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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 | pm2.1 895 | . . 3 ⊢ (¬ 𝐴 = 0ℎ ∨ 𝐴 = 0ℎ) | |
2 | df-ne 2937 | . . . . . 6 ⊢ (𝐴 ≠ 0ℎ ↔ ¬ 𝐴 = 0ℎ) | |
3 | ax-his4 30888 | . . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 0 < (𝐴 ·ih 𝐴)) | |
4 | 2, 3 | sylan2br 594 | . . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → 0 < (𝐴 ·ih 𝐴)) |
5 | 4 | ex 412 | . . . 4 ⊢ (𝐴 ∈ ℋ → (¬ 𝐴 = 0ℎ → 0 < (𝐴 ·ih 𝐴))) |
6 | oveq1 7421 | . . . . . . 7 ⊢ (𝐴 = 0ℎ → (𝐴 ·ih 𝐴) = (0ℎ ·ih 𝐴)) | |
7 | hi01 30899 | . . . . . . 7 ⊢ (𝐴 ∈ ℋ → (0ℎ ·ih 𝐴) = 0) | |
8 | 6, 7 | sylan9eqr 2790 | . . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 = 0ℎ) → (𝐴 ·ih 𝐴) = 0) |
9 | 8 | eqcomd 2734 | . . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 = 0ℎ) → 0 = (𝐴 ·ih 𝐴)) |
10 | 9 | ex 412 | . . . 4 ⊢ (𝐴 ∈ ℋ → (𝐴 = 0ℎ → 0 = (𝐴 ·ih 𝐴))) |
11 | 5, 10 | orim12d 963 | . . 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 11240 | . . 3 ⊢ 0 ∈ ℝ | |
14 | hiidrcl 30898 | . . 3 ⊢ (𝐴 ∈ ℋ → (𝐴 ·ih 𝐴) ∈ ℝ) | |
15 | leloe 11324 | . . 3 ⊢ ((0 ∈ ℝ ∧ (𝐴 ·ih 𝐴) ∈ ℝ) → (0 ≤ (𝐴 ·ih 𝐴) ↔ (0 < (𝐴 ·ih 𝐴) ∨ 0 = (𝐴 ·ih 𝐴)))) | |
16 | 13, 14, 15 | sylancr 586 | . 2 ⊢ (𝐴 ∈ ℋ → (0 ≤ (𝐴 ·ih 𝐴) ↔ (0 < (𝐴 ·ih 𝐴) ∨ 0 = (𝐴 ·ih 𝐴)))) |
17 | 12, 16 | mpbird 257 | 1 ⊢ (𝐴 ∈ ℋ → 0 ≤ (𝐴 ·ih 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 846 = wceq 1534 ∈ wcel 2099 ≠ wne 2936 class class class wbr 5142 (class class class)co 7414 ℝcr 11131 0cc0 11132 < clt 11272 ≤ cle 11273 ℋchba 30722 ·ih csp 30725 0ℎc0v 30727 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 ax-hv0cl 30806 ax-hvmul0 30813 ax-hfi 30882 ax-his1 30885 ax-his3 30887 ax-his4 30888 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-po 5584 df-so 5585 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-div 11896 df-2 12299 df-cj 15072 df-re 15073 df-im 15074 |
This theorem is referenced by: normlem5 30917 normlem6 30918 normlem7 30919 normf 30926 normge0 30929 normgt0 30930 normsqi 30935 norm-ii-i 30940 norm-iii-i 30942 bcsiALT 30982 pjhthlem1 31194 cnlnadjlem7 31876 branmfn 31908 leopsq 31932 idleop 31934 |
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