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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 11701 | . . . 4 ⊢ 2 = (1 + 1) | |
2 | 1 | oveq1i 7166 | . . 3 ⊢ (2 ·ℎ 𝐴) = ((1 + 1) ·ℎ 𝐴) |
3 | ax-1cn 10595 | . . . 4 ⊢ 1 ∈ ℂ | |
4 | ax-hvdistr2 28786 | . . . 4 ⊢ ((1 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝐴 ∈ ℋ) → ((1 + 1) ·ℎ 𝐴) = ((1 ·ℎ 𝐴) +ℎ (1 ·ℎ 𝐴))) | |
5 | 3, 3, 4 | mp3an12 1447 | . . 3 ⊢ (𝐴 ∈ ℋ → ((1 + 1) ·ℎ 𝐴) = ((1 ·ℎ 𝐴) +ℎ (1 ·ℎ 𝐴))) |
6 | 2, 5 | syl5eq 2868 | . 2 ⊢ (𝐴 ∈ ℋ → (2 ·ℎ 𝐴) = ((1 ·ℎ 𝐴) +ℎ (1 ·ℎ 𝐴))) |
7 | ax-hvdistr1 28785 | . . . 4 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (1 ·ℎ (𝐴 +ℎ 𝐴)) = ((1 ·ℎ 𝐴) +ℎ (1 ·ℎ 𝐴))) | |
8 | 3, 7 | mp3an1 1444 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (1 ·ℎ (𝐴 +ℎ 𝐴)) = ((1 ·ℎ 𝐴) +ℎ (1 ·ℎ 𝐴))) |
9 | 8 | anidms 569 | . 2 ⊢ (𝐴 ∈ ℋ → (1 ·ℎ (𝐴 +ℎ 𝐴)) = ((1 ·ℎ 𝐴) +ℎ (1 ·ℎ 𝐴))) |
10 | hvaddcl 28789 | . . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (𝐴 +ℎ 𝐴) ∈ ℋ) | |
11 | 10 | anidms 569 | . . 3 ⊢ (𝐴 ∈ ℋ → (𝐴 +ℎ 𝐴) ∈ ℋ) |
12 | ax-hvmulid 28783 | . . 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 2862 | 1 ⊢ (𝐴 ∈ ℋ → (2 ·ℎ 𝐴) = (𝐴 +ℎ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 (class class class)co 7156 ℂcc 10535 1c1 10538 + caddc 10540 2c2 11693 ℋchba 28696 +ℎ cva 28697 ·ℎ csm 28698 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 ax-1cn 10595 ax-hfvadd 28777 ax-hvmulid 28783 ax-hvdistr1 28785 ax-hvdistr2 28786 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fv 6363 df-ov 7159 df-2 11701 |
This theorem is referenced by: hvsubcan2i 28841 mayete3i 29505 |
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