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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 27244 | . . . . 5 ⊢ 0ℎ ∈ ℋ | |
2 | ax-hvmul0 27251 | . . . . 5 ⊢ (0ℎ ∈ ℋ → (0 ·ℎ 0ℎ) = 0ℎ) | |
3 | 1, 2 | ax-mp 5 | . . . 4 ⊢ (0 ·ℎ 0ℎ) = 0ℎ |
4 | 3 | oveq1i 6559 | . . 3 ⊢ ((0 ·ℎ 0ℎ) ·ih 𝐴) = (0ℎ ·ih 𝐴) |
5 | 0cn 9911 | . . . 4 ⊢ 0 ∈ ℂ | |
6 | ax-his3 27325 | . . . 4 ⊢ ((0 ∈ ℂ ∧ 0ℎ ∈ ℋ ∧ 𝐴 ∈ ℋ) → ((0 ·ℎ 0ℎ) ·ih 𝐴) = (0 · (0ℎ ·ih 𝐴))) | |
7 | 5, 1, 6 | mp3an12 1406 | . . 3 ⊢ (𝐴 ∈ ℋ → ((0 ·ℎ 0ℎ) ·ih 𝐴) = (0 · (0ℎ ·ih 𝐴))) |
8 | 4, 7 | syl5eqr 2658 | . 2 ⊢ (𝐴 ∈ ℋ → (0ℎ ·ih 𝐴) = (0 · (0ℎ ·ih 𝐴))) |
9 | hicl 27321 | . . . 4 ⊢ ((0ℎ ∈ ℋ ∧ 𝐴 ∈ ℋ) → (0ℎ ·ih 𝐴) ∈ ℂ) | |
10 | 1, 9 | mpan 702 | . . 3 ⊢ (𝐴 ∈ ℋ → (0ℎ ·ih 𝐴) ∈ ℂ) |
11 | 10 | mul02d 10113 | . 2 ⊢ (𝐴 ∈ ℋ → (0 · (0ℎ ·ih 𝐴)) = 0) |
12 | 8, 11 | eqtrd 2644 | 1 ⊢ (𝐴 ∈ ℋ → (0ℎ ·ih 𝐴) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 (class class class)co 6549 ℂcc 9813 0cc0 9815 · cmul 9820 ℋchil 27160 ·ℎ csm 27162 ·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-hv0cl 27244 ax-hvmul0 27251 ax-hfi 27320 ax-his3 27325 |
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-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-ov 6552 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-ltxr 9958 |
This theorem is referenced by: hi02 27338 hiidge0 27339 his6 27340 hial0 27343 normgt0 27368 norm0 27369 ocsh 27526 0hmop 28226 adj0 28237 lnopeq0i 28250 leop3 28368 leoprf2 28370 leoprf 28371 idleop 28374 |
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