Lemma·L46
The embedding preserves addition and multiplication
The embedding of the naturals into the integers turns sums into sums and products into products.
For
, with
from D057:
In words
Embedding two natural numbers and then adding them as integers gives the same result as adding them as naturals and then embedding; likewise for multiplying.
Never needed: F03 · F04 · F05 · F06 · F08 · F10 · F11 · F12 · F13 · A03 · A04 · A05 · A07 · A08 (computed from the citation graph, not asserted).
Proof
- 1
- 2
∎
Remarks
Makes
a ring homomorphism from
into
(T23); combined with injectivity (L42) and order-preservation (L49),
sits inside
as a faithful, structure-preserving copy. Used throughout Bezout's identity to move the division algorithm's equation from
into
.