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Mirrors > Home > ILE Home > Th. List > ax-mulrcl | GIF version |
Description: Closure law for multiplication in the real subfield of complex numbers. Axiom for real and complex numbers, justified by theorem axmulrcl 6753. Proofs should normally use remulcl 6807 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.) |
Ref | Expression |
---|---|
ax-mulrcl | ⊢ ((A ∈ ℝ ∧ B ∈ ℝ) → (A · B) ∈ ℝ) |
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 | 3, 5 | wa 97 | . 2 wff (A ∈ ℝ ∧ B ∈ ℝ) |
7 | cmul 6716 | . . . 4 class · | |
8 | 1, 4, 7 | co 5455 | . . 3 class (A · B) |
9 | 8, 2 | wcel 1390 | . 2 wff (A · B) ∈ ℝ |
10 | 6, 9 | wi 4 | 1 wff ((A ∈ ℝ ∧ B ∈ ℝ) → (A · B) ∈ ℝ) |
Colors of variables: wff set class |
This axiom is referenced by: remulcl 6807 |
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