Description: Extended real closure of the # function. (Contributed by Mario Carneiro, 22-Apr-2015)
Ref | Expression | ||
---|---|---|---|
Assertion | hashxrcl | ⊢ ( 𝐴 ∈ 𝑉 → ( ♯ ‘ 𝐴 ) ∈ ℝ^{*} ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0ssre | ⊢ ℕ_{0} ⊆ ℝ | |
2 | ressxr | ⊢ ℝ ⊆ ℝ^{*} | |
3 | 1 2 | sstri | ⊢ ℕ_{0} ⊆ ℝ^{*} |
4 | pnfxr | ⊢ +∞ ∈ ℝ^{*} | |
5 | snssi | ⊢ ( +∞ ∈ ℝ^{*} → { +∞ } ⊆ ℝ^{*} ) | |
6 | 4 5 | ax-mp | ⊢ { +∞ } ⊆ ℝ^{*} |
7 | 3 6 | unssi | ⊢ ( ℕ_{0} ∪ { +∞ } ) ⊆ ℝ^{*} |
8 | elex | ⊢ ( 𝐴 ∈ 𝑉 → 𝐴 ∈ V ) | |
9 | hashf | ⊢ ♯ : V ⟶ ( ℕ_{0} ∪ { +∞ } ) | |
10 | 9 | ffvelrni | ⊢ ( 𝐴 ∈ V → ( ♯ ‘ 𝐴 ) ∈ ( ℕ_{0} ∪ { +∞ } ) ) |
11 | 8 10 | syl | ⊢ ( 𝐴 ∈ 𝑉 → ( ♯ ‘ 𝐴 ) ∈ ( ℕ_{0} ∪ { +∞ } ) ) |
12 | 7 11 | sseldi | ⊢ ( 𝐴 ∈ 𝑉 → ( ♯ ‘ 𝐴 ) ∈ ℝ^{*} ) |