P. 122 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 12101-12200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	fimaxre2 12101*	A nonempty finite set of real numbers has an upper bound. (Contributed by Jeff Madsen, 27-May-2011.) (Revised by Mario Carneiro, 13-Feb-2014.)
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ∈ Fin) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)
Theorem	fimaxre3 12102*	A nonempty finite set of real numbers has a maximum (image set version). (Contributed by Mario Carneiro, 13-Feb-2014.)
⊢ ((𝐴 ∈ Fin ∧ ∀𝑦 ∈ 𝐴 𝐵 ∈ ℝ) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝐵 ≤ 𝑥)
Theorem	fiminre 12103*	A nonempty finite set of real numbers has a minimum. Analogous to fimaxre 12100. (Contributed by AV, 9-Aug-2020.) (Proof shortened by Steven Nguyen, 3-Jun-2023.)
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦)
Theorem	fiminre2 12104*	A nonempty finite set of real numbers is bounded below. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ∈ Fin) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦)
Theorem	negfi 12105*	The negation of a finite set of real numbers is finite. (Contributed by AV, 9-Aug-2020.)
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ∈ Fin) → {𝑛 ∈ ℝ ∣ -𝑛 ∈ 𝐴} ∈ Fin)
Theorem	lbreu 12106*	If a set of reals contains a lower bound, it contains a unique lower bound. (Contributed by NM, 9-Oct-2005.)
⊢ ((𝑆 ⊆ ℝ ∧ ∃𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) → ∃!𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦)
Theorem	lbcl 12107*	If a set of reals contains a lower bound, it contains a unique lower bound that belongs to the set. (Contributed by NM, 9-Oct-2005.) (Revised by Mario Carneiro, 24-Dec-2016.)
⊢ ((𝑆 ⊆ ℝ ∧ ∃𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) → (℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) ∈ 𝑆)
Theorem	lble 12108*	If a set of reals contains a lower bound, the lower bound is less than or equal to all members of the set. (Contributed by NM, 9-Oct-2005.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
⊢ ((𝑆 ⊆ ℝ ∧ ∃𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ 𝐴 ∈ 𝑆) → (℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) ≤ 𝐴)
Theorem	lbinf 12109*	If a set of reals contains a lower bound, the lower bound is its infimum. (Contributed by NM, 9-Oct-2005.) (Revised by AV, 4-Sep-2020.)
⊢ ((𝑆 ⊆ ℝ ∧ ∃𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) → inf(𝑆, ℝ, < ) = (℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦))
Theorem	lbinfcl 12110*	If a set of reals contains a lower bound, it contains its infimum. (Contributed by NM, 11-Oct-2005.) (Revised by AV, 4-Sep-2020.)
⊢ ((𝑆 ⊆ ℝ ∧ ∃𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) → inf(𝑆, ℝ, < ) ∈ 𝑆)
Theorem	lbinfle 12111*	If a set of reals contains a lower bound, its infimum is less than or equal to all members of the set. (Contributed by NM, 11-Oct-2005.) (Revised by AV, 4-Sep-2020.)
⊢ ((𝑆 ⊆ ℝ ∧ ∃𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ 𝐴 ∈ 𝑆) → inf(𝑆, ℝ, < ) ≤ 𝐴)
Theorem	sup2 12112*	A nonempty, bounded-above set of reals has a supremum. Stronger version of completeness axiom (it has a slightly weaker antecedent). (Contributed by NM, 19-Jan-1997.)
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 (𝑦 < 𝑥 ∨ 𝑦 = 𝑥)) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
Theorem	sup3 12113*	A version of the completeness axiom for reals. (Contributed by NM, 12-Oct-2004.)
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
Theorem	infm3lem 12114*	Lemma for infm3 12115. (Contributed by NM, 14-Jun-2005.)
⊢ (𝑥 ∈ ℝ → ∃𝑦 ∈ ℝ 𝑥 = -𝑦)
Theorem	infm3 12115*	The completeness axiom for reals in terms of infimum: a nonempty, bounded-below set of reals has an infimum. (This theorem is the dual of sup3 12113.) (Contributed by NM, 14-Jun-2005.)
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
Theorem	suprcl 12116*	Closure of supremum of a nonempty bounded set of reals. (Contributed by NM, 12-Oct-2004.)
