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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 10417 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (𝐴 · 0) = 0) | |
2 | 1 | oveq1d 6808 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((𝐴 · 0) ·ℎ 0ℎ) = (0 ·ℎ 0ℎ)) |
3 | ax-hv0cl 28200 | . . . . 5 ⊢ 0ℎ ∈ ℋ | |
4 | ax-hvmul0 28207 | . . . . 5 ⊢ (0ℎ ∈ ℋ → (0 ·ℎ 0ℎ) = 0ℎ) | |
5 | 3, 4 | ax-mp 5 | . . . 4 ⊢ (0 ·ℎ 0ℎ) = 0ℎ |
6 | 2, 5 | syl6eq 2821 | . . 3 ⊢ (𝐴 ∈ ℂ → ((𝐴 · 0) ·ℎ 0ℎ) = 0ℎ) |
7 | 0cn 10234 | . . . 4 ⊢ 0 ∈ ℂ | |
8 | ax-hvmulass 28204 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 0 ∈ ℂ ∧ 0ℎ ∈ ℋ) → ((𝐴 · 0) ·ℎ 0ℎ) = (𝐴 ·ℎ (0 ·ℎ 0ℎ))) | |
9 | 7, 3, 8 | mp3an23 1564 | . . 3 ⊢ (𝐴 ∈ ℂ → ((𝐴 · 0) ·ℎ 0ℎ) = (𝐴 ·ℎ (0 ·ℎ 0ℎ))) |
10 | 6, 9 | eqtr3d 2807 | . 2 ⊢ (𝐴 ∈ ℂ → 0ℎ = (𝐴 ·ℎ (0 ·ℎ 0ℎ))) |
11 | 5 | oveq2i 6804 | . 2 ⊢ (𝐴 ·ℎ (0 ·ℎ 0ℎ)) = (𝐴 ·ℎ 0ℎ) |
12 | 10, 11 | syl6req 2822 | 1 ⊢ (𝐴 ∈ ℂ → (𝐴 ·ℎ 0ℎ) = 0ℎ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1631 ∈ wcel 2145 (class class class)co 6793 ℂcc 10136 0cc0 10138 · cmul 10143 ℋchil 28116 ·ℎ csm 28118 0ℎc0v 28121 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-hv0cl 28200 ax-hvmulass 28204 ax-hvmul0 28207 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-br 4787 df-opab 4847 df-mpt 4864 df-id 5157 df-po 5170 df-so 5171 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-ov 6796 df-er 7896 df-en 8110 df-dom 8111 df-sdom 8112 df-pnf 10278 df-mnf 10279 df-ltxr 10281 |
This theorem is referenced by: hvmul0or 28222 hvsub0 28273 hsn0elch 28445 pjssmii 28880 |
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