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Mirrors > Home > HSE Home > Th. List > hiidge0 | Structured version Visualization version GIF version |
Description: Inner product with self is not negative. (Contributed by NM, 29-May-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hiidge0 | ⊢ (𝐴 ∈ ℋ → 0 ≤ (𝐴 ·ih 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm2.1 432 | . . 3 ⊢ (¬ 𝐴 = 0ℎ ∨ 𝐴 = 0ℎ) | |
2 | df-ne 2824 | . . . . . 6 ⊢ (𝐴 ≠ 0ℎ ↔ ¬ 𝐴 = 0ℎ) | |
3 | ax-his4 28070 | . . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 0 < (𝐴 ·ih 𝐴)) | |
4 | 2, 3 | sylan2br 492 | . . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → 0 < (𝐴 ·ih 𝐴)) |
5 | 4 | ex 449 | . . . 4 ⊢ (𝐴 ∈ ℋ → (¬ 𝐴 = 0ℎ → 0 < (𝐴 ·ih 𝐴))) |
6 | oveq1 6697 | . . . . . . 7 ⊢ (𝐴 = 0ℎ → (𝐴 ·ih 𝐴) = (0ℎ ·ih 𝐴)) | |
7 | hi01 28081 | . . . . . . 7 ⊢ (𝐴 ∈ ℋ → (0ℎ ·ih 𝐴) = 0) | |
8 | 6, 7 | sylan9eqr 2707 | . . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 = 0ℎ) → (𝐴 ·ih 𝐴) = 0) |
9 | 8 | eqcomd 2657 | . . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 = 0ℎ) → 0 = (𝐴 ·ih 𝐴)) |
10 | 9 | ex 449 | . . . 4 ⊢ (𝐴 ∈ ℋ → (𝐴 = 0ℎ → 0 = (𝐴 ·ih 𝐴))) |
11 | 5, 10 | orim12d 901 | . . 3 ⊢ (𝐴 ∈ ℋ → ((¬ 𝐴 = 0ℎ ∨ 𝐴 = 0ℎ) → (0 < (𝐴 ·ih 𝐴) ∨ 0 = (𝐴 ·ih 𝐴)))) |
12 | 1, 11 | mpi 20 | . 2 ⊢ (𝐴 ∈ ℋ → (0 < (𝐴 ·ih 𝐴) ∨ 0 = (𝐴 ·ih 𝐴))) |
13 | 0re 10078 | . . 3 ⊢ 0 ∈ ℝ | |
14 | hiidrcl 28080 | . . 3 ⊢ (𝐴 ∈ ℋ → (𝐴 ·ih 𝐴) ∈ ℝ) | |
15 | leloe 10162 | . . 3 ⊢ ((0 ∈ ℝ ∧ (𝐴 ·ih 𝐴) ∈ ℝ) → (0 ≤ (𝐴 ·ih 𝐴) ↔ (0 < (𝐴 ·ih 𝐴) ∨ 0 = (𝐴 ·ih 𝐴)))) | |
16 | 13, 14, 15 | sylancr 696 | . 2 ⊢ (𝐴 ∈ ℋ → (0 ≤ (𝐴 ·ih 𝐴) ↔ (0 < (𝐴 ·ih 𝐴) ∨ 0 = (𝐴 ·ih 𝐴)))) |
17 | 12, 16 | mpbird 247 | 1 ⊢ (𝐴 ∈ ℋ → 0 ≤ (𝐴 ·ih 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∨ wo 382 ∧ wa 383 = wceq 1523 ∈ wcel 2030 ≠ wne 2823 class class class wbr 4685 (class class class)co 6690 ℝcr 9973 0cc0 9974 < clt 10112 ≤ cle 10113 ℋchil 27904 ·ih csp 27907 0ℎc0v 27909 |
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-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 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 ax-hv0cl 27988 ax-hvmul0 27995 ax-hfi 28064 ax-his1 28067 ax-his3 28069 ax-his4 28070 |
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-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-po 5064 df-so 5065 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-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-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-div 10723 df-2 11117 df-cj 13883 df-re 13884 df-im 13885 |
This theorem is referenced by: normlem5 28099 normlem6 28100 normlem7 28101 normf 28108 normge0 28111 normgt0 28112 normsqi 28117 norm-ii-i 28122 norm-iii-i 28124 bcsiALT 28164 pjhthlem1 28378 cnlnadjlem7 29060 branmfn 29092 leopsq 29116 idleop 29118 |
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