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → sup(𝐴, ℝ, < ) ∈ ℝ)
Theorem	suprub 12117*	A member of a nonempty bounded set of reals is less than or equal to the set's upper bound. (Contributed by NM, 12-Oct-2004.)
⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ 𝐵 ∈ 𝐴) → 𝐵 ≤ sup(𝐴, ℝ, < ))
Theorem	suprubd 12118*	Natural deduction form of suprubd 12118. (Contributed by Stanislas Polu, 9-Mar-2020.)
⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐴 ≠ ∅)    &   ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐴)    ⇒   ⊢ (𝜑 → 𝐵 ≤ sup(𝐴, ℝ, < ))
Theorem	suprcld 12119*	Natural deduction form of suprcl 12116. (Contributed by Stanislas Polu, 9-Mar-2020.)
⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐴 ≠ ∅)    &   ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)    ⇒   ⊢ (𝜑 → sup(𝐴, ℝ, < ) ∈ ℝ)
Theorem	suprlub 12120*	The supremum of a nonempty bounded set of reals is the least upper bound. (Contributed by NM, 15-Nov-2004.) (Revised by Mario Carneiro, 6-Sep-2014.)
⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ 𝐵 ∈ ℝ) → (𝐵 < sup(𝐴, ℝ, < ) ↔ ∃𝑧 ∈ 𝐴 𝐵 < 𝑧))
Theorem	suprnub 12121*	An upper bound is not less than the supremum of a nonempty bounded set of reals. (Contributed by NM, 15-Nov-2004.) (Revised by Mario Carneiro, 6-Sep-2014.)
⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ 𝐵 ∈ ℝ) → (¬ 𝐵 < sup(𝐴, ℝ, < ) ↔ ∀𝑧 ∈ 𝐴 ¬ 𝐵 < 𝑧))
Theorem	suprleub 12122*	The supremum of a nonempty bounded set of reals is less than or equal to an upper bound. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 6-Sep-2014.)
⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ 𝐵 ∈ ℝ) → (sup(𝐴, ℝ, < ) ≤ 𝐵 ↔ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝐵))
Theorem	supaddc 12123*	The supremum function distributes over addition in a sense similar to that in supmul1 12125. (Contributed by Brendan Leahy, 25-Sep-2017.)
⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐴 ≠ ∅)    &   ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ 𝐶 = {𝑧 ∣ ∃𝑣 ∈ 𝐴 𝑧 = (𝑣 + 𝐵)}    ⇒   ⊢ (𝜑 → (sup(𝐴, ℝ, < ) + 𝐵) = sup(𝐶, ℝ, < ))
Theorem	supadd 12124*	The supremum function distributes over addition in a sense similar to that in supmul 12128. (Contributed by Brendan Leahy, 26-Sep-2017.)
⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐴 ≠ ∅)    &   ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)    &   ⊢ (𝜑 → 𝐵 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐵 ≠ ∅)    &   ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐵 𝑦 ≤ 𝑥)    &   ⊢ 𝐶 = {𝑧 ∣ ∃𝑣 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑣 + 𝑏)}    ⇒   ⊢ (𝜑 → (sup(𝐴, ℝ, < ) + sup(𝐵, ℝ, < )) = sup(𝐶, ℝ, < ))
Theorem	supmul1 12125*	The supremum function distributes over multiplication, in the sense that 𝐴 · (sup𝐵) = sup(𝐴 · 𝐵), where 𝐴 · 𝐵 is shorthand for {𝐴 · 𝑏 ∣ 𝑏 ∈ 𝐵} and is defined as 𝐶 below. This is the simple version, with only one set argument; see supmul 12128 for the more general case with two set arguments. (Contributed by Mario Carneiro, 5-Jul-2013.)
⊢ 𝐶 = {𝑧 ∣ ∃𝑣 ∈ 𝐵 𝑧 = (𝐴 · 𝑣)}    &   ⊢ (𝜑 ↔ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ ∀𝑥 ∈ 𝐵 0 ≤ 𝑥) ∧ (𝐵 ⊆ ℝ ∧ 𝐵 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐵 𝑦 ≤ 𝑥)))    ⇒   ⊢ (𝜑 → (𝐴 · sup(𝐵, ℝ, < )) = sup(𝐶, ℝ, < ))
Theorem	supmullem1 12126*	Lemma for supmul 12128. (Contributed by Mario Carneiro, 5-Jul-2013.)
