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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 11117 | . . . 4 ⊢ 2 = (1 + 1) | |
2 | 1 | oveq1i 6700 | . . 3 ⊢ (2 ·ℎ 𝐴) = ((1 + 1) ·ℎ 𝐴) |
3 | ax-1cn 10032 | . . . 4 ⊢ 1 ∈ ℂ | |
4 | ax-hvdistr2 27994 | . . . 4 ⊢ ((1 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝐴 ∈ ℋ) → ((1 + 1) ·ℎ 𝐴) = ((1 ·ℎ 𝐴) +ℎ (1 ·ℎ 𝐴))) | |
5 | 3, 3, 4 | mp3an12 1454 | . . 3 ⊢ (𝐴 ∈ ℋ → ((1 + 1) ·ℎ 𝐴) = ((1 ·ℎ 𝐴) +ℎ (1 ·ℎ 𝐴))) |
6 | 2, 5 | syl5eq 2697 | . 2 ⊢ (𝐴 ∈ ℋ → (2 ·ℎ 𝐴) = ((1 ·ℎ 𝐴) +ℎ (1 ·ℎ 𝐴))) |
7 | ax-hvdistr1 27993 | . . . 4 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (1 ·ℎ (𝐴 +ℎ 𝐴)) = ((1 ·ℎ 𝐴) +ℎ (1 ·ℎ 𝐴))) | |
8 | 3, 7 | mp3an1 1451 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (1 ·ℎ (𝐴 +ℎ 𝐴)) = ((1 ·ℎ 𝐴) +ℎ (1 ·ℎ 𝐴))) |
9 | 8 | anidms 678 | . 2 ⊢ (𝐴 ∈ ℋ → (1 ·ℎ (𝐴 +ℎ 𝐴)) = ((1 ·ℎ 𝐴) +ℎ (1 ·ℎ 𝐴))) |
10 | hvaddcl 27997 | . . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (𝐴 +ℎ 𝐴) ∈ ℋ) | |
11 | 10 | anidms 678 | . . 3 ⊢ (𝐴 ∈ ℋ → (𝐴 +ℎ 𝐴) ∈ ℋ) |
12 | ax-hvmulid 27991 | . . 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 2691 | 1 ⊢ (𝐴 ∈ ℋ → (2 ·ℎ 𝐴) = (𝐴 +ℎ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1523 ∈ wcel 2030 (class class class)co 6690 ℂcc 9972 1c1 9975 + caddc 9977 2c2 11108 ℋchil 27904 +ℎ cva 27905 ·ℎ csm 27906 |
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-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pr 4936 ax-1cn 10032 ax-hfvadd 27985 ax-hvmulid 27991 ax-hvdistr1 27993 ax-hvdistr2 27994 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 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-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-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-fv 5934 df-ov 6693 df-2 11117 |
This theorem is referenced by: hvsubcan2i 28049 mayete3i 28715 |
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