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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 7752 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶))) | |
5 | 1, 2, 3, 4 | syl3anc 1216 | 1 ⊢ (𝜑 → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∈ wcel 1480 (class class class)co 5774 ℂcc 7618 + caddc 7623 · cmul 7625 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-distr 7724 |
This theorem depends on definitions: df-bi 116 df-3an 964 |
This theorem is referenced by: subdi 8147 mulreim 8366 apadd1 8370 conjmulap 8489 cju 8719 flhalf 10075 modqcyc 10132 addmodlteq 10171 binom2 10403 binom3 10409 sqoddm1div8 10444 bcpasc 10512 remim 10632 mulreap 10636 readd 10641 remullem 10643 imadd 10649 cjadd 10656 bdtrilem 11010 fsummulc2 11217 binomlem 11252 tanval3ap 11421 sinadd 11443 tanaddap 11446 bezoutlemnewy 11684 dvdsmulgcd 11713 lcmgcdlem 11758 tangtx 12919 |
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