⊢ 𝐶 = {𝑧 ∣ ∃𝑣 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑣 · 𝑏)}    &   ⊢ (𝜑 ↔ ((∀𝑥 ∈ 𝐴 0 ≤ 𝑥 ∧ ∀𝑥 ∈ 𝐵 0 ≤ 𝑥) ∧ (𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ (𝐵 ⊆ ℝ ∧ 𝐵 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐵 𝑦 ≤ 𝑥)))    ⇒   ⊢ (𝜑 → ∀𝑤 ∈ 𝐶 𝑤 ≤ (sup(𝐴, ℝ, < ) · sup(𝐵, ℝ, < )))
Theorem	supmullem2 12127*	Lemma for supmul 12128. (Contributed by Mario Carneiro, 5-Jul-2013.)
⊢ 𝐶 = {𝑧 ∣ ∃𝑣 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑣 · 𝑏)}    &   ⊢ (𝜑 ↔ ((∀𝑥 ∈ 𝐴 0 ≤ 𝑥 ∧ ∀𝑥 ∈ 𝐵 0 ≤ 𝑥) ∧ (𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ (𝐵 ⊆ ℝ ∧ 𝐵 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐵 𝑦 ≤ 𝑥)))    ⇒   ⊢ (𝜑 → (𝐶 ⊆ ℝ ∧ 𝐶 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑤 ∈ 𝐶 𝑤 ≤ 𝑥))
Theorem	supmul 12128*	The supremum function distributes over multiplication, in the sense that (sup𝐴) · (sup𝐵) = sup(𝐴 · 𝐵), where 𝐴 · 𝐵 is shorthand for {𝑎 · 𝑏 ∣ 𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵} and is defined as 𝐶 below. We made use of this in our definition of multiplication in the Dedekind cut construction of the reals (see df-mp 10907). (Contributed by Mario Carneiro, 5-Jul-2013.) (Revised by Mario Carneiro, 6-Sep-2014.)
⊢ 𝐶 = {𝑧 ∣ ∃𝑣 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑣 · 𝑏)}    &   ⊢ (𝜑 ↔ ((∀𝑥 ∈ 𝐴 0 ≤ 𝑥 ∧ ∀𝑥 ∈ 𝐵 0 ≤ 𝑥) ∧ (𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ (𝐵 ⊆ ℝ ∧ 𝐵 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐵 𝑦 ≤ 𝑥)))    ⇒   ⊢ (𝜑 → (sup(𝐴, ℝ, < ) · sup(𝐵, ℝ, < )) = sup(𝐶, ℝ, < ))
Theorem	sup3ii 12129*	A version of the completeness axiom for reals. (Contributed by NM, 23-Aug-1999.)
⊢ (𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)    ⇒   ⊢ ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))
Theorem	suprclii 12130*	Closure of supremum of a nonempty bounded set of reals. (Contributed by NM, 12-Sep-1999.)
⊢ (𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)    ⇒   ⊢ sup(𝐴, ℝ, < ) ∈ ℝ
Theorem	suprubii 12131*	A member of a nonempty bounded set of reals is less than or equal to the set's upper bound. (Contributed by NM, 12-Sep-1999.)
⊢ (𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)    ⇒   ⊢ (𝐵 ∈ 𝐴 → 𝐵 ≤ sup(𝐴, ℝ, < ))
Theorem	suprlubii 12132*	The supremum of a nonempty bounded set of reals is the least upper bound. (Contributed by NM, 15-Oct-2004.) (Revised by Mario Carneiro, 6-Sep-2014.)
⊢ (𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)    ⇒   ⊢ (𝐵 ∈ ℝ → (𝐵 < sup(𝐴, ℝ, < ) ↔ ∃𝑧 ∈ 𝐴 𝐵 < 𝑧))
Theorem	suprnubii 12133*	An upper bound is not less than the supremum of a nonempty bounded set of reals. (Contributed by NM, 15-Oct-2004.) (Revised by Mario Carneiro, 6-Sep-2014.)
