P. 455 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 45401-45500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	uzssd3 45401	Subset relationship for two sets of upper integers. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ 𝑍 = (ℤ≥‘𝑀)    ⇒   ⊢ (𝑁 ∈ 𝑍 → (ℤ≥‘𝑁) ⊆ 𝑍)
Theorem	rexabsle2 45402*	An indexed set of absolute values of real numbers is bounded if and only if the original values are bounded above and below. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 (abs‘𝐵) ≤ 𝑦 ↔ (∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ∧ ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵)))
Theorem	infxrunb3rnmpt 45403*	The infimum of an unbounded-below set of extended reals is minus infinity. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*)    ⇒   ⊢ (𝜑 → (∀𝑦 ∈ ℝ ∃𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ↔ inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) = -∞))
Theorem	supxrre3rnmpt 45404*	The indexed supremum of a nonempty set of reals, is real if and only if it is bounded-above . (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝐴 ≠ ∅)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → (sup(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) ∈ ℝ ↔ ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦))
Theorem	uzublem 45405*	A set of reals, indexed by upper integers, is bound if and only if any upper part is bound. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑗𝜑    &   ⊢ Ⅎ𝑗𝑋    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑌 ∈ ℝ)    &   ⊢ 𝑊 = sup(ran (𝑗 ∈ (𝑀...𝐾) ↦ 𝐵), ℝ, < )    &   ⊢ 𝑋 = if(𝑊 ≤ 𝑌, 𝑌, 𝑊)    &   ⊢ (𝜑 → 𝐾 ∈ 𝑍)    &   ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → ∀𝑗 ∈ (ℤ≥‘𝐾)𝐵 ≤ 𝑌)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑥)
Theorem	uzub 45406*	A set of reals, indexed by upper integers, is bound if and only if any upper part is bound. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑗𝜑    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)𝐵 ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑥))
Theorem	ssrexr 45407	A subset of the reals is a subset of the extended reals. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (𝜑 → 𝐴 ⊆ ℝ)    ⇒   ⊢ (𝜑 → 𝐴 ⊆ ℝ*)
Theorem	supxrmnf2 45408	Removing minus infinity from a set does not affect its supremum. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (𝐴 ⊆ ℝ* → sup((𝐴 ∖ {-∞}), ℝ*, < ) = sup(𝐴, ℝ*, < ))
Theorem	supxrcli 45409	The supremum of an arbitrary set of extended reals is an extended real. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ 𝐴 ⊆ ℝ*    ⇒   ⊢ sup(𝐴, ℝ*, < ) ∈ ℝ*
Theorem	uzid3 45410	Membership of the least member in an upper set of integers. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ 𝑍 = (ℤ≥‘𝑀)    ⇒   ⊢ (𝑁 ∈ 𝑍 → 𝑁 ∈ (ℤ≥‘𝑁))
Theorem	infxrlesupxr 45411	The supremum of a nonempty set is greater than or equal to the infimum. The second condition is needed, see supxrltinfxr 45424. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (𝜑 → 𝐴 ⊆ ℝ*)    &   ⊢ (𝜑 → 𝐴 ≠ ∅)    ⇒   ⊢ (𝜑 → inf(𝐴, ℝ*, < ) ≤ sup(𝐴, ℝ*, < ))
Theorem	xnegeqd 45412	Equality of two extended numbers with -𝑒 in front of them. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → -𝑒𝐴 = -𝑒𝐵)
Theorem	xnegrecl 45413	The extended real negative of a real number is real. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (𝐴 ∈ ℝ → -𝑒𝐴 ∈ ℝ)
Theorem	xnegnegi 45414	Extended real version of negneg 11531. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ 𝐴 ∈ ℝ*    ⇒   ⊢ -𝑒-𝑒𝐴 = 𝐴
Theorem	xnegeqi 45415	Equality of two extended numbers with -𝑒 in front of them. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ -𝑒𝐴 = -𝑒𝐵
Theorem	nfxnegd 45416	Deduction version of nfxneg 45436. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (𝜑 → Ⅎ𝑥𝐴)    ⇒   ⊢ (𝜑 → Ⅎ𝑥-𝑒𝐴)
Theorem	xnegnegd 45417	Extended real version of negnegd 11583. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    ⇒   ⊢ (𝜑 → -𝑒-𝑒𝐴 = 𝐴)
Theorem	uzred 45418	An upper integer is a real number. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑍)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℝ)
