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Mirrors > Home > HSE Home > Th. List > hi01 | Structured version Visualization version GIF version |
Description: Inner product with the 0 vector. (Contributed by NM, 29-May-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hi01 | ⊢ (𝐴 ∈ ℋ → (0ℎ ·ih 𝐴) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-hv0cl 28786 | . . . . 5 ⊢ 0ℎ ∈ ℋ | |
2 | ax-hvmul0 28793 | . . . . 5 ⊢ (0ℎ ∈ ℋ → (0 ·ℎ 0ℎ) = 0ℎ) | |
3 | 1, 2 | ax-mp 5 | . . . 4 ⊢ (0 ·ℎ 0ℎ) = 0ℎ |
4 | 3 | oveq1i 7145 | . . 3 ⊢ ((0 ·ℎ 0ℎ) ·ih 𝐴) = (0ℎ ·ih 𝐴) |
5 | 0cn 10622 | . . . 4 ⊢ 0 ∈ ℂ | |
6 | ax-his3 28867 | . . . 4 ⊢ ((0 ∈ ℂ ∧ 0ℎ ∈ ℋ ∧ 𝐴 ∈ ℋ) → ((0 ·ℎ 0ℎ) ·ih 𝐴) = (0 · (0ℎ ·ih 𝐴))) | |
7 | 5, 1, 6 | mp3an12 1448 | . . 3 ⊢ (𝐴 ∈ ℋ → ((0 ·ℎ 0ℎ) ·ih 𝐴) = (0 · (0ℎ ·ih 𝐴))) |
8 | 4, 7 | syl5eqr 2847 | . 2 ⊢ (𝐴 ∈ ℋ → (0ℎ ·ih 𝐴) = (0 · (0ℎ ·ih 𝐴))) |
9 | hicl 28863 | . . . 4 ⊢ ((0ℎ ∈ ℋ ∧ 𝐴 ∈ ℋ) → (0ℎ ·ih 𝐴) ∈ ℂ) | |
10 | 1, 9 | mpan 689 | . . 3 ⊢ (𝐴 ∈ ℋ → (0ℎ ·ih 𝐴) ∈ ℂ) |
11 | 10 | mul02d 10827 | . 2 ⊢ (𝐴 ∈ ℋ → (0 · (0ℎ ·ih 𝐴)) = 0) |
12 | 8, 11 | eqtrd 2833 | 1 ⊢ (𝐴 ∈ ℋ → (0ℎ ·ih 𝐴) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 (class class class)co 7135 ℂcc 10524 0cc0 10526 · cmul 10531 ℋchba 28702 ·ℎ csm 28704 ·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-hv0cl 28786 ax-hvmul0 28793 ax-hfi 28862 ax-his3 28867 |
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-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-ov 7138 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-ltxr 10669 |
This theorem is referenced by: hi02 28880 hiidge0 28881 his6 28882 hial0 28885 normgt0 28910 norm0 28911 ocsh 29066 0hmop 29766 adj0 29777 lnopeq0i 29790 leop3 29908 leoprf2 29910 leoprf 29911 idleop 29914 |
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