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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 483 | . . 3 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴 ∈ 𝑉) → 𝑊 ∈ (NrmVec ∩ ℂVec)) | |
2 | elin 4090 | . . . . 5 ⊢ (𝑊 ∈ (NrmVec ∩ ℂVec) ↔ (𝑊 ∈ NrmVec ∧ 𝑊 ∈ ℂVec)) | |
3 | id 22 | . . . . . . 7 ⊢ (𝑊 ∈ ℂVec → 𝑊 ∈ ℂVec) | |
4 | 3 | cvsclm 23413 | . . . . . 6 ⊢ (𝑊 ∈ ℂVec → 𝑊 ∈ ℂMod) |
5 | eqid 2795 | . . . . . . 7 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
6 | eqid 2795 | . . . . . . 7 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
7 | 5, 6 | clmneg1 23369 | . . . . . 6 ⊢ (𝑊 ∈ ℂMod → -1 ∈ (Base‘(Scalar‘𝑊))) |
8 | 4, 7 | syl 17 | . . . . 5 ⊢ (𝑊 ∈ ℂVec → -1 ∈ (Base‘(Scalar‘𝑊))) |
9 | 2, 8 | simplbiim 505 | . . . 4 ⊢ (𝑊 ∈ (NrmVec ∩ ℂVec) → -1 ∈ (Base‘(Scalar‘𝑊))) |
10 | 9 | adantr 481 | . . 3 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴 ∈ 𝑉) → -1 ∈ (Base‘(Scalar‘𝑊))) |
11 | simpr 485 | . . 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 23439 | . . 3 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ -1 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝐴 ∈ 𝑉) → (𝑁‘(-1 · 𝐴)) = ((abs‘-1) · (𝑁‘𝐴))) |
16 | 1, 10, 11, 15 | syl3anc 1364 | . 2 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴 ∈ 𝑉) → (𝑁‘(-1 · 𝐴)) = ((abs‘-1) · (𝑁‘𝐴))) |
17 | ax-1cn 10441 | . . . . . 6 ⊢ 1 ∈ ℂ | |
18 | 17 | absnegi 14594 | . . . . 5 ⊢ (abs‘-1) = (abs‘1) |
19 | abs1 14491 | . . . . 5 ⊢ (abs‘1) = 1 | |
20 | 18, 19 | eqtri 2819 | . . . 4 ⊢ (abs‘-1) = 1 |
21 | 20 | oveq1i 7026 | . . 3 ⊢ ((abs‘-1) · (𝑁‘𝐴)) = (1 · (𝑁‘𝐴)) |
22 | nvcnlm 22988 | . . . . . . . . 9 ⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ NrmMod) | |
23 | nlmngp 22969 | . . . . . . . . 9 ⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ NrmGrp) | |
24 | 22, 23 | syl 17 | . . . . . . . 8 ⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ NrmGrp) |
25 | 24 | adantr 481 | . . . . . . 7 ⊢ ((𝑊 ∈ NrmVec ∧ 𝑊 ∈ ℂVec) → 𝑊 ∈ NrmGrp) |
26 | 2, 25 | sylbi 218 | . . . . . 6 ⊢ (𝑊 ∈ (NrmVec ∩ ℂVec) → 𝑊 ∈ NrmGrp) |
27 | 12, 13 | nmcl 22908 | . . . . . 6 ⊢ ((𝑊 ∈ NrmGrp ∧ 𝐴 ∈ 𝑉) → (𝑁‘𝐴) ∈ ℝ) |
28 | 26, 27 | sylan 580 | . . . . 5 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴 ∈ 𝑉) → (𝑁‘𝐴) ∈ ℝ) |
29 | 28 | recnd 10515 | . . . 4 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴 ∈ 𝑉) → (𝑁‘𝐴) ∈ ℂ) |
30 | 29 | mulid2d 10505 | . . 3 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴 ∈ 𝑉) → (1 · (𝑁‘𝐴)) = (𝑁‘𝐴)) |
31 | 21, 30 | syl5eq 2843 | . 2 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴 ∈ 𝑉) → ((abs‘-1) · (𝑁‘𝐴)) = (𝑁‘𝐴)) |
32 | 16, 31 | eqtrd 2831 | 1 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴 ∈ 𝑉) → (𝑁‘(-1 · 𝐴)) = (𝑁‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1522 ∈ wcel 2081 ∩ cin 3858 ‘cfv 6225 (class class class)co 7016 ℝcr 10382 1c1 10384 · cmul 10388 -cneg 10718 abscabs 14427 Basecbs 16312 Scalarcsca 16397 ·𝑠 cvsca 16398 normcnm 22869 NrmGrpcngp 22870 NrmModcnlm 22873 NrmVeccnvc 22874 ℂModcclm 23349 ℂVecccvs 23410 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 ax-pre-sup 10461 ax-addf 10462 ax-mulf 10463 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-int 4783 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-om 7437 df-1st 7545 df-2nd 7546 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-1o 7953 df-oadd 7957 df-er 8139 df-map 8258 df-en 8358 df-dom 8359 df-sdom 8360 df-fin 8361 df-sup 8752 df-inf 8753 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-div 11146 df-nn 11487 df-2 11548 df-3 11549 df-4 11550 df-5 11551 df-6 11552 df-7 11553 df-8 11554 df-9 11555 df-n0 11746 df-z 11830 df-dec 11948 df-uz 12094 df-q 12198 df-rp 12240 df-xneg 12357 df-xadd 12358 df-xmul 12359 df-fz 12743 df-seq 13220 df-exp 13280 df-cj 14292 df-re 14293 df-im 14294 df-sqrt 14428 df-abs 14429 df-struct 16314 df-ndx 16315 df-slot 16316 df-base 16318 df-sets 16319 df-ress 16320 df-plusg 16407 df-mulr 16408 df-starv 16409 df-tset 16413 df-ple 16414 df-ds 16416 df-unif 16417 df-0g 16544 df-topgen 16546 df-mgm 17681 df-sgrp 17723 df-mnd 17734 df-grp 17864 df-minusg 17865 df-mulg 17982 df-subg 18030 df-cmn 18635 df-mgp 18930 df-ur 18942 df-ring 18989 df-cring 18990 df-subrg 19223 df-psmet 20219 df-xmet 20220 df-met 20221 df-bl 20222 df-mopn 20223 df-cnfld 20228 df-top 21186 df-topon 21203 df-topsp 21225 df-bases 21238 df-xms 22613 df-ms 22614 df-nm 22875 df-ngp 22876 df-nlm 22879 df-nvc 22880 df-clm 23350 df-cvs 23411 |
This theorem is referenced by: (None) |
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