐴)) = (1 · (𝑁‘𝐴)) |
22 | nvcnlm 23299 | . . . . . . . . 9 ⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ NrmMod) | |
23 | nlmngp 23280 | . . . . . . . . 9 ⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ NrmGrp) | |
24 | 22, 23 | syl 17 | . . . . . . . 8 ⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ NrmGrp) |
25 | 24 | adantr 483 | . . . . . . 7 ⊢ ((𝑊 ∈ NrmVec ∧ 𝑊 ∈ ℂVec) → 𝑊 ∈ NrmGrp) |
26 | 2, 25 | sylbi 219 | . . . . . 6 ⊢ (𝑊 ∈ (NrmVec ∩ ℂVec) → 𝑊 ∈ NrmGrp) |
27 | 12, 13 | nmcl 23219 | . . . . . 6 ⊢ ((𝑊 ∈ NrmGrp ∧ 𝐴 ∈ 𝑉) → (𝑁‘𝐴) ∈ ℝ) |
28 | 26, 27 | sylan 582 | . . . . 5 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴 ∈ 𝑉) → (𝑁‘𝐴) ∈ ℝ) |
29 | 28 | recnd 10663 | . . . 4 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴 ∈ 𝑉) → (𝑁‘𝐴) ∈ ℂ) |
30 | 29 | mulid2d 10653 | . . 3 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴 ∈ 𝑉) → (1 · (𝑁‘𝐴)) = (𝑁‘𝐴)) |
31 | 21, 30 | syl5eq 2868 | . 2 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴 ∈ 𝑉) → ((abs‘-1) · (𝑁‘𝐴)) = (𝑁‘𝐴)) |
32 | 16, 31 | eqtrd 2856 | 1 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴 ∈ 𝑉) → (𝑁‘(-1 · 𝐴)) = (𝑁‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∩ cin 3934 ‘cfv 6349 (class class class)co 7150 ℝcr 10530 1c1 10532 · cmul 10536 -cneg 10865 abscabs 14587 Basecbs 16477 Scalarcsca 16562 ·𝑠 cvsca 16563 normcnm 23180 NrmGrpcngp 23181 NrmModcnlm 23184 NrmVeccnvc 23185 ℂModcclm 23660 ℂVecccvs 23721 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 ax-addf 10610 ax-mulf 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-sup 8900 df-inf 8901 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-q 12343 df-rp 12384 df-xneg 12501 df-xadd 12502 df-xmul 12503 df-fz 12887 df-seq 13364 df-exp 13424 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-starv 16574 df-tset 16578 df-ple 16579 df-ds 16581 df-unif 16582 df-0g 16709 df-topgen 16711 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-grp 18100 df-minusg 18101 df-mulg 18219 df-subg 18270 df-cmn 18902 df-mgp 19234 df-ur 19246 df-ring 19293 df-cring 19294 df-subrg 19527 df-psmet 20531 df-xmet 20532 df-met 20533 df-bl 20534 df-mopn 20535 df-cnfld 20540 df-top 21496 df-topon 21513 df-topsp 21535 df-bases 21548 df-xms 22924 df-ms 22925 df-nm 23186 df-ngp 23187 df-nlm 23190 df-nvc 23191 df-clm 23661 df-cvs 23722 |
This theorem is referenced by: (None) |
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