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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 11706 | . . 3 ⊢ 2 ∈ ℂ | |
2 | mulcom 10617 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 2 ∈ ℂ) → (𝐴 · 2) = (2 · 𝐴)) | |
3 | 1, 2 | mpan2 689 | . 2 ⊢ (𝐴 ∈ ℂ → (𝐴 · 2) = (2 · 𝐴)) |
4 | 2times 11767 | . 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 1533 ∈ wcel 2110 (class class class)co 7150 ℂcc 10529 + caddc 10534 · cmul 10536 2c2 11686 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-mulcl 10593 ax-mulcom 10595 ax-mulass 10597 ax-distr 10598 ax-1rid 10601 ax-cnre 10604 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 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 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-iota 6308 df-fv 6357 df-ov 7153 df-2 11694 |
This theorem is referenced by: times2i 11770 avglt1 11869 times2d 11875 |
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