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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 2651 | . . . . . 6 ⊢ (*𝑟‘𝐹) = (*𝑟‘𝐹) | |
6 | ip0l.z | . . . . . 6 ⊢ 𝑍 = (0g‘𝐹) | |
7 | 1, 2, 3, 4, 5, 6 | isphl 20021 | . . . . 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 1098 | . . . 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 1082 | . . . . 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 2981 | . . . 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 6699 | . . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑥 = 𝐴) → (𝑥 , 𝑥) = (𝐴 , 𝐴)) | |
13 | 12 | anidms 678 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 , 𝑥) = (𝐴 , 𝐴)) |
14 | 13 | eqeq1d 2653 | . . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑥 , 𝑥) = 𝑍 ↔ (𝐴 , 𝐴) = 𝑍)) |
15 | eqeq1 2655 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 = 0 ↔ 𝐴 = 0 )) | |
16 | 14, 15 | imbi12d 333 | . . . 4 ⊢ (𝑥 = 𝐴 → (((𝑥 , 𝑥) = 𝑍 → 𝑥 = 0 ) ↔ ((𝐴 , 𝐴) = 𝑍 → 𝐴 = 0 ))) |
17 | 16 | rspccva 3339 | . . 3 ⊢ ((∀𝑥 ∈ 𝑉 ((𝑥 , 𝑥) = 𝑍 → 𝑥 = 0 ) ∧ 𝐴 ∈ 𝑉) → ((𝐴 , 𝐴) = 𝑍 → 𝐴 = 0 )) |
18 | 11, 17 | sylan 487 | . 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉) → ((𝐴 , 𝐴) = 𝑍 → 𝐴 = 0 )) |
19 | 2, 3, 1, 6, 4 | ip0l 20029 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉) → ( 0 , 𝐴) = 𝑍) |
20 | oveq1 6697 | . . . 4 ⊢ (𝐴 = 0 → (𝐴 , 𝐴) = ( 0 , 𝐴)) | |
21 | 20 | eqeq1d 2653 | . . 3 ⊢ (𝐴 = 0 → ((𝐴 , 𝐴) = 𝑍 ↔ ( 0 , 𝐴) = 𝑍)) |
22 | 19, 21 | syl5ibrcom 237 | . 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉) → (𝐴 = 0 → (𝐴 , 𝐴) = 𝑍)) |
23 | 18, 22 | impbid 202 | 1 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉) → ((𝐴 , 𝐴) = 𝑍 ↔ 𝐴 = 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 ∧ w3a 1054 = wceq 1523 ∈ wcel 2030 ∀wral 2941 ↦ cmpt 4762 ‘cfv 5926 (class class class)co 6690 Basecbs 15904 *𝑟cstv 15990 Scalarcsca 15991 ·𝑖cip 15993 0gc0g 16147 *-Ringcsr 18892 LMHom clmhm 19067 LVecclvec 19150 ringLModcrglmod 19217 PreHilcphl 20017 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-7 11122 df-8 11123 df-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-plusg 16001 df-sca 16004 df-vsca 16005 df-ip 16006 df-0g 16149 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-grp 17472 df-ghm 17705 df-lmod 18913 df-lmhm 19070 df-lvec 19151 df-sra 19220 df-rgmod 19221 df-phl 20019 |
This theorem is referenced by: ip2eq 20046 ocvin 20066 lsmcss 20084 obsne0 20117 cphipeq0 23050 ipcau2 23079 tchcph 23082 |
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