Axiom ax-pre-mulext 6781

Description: Strong extensionality of multiplication (expressed in terms of <_ℝ). Axiom for real and complex numbers, justified by theorem axpre-mulext 6752

(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 6690	. . . 4 class ℝ
3	1, 2	wcel 1390	. . 3 wff A ∈ ℝ
4		cB	. . . 4 class B
5	4, 2	wcel 1390	. . 3 wff B ∈ ℝ
6		cC	. . . 4 class 𝐶
7	6, 2	wcel 1390	. . 3 wff 𝐶 ∈ ℝ
8	3, 5, 7	w3a 884	. 2 wff (A ∈ ℝ ∧ B ∈ ℝ ∧ 𝐶 ∈ ℝ)
9		cmul 6696	. . . . 5 class ·
10	1, 6, 9	co 5455	. . . 4 class (A · 𝐶)
11	4, 6, 9	co 5455	. . . 4 class (B · 𝐶)
12		cltrr 6695	. . . 4 class <ℝ
13	10, 11, 12	wbr 3755	. . 3 wff (A · 𝐶) <ℝ (B · 𝐶)
14	1, 4, 12	wbr 3755	. . . 4 wff A <ℝ B
15	4, 1, 12	wbr 3755	. . . 4 wff B <ℝ A
16	14, 15	wo 628	. . 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  7363