Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > ipeq0 | 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.) (Revised by Mario Carneiro, 7-Oct-2015.) |
Ref | Expression |
---|---|
phlsrng.f | ⊢ 𝐹 = (Scalar‘𝑊) |
phllmhm.h | ⊢ , = (·𝑖‘𝑊) |
phllmhm.v | ⊢ 𝑉 = (Base‘𝑊) |
ip0l.z | ⊢ 𝑍 = (0g‘𝐹) |
ip0l.o | ⊢ 0 = (0g‘𝑊) |
Ref | Expression |
---|---|
ipeq0 | ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉) → ((𝐴 , 𝐴) = 𝑍 ↔ 𝐴 = 0 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | phllmhm.v | . . . . . 6 ⊢ 𝑉 = (Base‘𝑊) | |
2 | phlsrng.f | . . . . . 6 ⊢ 𝐹 = (Scalar‘𝑊) | |
3 | phllmhm.h | . . . . . 6 ⊢ , = (·𝑖‘𝑊) | |
4 | ip0l.o | . . . . . 6 ⊢ 0 = (0g‘𝑊) | |
5 | eqid 2818 | . . . . . 6 ⊢ (*𝑟‘𝐹) = (*𝑟‘𝐹) | |
6 | ip0l.z | . . . . . 6 ⊢ 𝑍 = (0g‘𝐹) | |
7 | 1, 2, 3, 4, 5, 6 | isphl 20700 | . . . . 5 ⊢ (𝑊 ∈ PreHil ↔ (𝑊 ∈ LVec ∧ 𝐹 ∈ *-Ring ∧ ∀𝑥 ∈ 𝑉 ((𝑦 ∈ 𝑉 ↦ (𝑦 , 𝑥)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ∧ ((𝑥 , 𝑥) = 𝑍 → 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑉 ((*𝑟‘𝐹)‘(𝑥 , 𝑦)) = (𝑦 , 𝑥)))) |
8 | 7 | simp3bi 1139 | . . . 4 ⊢ (𝑊 ∈ PreHil → ∀𝑥 ∈ 𝑉 ((𝑦 ∈ 𝑉 ↦ (𝑦 , 𝑥)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ∧ ((𝑥 , 𝑥) = 𝑍 → 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑉 ((*𝑟‘𝐹)‘(𝑥 , 𝑦)) = (𝑦 , 𝑥))) |
9 | simp2 1129 | . . . . 5 ⊢ (((𝑦 ∈ 𝑉 ↦ (𝑦 , 𝑥)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ∧ ((𝑥 , 𝑥) = 𝑍 → 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑉 ((*𝑟‘𝐹)‘(𝑥 , 𝑦)) = (𝑦 , 𝑥)) → ((𝑥 , 𝑥) = 𝑍 → 𝑥 = 0 )) | |
10 | 9 | ralimi 3157 | . . . 4 ⊢ (∀𝑥 ∈ 𝑉 ((𝑦 ∈ 𝑉 ↦ (𝑦 , 𝑥)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ∧ ((𝑥 , 𝑥) = 𝑍 → 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑉 ((*𝑟‘𝐹)‘(𝑥 , 𝑦)) = (𝑦 , 𝑥)) → ∀𝑥 ∈ 𝑉 ((𝑥 , 𝑥) = 𝑍 → 𝑥 = 0 )) |
11 | 8, 10 | syl 17 | . . 3 ⊢ (𝑊 ∈ PreHil → ∀𝑥 ∈ 𝑉 ((𝑥 , 𝑥) = 𝑍 → 𝑥 = 0 )) |
12 | oveq12 7154 | . . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑥 = 𝐴) → (𝑥 , 𝑥) = (𝐴 , 𝐴)) | |
13 | 12 | anidms 567 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 , 𝑥) = (𝐴 , 𝐴)) |
14 | 13 | eqeq1d 2820 | . . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑥 , 𝑥) = 𝑍 ↔ (𝐴 , 𝐴) = 𝑍)) |
15 | eqeq1 2822 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 = 0 ↔ 𝐴 = 0 )) | |
16 | 14, 15 | imbi12d 346 | . . . 4 ⊢ (𝑥 = 𝐴 → (((𝑥 , 𝑥) = 𝑍 → 𝑥 = 0 ) ↔ ((𝐴 , 𝐴) = 𝑍 → 𝐴 = 0 ))) |
17 | 16 | rspccva 3619 | . . 3 ⊢ ((∀𝑥 ∈ 𝑉 ((𝑥 , 𝑥) = 𝑍 → 𝑥 = 0 ) ∧ 𝐴 ∈ 𝑉) → ((𝐴 , 𝐴) = 𝑍 → 𝐴 = 0 )) |
18 | 11, 17 | sylan 580 | . 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉) → ((𝐴 , 𝐴) = 𝑍 → 𝐴 = 0 )) |
19 | 2, 3, 1, 6, 4 | ip0l 20708 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉) → ( 0 , 𝐴) = 𝑍) |
20 | oveq1 7152 | . . . 4 ⊢ (𝐴 = 0 → (𝐴 , 𝐴) = ( 0 , 𝐴)) | |
21 | 20 | eqeq1d 2820 | . . 3 ⊢ (𝐴 = 0 → ((𝐴 , 𝐴) = 𝑍 ↔ ( 0 , 𝐴) = 𝑍)) |
22 | 19, 21 | syl5ibrcom 248 | . 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉) → (𝐴 = 0 → (𝐴 , 𝐴) = 𝑍)) |
23 | 18, 22 | impbid 213 | 1 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉) → ((𝐴 , 𝐴) = 𝑍 ↔ 𝐴 = 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ∀wral 3135 ↦ cmpt 5137 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 *𝑟cstv 16555 Scalarcsca 16556 ·𝑖cip 16558 0gc0g 16701 *-Ringcsr 19544 LMHom clmhm 19720 LVecclvec 19803 ringLModcrglmod 19870 PreHilcphl 20696 |
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 |
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-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-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 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-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-plusg 16566 df-sca 16569 df-vsca 16570 df-ip 16571 df-0g 16703 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-grp 18044 df-ghm 18294 df-lmod 19565 df-lmhm 19723 df-lvec 19804 df-sra 19873 df-rgmod 19874 df-phl 20698 |
This theorem is referenced by: ip2eq 20725 phlssphl 20731 ocvin 20746 lsmcss 20764 obsne0 20797 cphipeq0 23735 ipcau2 23764 tcphcph 23767 |
Copyright terms: Public domain | W3C validator |