⊢ (𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)    ⇒   ⊢ (𝐵 ∈ ℝ → (¬ 𝐵 < sup(𝐴, ℝ, < ) ↔ ∀𝑧 ∈ 𝐴 ¬ 𝐵 < 𝑧))
Theorem	suprleubii 12134*	The supremum of a nonempty bounded set of reals is less than or equal to an upper bound. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 6-Sep-2014.)
⊢ (𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)    ⇒   ⊢ (𝐵 ∈ ℝ → (sup(𝐴, ℝ, < ) ≤ 𝐵 ↔ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝐵))
Theorem	riotaneg 12135*	The negative of the unique real such that 𝜑. (Contributed by NM, 13-Jun-2005.)
⊢ (𝑥 = -𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃!𝑥 ∈ ℝ 𝜑 → (℩𝑥 ∈ ℝ 𝜑) = -(℩𝑦 ∈ ℝ 𝜓))
Theorem	negiso 12136	Negation is an order anti-isomorphism of the real numbers, which is its own inverse. (Contributed by Mario Carneiro, 24-Dec-2016.)
⊢ 𝐹 = (𝑥 ∈ ℝ ↦ -𝑥)    ⇒   ⊢ (𝐹 Isom < , ◡ < (ℝ, ℝ) ∧ ◡𝐹 = 𝐹)
Theorem	dfinfre 12137*	The infimum of a set of reals 𝐴. (Contributed by NM, 9-Oct-2005.) (Revised by AV, 4-Sep-2020.)
⊢ (𝐴 ⊆ ℝ → inf(𝐴, ℝ, < ) = ∪ {𝑥 ∈ ℝ ∣ (∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))})
Theorem	infrecl 12138*	Closure of infimum of a nonempty bounded set of reals. (Contributed by NM, 8-Oct-2005.) (Revised by AV, 4-Sep-2020.)
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → inf(𝐴, ℝ, < ) ∈ ℝ)
Theorem	infrenegsup 12139*	The infimum of a set of reals 𝐴 is the negative of the supremum of the negatives of its elements. The antecedent ensures that 𝐴 is nonempty and has a lower bound. (Contributed by NM, 14-Jun-2005.) (Revised by AV, 4-Sep-2020.)
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → inf(𝐴, ℝ, < ) = -sup({𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}, ℝ, < ))
Theorem	infregelb 12140*	Any lower bound of a nonempty set of real numbers is less than or equal to its infimum. (Contributed by Jeff Hankins, 1-Sep-2013.) (Revised by AV, 4-Sep-2020.) (Proof modification is discouraged.)
⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) ∧ 𝐵 ∈ ℝ) → (𝐵 ≤ inf(𝐴, ℝ, < ) ↔ ∀𝑧 ∈ 𝐴 𝐵 ≤ 𝑧))
Theorem	infrelb 12141*	If a nonempty set of real numbers has a lower bound, its infimum is less than or equal to any of its elements. (Contributed by Jeff Hankins, 15-Sep-2013.) (Revised by AV, 4-Sep-2020.)
⊢ ((𝐵 ⊆ ℝ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦 ∧ 𝐴 ∈ 𝐵) → inf(𝐵, ℝ, < ) ≤ 𝐴)
Theorem	infrefilb 12142	The infimum of a finite set of reals is less than or equal to any of its elements. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ ((𝐵 ⊆ ℝ ∧ 𝐵 ∈ Fin ∧ 𝐴 ∈ 𝐵) → inf(𝐵, ℝ, < ) ≤ 𝐴)
Theorem	supfirege 12143	The supremum of a finite set of real numbers is greater than or equal to all the real numbers of the set. (Contributed by AV, 1-Oct-2019.)
⊢ (𝜑 → 𝐵 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ Fin)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑆 = sup(𝐵, ℝ, < ))    ⇒   ⊢ (𝜑 → 𝐶 ≤ 𝑆)
5.3.9  Imaginary and complex number properties
Theorem	neg1cn 12144	-1 is a complex number. (Contributed by David A. Wheeler, 7-Jul-2016.)
⊢ -1 ∈ ℂ
Theorem	neg1rr 12145	-1 is a real number. (Contributed by David A. Wheeler, 5-Dec-2018.)
