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Theorem adddii 6835
Description: Distributive law (left-distributivity). (Contributed by NM, 23-Nov-1994.)
Hypotheses
Ref Expression
axi.1 A
axi.2 B
axi.3 𝐶
Assertion
Ref Expression
adddii (A · (B + 𝐶)) = ((A · B) + (A · 𝐶))

Proof of Theorem adddii
StepHypRef Expression
1 axi.1 . 2 A
2 axi.2 . 2 B
3 axi.3 . 2 𝐶
4 adddi 6811 . 2 ((A B 𝐶 ℂ) → (A · (B + 𝐶)) = ((A · B) + (A · 𝐶)))
51, 2, 3, 4mp3an 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  cc 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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