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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 29116 | . . . . 5 ⊢ 0ℎ ∈ ℋ | |
2 | ax-hvmul0 29123 | . . . . 5 ⊢ (0ℎ ∈ ℋ → (0 ·ℎ 0ℎ) = 0ℎ) | |
3 | 1, 2 | ax-mp 5 | . . . 4 ⊢ (0 ·ℎ 0ℎ) = 0ℎ |
4 | 3 | oveq1i 7245 | . . 3 ⊢ ((0 ·ℎ 0ℎ) ·ih 𝐴) = (0ℎ ·ih 𝐴) |
5 | 0cn 10855 | . . . 4 ⊢ 0 ∈ ℂ | |
6 | ax-his3 29197 | . . . 4 ⊢ ((0 ∈ ℂ ∧ 0ℎ ∈ ℋ ∧ 𝐴 ∈ ℋ) → ((0 ·ℎ 0ℎ) ·ih 𝐴) = (0 · (0ℎ ·ih 𝐴))) | |
7 | 5, 1, 6 | mp3an12 1453 | . . 3 ⊢ (𝐴 ∈ ℋ → ((0 ·ℎ 0ℎ) ·ih 𝐴) = (0 · (0ℎ ·ih 𝐴))) |
8 | 4, 7 | eqtr3id 2794 | . 2 ⊢ (𝐴 ∈ ℋ → (0ℎ ·ih 𝐴) = (0 · (0ℎ ·ih 𝐴))) |
9 | hicl 29193 | . . . 4 ⊢ ((0ℎ ∈ ℋ ∧ 𝐴 ∈ ℋ) → (0ℎ ·ih 𝐴) ∈ ℂ) | |
10 | 1, 9 | mpan 690 | . . 3 ⊢ (𝐴 ∈ ℋ → (0ℎ ·ih 𝐴) ∈ ℂ) |
11 | 10 | mul02d 11060 | . 2 ⊢ (𝐴 ∈ ℋ → (0 · (0ℎ ·ih 𝐴)) = 0) |
12 | 8, 11 | eqtrd 2779 | 1 ⊢ (𝐴 ∈ ℋ → (0ℎ ·ih 𝐴) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2112 (class class class)co 7235 ℂcc 10757 0cc0 10759 · cmul 10764 ℋchba 29032 ·ℎ csm 29034 ·ih csp 29035 0ℎc0v 29037 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2710 ax-sep 5209 ax-nul 5216 ax-pow 5275 ax-pr 5339 ax-un 7545 ax-resscn 10816 ax-1cn 10817 ax-icn 10818 ax-addcl 10819 ax-addrcl 10820 ax-mulcl 10821 ax-mulrcl 10822 ax-mulcom 10823 ax-addass 10824 ax-mulass 10825 ax-distr 10826 ax-i2m1 10827 ax-1ne0 10828 ax-1rid 10829 ax-rnegex 10830 ax-rrecex 10831 ax-cnre 10832 ax-pre-lttri 10833 ax-pre-lttrn 10834 ax-pre-ltadd 10835 ax-hv0cl 29116 ax-hvmul0 29123 ax-hfi 29192 ax-his3 29197 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2818 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3425 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4255 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5153 df-id 5472 df-po 5486 df-so 5487 df-xp 5575 df-rel 5576 df-cnv 5577 df-co 5578 df-dm 5579 df-rn 5580 df-res 5581 df-ima 5582 df-iota 6359 df-fun 6403 df-fn 6404 df-f 6405 df-f1 6406 df-fo 6407 df-f1o 6408 df-fv 6409 df-ov 7238 df-er 8415 df-en 8651 df-dom 8652 df-sdom 8653 df-pnf 10899 df-mnf 10900 df-ltxr 10902 |
This theorem is referenced by: hi02 29210 hiidge0 29211 his6 29212 hial0 29215 normgt0 29240 norm0 29241 ocsh 29396 0hmop 30096 adj0 30107 lnopeq0i 30120 leop3 30238 leoprf2 30240 leoprf 30241 idleop 30244 |
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