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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 10533 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (𝐴 · 0) = 0) | |
2 | 1 | oveq1d 6919 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((𝐴 · 0) ·ℎ 0ℎ) = (0 ·ℎ 0ℎ)) |
3 | ax-hv0cl 28414 | . . . . 5 ⊢ 0ℎ ∈ ℋ | |
4 | ax-hvmul0 28421 | . . . . 5 ⊢ (0ℎ ∈ ℋ → (0 ·ℎ 0ℎ) = 0ℎ) | |
5 | 3, 4 | ax-mp 5 | . . . 4 ⊢ (0 ·ℎ 0ℎ) = 0ℎ |
6 | 2, 5 | syl6eq 2876 | . . 3 ⊢ (𝐴 ∈ ℂ → ((𝐴 · 0) ·ℎ 0ℎ) = 0ℎ) |
7 | 0cn 10347 | . . . 4 ⊢ 0 ∈ ℂ | |
8 | ax-hvmulass 28418 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 0 ∈ ℂ ∧ 0ℎ ∈ ℋ) → ((𝐴 · 0) ·ℎ 0ℎ) = (𝐴 ·ℎ (0 ·ℎ 0ℎ))) | |
9 | 7, 3, 8 | mp3an23 1583 | . . 3 ⊢ (𝐴 ∈ ℂ → ((𝐴 · 0) ·ℎ 0ℎ) = (𝐴 ·ℎ (0 ·ℎ 0ℎ))) |
10 | 6, 9 | eqtr3d 2862 | . 2 ⊢ (𝐴 ∈ ℂ → 0ℎ = (𝐴 ·ℎ (0 ·ℎ 0ℎ))) |
11 | 5 | oveq2i 6915 | . 2 ⊢ (𝐴 ·ℎ (0 ·ℎ 0ℎ)) = (𝐴 ·ℎ 0ℎ) |
12 | 10, 11 | syl6req 2877 | 1 ⊢ (𝐴 ∈ ℂ → (𝐴 ·ℎ 0ℎ) = 0ℎ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ∈ wcel 2166 (class class class)co 6904 ℂcc 10249 0cc0 10251 · cmul 10256 ℋchba 28330 ·ℎ csm 28332 0ℎc0v 28335 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-sep 5004 ax-nul 5012 ax-pow 5064 ax-pr 5126 ax-un 7208 ax-resscn 10308 ax-1cn 10309 ax-icn 10310 ax-addcl 10311 ax-addrcl 10312 ax-mulcl 10313 ax-mulrcl 10314 ax-mulcom 10315 ax-addass 10316 ax-mulass 10317 ax-distr 10318 ax-i2m1 10319 ax-1ne0 10320 ax-1rid 10321 ax-rnegex 10322 ax-rrecex 10323 ax-cnre 10324 ax-pre-lttri 10325 ax-pre-lttrn 10326 ax-pre-ltadd 10327 ax-hv0cl 28414 ax-hvmulass 28418 ax-hvmul0 28421 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ne 2999 df-nel 3102 df-ral 3121 df-rex 3122 df-rab 3125 df-v 3415 df-sbc 3662 df-csb 3757 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-nul 4144 df-if 4306 df-pw 4379 df-sn 4397 df-pr 4399 df-op 4403 df-uni 4658 df-br 4873 df-opab 4935 df-mpt 4952 df-id 5249 df-po 5262 df-so 5263 df-xp 5347 df-rel 5348 df-cnv 5349 df-co 5350 df-dm 5351 df-rn 5352 df-res 5353 df-ima 5354 df-iota 6085 df-fun 6124 df-fn 6125 df-f 6126 df-f1 6127 df-fo 6128 df-f1o 6129 df-fv 6130 df-ov 6907 df-er 8008 df-en 8222 df-dom 8223 df-sdom 8224 df-pnf 10392 df-mnf 10393 df-ltxr 10395 |
This theorem is referenced by: hvmul0or 28436 hvsub0 28487 hsn0elch 28659 pjssmii 29094 |
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