Metamath Proof Explorer


Theorem hoissrrn2

Description: A half-open interval is a subset of R^n . (Contributed by Glauco Siliprandi, 21-Nov-2020)

Ref Expression
Hypotheses hoissrrn2.kph 𝑘 𝜑
hoissrrn2.a ( ( 𝜑𝑘𝑋 ) → 𝐴 ∈ ℝ )
hoissrrn2.b ( ( 𝜑𝑘𝑋 ) → 𝐵 ∈ ℝ* )
Assertion hoissrrn2 ( 𝜑X 𝑘𝑋 ( 𝐴 [,) 𝐵 ) ⊆ ( ℝ ↑m 𝑋 ) )

Proof

Step Hyp Ref Expression
1 hoissrrn2.kph 𝑘 𝜑
2 hoissrrn2.a ( ( 𝜑𝑘𝑋 ) → 𝐴 ∈ ℝ )
3 hoissrrn2.b ( ( 𝜑𝑘𝑋 ) → 𝐵 ∈ ℝ* )
4 ovex ( 𝐴 [,) 𝐵 ) ∈ V
5 4 rgenw 𝑘𝑋 ( 𝐴 [,) 𝐵 ) ∈ V
6 ixpssmapg ( ∀ 𝑘𝑋 ( 𝐴 [,) 𝐵 ) ∈ V → X 𝑘𝑋 ( 𝐴 [,) 𝐵 ) ⊆ ( 𝑘𝑋 ( 𝐴 [,) 𝐵 ) ↑m 𝑋 ) )
7 5 6 ax-mp X 𝑘𝑋 ( 𝐴 [,) 𝐵 ) ⊆ ( 𝑘𝑋 ( 𝐴 [,) 𝐵 ) ↑m 𝑋 )
8 7 a1i ( 𝜑X 𝑘𝑋 ( 𝐴 [,) 𝐵 ) ⊆ ( 𝑘𝑋 ( 𝐴 [,) 𝐵 ) ↑m 𝑋 ) )
9 reex ℝ ∈ V
10 9 a1i ( 𝜑 → ℝ ∈ V )
11 icossre ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ* ) → ( 𝐴 [,) 𝐵 ) ⊆ ℝ )
12 2 3 11 syl2anc ( ( 𝜑𝑘𝑋 ) → ( 𝐴 [,) 𝐵 ) ⊆ ℝ )
13 12 ex ( 𝜑 → ( 𝑘𝑋 → ( 𝐴 [,) 𝐵 ) ⊆ ℝ ) )
14 1 13 ralrimi ( 𝜑 → ∀ 𝑘𝑋 ( 𝐴 [,) 𝐵 ) ⊆ ℝ )
15 iunss ( 𝑘𝑋 ( 𝐴 [,) 𝐵 ) ⊆ ℝ ↔ ∀ 𝑘𝑋 ( 𝐴 [,) 𝐵 ) ⊆ ℝ )
16 14 15 sylibr ( 𝜑 𝑘𝑋 ( 𝐴 [,) 𝐵 ) ⊆ ℝ )
17 mapss ( ( ℝ ∈ V ∧ 𝑘𝑋 ( 𝐴 [,) 𝐵 ) ⊆ ℝ ) → ( 𝑘𝑋 ( 𝐴 [,) 𝐵 ) ↑m 𝑋 ) ⊆ ( ℝ ↑m 𝑋 ) )
18 10 16 17 syl2anc ( 𝜑 → ( 𝑘𝑋 ( 𝐴 [,) 𝐵 ) ↑m 𝑋 ) ⊆ ( ℝ ↑m 𝑋 ) )
19 8 18 sstrd ( 𝜑X 𝑘𝑋 ( 𝐴 [,) 𝐵 ) ⊆ ( ℝ ↑m 𝑋 ) )