Axiom ax-pre-mulext 7997

Description: Strong extensionality of multiplication (expressed in terms of <_ℝ). Axiom for real and complex numbers, justified by Theorem axpre-mulext 7955

(Contributed by Jim Kingdon, 18-Feb-2020.)

Assertion

Ref	Expression
ax-pre-mulext	⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 · 𝐶) <ℝ (𝐵 · 𝐶) → (𝐴 <ℝ 𝐵 ∨ 𝐵 <ℝ 𝐴)))

Detailed syntax breakdown of Axiom ax-pre-mulext

Step	Hyp	Ref	Expression
1		cA	. . . 4 class 𝐴
2		cr 7878	. . . 4 class ℝ
3	1, 2	wcel 2167	. . 3 wff 𝐴 ∈ ℝ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 2167	. . 3 wff 𝐵 ∈ ℝ
6		cC	. . . 4 class 𝐶
7	6, 2	wcel 2167	. . 3 wff 𝐶 ∈ ℝ
8	3, 5, 7	w3a 980	. 2 wff (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)
9		cmul 7884	. . . . 5 class ·
10	1, 6, 9	co 5922	. . . 4 class (𝐴 · 𝐶)
11	4, 6, 9	co 5922	. . . 4 class (𝐵 · 𝐶)
12		cltrr 7883	. . . 4 class <ℝ
13	10, 11, 12	wbr 4033	. . 3 wff (𝐴 · 𝐶) <ℝ (𝐵 · 𝐶)
14	1, 4, 12	wbr 4033	. . . 4 wff 𝐴 <ℝ 𝐵
15	4, 1, 12	wbr 4033	. . . 4 wff 𝐵 <ℝ 𝐴
16	14, 15	wo 709	. . 3 wff (𝐴 <ℝ 𝐵 ∨ 𝐵 <ℝ 𝐴)
17	13, 16	wi 4	. 2 wff ((𝐴 · 𝐶) <ℝ (𝐵 · 𝐶) → (𝐴 <ℝ 𝐵 ∨ 𝐵 <ℝ 𝐴))
18	8, 17	wi 4	1 wff ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 · 𝐶) <ℝ (𝐵 · 𝐶) → (𝐴 <ℝ 𝐵 ∨ 𝐵 <ℝ 𝐴)))

This axiom is referenced by:  remulext1  8626