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Mirrors > Home > ILE Home > Th. List > remulcli | GIF version |
Description: Closure law for multiplication of reals. (Contributed by NM, 17-Jan-1997.) |
Ref | Expression |
---|---|
recni.1 | ⊢ 𝐴 ∈ ℝ |
axri.2 | ⊢ 𝐵 ∈ ℝ |
Ref | Expression |
---|---|
remulcli | ⊢ (𝐴 · 𝐵) ∈ ℝ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | recni.1 | . 2 ⊢ 𝐴 ∈ ℝ | |
2 | axri.2 | . 2 ⊢ 𝐵 ∈ ℝ | |
3 | remulcl 7468 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) ∈ ℝ) | |
4 | 1, 2, 3 | mp2an 417 | 1 ⊢ (𝐴 · 𝐵) ∈ ℝ |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1438 (class class class)co 5652 ℝcr 7347 · cmul 7353 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia3 106 ax-mulrcl 7442 |
This theorem is referenced by: addltmul 8650 nn0lele2xi 8722 numltc 8900 ef01bndlem 11043 cos2bnd 11047 sin4lt0 11053 |
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