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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 2821 | . . . . . 6 ⊢ (*𝑟‘𝐹) = (*𝑟‘𝐹) | |
6 | ip0l.z | . . . . . 6 ⊢ 𝑍 = (0g‘𝐹) | |
7 | 1, 2, 3, 4, 5, 6 | isphl 20771 | . . . . 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 1143 | . . . 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 1133 | . . . . 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 3160 | . . . 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 7164 | . . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑥 = 𝐴) → (𝑥 , 𝑥) = (𝐴 , 𝐴)) | |
13 | 12 | anidms 569 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 , 𝑥) = (𝐴 , 𝐴)) |
14 | 13 | eqeq1d 2823 | . . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑥 , 𝑥) = 𝑍 ↔ (𝐴 , 𝐴) = 𝑍)) |
15 | eqeq1 2825 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 = 0 ↔ 𝐴 = 0 )) | |
16 | 14, 15 | imbi12d 347 | . . . 4 ⊢ (𝑥 = 𝐴 → (((𝑥 , 𝑥) = 𝑍 → 𝑥 = 0 ) ↔ ((𝐴 , 𝐴) = 𝑍 → 𝐴 = 0 ))) |
17 | 16 | rspccva 3621 | . . 3 ⊢ ((∀𝑥 ∈ 𝑉 ((𝑥 , 𝑥) = 𝑍 → 𝑥 = 0 ) ∧ 𝐴 ∈ 𝑉) → ((𝐴 , 𝐴) = 𝑍 → 𝐴 = 0 )) |
18 | 11, 17 | sylan 582 | . 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉) → ((𝐴 , 𝐴) = 𝑍 → 𝐴 = 0 )) |
19 | 2, 3, 1, 6, 4 | ip0l 20779 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉) → ( 0 , 𝐴) = 𝑍) |
20 | oveq1 7162 | . . . 4 ⊢ (𝐴 = 0 → (𝐴 , 𝐴) = ( 0 , 𝐴)) | |
21 | 20 | eqeq1d 2823 | . . 3 ⊢ (𝐴 = 0 → ((𝐴 , 𝐴) = 𝑍 ↔ ( 0 , 𝐴) = 𝑍)) |
22 | 19, 21 | syl5ibrcom 249 | . 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉) → (𝐴 = 0 → (𝐴 , 𝐴) = 𝑍)) |
23 | 18, 22 | impbid 214 | 1 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉) → ((𝐴 , 𝐴) = 𝑍 ↔ 𝐴 = 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ↦ cmpt 5145 ‘cfv 6354 (class class class)co 7155 Basecbs 16482 *𝑟cstv 16566 Scalarcsca 16567 ·𝑖cip 16569 0gc0g 16712 *-Ringcsr 19614 LMHom clmhm 19790 LVecclvec 19873 ringLModcrglmod 19940 PreHilcphl 20767 |
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-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 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 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-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-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-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-2 11699 df-3 11700 df-4 11701 df-5 11702 df-6 11703 df-7 11704 df-8 11705 df-ndx 16485 df-slot 16486 df-base 16488 df-sets 16489 df-plusg 16577 df-sca 16580 df-vsca 16581 df-ip 16582 df-0g 16714 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-grp 18105 df-ghm 18355 df-lmod 19635 df-lmhm 19793 df-lvec 19874 df-sra 19943 df-rgmod 19944 df-phl 20769 |
This theorem is referenced by: ip2eq 20796 phlssphl 20802 ocvin 20817 lsmcss 20835 obsne0 20868 cphipeq0 23807 ipcau2 23836 tcphcph 23839 |
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