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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 28782 | . . . . 5 ⊢ 0ℎ ∈ ℋ | |
2 | ax-hvmul0 28789 | . . . . 5 ⊢ (0ℎ ∈ ℋ → (0 ·ℎ 0ℎ) = 0ℎ) | |
3 | 1, 2 | ax-mp 5 | . . . 4 ⊢ (0 ·ℎ 0ℎ) = 0ℎ |
4 | 3 | oveq1i 7168 | . . 3 ⊢ ((0 ·ℎ 0ℎ) ·ih 𝐴) = (0ℎ ·ih 𝐴) |
5 | 0cn 10635 | . . . 4 ⊢ 0 ∈ ℂ | |
6 | ax-his3 28863 | . . . 4 ⊢ ((0 ∈ ℂ ∧ 0ℎ ∈ ℋ ∧ 𝐴 ∈ ℋ) → ((0 ·ℎ 0ℎ) ·ih 𝐴) = (0 · (0ℎ ·ih 𝐴))) | |
7 | 5, 1, 6 | mp3an12 1447 | . . 3 ⊢ (𝐴 ∈ ℋ → ((0 ·ℎ 0ℎ) ·ih 𝐴) = (0 · (0ℎ ·ih 𝐴))) |
8 | 4, 7 | syl5eqr 2872 | . 2 ⊢ (𝐴 ∈ ℋ → (0ℎ ·ih 𝐴) = (0 · (0ℎ ·ih 𝐴))) |
9 | hicl 28859 | . . . 4 ⊢ ((0ℎ ∈ ℋ ∧ 𝐴 ∈ ℋ) → (0ℎ ·ih 𝐴) ∈ ℂ) | |
10 | 1, 9 | mpan 688 | . . 3 ⊢ (𝐴 ∈ ℋ → (0ℎ ·ih 𝐴) ∈ ℂ) |
11 | 10 | mul02d 10840 | . 2 ⊢ (𝐴 ∈ ℋ → (0 · (0ℎ ·ih 𝐴)) = 0) |
12 | 8, 11 | eqtrd 2858 | 1 ⊢ (𝐴 ∈ ℋ → (0ℎ ·ih 𝐴) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 (class class class)co 7158 ℂcc 10537 0cc0 10539 · cmul 10544 ℋchba 28698 ·ℎ csm 28700 ·ih csp 28701 0ℎc0v 28703 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-hv0cl 28782 ax-hvmul0 28789 ax-hfi 28858 ax-his3 28863 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-ltxr 10682 |
This theorem is referenced by: hi02 28876 hiidge0 28877 his6 28878 hial0 28881 normgt0 28906 norm0 28907 ocsh 29062 0hmop 29762 adj0 29773 lnopeq0i 29786 leop3 29904 leoprf2 29906 leoprf 29907 idleop 29910 |
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