⊢ -1 ∈ ℝ
Theorem	neg1ne0 12146	-1 is nonzero. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ -1 ≠ 0
Theorem	neg1lt0 12147	-1 is less than 0. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ -1 < 0
Theorem	negneg1e1 12148	--1 is 1. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ --1 = 1
Theorem	inelr 12149	The imaginary unit i is not a real number. (Contributed by NM, 6-May-1999.)
⊢ ¬ i ∈ ℝ
Theorem	rimul 12150	A real number times the imaginary unit is real only if the number is 0. (Contributed by NM, 28-May-1999.) (Revised by Mario Carneiro, 27-May-2016.)
⊢ ((𝐴 ∈ ℝ ∧ (i · 𝐴) ∈ ℝ) → 𝐴 = 0)
Theorem	cru 12151	The representation of complex numbers in terms of real and imaginary parts is unique. Proposition 10-1.3 of [Gleason] p. 130. (Contributed by NM, 9-May-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐴 + (i · 𝐵)) = (𝐶 + (i · 𝐷)) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
Theorem	crne0 12152	The real representation of complex numbers is nonzero iff one of its terms is nonzero. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Mario Carneiro, 27-May-2016.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 ≠ 0 ∨ 𝐵 ≠ 0) ↔ (𝐴 + (i · 𝐵)) ≠ 0))
Theorem	creur 12153*	The real part of a complex number is unique. Proposition 10-1.3 of [Gleason] p. 130. (Contributed by NM, 9-May-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
⊢ (𝐴 ∈ ℂ → ∃!𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦)))
Theorem	creui 12154*	The imaginary part of a complex number is unique. Proposition 10-1.3 of [Gleason] p. 130. (Contributed by NM, 9-May-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
⊢ (𝐴 ∈ ℂ → ∃!𝑦 ∈ ℝ ∃𝑥 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦)))
Theorem	cju 12155*	The complex conjugate of a complex number is unique. (Contributed by Mario Carneiro, 6-Nov-2013.)
⊢ (𝐴 ∈ ℂ → ∃!𝑥 ∈ ℂ ((𝐴 + 𝑥) ∈ ℝ ∧ (i · (𝐴 − 𝑥)) ∈ ℝ))
5.3.10  Function operation analogue theorems
Theorem	ofsubeq0 12156	Function analogue of subeq0 11420. (Contributed by Mario Carneiro, 24-Jul-2014.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶ℂ) → ((𝐹 ∘f − 𝐺) = (𝐴 × {0}) ↔ 𝐹 = 𝐺))
Theorem	ofnegsub 12157	Function analogue of negsub 11442. (Contributed by Mario Carneiro, 24-Jul-2014.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶ℂ) → (𝐹 ∘f + ((𝐴 × {-1}) ∘f · 𝐺)) = (𝐹 ∘f − 𝐺))
Theorem	ofsubge0 12158	Function analogue of subge0 11663. (Contributed by Mario Carneiro, 24-Jul-2014.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶ℝ ∧ 𝐺:𝐴⟶ℝ) → ((𝐴 × {0}) ∘r ≤ (𝐹 ∘f − 𝐺) ↔ 𝐺 ∘r ≤ 𝐹))
5.3.11  Indicator Functions According to Wikipedia (https://en.wikipedia.org/wiki/Indicator_function, "Indicator function", 11-Apr-2026): "In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if 𝐴 is a subset of some set 𝑋, then the indicator function of 𝐴 is the function 𝟭A defined by 𝟭A ( x ) = 1 if 𝑥 ∈ 𝐴, and 𝟭A ( x ) = 0 otherwise." See also definition in [Lang2] p. 3: "The characteristic function of a subset S' of S is the function 𝜒 such that 𝜒(x) = 1 if 𝑥 ∈ 𝑆' and 𝜒(x) = 0 if 𝑥 ∉ 𝑆'".
Syntax	cind 12159	Extend class notation with the indicator function generator.
class 𝟭
Definition	df-ind 12160*	Define the indicator function generator. It generates an indicator function ((𝟭‘𝑂)‘𝐴) for a given domain 𝑂 and a given subset 𝐴 of the domain, see indval 12162. In contrast to the definitions and notations in Wikipedia and [Lang2] p. 3, the domain and the subset are always mentioned explicitly. (Contributed by Thierry Arnoux, 20-Jan-2017.)
