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Axiom ax-pre-mulext 7000
Description: Strong extensionality of multiplication (expressed in terms of <). Axiom for real and complex numbers, justified by theorem axpre-mulext 6960

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

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

Detailed syntax breakdown of Axiom ax-pre-mulext
StepHypRef Expression
1 cA . . . 4 class 𝐴
2 cr 6886 . . . 4 class
31, 2wcel 1393 . . 3 wff 𝐴 ∈ ℝ
4 cB . . . 4 class 𝐵
54, 2wcel 1393 . . 3 wff 𝐵 ∈ ℝ
6 cC . . . 4 class 𝐶
76, 2wcel 1393 . . 3 wff 𝐶 ∈ ℝ
83, 5, 7w3a 885 . 2 wff (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)
9 cmul 6892 . . . . 5 class ·
101, 6, 9co 5512 . . . 4 class (𝐴 · 𝐶)
114, 6, 9co 5512 . . . 4 class (𝐵 · 𝐶)
12 cltrr 6891 . . . 4 class <
1310, 11, 12wbr 3764 . . 3 wff (𝐴 · 𝐶) < (𝐵 · 𝐶)
141, 4, 12wbr 3764 . . . 4 wff 𝐴 < 𝐵
154, 1, 12wbr 3764 . . . 4 wff 𝐵 < 𝐴
1614, 15wo 629 . . 3 wff (𝐴 < 𝐵𝐵 < 𝐴)
1713, 16wi 4 . 2 wff ((𝐴 · 𝐶) < (𝐵 · 𝐶) → (𝐴 < 𝐵𝐵 < 𝐴))
188, 17wi 4 1 wff ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 · 𝐶) < (𝐵 · 𝐶) → (𝐴 < 𝐵𝐵 < 𝐴)))
Colors of variables: wff set class
This axiom is referenced by:  remulext1  7588
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