Theorem	xnegcli 45419	Closure of extended real negative. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ 𝐴 ∈ ℝ*    ⇒   ⊢ -𝑒𝐴 ∈ ℝ*
Theorem	supminfrnmpt 45420*	The indexed supremum of a bounded-above set of reals is the negation of the indexed infimum of that set's image under negation. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝐴 ≠ ∅)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦)    ⇒   ⊢ (𝜑 → sup(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ, < ) = -inf(ran (𝑥 ∈ 𝐴 ↦ -𝐵), ℝ, < ))
Theorem	infxrpnf 45421	Adding plus infinity to a set does not affect its infimum. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (𝐴 ⊆ ℝ* → inf((𝐴 ∪ {+∞}), ℝ*, < ) = inf(𝐴, ℝ*, < ))
Theorem	infxrrnmptcl 45422*	The infimum of an arbitrary indexed set of extended reals is an extended real. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ Ⅎ𝑥𝜑    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*)    ⇒   ⊢ (𝜑 → inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) ∈ ℝ*)
Theorem	leneg2d 45423	Negative of one side of 'less than or equal to'. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝐴 ≤ -𝐵 ↔ 𝐵 ≤ -𝐴))
Theorem	supxrltinfxr 45424	The supremum of the empty set is strictly smaller than the infimum of the empty set. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ sup(∅, ℝ*, < ) < inf(∅, ℝ*, < )
Theorem	max1d 45425	A number is less than or equal to the maximum of it and another. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → 𝐴 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴))
Theorem	supxrleubrnmptf 45426	The supremum of a nonempty bounded indexed set of extended reals is less than or equal to an upper bound. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐶    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ*)    ⇒   ⊢ (𝜑 → (sup(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) ≤ 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝐶))
Theorem	nleltd 45427	'Not less than or equal to' implies 'grater than'. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → ¬ 𝐵 ≤ 𝐴)    ⇒   ⊢ (𝜑 → 𝐴 < 𝐵)
Theorem	zxrd 45428	An integer is an extended real number. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (𝜑 → 𝐴 ∈ ℤ)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℝ*)
Theorem	infxrgelbrnmpt 45429*	The infimum of an indexed set of extended reals is greater than or equal to a lower bound. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ Ⅎ𝑥𝜑    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ*)    ⇒   ⊢ (𝜑 → (𝐶 ≤ inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) ↔ ∀𝑥 ∈ 𝐴 𝐶 ≤ 𝐵))
Theorem	rphalfltd 45430	Half of a positive real is less than the original number. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (𝐴 / 2) < 𝐴)
Theorem	uzssz2 45431	An upper set of integers is a subset of all integers. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ 𝑍 = (ℤ≥‘𝑀)    ⇒   ⊢ 𝑍 ⊆ ℤ
Theorem	leneg3d 45432	Negative of one side of 'less than or equal to'. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → (-𝐴 ≤ 𝐵 ↔ -𝐵 ≤ 𝐴))
Theorem	max2d 45433	A number is less than or equal to the maximum of it and another. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → 𝐵 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴))
Theorem	uzn0bi 45434	The upper integers function needs to be applied to an integer, in order to return a nonempty set. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ ((ℤ≥‘𝑀) ≠ ∅ ↔ 𝑀 ∈ ℤ)
Theorem	xnegrecl2 45435	If the extended real negative is real, then the number itself is real. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ ((𝐴 ∈ ℝ* ∧ -𝑒𝐴 ∈ ℝ) → 𝐴 ∈ ℝ)
Theorem	nfxneg 45436	Bound-variable hypothesis builder for the negative of an extended real number. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥-𝑒𝐴
Theorem	uzxrd 45437	An upper integer is an extended real. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑍)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℝ*)
