Theorem infxrpnf2

Description: Removing plus infinity from a set does not affect its infimum. (Contributed by Glauco Siliprandi, 2-Jan-2022)

	Ref	Expression
Assertion	infxrpnf2	⊢ ( 𝐴 ⊆ ℝ* → inf ( ( 𝐴 ∖ { +∞ } ) , ℝ* , < ) = inf ( 𝐴 , ℝ* , < ) )

Proof

Step	Hyp	Ref	Expression
1		ssdifss	⊢ ( 𝐴 ⊆ ℝ* → ( 𝐴 ∖ { +∞ } ) ⊆ ℝ* )
2		infxrpnf	⊢ ( ( 𝐴 ∖ { +∞ } ) ⊆ ℝ* → inf ( ( ( 𝐴 ∖ { +∞ } ) ∪ { +∞ } ) , ℝ* , < ) = inf ( ( 𝐴 ∖ { +∞ } ) , ℝ* , < ) )
3	1 2	syl	⊢ ( 𝐴 ⊆ ℝ* → inf ( ( ( 𝐴 ∖ { +∞ } ) ∪ { +∞ } ) , ℝ* , < ) = inf ( ( 𝐴 ∖ { +∞ } ) , ℝ* , < ) )
4	3	adantr	⊢ ( ( 𝐴 ⊆ ℝ* ∧ +∞ ∈ 𝐴 ) → inf ( ( ( 𝐴 ∖ { +∞ } ) ∪ { +∞ } ) , ℝ* , < ) = inf ( ( 𝐴 ∖ { +∞ } ) , ℝ* , < ) )
5		difsnid	⊢ ( +∞ ∈ 𝐴 → ( ( 𝐴 ∖ { +∞ } ) ∪ { +∞ } ) = 𝐴 )
6	5	infeq1d	⊢ ( +∞ ∈ 𝐴 → inf ( ( ( 𝐴 ∖ { +∞ } ) ∪ { +∞ } ) , ℝ* , < ) = inf ( 𝐴 , ℝ* , < ) )
7	6	adantl	⊢ ( ( 𝐴 ⊆ ℝ* ∧ +∞ ∈ 𝐴 ) → inf ( ( ( 𝐴 ∖ { +∞ } ) ∪ { +∞ } ) , ℝ* , < ) = inf ( 𝐴 , ℝ* , < ) )
8	4 7	eqtr3d	⊢ ( ( 𝐴 ⊆ ℝ* ∧ +∞ ∈ 𝐴 ) → inf ( ( 𝐴 ∖ { +∞ } ) , ℝ* , < ) = inf ( 𝐴 , ℝ* , < ) )
9		difsn	⊢ ( ¬ +∞ ∈ 𝐴 → ( 𝐴 ∖ { +∞ } ) = 𝐴 )
10	9	infeq1d	⊢ ( ¬ +∞ ∈ 𝐴 → inf ( ( 𝐴 ∖ { +∞ } ) , ℝ* , < ) = inf ( 𝐴 , ℝ* , < ) )
11	10	adantl	⊢ ( ( 𝐴 ⊆ ℝ* ∧ ¬ +∞ ∈ 𝐴 ) → inf ( ( 𝐴 ∖ { +∞ } ) , ℝ* , < ) = inf ( 𝐴 , ℝ* , < ) )
12	8 11	pm2.61dan	⊢ ( 𝐴 ⊆ ℝ* → inf ( ( 𝐴 ∖ { +∞ } ) , ℝ* , < ) = inf ( 𝐴 , ℝ* , < ) )