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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 11079 | . . . 4 ⊢ 2 = (1 + 1) | |
2 | 1 | oveq1i 6660 | . . 3 ⊢ (2 ·ℎ 𝐴) = ((1 + 1) ·ℎ 𝐴) |
3 | ax-1cn 9994 | . . . 4 ⊢ 1 ∈ ℂ | |
4 | ax-hvdistr2 27866 | . . . 4 ⊢ ((1 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝐴 ∈ ℋ) → ((1 + 1) ·ℎ 𝐴) = ((1 ·ℎ 𝐴) +ℎ (1 ·ℎ 𝐴))) | |
5 | 3, 3, 4 | mp3an12 1414 | . . 3 ⊢ (𝐴 ∈ ℋ → ((1 + 1) ·ℎ 𝐴) = ((1 ·ℎ 𝐴) +ℎ (1 ·ℎ 𝐴))) |
6 | 2, 5 | syl5eq 2668 | . 2 ⊢ (𝐴 ∈ ℋ → (2 ·ℎ 𝐴) = ((1 ·ℎ 𝐴) +ℎ (1 ·ℎ 𝐴))) |
7 | ax-hvdistr1 27865 | . . . 4 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (1 ·ℎ (𝐴 +ℎ 𝐴)) = ((1 ·ℎ 𝐴) +ℎ (1 ·ℎ 𝐴))) | |
8 | 3, 7 | mp3an1 1411 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (1 ·ℎ (𝐴 +ℎ 𝐴)) = ((1 ·ℎ 𝐴) +ℎ (1 ·ℎ 𝐴))) |
9 | 8 | anidms 677 | . 2 ⊢ (𝐴 ∈ ℋ → (1 ·ℎ (𝐴 +ℎ 𝐴)) = ((1 ·ℎ 𝐴) +ℎ (1 ·ℎ 𝐴))) |
10 | hvaddcl 27869 | . . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (𝐴 +ℎ 𝐴) ∈ ℋ) | |
11 | 10 | anidms 677 | . . 3 ⊢ (𝐴 ∈ ℋ → (𝐴 +ℎ 𝐴) ∈ ℋ) |
12 | ax-hvmulid 27863 | . . 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 2662 | 1 ⊢ (𝐴 ∈ ℋ → (2 ·ℎ 𝐴) = (𝐴 +ℎ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 (class class class)co 6650 ℂcc 9934 1c1 9937 + caddc 9939 2c2 11070 ℋchil 27776 +ℎ cva 27777 ·ℎ csm 27778 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 ax-1cn 9994 ax-hfvadd 27857 ax-hvmulid 27863 ax-hvdistr1 27865 ax-hvdistr2 27866 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-ov 6653 df-2 11079 |
This theorem is referenced by: hvsubcan2i 27921 mayete3i 28587 |
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