Theorem	infxrpnf2 45438	Removing plus infinity from a set does not affect its infimum. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (𝐴 ⊆ ℝ* → inf((𝐴 ∖ {+∞}), ℝ*, < ) = inf(𝐴, ℝ*, < ))
Theorem	supminfxr 45439*	The extended real suprema of a set of reals is the extended real negative of the extended real infima of that set's image under negation. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (𝜑 → 𝐴 ⊆ ℝ)    ⇒   ⊢ (𝜑 → sup(𝐴, ℝ*, < ) = -𝑒inf({𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}, ℝ*, < ))
Theorem	infrpgernmpt 45440*	The infimum of a nonempty, bounded below, indexed subset of extended reals can be approximated from above by an element of the set. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝐴 ≠ ∅)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝐵 ≤ (inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) +𝑒 𝐶))
Theorem	xnegre 45441	An extended real is real if and only if its extended negative is real. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (𝐴 ∈ ℝ* → (𝐴 ∈ ℝ ↔ -𝑒𝐴 ∈ ℝ))
Theorem	xnegrecl2d 45442	If the extended real negative is real, then the number itself is real. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → -𝑒𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℝ)
Theorem	uzxr 45443	An upper integer is an extended real. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (𝐴 ∈ (ℤ≥‘𝑀) → 𝐴 ∈ ℝ*)
Theorem	supminfxr2 45444*	The extended real suprema of a set of extended reals is the extended real negative of the extended real infima of that set's image under extended real negation. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (𝜑 → 𝐴 ⊆ ℝ*)    ⇒   ⊢ (𝜑 → sup(𝐴, ℝ*, < ) = -𝑒inf({𝑥 ∈ ℝ* ∣ -𝑒𝑥 ∈ 𝐴}, ℝ*, < ))
Theorem	xnegred 45445	An extended real is real if and only if its extended negative is real. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    ⇒   ⊢ (𝜑 → (𝐴 ∈ ℝ ↔ -𝑒𝐴 ∈ ℝ))
Theorem	supminfxrrnmpt 45446*	The indexed supremum of a set of reals is the negation of the indexed infimum of that set's image under negation. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ Ⅎ𝑥𝜑    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*)    ⇒   ⊢ (𝜑 → sup(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) = -𝑒inf(ran (𝑥 ∈ 𝐴 ↦ -𝑒𝐵), ℝ*, < ))
Theorem	min1d 45447	The minimum of two numbers is less than or equal to the first. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → if(𝐴 ≤ 𝐵, 𝐴, 𝐵) ≤ 𝐴)
Theorem	min2d 45448	The minimum of two numbers is less than or equal to the second. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → if(𝐴 ≤ 𝐵, 𝐴, 𝐵) ≤ 𝐵)
Theorem	xrnpnfmnf 45449	An extended real that is neither real nor plus infinity, is minus infinity. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → ¬ 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≠ +∞)    ⇒   ⊢ (𝜑 → 𝐴 = -∞)
Theorem	uzsscn 45450	An upper set of integers is a subset of the complex numbers. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
⊢ (ℤ≥‘𝑀) ⊆ ℂ
Theorem	absimnre 45451	The absolute value of the imaginary part of a non-real, complex number, is strictly positive. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → ¬ 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → (abs‘(ℑ‘𝐴)) ∈ ℝ+)
Theorem	uzsscn2 45452	An upper set of integers is a subset of the complex numbers. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
⊢ 𝑍 = (ℤ≥‘𝑀)    ⇒   ⊢ 𝑍 ⊆ ℂ
Theorem	xrtgcntopre 45453	The standard topologies on the extended reals and on the complex numbers, coincide when restricted to the reals. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
⊢ ((ordTop‘ ≤ ) ↾t ℝ) = ((TopOpen‘ℂfld) ↾t ℝ)
Theorem	absimlere 45454	The absolute value of the imaginary part of a complex number is a lower bound of the distance to any real number. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → (abs‘(ℑ‘𝐴)) ≤ (abs‘(𝐵 − 𝐴)))
Theorem	rpssxr 45455	The positive reals are a subset of the extended reals. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
⊢ ℝ+ ⊆ ℝ*
Theorem	monoordxrv 45456*	Ordering relation for a monotonic sequence, increasing case. (Contributed by Glauco Siliprandi, 13-Feb-2022.)
