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Mirrors > Home > HSE Home > Th. List > hv2times | Structured version Visualization version GIF version |
Description: Two times a vector. (Contributed by NM, 22-Jun-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hv2times | ⊢ (𝐴 ∈ ℋ → (2 ·ℎ 𝐴) = (𝐴 +ℎ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-2 11688 | . . . 4 ⊢ 2 = (1 + 1) | |
2 | 1 | oveq1i 7145 | . . 3 ⊢ (2 ·ℎ 𝐴) = ((1 + 1) ·ℎ 𝐴) |
3 | ax-1cn 10584 | . . . 4 ⊢ 1 ∈ ℂ | |
4 | ax-hvdistr2 28792 | . . . 4 ⊢ ((1 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝐴 ∈ ℋ) → ((1 + 1) ·ℎ 𝐴) = ((1 ·ℎ 𝐴) +ℎ (1 ·ℎ 𝐴))) | |
5 | 3, 3, 4 | mp3an12 1448 | . . 3 ⊢ (𝐴 ∈ ℋ → ((1 + 1) ·ℎ 𝐴) = ((1 ·ℎ 𝐴) +ℎ (1 ·ℎ 𝐴))) |
6 | 2, 5 | syl5eq 2845 | . 2 ⊢ (𝐴 ∈ ℋ → (2 ·ℎ 𝐴) = ((1 ·ℎ 𝐴) +ℎ (1 ·ℎ 𝐴))) |
7 | ax-hvdistr1 28791 | . . . 4 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (1 ·ℎ (𝐴 +ℎ 𝐴)) = ((1 ·ℎ 𝐴) +ℎ (1 ·ℎ 𝐴))) | |
8 | 3, 7 | mp3an1 1445 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (1 ·ℎ (𝐴 +ℎ 𝐴)) = ((1 ·ℎ 𝐴) +ℎ (1 ·ℎ 𝐴))) |
9 | 8 | anidms 570 | . 2 ⊢ (𝐴 ∈ ℋ → (1 ·ℎ (𝐴 +ℎ 𝐴)) = ((1 ·ℎ 𝐴) +ℎ (1 ·ℎ 𝐴))) |
10 | hvaddcl 28795 | . . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (𝐴 +ℎ 𝐴) ∈ ℋ) | |
11 | 10 | anidms 570 | . . 3 ⊢ (𝐴 ∈ ℋ → (𝐴 +ℎ 𝐴) ∈ ℋ) |
12 | ax-hvmulid 28789 | . . 3 ⊢ ((𝐴 +ℎ 𝐴) ∈ ℋ → (1 ·ℎ (𝐴 +ℎ 𝐴)) = (𝐴 +ℎ 𝐴)) | |
13 | 11, 12 | syl 17 | . 2 ⊢ (𝐴 ∈ ℋ → (1 ·ℎ (𝐴 +ℎ 𝐴)) = (𝐴 +ℎ 𝐴)) |
14 | 6, 9, 13 | 3eqtr2d 2839 | 1 ⊢ (𝐴 ∈ ℋ → (2 ·ℎ 𝐴) = (𝐴 +ℎ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 (class class class)co 7135 ℂcc 10524 1c1 10527 + caddc 10529 2c2 11680 ℋchba 28702 +ℎ cva 28703 ·ℎ csm 28704 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 ax-1cn 10584 ax-hfvadd 28783 ax-hvmulid 28789 ax-hvdistr1 28791 ax-hvdistr2 28792 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-ov 7138 df-2 11688 |
This theorem is referenced by: hvsubcan2i 28847 mayete3i 29511 |
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