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Mirrors > Home > MPE Home > Th. List > cphipeq0 | 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. Complex version of ipeq0 20710. (Contributed by Mario Carneiro, 16-Oct-2015.) |
Ref | Expression |
---|---|
cphipcj.h | ⊢ , = (·𝑖‘𝑊) |
cphipcj.v | ⊢ 𝑉 = (Base‘𝑊) |
cphip0l.z | ⊢ 0 = (0g‘𝑊) |
Ref | Expression |
---|---|
cphipeq0 | ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉) → ((𝐴 , 𝐴) = 0 ↔ 𝐴 = 0 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cphclm 23720 | . . . . 5 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ ℂMod) | |
2 | eqid 2818 | . . . . . 6 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
3 | 2 | clm0 23603 | . . . . 5 ⊢ (𝑊 ∈ ℂMod → 0 = (0g‘(Scalar‘𝑊))) |
4 | 1, 3 | syl 17 | . . . 4 ⊢ (𝑊 ∈ ℂPreHil → 0 = (0g‘(Scalar‘𝑊))) |
5 | 4 | adantr 481 | . . 3 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉) → 0 = (0g‘(Scalar‘𝑊))) |
6 | 5 | eqeq2d 2829 | . 2 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉) → ((𝐴 , 𝐴) = 0 ↔ (𝐴 , 𝐴) = (0g‘(Scalar‘𝑊)))) |
7 | cphphl 23702 | . . 3 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ PreHil) | |
8 | cphipcj.h | . . . 4 ⊢ , = (·𝑖‘𝑊) | |
9 | cphipcj.v | . . . 4 ⊢ 𝑉 = (Base‘𝑊) | |
10 | eqid 2818 | . . . 4 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊)) | |
11 | cphip0l.z | . . . 4 ⊢ 0 = (0g‘𝑊) | |
12 | 2, 8, 9, 10, 11 | ipeq0 20710 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉) → ((𝐴 , 𝐴) = (0g‘(Scalar‘𝑊)) ↔ 𝐴 = 0 )) |
13 | 7, 12 | sylan 580 | . 2 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉) → ((𝐴 , 𝐴) = (0g‘(Scalar‘𝑊)) ↔ 𝐴 = 0 )) |
14 | 6, 13 | bitrd 280 | 1 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉) → ((𝐴 , 𝐴) = 0 ↔ 𝐴 = 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ‘cfv 6348 (class class class)co 7145 0cc0 10525 Basecbs 16471 Scalarcsca 16556 ·𝑖cip 16558 0gc0g 16701 PreHilcphl 20696 ℂModcclm 23593 ℂPreHilccph 23697 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-addf 10604 ax-mulf 10605 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-tpos 7881 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12881 df-seq 13358 df-exp 13418 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-starv 16568 df-sca 16569 df-vsca 16570 df-ip 16571 df-tset 16572 df-ple 16573 df-ds 16575 df-unif 16576 df-0g 16703 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-grp 18044 df-subg 18214 df-ghm 18294 df-cmn 18837 df-mgp 19169 df-ur 19181 df-ring 19228 df-cring 19229 df-oppr 19302 df-dvdsr 19320 df-unit 19321 df-drng 19433 df-subrg 19462 df-lmod 19565 df-lmhm 19723 df-lvec 19804 df-sra 19873 df-rgmod 19874 df-cnfld 20474 df-phl 20698 df-nlm 23123 df-clm 23594 df-cph 23699 |
This theorem is referenced by: (None) |
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