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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 2782 | . . . . . 6 ⊢ (𝐴 ≠ 0ℎ ↔ ¬ 𝐴 = 0ℎ) | |
3 | ax-his4 27326 | . . . . . 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 6556 | . . . . . . 7 ⊢ (𝐴 = 0ℎ → (𝐴 ·ih 𝐴) = (0ℎ ·ih 𝐴)) | |
7 | hi01 27337 | . . . . . . 7 ⊢ (𝐴 ∈ ℋ → (0ℎ ·ih 𝐴) = 0) | |
8 | 6, 7 | sylan9eqr 2666 | . . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 = 0ℎ) → (𝐴 ·ih 𝐴) = 0) |
9 | 8 | eqcomd 2616 | . . . . 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 879 | . . 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 9919 | . . 3 ⊢ 0 ∈ ℝ | |
14 | hiidrcl 27336 | . . 3 ⊢ (𝐴 ∈ ℋ → (𝐴 ·ih 𝐴) ∈ ℝ) | |
15 | leloe 10003 | . . 3 ⊢ ((0 ∈ ℝ ∧ (𝐴 ·ih 𝐴) ∈ ℝ) → (0 ≤ (𝐴 ·ih 𝐴) ↔ (0 < (𝐴 ·ih 𝐴) ∨ 0 = (𝐴 ·ih 𝐴)))) | |
16 | 13, 14, 15 | sylancr 694 | . 2 ⊢ (𝐴 ∈ ℋ → (0 ≤ (𝐴 ·ih 𝐴) ↔ (0 < (𝐴 ·ih 𝐴) ∨ 0 = (𝐴 ·ih 𝐴)))) |
17 | 12, 16 | mpbird 246 | 1 ⊢ (𝐴 ∈ ℋ → 0 ≤ (𝐴 ·ih 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 class class class wbr 4583 (class class class)co 6549 ℝcr 9814 0cc0 9815 < clt 9953 ≤ cle 9954 ℋchil 27160 ·ih csp 27163 0ℎc0v 27165 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-hv0cl 27244 ax-hvmul0 27251 ax-hfi 27320 ax-his1 27323 ax-his3 27325 ax-his4 27326 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-2 10956 df-cj 13687 df-re 13688 df-im 13689 |
This theorem is referenced by: normlem5 27355 normlem6 27356 normlem7 27357 normf 27364 normge0 27367 normgt0 27368 normsqi 27373 norm-ii-i 27378 norm-iii-i 27380 bcsiALT 27420 pjhthlem1 27634 cnlnadjlem7 28316 branmfn 28348 leopsq 28372 idleop 28374 |
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