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PROOF 10

Prove.

What an extension preserves determines the extension

On each number system.

What you may use

A1a + 0 = 0 + a = a
additive identity
A2there exists −a with a + (−a) = 0
additive inverse
A3a × 1 = a
multiplicative identity
A4a ≠ 0 ⟹ there exists b with ab = 1
multiplicative inverse
A5(ab)c = a(bc)
associativity of multiplication
A6ab = ba
commutativity of multiplication
A7a(b + c) = ab + ac
distributive law
A80 ≠ 1
0 and 1 are distinct numbers
A9a × 0 = 0
lemma from Proofs 01 and 02
A10(−a)(−a) = a × a
product of negatives (Proof 01)
A11exactly one of a<b, a=b, b<a
trichotomy of order
A12a < b ⟹ a + c < b + c
order and addition
A130<a, 0<b ⟹ 0 < ab
order and multiplication
A14a<b, b<c ⟹ a<c
transitivity of order
A15∅ ≠ S ⟹ S has a least element
well-ordering (property of ℕ)
Ea = b ⟹ apply the same operation to both sides
properties of equality

Nothing outside this list may be cited.

Definition D1

An ordered field is a number system satisfying A1–A8, with a relation < on it satisfying A11–A14.

Proof.