![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > 1p1times | Structured version Visualization version GIF version |
Description: Two times a number. (Contributed by NM, 18-May-1999.) (Revised by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
1p1times | ⊢ (𝐴 ∈ ℂ → ((1 + 1) · 𝐴) = (𝐴 + 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1cnd 10625 | . 2 ⊢ (𝐴 ∈ ℂ → 1 ∈ ℂ) | |
2 | id 22 | . 2 ⊢ (𝐴 ∈ ℂ → 𝐴 ∈ ℂ) | |
3 | mulid2 10629 | . . 3 ⊢ (𝐴 ∈ ℂ → (1 · 𝐴) = 𝐴) | |
4 | 3, 3 | oveq12d 7153 | . 2 ⊢ (𝐴 ∈ ℂ → ((1 · 𝐴) + (1 · 𝐴)) = (𝐴 + 𝐴)) |
5 | 1, 2, 1, 4 | joinlmuladdmuld 10657 | 1 ⊢ (𝐴 ∈ ℂ → ((1 + 1) · 𝐴) = (𝐴 + 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 (class class class)co 7135 ℂcc 10524 1c1 10527 + caddc 10529 · cmul 10531 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-mulcl 10588 ax-mulcom 10590 ax-mulass 10592 ax-distr 10593 ax-1rid 10596 ax-cnre 10599 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-ral 3111 df-rex 3112 df-v 3443 df-un 3886 df-in 3888 df-ss 3898 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-iota 6283 df-fv 6332 df-ov 7138 |
This theorem is referenced by: addcom 10815 addcomd 10831 eqneg 11349 2times 11761 |
Copyright terms: Public domain | W3C validator |