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℝ*)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘𝑘) ≤ (𝐹‘(𝑘 + 1)))    ⇒   ⊢ (𝜑 → (𝐹‘𝑀) ≤ (𝐹‘𝑁))
Theorem	monoordxr 45457*	Ordering relation for a monotonic sequence, increasing case. (Contributed by Glauco Siliprandi, 13-Feb-2022.)
⊢ Ⅎ𝑘𝜑    &   ⊢ Ⅎ𝑘𝐹    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℝ*)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘𝑘) ≤ (𝐹‘(𝑘 + 1)))    ⇒   ⊢ (𝜑 → (𝐹‘𝑀) ≤ (𝐹‘𝑁))
Theorem	monoord2xrv 45458*	Ordering relation for a monotonic sequence, decreasing case. (Contributed by Glauco Siliprandi, 13-Feb-2022.)
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℝ*)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘))    ⇒   ⊢ (𝜑 → (𝐹‘𝑁) ≤ (𝐹‘𝑀))
Theorem	monoord2xr 45459*	Ordering relation for a monotonic sequence, decreasing case. (Contributed by Glauco Siliprandi, 13-Feb-2022.)
⊢ Ⅎ𝑘𝜑    &   ⊢ Ⅎ𝑘𝐹    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℝ*)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘))    ⇒   ⊢ (𝜑 → (𝐹‘𝑁) ≤ (𝐹‘𝑀))
Theorem	xrpnf 45460*	An extended real is plus infinity iff it's larger than all real numbers. (Contributed by Glauco Siliprandi, 13-Feb-2022.)
⊢ (𝐴 ∈ ℝ* → (𝐴 = +∞ ↔ ∀𝑥 ∈ ℝ 𝑥 ≤ 𝐴))
Theorem	xlenegcon1 45461	Extended real version of lenegcon1 11739. (Contributed by Glauco Siliprandi, 23-Apr-2023.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (-𝑒𝐴 ≤ 𝐵 ↔ -𝑒𝐵 ≤ 𝐴))
Theorem	xlenegcon2 45462	Extended real version of lenegcon2 11740. (Contributed by Glauco Siliprandi, 23-Apr-2023.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ≤ -𝑒𝐵 ↔ 𝐵 ≤ -𝑒𝐴))
Theorem	pimxrneun 45463	The preimage of a set of extended reals that does not contain a value 𝐶 is the union of the preimage of the elements smaller than 𝐶 and the preimage of the subset of elements larger than 𝐶. (Contributed by Glauco Siliprandi, 21-Dec-2024.)
⊢ Ⅎ𝑥𝜑    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℝ*)    ⇒   ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 ≠ 𝐶} = ({𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶} ∪ {𝑥 ∈ 𝐴 ∣ 𝐶 < 𝐵}))
Theorem	caucvgbf 45464*	A function is convergent if and only if it is Cauchy. Theorem 12-5.3 of [Gleason] p. 180. (Contributed by Glauco Siliprandi, 15-Feb-2025.)
⊢ Ⅎ𝑗𝐹    &   ⊢ Ⅎ𝑘𝐹    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    ⇒   ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) → (𝐹 ∈ dom ⇝ ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥)))
Theorem	cvgcau 45465*	A convergent function is Cauchy. (Contributed by Glauco Siliprandi, 15-Feb-2025.)
⊢ Ⅎ𝑗𝐹    &   ⊢ Ⅎ𝑘𝐹    &   ⊢ (𝜑 → 𝑀 ∈ 𝑍)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹 ∈ dom ⇝ )    &   ⊢ (𝜑 → 𝑋 ∈ ℝ+)    ⇒   ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑋))
Theorem	cvgcaule 45466*	A convergent function is Cauchy. (Contributed by Glauco Siliprandi, 15-Feb-2025.)
⊢ Ⅎ𝑗𝐹    &   ⊢ Ⅎ𝑘𝐹    &   ⊢ (𝜑 → 𝑀 ∈ 𝑍)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹 ∈ dom ⇝ )    &   ⊢ (𝜑 → 𝑋 ∈ ℝ+)    ⇒   ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) ≤ 𝑋))
Theorem	rexanuz2nf 45467*	A simple counterexample related to theorem rexanuz2 15366, demonstrating the necessity of its disjoint variable constraints. Here, 𝑗 appears free in 𝜑, showing that without these constraints, rexanuz2 15366 and similar theorems would not hold (see rexanre 15363 and rexanuz 15362). (Contributed by Glauco Siliprandi, 15-Feb-2025.)
