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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 3908 | . . . . 5 ⊢ (𝑊 ∈ (NrmVec ∩ ℂVec) ↔ (𝑊 ∈ NrmVec ∧ 𝑊 ∈ ℂVec)) | |
3 | id 22 | . . . . . . 7 ⊢ (𝑊 ∈ ℂVec → 𝑊 ∈ ℂVec) | |
4 | 3 | cvsclm 24287 | . . . . . 6 ⊢ (𝑊 ∈ ℂVec → 𝑊 ∈ ℂMod) |
5 | eqid 2740 | . . . . . . 7 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
6 | eqid 2740 | . . . . . . 7 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
7 | 5, 6 | clmneg1 24243 | . . . . . 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 24314 | . . 3 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ -1 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝐴 ∈ 𝑉) → (𝑁‘(-1 · 𝐴)) = ((abs‘-1) · (𝑁‘𝐴))) |
16 | 1, 10, 11, 15 | syl3anc 1370 | . 2 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴 ∈ 𝑉) → (𝑁‘(-1 · 𝐴)) = ((abs‘-1) · (𝑁‘𝐴))) |
17 | ax-1cn 10930 | . . . . . 6 ⊢ 1 ∈ ℂ | |
18 | 17 | absnegi 15110 | . . . . 5 ⊢ (abs‘-1) = (abs‘1) |
19 | abs1 15007 | . . . . 5 ⊢ (abs‘1) = 1 | |
20 | 18, 19 | eqtri 2768 | . . . 4 ⊢ (abs‘-1) = 1 |
21 | 20 | oveq1i 7281 | . . 3 ⊢ ((abs‘-1) · (𝑁‘𝐴)) = (1 · (𝑁‘𝐴)) |
22 | nvcnlm 23858 | . . . . . . . . 9 ⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ NrmMod) | |
23 | nlmngp 23839 | . . . . . . . . 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 216 | . . . . . 6 ⊢ (𝑊 ∈ (NrmVec ∩ ℂVec) → 𝑊 ∈ NrmGrp) |
27 | 12, 13 | nmcl 23770 | . . . . . 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 11004 | . . . 4 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴 ∈ 𝑉) → (𝑁‘𝐴) ∈ ℂ) |
30 | 29 | mulid2d 10994 | . . 3 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴 ∈ 𝑉) → (1 · (𝑁‘𝐴)) = (𝑁‘𝐴)) |
31 | 21, 30 | eqtrid 2792 | . 2 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴 ∈ 𝑉) → ((abs‘-1) · (𝑁‘𝐴)) = (𝑁‘𝐴)) |
32 | 16, 31 | eqtrd 2780 | 1 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴 ∈ 𝑉) → (𝑁‘(-1 · 𝐴)) = (𝑁‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1542 ∈ wcel 2110 ∩ cin 3891 ‘cfv 6432 (class class class)co 7271 ℝcr 10871 1c1 10873 · cmul 10877 -cneg 11206 abscabs 14943 Basecbs 16910 Scalarcsca 16963 ·𝑠 cvsca 16964 normcnm 23730 NrmGrpcngp 23731 NrmModcnlm 23734 NrmVeccnvc 23735 ℂModcclm 24223 ℂVecccvs 24284 |
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 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 ax-pre-sup 10950 ax-addf 10951 ax-mulf 10952 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-om 7707 df-1st 7824 df-2nd 7825 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-1o 8288 df-er 8481 df-map 8600 df-en 8717 df-dom 8718 df-sdom 8719 df-fin 8720 df-sup 9179 df-inf 9180 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12437 df-uz 12582 df-q 12688 df-rp 12730 df-xneg 12847 df-xadd 12848 df-xmul 12849 df-fz 13239 df-seq 13720 df-exp 13781 df-cj 14808 df-re 14809 df-im 14810 df-sqrt 14944 df-abs 14945 df-struct 16846 df-sets 16863 df-slot 16881 df-ndx 16893 df-base 16911 df-ress 16940 df-plusg 16973 df-mulr 16974 df-starv 16975 df-tset 16979 df-ple 16980 df-ds 16982 df-unif 16983 df-0g 17150 df-topgen 17152 df-mgm 18324 df-sgrp 18373 df-mnd 18384 df-grp 18578 df-minusg 18579 df-mulg 18699 df-subg 18750 df-cmn 19386 df-mgp 19719 df-ur 19736 df-ring 19783 df-cring 19784 df-subrg 20020 df-psmet 20587 df-xmet 20588 df-met 20589 df-bl 20590 df-mopn 20591 df-cnfld 20596 df-top 22041 df-topon 22058 df-topsp 22080 df-bases 22094 df-xms 23471 df-ms 23472 df-nm 23736 df-ngp 23737 df-nlm 23740 df-nvc 23741 df-clm 24224 df-cvs 24285 |
This theorem is referenced by: (None) |
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