Theorem xrsle

Description: The ordering of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015)

	Ref	Expression
Assertion	xrsle	⊢ ≤ = ( le ‘ ℝ*𝑠 )

Proof

Step	Hyp	Ref	Expression
1		xrex	⊢ ℝ* ∈ V
2	1 1	xpex	⊢ ( ℝ* × ℝ* ) ∈ V
3		lerelxr	⊢ ≤ ⊆ ( ℝ* × ℝ* )
4	2 3	ssexi	⊢ ≤ ∈ V
5		df-xrs	⊢ ℝ*𝑠 = ( { ⟨ ( Base ‘ ndx ) , ℝ* ⟩ , ⟨ ( +g ‘ ndx ) , +e ⟩ , ⟨ ( .r ‘ ndx ) , ·e ⟩ } ∪ { ⟨ ( TopSet ‘ ndx ) , ( ordTop ‘ ≤ ) ⟩ , ⟨ ( le ‘ ndx ) , ≤ ⟩ , ⟨ ( dist ‘ ndx ) , ( 𝑥 ∈ ℝ* , 𝑦 ∈ ℝ* ↦ if ( 𝑥 ≤ 𝑦 , ( 𝑦 +e -e 𝑥 ) , ( 𝑥 +e -e 𝑦 ) ) ) ⟩ } )
6	5	odrngle	⊢ ( ≤ ∈ V → ≤ = ( le ‘ ℝ*𝑠 ) )
7	4 6	ax-mp	⊢ ≤ = ( le ‘ ℝ*𝑠 )