⊢ 𝑍 = ℕ0    &   ⊢ (𝜑 ↔ (𝑗 = 0 ∧ 𝑗 ≤ 𝑘))    &   ⊢ (𝜓 ↔ 0 < 𝑘)    ⇒   ⊢ ¬ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜑 ∧ 𝜓) ↔ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ∧ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜓))
21.43.4  Real intervals
Theorem	gtnelioc 45468	A real number larger than the upper bound of a left-open right-closed interval is not an element of the interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 < 𝐶)    ⇒   ⊢ (𝜑 → ¬ 𝐶 ∈ (𝐴(,]𝐵))
Theorem	ioossioc 45469	An open interval is a subset of its right closure. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝐴(,)𝐵) ⊆ (𝐴(,]𝐵)
Theorem	ioondisj2 45470	A condition for two open intervals not to be disjoint. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵) ∧ (𝐶 ∈ ℝ* ∧ 𝐷 ∈ ℝ* ∧ 𝐶 < 𝐷)) ∧ (𝐴 < 𝐷 ∧ 𝐷 ≤ 𝐵)) → ((𝐴(,)𝐵) ∩ (𝐶(,)𝐷)) ≠ ∅)
Theorem	ioondisj1 45471	A condition for two open intervals not to be disjoint. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵) ∧ (𝐶 ∈ ℝ* ∧ 𝐷 ∈ ℝ* ∧ 𝐶 < 𝐷)) ∧ (𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)) → ((𝐴(,)𝐵) ∩ (𝐶(,)𝐷)) ≠ ∅)
Theorem	ioogtlb 45472	An element of a closed interval is greater than its lower bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴(,)𝐵)) → 𝐴 < 𝐶)
Theorem	evthiccabs 45473*	Extreme Value Theorem on y closed interval, for the absolute value of y continuous function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘(𝐹‘𝑦)) ≤ (abs‘(𝐹‘𝑥)) ∧ ∃𝑧 ∈ (𝐴[,]𝐵)∀𝑤 ∈ (𝐴[,]𝐵)(abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑤))))
Theorem	ltnelicc 45474	A real number smaller than the lower bound of a closed interval is not an element of the interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐶 < 𝐴)    ⇒   ⊢ (𝜑 → ¬ 𝐶 ∈ (𝐴[,]𝐵))
Theorem	eliood 45475	Membership in an open real interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 𝐶)    &   ⊢ (𝜑 → 𝐶 < 𝐵)    ⇒   ⊢ (𝜑 → 𝐶 ∈ (𝐴(,)𝐵))
Theorem	iooabslt 45476	An upper bound for the distance from the center of an open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ((𝐴 − 𝐵)(,)(𝐴 + 𝐵)))    ⇒   ⊢ (𝜑 → (abs‘(𝐴 − 𝐶)) < 𝐵)
Theorem	gtnelicc 45477	A real number greater than the upper bound of a closed interval is not an element of the interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 < 𝐶)    ⇒   ⊢ (𝜑 → ¬ 𝐶 ∈ (𝐴[,]𝐵))
Theorem	iooinlbub 45478	An open interval has empty intersection with its bounds. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((𝐴(,)𝐵) ∩ {𝐴, 𝐵}) = ∅
Theorem	iocgtlb 45479	An element of a left-open right-closed interval is larger than its lower bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴(,]𝐵)) → 𝐴 < 𝐶)
Theorem	iocleub 45480	An element of a left-open right-closed interval is smaller than or equal to its upper bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴(,]𝐵)) → 𝐶 ≤ 𝐵)
Theorem	eliccd 45481	Membership in a closed real interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐶)    &   ⊢ (𝜑 → 𝐶 ≤ 𝐵)    ⇒   ⊢ (𝜑 → 𝐶 ∈ (𝐴[,]𝐵))
Theorem	eliccre 45482	A member of a closed interval of reals is real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐶 ∈ ℝ)
Theorem	eliooshift 45483	Element of an open interval shifted by a displacement. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐷 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝐴 ∈ (𝐵(,)𝐶) ↔ (𝐴 + 𝐷) ∈ ((𝐵 + 𝐷)(,)(𝐶 + 𝐷))))
