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Mirrors > Home > ILE Home > Th. List > ax-pre-mulext | GIF version |
Description: Strong extensionality of
multiplication (expressed in terms of <ℝ).
Axiom for real and complex numbers, justified by theorem axpre-mulext 6772
(Contributed by Jim Kingdon, 18-Feb-2020.) |
Ref | Expression |
---|---|
ax-pre-mulext | ⊢ ((A ∈ ℝ ∧ B ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((A · 𝐶) <ℝ (B · 𝐶) → (A <ℝ B ∨ B <ℝ A))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cA | . . . 4 class A | |
2 | cr 6710 | . . . 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 6716 | . . . . 5 class · | |
10 | 1, 6, 9 | co 5455 | . . . 4 class (A · 𝐶) |
11 | 4, 6, 9 | co 5455 | . . . 4 class (B · 𝐶) |
12 | cltrr 6715 | . . . 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))) |
Colors of variables: wff set class |
This axiom is referenced by: remulext1 7383 |
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