⊢ 𝟭 = (𝑜 ∈ V ↦ (𝑎 ∈ 𝒫 𝑜 ↦ (𝑥 ∈ 𝑜 ↦ if(𝑥 ∈ 𝑎, 1, 0))))
Theorem	indv 12161*	Value of the indicator function generator with domain 𝑂. (Contributed by Thierry Arnoux, 23-Aug-2017.)
⊢ (𝑂 ∈ 𝑉 → (𝟭‘𝑂) = (𝑎 ∈ 𝒫 𝑂 ↦ (𝑥 ∈ 𝑂 ↦ if(𝑥 ∈ 𝑎, 1, 0))))
Theorem	indval 12162*	Value of the indicator function generator for a set 𝐴 and a domain 𝑂, i.e., an indicator function for a given domain 𝑂 and a given subset 𝐴 of the domain. (Contributed by Thierry Arnoux, 2-Feb-2017.)
⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂) → ((𝟭‘𝑂)‘𝐴) = (𝑥 ∈ 𝑂 ↦ if(𝑥 ∈ 𝐴, 1, 0)))
Theorem	indval0 12163	The indicator function generator does not generate a (meaningful) indicator function for a class which is not a subset of the domain. (Contributed by AV, 11-Apr-2026.)
⊢ (¬ 𝐴 ⊆ 𝑂 → ((𝟭‘𝑂)‘𝐴) = ∅)
Theorem	indval2 12164	Alternate value of the indicator function generator. (Contributed by Thierry Arnoux, 2-Feb-2017.)
⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂) → ((𝟭‘𝑂)‘𝐴) = ((𝐴 × {1}) ∪ ((𝑂 ∖ 𝐴) × {0})))
Theorem	indf 12165	An indicator function as a function with domain and codomain. (Contributed by Thierry Arnoux, 13-Aug-2017.)
⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂) → ((𝟭‘𝑂)‘𝐴):𝑂⟶{0, 1})
Theorem	indfval 12166	Value of the indicator function. (Contributed by Thierry Arnoux, 13-Aug-2017.)
⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂 ∧ 𝑋 ∈ 𝑂) → (((𝟭‘𝑂)‘𝐴)‘𝑋) = if(𝑋 ∈ 𝐴, 1, 0))
Theorem	fvindre 12167	The range of the indicator function is a subset of ℝ. (Contributed by AV, 10-Apr-2026.)
⊢ (((𝑂 ∈ Fin ∧ 𝐴 ⊆ 𝑂) ∧ 𝑋 ∈ 𝑂) → (((𝟭‘𝑂)‘𝐴)‘𝑋) ∈ ℝ)
Theorem	ind1 12168	Value of the indicator function where it is 1. (Contributed by Thierry Arnoux, 14-Aug-2017.)
⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂 ∧ 𝑋 ∈ 𝐴) → (((𝟭‘𝑂)‘𝐴)‘𝑋) = 1)
Theorem	ind0 12169	Value of the indicator function where it is 0. (Contributed by Thierry Arnoux, 14-Aug-2017.)
⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂 ∧ 𝑋 ∈ (𝑂 ∖ 𝐴)) → (((𝟭‘𝑂)‘𝐴)‘𝑋) = 0)
Theorem	ind1a 12170	Value of the indicator function where it is 1. (Contributed by Thierry Arnoux, 22-Aug-2017.)
⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂 ∧ 𝑋 ∈ 𝑂) → ((((𝟭‘𝑂)‘𝐴)‘𝑋) = 1 ↔ 𝑋 ∈ 𝐴))
Theorem	indconst0 12171	Indicator of the empty set. (Contributed by Thierry Arnoux, 25-Jan-2026.)
⊢ (𝑂 ∈ 𝑉 → ((𝟭‘𝑂)‘∅) = (𝑂 × {0}))
Theorem	indconst1 12172	Indicator of the whole set. (Contributed by Thierry Arnoux, 25-Jan-2026.)
⊢ (𝑂 ∈ 𝑉 → ((𝟭‘𝑂)‘𝑂) = (𝑂 × {1}))
Theorem	indpi1 12173	Preimage of the singleton {1} by the indicator function. See i1f1lem 25656. (Contributed by Thierry Arnoux, 21-Aug-2017.)
⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂) → (◡((𝟭‘𝑂)‘𝐴) “ {1}) = 𝐴)
5.4  Integer sets
5.4.1  Positive integers (as a subset of complex numbers)
Syntax	cn 12174	Extend class notation to include the class of positive integers.
class ℕ
Definition	df-nn 12175	Define the set of positive integers. Some authors, especially in analysis books, call these the natural numbers, whereas other authors choose to include 0 in their definition of natural numbers. Note that ℕ is a subset of complex numbers (nnsscn 12179), in contrast to the more elementary ordinal natural numbers ω, df-om 7818). See nnind 12192 for the principle of mathematical induction. See df-n0 12438 for the set of nonnegative integers ℕ0. See dfn2 12450 for ℕ defined in terms of ℕ0. This is a technical definition that helps us avoid the Axiom of Infinity ax-inf2 9562 in certain proofs. For a more conventional and intuitive definition ("the smallest set of reals containing 1 as well as the successor of every member") see dfnn3 12188 (or its slight variant dfnn2 12187). (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 3-May-2014.)
⊢ ℕ = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) “ ω)
Theorem	nnexALT 12176	Alternate proof of nnex 12180, more direct, that makes use of ax-rep 5213. (Contributed by Mario Carneiro, 3-May-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ℕ ∈ V
Theorem	peano5nni 12177*	Peano's inductive postulate. Theorem I.36 (principle of mathematical induction) of [Apostol] p. 34. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 17-Nov-2014.)
⊢ ((1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 + 1) ∈ 𝐴) → ℕ ⊆ 𝐴)
Theorem	nnssre 12178	The positive integers are a subset of the reals. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 16-Jun-2013.)
⊢ ℕ ⊆ ℝ
Theorem	nnsscn 12179	The positive integers are a subset of the complex numbers. Remark: this could also be proven from nnssre 12178 and ax-resscn 11095 at the cost of using more axioms. (Contributed by NM, 2-Aug-2004.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 4-Oct-2022.)
⊢ ℕ ⊆ ℂ
Theorem	nnex 12180	The set of positive integers exists. (Contributed by NM, 3-Oct-1999.) (Revised by Mario Carneiro, 17-Nov-2014.)
⊢ ℕ ∈ V
Theorem	nnre 12181	A positive integer is a real number. (Contributed by NM, 18-Aug-1999.)
⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ)
Theorem	nncn 12182	A positive integer is a complex number. (Contributed by NM, 18-Aug-1999.)
⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℂ)
Theorem	nnrei 12183	A positive integer is a real number. (Contributed by NM, 18-Aug-1999.)
⊢ 𝐴 ∈ ℕ    ⇒   ⊢ 𝐴 ∈ ℝ
Theorem	nncni 12184	A positive integer is a complex number. (Contributed by NM, 18-Aug-1999.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 4-Oct-2022.)
⊢ 𝐴 ∈ ℕ    ⇒   ⊢ 𝐴 ∈ ℂ
Theorem	1nn 12185	Peano postulate: 1 is a positive integer. (Contributed by NM, 11-Jan-1997.) (Revised by Mario Carneiro, 17-Nov-2014.)
⊢ 1 ∈ ℕ
Theorem	peano2nn 12186	Peano postulate: a successor of a positive integer is a positive integer. (Contributed by NM, 11-Jan-1997.) (Revised by Mario Carneiro, 17-Nov-2014.)
⊢ (𝐴 ∈ ℕ → (𝐴 + 1) ∈ ℕ)
Theorem	dfnn2 12187*	Alternate definition of the set of positive integers. This was our original definition, before the current df-nn 12175 replaced it. This definition requires the axiom of infinity to ensure it has the properties we expect. (Contributed by Jeff Hankins, 12-Sep-2013.) (Revised by Mario Carneiro, 3-May-2014.)
⊢ ℕ = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}
Theorem	dfnn3 12188*	Alternate definition of the set of positive integers. Definition of positive integers in [Apostol] p. 22. (Contributed by NM, 3-Jul-2005.)
