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Lemma·L13

Distributivity

Multiplication spreads over addition.

for all , with and from D027 and D028; by commutativity also .
In words
Multiplying m by the sum of n and k gives the same result as multiplying m by each summand and adding the two products, for all natural numbers m, n, k; the mirrored law, with the factor m on the right, holds too.
Rests onF02F09F14A01A06
Never needed: F03 · F04 · F05 · F06 · F08 · F10 · F11 · F12 · F13 · A02 · A03 · A04 · A05 · A07 · A08 (computed from the citation graph, not asserted).

Proof

  1. 1
    Fix and ; induct on (T05, Separation).
  2. 2
    Base: , by D027 and D028.
  3. 3
    Step: assume . Then by D027, D028, and the hypothesis. Regrouping with associativity: by D028 again.
  4. 4
    T05 closes the induction; the mirrored law follows by three applications of L12 with substitutivity.
∎

Remarks

Distributivity is the law that entangles the two operations, and the reason multiplication by a fixed respects everything additive. It powers associativity of multiplication, the monotonicity facts of L18, and every algebraic manipulation in the division algorithm and divisibility theory.
Wikipedia ↗Nat.left_distrib ↗

Used by

L14L18L19L40L51T10T23
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