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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 27988 | . . . . 5 ⊢ 0ℎ ∈ ℋ | |
2 | ax-hvmul0 27995 | . . . . 5 ⊢ (0ℎ ∈ ℋ → (0 ·ℎ 0ℎ) = 0ℎ) | |
3 | 1, 2 | ax-mp 5 | . . . 4 ⊢ (0 ·ℎ 0ℎ) = 0ℎ |
4 | 3 | oveq1i 6700 | . . 3 ⊢ ((0 ·ℎ 0ℎ) ·ih 𝐴) = (0ℎ ·ih 𝐴) |
5 | 0cn 10070 | . . . 4 ⊢ 0 ∈ ℂ | |
6 | ax-his3 28069 | . . . 4 ⊢ ((0 ∈ ℂ ∧ 0ℎ ∈ ℋ ∧ 𝐴 ∈ ℋ) → ((0 ·ℎ 0ℎ) ·ih 𝐴) = (0 · (0ℎ ·ih 𝐴))) | |
7 | 5, 1, 6 | mp3an12 1454 | . . 3 ⊢ (𝐴 ∈ ℋ → ((0 ·ℎ 0ℎ) ·ih 𝐴) = (0 · (0ℎ ·ih 𝐴))) |
8 | 4, 7 | syl5eqr 2699 | . 2 ⊢ (𝐴 ∈ ℋ → (0ℎ ·ih 𝐴) = (0 · (0ℎ ·ih 𝐴))) |
9 | hicl 28065 | . . . 4 ⊢ ((0ℎ ∈ ℋ ∧ 𝐴 ∈ ℋ) → (0ℎ ·ih 𝐴) ∈ ℂ) | |
10 | 1, 9 | mpan 706 | . . 3 ⊢ (𝐴 ∈ ℋ → (0ℎ ·ih 𝐴) ∈ ℂ) |
11 | 10 | mul02d 10272 | . 2 ⊢ (𝐴 ∈ ℋ → (0 · (0ℎ ·ih 𝐴)) = 0) |
12 | 8, 11 | eqtrd 2685 | 1 ⊢ (𝐴 ∈ ℋ → (0ℎ ·ih 𝐴) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1523 ∈ wcel 2030 (class class class)co 6690 ℂcc 9972 0cc0 9974 · cmul 9979 ℋchil 27904 ·ℎ csm 27906 ·ih csp 27907 0ℎc0v 27909 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-hv0cl 27988 ax-hvmul0 27995 ax-hfi 28064 ax-his3 28069 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-po 5064 df-so 5065 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-ov 6693 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-pnf 10114 df-mnf 10115 df-ltxr 10117 |
This theorem is referenced by: hi02 28082 hiidge0 28083 his6 28084 hial0 28087 normgt0 28112 norm0 28113 ocsh 28270 0hmop 28970 adj0 28981 lnopeq0i 28994 leop3 29112 leoprf2 29114 leoprf 29115 idleop 29118 |
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