⊢ ℕ = ∩ {𝑥 ∣ (𝑥 ⊆ ℝ ∧ 1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}
Theorem	nnred 12189	A positive integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℝ)
Theorem	nncnd 12190	A positive integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℂ)
Theorem	peano2nnd 12191	Peano postulate: a successor of a positive integer is a positive integer. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    ⇒   ⊢ (𝜑 → (𝐴 + 1) ∈ ℕ)
5.4.2  Principle of mathematical induction
Theorem	nnind 12192*	Principle of Mathematical Induction (inference schema). The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. See nnaddcl 12197 for an example of its use. See nn0ind 12624 for induction on nonnegative integers and uzind 12621, uzind4 12856 for induction on an arbitrary upper set of integers. See indstr 12866 for strong induction. See also nnindALT 12193. This is an alternative for Metamath 100 proof #74. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 16-Jun-2013.)
⊢ (𝑥 = 1 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜃))    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))    &   ⊢ 𝜓    &   ⊢ (𝑦 ∈ ℕ → (𝜒 → 𝜃))    ⇒   ⊢ (𝐴 ∈ ℕ → 𝜏)
Theorem	nnindALT 12193*	Principle of Mathematical Induction (inference schema). The last four hypotheses give us the substitution instances we need; the first two are the induction step and the basis. This ALT version of nnind 12192 has a different hypothesis order. It may be easier to use with the Metamath program Proof Assistant, because "MM-PA> ASSIGN LAST" will be applied to the substitution instances first. We may eventually use this one as the official version. You may use either version. After the proof is complete, the ALT version can be changed to the non-ALT version with "MM-PA> MINIMIZE_WITH nnind / MAYGROW";. (Contributed by NM, 7-Dec-2005.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (𝑦 ∈ ℕ → (𝜒 → 𝜃))    &   ⊢ 𝜓    &   ⊢ (𝑥 = 1 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜃))    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))    ⇒   ⊢ (𝐴 ∈ ℕ → 𝜏)
Theorem	nnindd 12194*	Principle of Mathematical Induction (inference schema) on integers, a deduction version. (Contributed by Thierry Arnoux, 19-Jul-2020.)
⊢ (𝑥 = 1 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜃))    &   ⊢ (𝑥 = (𝑦 + 1) → (𝜓 ↔ 𝜏))    &   ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜂))    &   ⊢ (𝜑 → 𝜒)    &   ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ 𝜃) → 𝜏)    ⇒   ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ) → 𝜂)
Theorem	nn1m1nn 12195	Every positive integer is one or a successor. (Contributed by Mario Carneiro, 16-May-2014.)
⊢ (𝐴 ∈ ℕ → (𝐴 = 1 ∨ (𝐴 − 1) ∈ ℕ))
Theorem	nn1suc 12196*	If a statement holds for 1 and also holds for a successor, it holds for all positive integers. The first three hypotheses give us the substitution instances we need; the last two show that it holds for 1 and for a successor. (Contributed by NM, 11-Oct-2004.) (Revised by Mario Carneiro, 16-May-2014.)
⊢ (𝑥 = 1 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜃))    &   ⊢ 𝜓    &   ⊢ (𝑦 ∈ ℕ → 𝜒)    ⇒   ⊢ (𝐴 ∈ ℕ → 𝜃)
Theorem	nnaddcl 12197	Closure of addition of positive integers, proved by induction on the second addend. (Contributed by NM, 12-Jan-1997.)
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 + 𝐵) ∈ ℕ)
Theorem	nnmulcl 12198	Closure of multiplication of positive integers. (Contributed by NM, 12-Jan-1997.) Remove dependency on ax-mulcom 11102 and ax-mulass 11104. (Revised by Steven Nguyen, 24-Sep-2022.)
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 · 𝐵) ∈ ℕ)
Theorem	nnmulcli 12199	Closure of multiplication of positive integers. (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 𝐴 ∈ ℕ    &   ⊢ 𝐵 ∈ ℕ    ⇒   ⊢ (𝐴 · 𝐵) ∈ ℕ
Theorem	nnadd1com 12200	Addition with 1 is commutative for natural numbers. (Contributed by Steven Nguyen, 9-Dec-2022.)
⊢ (𝐴 ∈ ℕ → (𝐴 + 1) = (1 + 𝐴))