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Mirrors > Home > HSE Home > Th. List > hvmul0 | Structured version Visualization version GIF version |
Description: Scalar multiplication with the zero vector. (Contributed by NM, 30-May-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hvmul0 | ⊢ (𝐴 ∈ ℂ → (𝐴 ·ℎ 0ℎ) = 0ℎ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mul01 10253 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (𝐴 · 0) = 0) | |
2 | 1 | oveq1d 6705 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((𝐴 · 0) ·ℎ 0ℎ) = (0 ·ℎ 0ℎ)) |
3 | ax-hv0cl 27988 | . . . . 5 ⊢ 0ℎ ∈ ℋ | |
4 | ax-hvmul0 27995 | . . . . 5 ⊢ (0ℎ ∈ ℋ → (0 ·ℎ 0ℎ) = 0ℎ) | |
5 | 3, 4 | ax-mp 5 | . . . 4 ⊢ (0 ·ℎ 0ℎ) = 0ℎ |
6 | 2, 5 | syl6eq 2701 | . . 3 ⊢ (𝐴 ∈ ℂ → ((𝐴 · 0) ·ℎ 0ℎ) = 0ℎ) |
7 | 0cn 10070 | . . . 4 ⊢ 0 ∈ ℂ | |
8 | ax-hvmulass 27992 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 0 ∈ ℂ ∧ 0ℎ ∈ ℋ) → ((𝐴 · 0) ·ℎ 0ℎ) = (𝐴 ·ℎ (0 ·ℎ 0ℎ))) | |
9 | 7, 3, 8 | mp3an23 1456 | . . 3 ⊢ (𝐴 ∈ ℂ → ((𝐴 · 0) ·ℎ 0ℎ) = (𝐴 ·ℎ (0 ·ℎ 0ℎ))) |
10 | 6, 9 | eqtr3d 2687 | . 2 ⊢ (𝐴 ∈ ℂ → 0ℎ = (𝐴 ·ℎ (0 ·ℎ 0ℎ))) |
11 | 5 | oveq2i 6701 | . 2 ⊢ (𝐴 ·ℎ (0 ·ℎ 0ℎ)) = (𝐴 ·ℎ 0ℎ) |
12 | 10, 11 | syl6req 2702 | 1 ⊢ (𝐴 ∈ ℂ → (𝐴 ·ℎ 0ℎ) = 0ℎ) |
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 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-hvmulass 27992 ax-hvmul0 27995 |
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-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: hvmul0or 28010 hvsub0 28061 hsn0elch 28233 pjssmii 28668 |
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