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Mirrors > Home > MPE Home > Th. List > ipz | Structured version Visualization version GIF version |
Description: The inner product of a vector with itself is zero iff the vector is zero. Part of Definition 3.1-1 of [Kreyszig] p. 129. (Contributed by NM, 24-Jan-2008.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dip0r.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
dip0r.5 | ⊢ 𝑍 = (0vec‘𝑈) |
dip0r.7 | ⊢ 𝑃 = (·𝑖OLD‘𝑈) |
Ref | Expression |
---|---|
ipz | ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((𝐴𝑃𝐴) = 0 ↔ 𝐴 = 𝑍)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dip0r.1 | . . . 4 ⊢ 𝑋 = (BaseSet‘𝑈) | |
2 | eqid 2821 | . . . 4 ⊢ (normCV‘𝑈) = (normCV‘𝑈) | |
3 | dip0r.7 | . . . 4 ⊢ 𝑃 = (·𝑖OLD‘𝑈) | |
4 | 1, 2, 3 | ipidsq 28486 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴𝑃𝐴) = (((normCV‘𝑈)‘𝐴)↑2)) |
5 | 4 | eqeq1d 2823 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((𝐴𝑃𝐴) = 0 ↔ (((normCV‘𝑈)‘𝐴)↑2) = 0)) |
6 | 1, 2 | nvcl 28437 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((normCV‘𝑈)‘𝐴) ∈ ℝ) |
7 | 6 | recnd 10668 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((normCV‘𝑈)‘𝐴) ∈ ℂ) |
8 | sqeq0 13485 | . . 3 ⊢ (((normCV‘𝑈)‘𝐴) ∈ ℂ → ((((normCV‘𝑈)‘𝐴)↑2) = 0 ↔ ((normCV‘𝑈)‘𝐴) = 0)) | |
9 | 7, 8 | syl 17 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((((normCV‘𝑈)‘𝐴)↑2) = 0 ↔ ((normCV‘𝑈)‘𝐴) = 0)) |
10 | dip0r.5 | . . 3 ⊢ 𝑍 = (0vec‘𝑈) | |
11 | 1, 10, 2 | nvz 28445 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (((normCV‘𝑈)‘𝐴) = 0 ↔ 𝐴 = 𝑍)) |
12 | 5, 9, 11 | 3bitrd 307 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((𝐴𝑃𝐴) = 0 ↔ 𝐴 = 𝑍)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ‘cfv 6354 (class class class)co 7155 ℂcc 10534 0cc0 10536 2c2 11691 ↑cexp 13428 NrmCVeccnv 28360 BaseSetcba 28362 0veccn0v 28364 normCVcnmcv 28366 ·𝑖OLDcdip 28476 |
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-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-inf2 9103 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 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 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-se 5514 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-isom 6363 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-sup 8905 df-oi 8973 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-2 11699 df-3 11700 df-4 11701 df-n0 11897 df-z 11981 df-uz 12243 df-rp 12389 df-fz 12892 df-fzo 13033 df-seq 13369 df-exp 13429 df-hash 13690 df-cj 14457 df-re 14458 df-im 14459 df-sqrt 14593 df-abs 14594 df-clim 14844 df-sum 15042 df-grpo 28269 df-gid 28270 df-ginv 28271 df-ablo 28321 df-vc 28335 df-nv 28368 df-va 28371 df-ba 28372 df-sm 28373 df-0v 28374 df-nmcv 28376 df-dip 28477 |
This theorem is referenced by: ip2eqi 28632 |
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