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Mirrors > Home > MPE Home > Th. List > times2 | Structured version Visualization version GIF version |
Description: A number times 2. (Contributed by NM, 16-Oct-2007.) |
Ref | Expression |
---|---|
times2 | ⊢ (𝐴 ∈ ℂ → (𝐴 · 2) = (𝐴 + 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2cn 11713 | . . 3 ⊢ 2 ∈ ℂ | |
2 | mulcom 10623 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 2 ∈ ℂ) → (𝐴 · 2) = (2 · 𝐴)) | |
3 | 1, 2 | mpan2 689 | . 2 ⊢ (𝐴 ∈ ℂ → (𝐴 · 2) = (2 · 𝐴)) |
4 | 2times 11774 | . 2 ⊢ (𝐴 ∈ ℂ → (2 · 𝐴) = (𝐴 + 𝐴)) | |
5 | 3, 4 | eqtrd 2856 | 1 ⊢ (𝐴 ∈ ℂ → (𝐴 · 2) = (𝐴 + 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 (class class class)co 7156 ℂcc 10535 + caddc 10540 · cmul 10542 2c2 11693 |
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-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-mulcl 10599 ax-mulcom 10601 ax-mulass 10603 ax-distr 10604 ax-1rid 10607 ax-cnre 10610 |
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-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 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-br 5067 df-iota 6314 df-fv 6363 df-ov 7159 df-2 11701 |
This theorem is referenced by: times2i 11777 avglt1 11876 times2d 11882 |
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