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Mirrors > Home > MPE Home > Th. List > ncvsdif | Structured version Visualization version GIF version |
Description: The norm of the difference between two vectors. (Contributed by NM, 1-Dec-2006.) (Revised by AV, 8-Oct-2021.) |
Ref | Expression |
---|---|
ncvsprp.v | ⊢ 𝑉 = (Base‘𝑊) |
ncvsprp.n | ⊢ 𝑁 = (norm‘𝑊) |
ncvsprp.s | ⊢ · = ( ·𝑠 ‘𝑊) |
ncvsdif.p | ⊢ + = (+g‘𝑊) |
Ref | Expression |
---|---|
ncvsdif | ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝑁‘(𝐴 + (-1 · 𝐵))) = (𝑁‘(𝐵 + (-1 · 𝐴)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elin 3960 | . . . . 5 ⊢ (𝑊 ∈ (NrmVec ∩ ℂVec) ↔ (𝑊 ∈ NrmVec ∧ 𝑊 ∈ ℂVec)) | |
2 | id 22 | . . . . . 6 ⊢ (𝑊 ∈ ℂVec → 𝑊 ∈ ℂVec) | |
3 | 2 | cvsclm 25097 | . . . . 5 ⊢ (𝑊 ∈ ℂVec → 𝑊 ∈ ℂMod) |
4 | 1, 3 | simplbiim 503 | . . . 4 ⊢ (𝑊 ∈ (NrmVec ∩ ℂVec) → 𝑊 ∈ ℂMod) |
5 | ncvsprp.v | . . . . . 6 ⊢ 𝑉 = (Base‘𝑊) | |
6 | ncvsdif.p | . . . . . 6 ⊢ + = (+g‘𝑊) | |
7 | eqid 2725 | . . . . . 6 ⊢ (-g‘𝑊) = (-g‘𝑊) | |
8 | eqid 2725 | . . . . . 6 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
9 | ncvsprp.s | . . . . . 6 ⊢ · = ( ·𝑠 ‘𝑊) | |
10 | 5, 6, 7, 8, 9 | clmvsubval 25080 | . . . . 5 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴(-g‘𝑊)𝐵) = (𝐴 + (-1 · 𝐵))) |
11 | 10 | eqcomd 2731 | . . . 4 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴 + (-1 · 𝐵)) = (𝐴(-g‘𝑊)𝐵)) |
12 | 4, 11 | syl3an1 1160 | . . 3 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴 + (-1 · 𝐵)) = (𝐴(-g‘𝑊)𝐵)) |
13 | 12 | fveq2d 6900 | . 2 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝑁‘(𝐴 + (-1 · 𝐵))) = (𝑁‘(𝐴(-g‘𝑊)𝐵))) |
14 | nvcnlm 24657 | . . . . . 6 ⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ NrmMod) | |
15 | nlmngp 24638 | . . . . . 6 ⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ NrmGrp) | |
16 | 14, 15 | syl 17 | . . . . 5 ⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ NrmGrp) |
17 | 16 | adantr 479 | . . . 4 ⊢ ((𝑊 ∈ NrmVec ∧ 𝑊 ∈ ℂVec) → 𝑊 ∈ NrmGrp) |
18 | 1, 17 | sylbi 216 | . . 3 ⊢ (𝑊 ∈ (NrmVec ∩ ℂVec) → 𝑊 ∈ NrmGrp) |
19 | ncvsprp.n | . . . 4 ⊢ 𝑁 = (norm‘𝑊) | |
20 | 5, 19, 7 | nmsub 24576 | . . 3 ⊢ ((𝑊 ∈ NrmGrp ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝑁‘(𝐴(-g‘𝑊)𝐵)) = (𝑁‘(𝐵(-g‘𝑊)𝐴))) |
21 | 18, 20 | syl3an1 1160 | . 2 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝑁‘(𝐴(-g‘𝑊)𝐵)) = (𝑁‘(𝐵(-g‘𝑊)𝐴))) |
22 | 4 | 3ad2ant1 1130 | . . . 4 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝑊 ∈ ℂMod) |
23 | simp3 1135 | . . . 4 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ 𝑉) | |
24 | simp2 1134 | . . . 4 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ 𝑉) | |
25 | 5, 6, 7, 8, 9 | clmvsubval 25080 | . . . 4 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → (𝐵(-g‘𝑊)𝐴) = (𝐵 + (-1 · 𝐴))) |
26 | 22, 23, 24, 25 | syl3anc 1368 | . . 3 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐵(-g‘𝑊)𝐴) = (𝐵 + (-1 · 𝐴))) |
27 | 26 | fveq2d 6900 | . 2 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝑁‘(𝐵(-g‘𝑊)𝐴)) = (𝑁‘(𝐵 + (-1 · 𝐴)))) |
28 | 13, 21, 27 | 3eqtrd 2769 | 1 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝑁‘(𝐴 + (-1 · 𝐵))) = (𝑁‘(𝐵 + (-1 · 𝐴)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∩ cin 3943 ‘cfv 6549 (class class class)co 7419 1c1 11141 -cneg 11477 Basecbs 17183 +gcplusg 17236 Scalarcsca 17239 ·𝑠 cvsca 17240 -gcsg 18900 normcnm 24529 NrmGrpcngp 24530 NrmModcnlm 24533 NrmVeccnvc 24534 ℂModcclm 25033 ℂVecccvs 25094 |
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 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-pre-sup 11218 ax-addf 11219 |
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 9467 df-inf 9468 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12506 df-z 12592 df-dec 12711 df-uz 12856 df-q 12966 df-rp 13010 df-xneg 13127 df-xadd 13128 df-xmul 13129 df-fz 13520 df-struct 17119 df-sets 17136 df-slot 17154 df-ndx 17166 df-base 17184 df-ress 17213 df-plusg 17249 df-mulr 17250 df-starv 17251 df-tset 17255 df-ple 17256 df-ds 17258 df-unif 17259 df-0g 17426 df-topgen 17428 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-grp 18901 df-minusg 18902 df-sbg 18903 df-subg 19086 df-cmn 19749 df-mgp 20087 df-ur 20134 df-ring 20187 df-cring 20188 df-subrg 20520 df-lmod 20757 df-psmet 21288 df-xmet 21289 df-met 21290 df-bl 21291 df-mopn 21292 df-cnfld 21297 df-top 22840 df-topon 22857 df-topsp 22879 df-bases 22893 df-xms 24270 df-ms 24271 df-nm 24535 df-ngp 24536 df-nlm 24539 df-nvc 24540 df-clm 25034 df-cvs 25095 |
This theorem is referenced by: (None) |
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