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Mirrors > Home > ILE Home > Th. List > adddii | GIF version |
Description: Distributive law (left-distributivity). (Contributed by NM, 23-Nov-1994.) |
Ref | Expression |
---|---|
axi.1 | ⊢ A ∈ ℂ |
axi.2 | ⊢ B ∈ ℂ |
axi.3 | ⊢ 𝐶 ∈ ℂ |
Ref | Expression |
---|---|
adddii | ⊢ (A · (B + 𝐶)) = ((A · B) + (A · 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | axi.1 | . 2 ⊢ A ∈ ℂ | |
2 | axi.2 | . 2 ⊢ B ∈ ℂ | |
3 | axi.3 | . 2 ⊢ 𝐶 ∈ ℂ | |
4 | adddi 6811 | . 2 ⊢ ((A ∈ ℂ ∧ B ∈ ℂ ∧ 𝐶 ∈ ℂ) → (A · (B + 𝐶)) = ((A · B) + (A · 𝐶))) | |
5 | 1, 2, 3, 4 | mp3an 1231 | 1 ⊢ (A · (B + 𝐶)) = ((A · B) + (A · 𝐶)) |
Colors of variables: wff set class |
Syntax hints: = wceq 1242 ∈ wcel 1390 (class class class)co 5455 ℂcc 6709 + caddc 6714 · cmul 6716 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-distr 6787 |
This theorem depends on definitions: df-bi 110 df-3an 886 |
This theorem is referenced by: 3t3e9 7850 numltc 8163 numsucc 8169 numma 8174 4t3lem 8214 decbin2 8247 binom2i 9013 |
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