Theorem	eliocd 45484	Membership in a left-open right-closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐴 < 𝐶)    &   ⊢ (𝜑 → 𝐶 ≤ 𝐵)    ⇒   ⊢ (𝜑 → 𝐶 ∈ (𝐴(,]𝐵))
Theorem	icoltub 45485	An element of a left-closed right-open interval is less than its upper bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐶 < 𝐵)
Theorem	eliocre 45486	A member of a left-open right-closed interval of reals is real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ (𝐴(,]𝐵)) → 𝐶 ∈ ℝ)
Theorem	iooltub 45487	An element of an open interval is less than its upper bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴(,)𝐵)) → 𝐶 < 𝐵)
Theorem	ioontr 45488	The interior of an interval in the standard topology on ℝ is the open interval itself. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((int‘(topGen‘ran (,)))‘(𝐴(,)𝐵)) = (𝐴(,)𝐵)
Theorem	snunioo1 45489	The closure of one end of an open real interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵) → ((𝐴(,)𝐵) ∪ {𝐴}) = (𝐴[,)𝐵))
Theorem	lbioc 45490	A left-open right-closed interval does not contain its left endpoint. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ¬ 𝐴 ∈ (𝐴(,]𝐵)
Theorem	ioomidp 45491	The midpoint is an element of the open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → ((𝐴 + 𝐵) / 2) ∈ (𝐴(,)𝐵))
Theorem	iccdifioo 45492	If the open inverval is removed from the closed interval, only the bounds are left. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → ((𝐴[,]𝐵) ∖ (𝐴(,)𝐵)) = {𝐴, 𝐵})
Theorem	iccdifprioo 45493	An open interval is the closed interval without the bounds. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐴[,]𝐵) ∖ {𝐴, 𝐵}) = (𝐴(,)𝐵))
Theorem	ioossioobi 45494	Biconditional form of ioossioo 13456. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐷 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐶 < 𝐷)    ⇒   ⊢ (𝜑 → ((𝐶(,)𝐷) ⊆ (𝐴(,)𝐵) ↔ (𝐴 ≤ 𝐶 ∧ 𝐷 ≤ 𝐵)))
Theorem	iccshift 45495*	A closed interval shifted by a real number. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝑇 ∈ ℝ)    ⇒   ⊢ (𝜑 → ((𝐴 + 𝑇)[,](𝐵 + 𝑇)) = {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴[,]𝐵)𝑤 = (𝑧 + 𝑇)})
Theorem	iccsuble 45496	An upper bound to the distance of two elements in a closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ (𝐴[,]𝐵))    &   ⊢ (𝜑 → 𝐷 ∈ (𝐴[,]𝐵))    ⇒   ⊢ (𝜑 → (𝐶 − 𝐷) ≤ (𝐵 − 𝐴))
Theorem	iocopn 45497	A left-open right-closed interval is an open set of the standard topology restricted to an interval that contains the original interval and has the same upper bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ 𝐾 = (topGen‘ran (,))    &   ⊢ 𝐽 = (𝐾 ↾t (𝐴(,]𝐵))    &   ⊢ (𝜑 → 𝐴 ≤ 𝐶)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝐶(,]𝐵) ∈ 𝐽)
Theorem	eliccelioc 45498	Membership in a closed interval and in a left-open right-closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ*)    ⇒   ⊢ (𝜑 → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ (𝐴[,]𝐵) ∧ 𝐶 ≠ 𝐴)))
Theorem	iooshift 45499*	An open interval shifted by a real number. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝑇 ∈ ℝ)    ⇒   ⊢ (𝜑 → ((𝐴 + 𝑇)(,)(𝐵 + 𝑇)) = {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴(,)𝐵)𝑤 = (𝑧 + 𝑇)})
Theorem	iccintsng 45500	Intersection of two adiacent closed intervals is a singleton. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶)) → ((𝐴[,]𝐵) ∩ (𝐵[,]𝐶)) = {𝐵})