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Mirrors > Home > ILE Home > Th. List > adddid | GIF version |
Description: Distributive law (left-distributivity). (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
addcld.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
addcld.2 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
addassd.3 | ⊢ (𝜑 → 𝐶 ∈ ℂ) |
Ref | Expression |
---|---|
adddid | ⊢ (𝜑 → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | addcld.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | addcld.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
3 | addassd.3 | . 2 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
4 | adddi 7377 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶))) | |
5 | 1, 2, 3, 4 | syl3anc 1170 | 1 ⊢ (𝜑 → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1285 ∈ wcel 1434 (class class class)co 5591 ℂcc 7251 + caddc 7256 · cmul 7258 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-distr 7352 |
This theorem depends on definitions: df-bi 115 df-3an 922 |
This theorem is referenced by: subdi 7766 mulreim 7981 apadd1 7985 conjmulap 8094 cju 8315 flhalf 9598 modqcyc 9655 addmodlteq 9694 binom2 9901 binom3 9906 sqoddm1div8 9941 bcpasc 10009 remim 10121 mulreap 10125 readd 10130 remullem 10132 imadd 10138 cjadd 10145 bezoutlemnewy 10765 dvdsmulgcd 10794 lcmgcdlem 10839 |
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