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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 | pm2.1 894 | . . 3 ⊢ (¬ 𝐴 = 0ℎ ∨ 𝐴 = 0ℎ) | |
2 | df-ne 2988 | . . . . . 6 ⊢ (𝐴 ≠ 0ℎ ↔ ¬ 𝐴 = 0ℎ) | |
3 | ax-his4 28868 | . . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 0 < (𝐴 ·ih 𝐴)) | |
4 | 2, 3 | sylan2br 597 | . . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → 0 < (𝐴 ·ih 𝐴)) |
5 | 4 | ex 416 | . . . 4 ⊢ (𝐴 ∈ ℋ → (¬ 𝐴 = 0ℎ → 0 < (𝐴 ·ih 𝐴))) |
6 | oveq1 7142 | . . . . . . 7 ⊢ (𝐴 = 0ℎ → (𝐴 ·ih 𝐴) = (0ℎ ·ih 𝐴)) | |
7 | hi01 28879 | . . . . . . 7 ⊢ (𝐴 ∈ ℋ → (0ℎ ·ih 𝐴) = 0) | |
8 | 6, 7 | sylan9eqr 2855 | . . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 = 0ℎ) → (𝐴 ·ih 𝐴) = 0) |
9 | 8 | eqcomd 2804 | . . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 = 0ℎ) → 0 = (𝐴 ·ih 𝐴)) |
10 | 9 | ex 416 | . . . 4 ⊢ (𝐴 ∈ ℋ → (𝐴 = 0ℎ → 0 = (𝐴 ·ih 𝐴))) |
11 | 5, 10 | orim12d 962 | . . 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 10632 | . . 3 ⊢ 0 ∈ ℝ | |
14 | hiidrcl 28878 | . . 3 ⊢ (𝐴 ∈ ℋ → (𝐴 ·ih 𝐴) ∈ ℝ) | |
15 | leloe 10716 | . . 3 ⊢ ((0 ∈ ℝ ∧ (𝐴 ·ih 𝐴) ∈ ℝ) → (0 ≤ (𝐴 ·ih 𝐴) ↔ (0 < (𝐴 ·ih 𝐴) ∨ 0 = (𝐴 ·ih 𝐴)))) | |
16 | 13, 14, 15 | sylancr 590 | . 2 ⊢ (𝐴 ∈ ℋ → (0 ≤ (𝐴 ·ih 𝐴) ↔ (0 < (𝐴 ·ih 𝐴) ∨ 0 = (𝐴 ·ih 𝐴)))) |
17 | 12, 16 | mpbird 260 | 1 ⊢ (𝐴 ∈ ℋ → 0 ≤ (𝐴 ·ih 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 844 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 class class class wbr 5030 (class class class)co 7135 ℝcr 10525 0cc0 10526 < clt 10664 ≤ cle 10665 ℋchba 28702 ·ih csp 28705 0ℎc0v 28707 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-hv0cl 28786 ax-hvmul0 28793 ax-hfi 28862 ax-his1 28865 ax-his3 28867 ax-his4 28868 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-2 11688 df-cj 14450 df-re 14451 df-im 14452 |
This theorem is referenced by: normlem5 28897 normlem6 28898 normlem7 28899 normf 28906 normge0 28909 normgt0 28910 normsqi 28915 norm-ii-i 28920 norm-iii-i 28922 bcsiALT 28962 pjhthlem1 29174 cnlnadjlem7 29856 branmfn 29888 leopsq 29912 idleop 29914 |
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