P. 114 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 11301-11400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	negfi 11301*	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	fiminre 11302*	A nonempty finite set of real numbers has a minimum. Analogous to fimaxre 11298. (Contributed by AV, 9-Aug-2020.)
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦)
Theorem	lbreu 11303*	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 11304*	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 11305*	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 11306*	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 11307*	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 11308*	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 11309*	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 11310*	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 11311*	Lemma for infm3 11312. (Contributed by NM, 14-Jun-2005.)
⊢ (𝑥 ∈ ℝ → ∃𝑦 ∈ ℝ 𝑥 = -𝑦)
Theorem	infm3 11312*	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 11310.) (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 11313*	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 11314*	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 11315*	Natural deduction form of suprubd 11315. (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 11316*	Natural deduction form of suprcl 11313. (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 11317*	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 11318*	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 11319*	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 11320*	The supremum function distributes over addition in a sense similar to that in supmul1 11322. (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 11321*	The supremum function distributes over addition in a sense similar to that in supmul 11325. (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 11322*	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 11325 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 11323*	Lemma for supmul 11325. (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 11324*	Lemma for supmul 11325. (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 11325*	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 10121). (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 11326*	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 11327*	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 11328*	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 11329*	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 11330*	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 11331*	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 11332*	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 11333	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 11334*	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 11335*	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 11336*	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 11337*	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 11338*	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	supfirege 11339	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	inelr 11340	The imaginary unit i is not a real number. (Contributed by NM, 6-May-1999.)
⊢ ¬ i ∈ ℝ
Theorem	rimul 11341	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 11342	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 11343	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 11344*	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 11345*	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 11346*	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 11347	Function analogue of subeq0 10628. (Contributed by Mario Carneiro, 24-Jul-2014.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶ℂ) → ((𝐹 ∘𝑓 − 𝐺) = (𝐴 × {0}) ↔ 𝐹 = 𝐺))
Theorem	ofnegsub 11348	Function analogue of negsub 10650. (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 11349	Function analogue of subge0 10865. (Contributed by Mario Carneiro, 24-Jul-2014.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶ℝ ∧ 𝐺:𝐴⟶ℝ) → ((𝐴 × {0}) ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝐺) ↔ 𝐺 ∘𝑟 ≤ 𝐹))
5.4  Integer sets
5.4.1  Positive integers (as a subset of complex numbers)
Syntax	cn 11350	Extend class notation to include the class of positive integers.
class ℕ
Definition	df-nn 11351	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 11355), in contrast to the more elementary ordinal natural numbers ω, df-om 7327). See nnind 11370 for the principle of mathematical induction. See df-n0 11619 for the set of nonnegative integers ℕ0. See dfn2 11633 for ℕ defined in terms of ℕ0. This is a technical definition that helps us avoid the Axiom of Infinity ax-inf2 8815 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 11366 (or its slight variant dfnn2 11365). (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 3-May-2014.)
⊢ ℕ = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) “ ω)
Theorem	nnexALT 11352	Alternate proof of nnex 11357, more direct, that makes use of ax-rep 4994. (Contributed by Mario Carneiro, 3-May-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ℕ ∈ V
Theorem	peano5nni 11353*	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 11354	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 11355	The positive integers are a subset of the complex numbers. Remark: this could also be proven from nnssre 11354 and ax-resscn 10309 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	nnsscnOLD 11356	Obsolete version of nnsscn 11355 as of 4-Oct-2022. The positive integers are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ℕ ⊆ ℂ
Theorem	nnex 11357	The set of positive integers exists. (Contributed by NM, 3-Oct-1999.) (Revised by Mario Carneiro, 17-Nov-2014.)
⊢ ℕ ∈ V
Theorem	nnre 11358	A positive integer is a real number. (Contributed by NM, 18-Aug-1999.)
⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ)
Theorem	nncn 11359	A positive integer is a complex number. (Contributed by NM, 18-Aug-1999.)
⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℂ)
Theorem	nnrei 11360	A positive integer is a real number. (Contributed by NM, 18-Aug-1999.)
⊢ 𝐴 ∈ ℕ    ⇒   ⊢ 𝐴 ∈ ℝ
Theorem	nncni 11361	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	nncniOLD 11362	Obsolete version of nncni 11361 as of 4-Oct-2022. A positive integer is a complex number. (Contributed by NM, 18-Aug-1999.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐴 ∈ ℕ    ⇒   ⊢ 𝐴 ∈ ℂ
Theorem	1nn 11363	Peano postulate: 1 is a positive integer. (Contributed by NM, 11-Jan-1997.) (Revised by Mario Carneiro, 17-Nov-2014.)
⊢ 1 ∈ ℕ
Theorem	peano2nn 11364	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 11365*	Alternate definition of the set of positive integers. This was our original definition, before the current df-nn 11351 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 11366*	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 11367	A positive integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℝ)
Theorem	nncnd 11368	A positive integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℂ)
Theorem	peano2nnd 11369	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 11370*	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 11374 for an example of its use. See nn0ind 11800 for induction on nonnegative integers and uzind 11797, uzind4 12028 for induction on an arbitrary upper set of integers. See indstr 12039 for strong induction. See also nnindALT 11371. 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 11371*	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 11370 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> MINIMIZEWITH 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	nn1m1nn 11372	Every positive integer is one or a successor. (Contributed by Mario Carneiro, 16-May-2014.)
⊢ (𝐴 ∈ ℕ → (𝐴 = 1 ∨ (𝐴 − 1) ∈ ℕ))
Theorem	nn1suc 11373*	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 11374	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 11375	Closure of multiplication of positive integers. (Contributed by NM, 12-Jan-1997.) Remove dependency on ax-mulcom 10316 and ax-mulass 10318. (Revised by Steven Nguyen, 24-Sep-2022.)
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 · 𝐵) ∈ ℕ)
Theorem	nnmulclOLD 11376	Obsolete version of nnmulcl 11375 as of 24-Sep-2022. Closure of multiplication of positive integers. (Contributed by NM, 12-Jan-1997.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 · 𝐵) ∈ ℕ)
Theorem	nnmulcli 11377	Closure of multiplication of positive integers. (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 𝐴 ∈ ℕ    &   ⊢ 𝐵 ∈ ℕ    ⇒   ⊢ (𝐴 · 𝐵) ∈ ℕ
Theorem	nnmtmip 11378	"Minus times minus is plus, The reason for this we need not discuss." (W. H. Auden, as quoted in M. Guillen "Bridges to Infinity", p. 64, see also Metamath Book, section 1.1.1, p. 5) This statement, formalized to "The product of two negative integers is a positive integer", is proved by the following theorem, therefore it actually need not be discussed anymore. "The reason for this" is that (-𝐴 · -𝐵) = (𝐴 · 𝐵) for all complex numbers 𝐴 and 𝐵 because of mul2neg 10793, 𝐴 and 𝐵 are complex numbers because of nncn 11359, and (𝐴 · 𝐵) ∈ ℕ because of nnmulcl 11375. This also holds for positive reals, see rpmtmip 12138. Note that the opposites -𝐴 and -𝐵 of the positive integers 𝐴 and 𝐵 are negative integers. (Contributed by AV, 23-Dec-2022.)
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (-𝐴 · -𝐵) ∈ ℕ)
Theorem	nn2ge 11379*	There exists a positive integer greater than or equal to any two others. (Contributed by NM, 18-Aug-1999.)
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ∃𝑥 ∈ ℕ (𝐴 ≤ 𝑥 ∧ 𝐵 ≤ 𝑥))
Theorem	nnge1 11380	A positive integer is one or greater. (Contributed by NM, 25-Aug-1999.)
⊢ (𝐴 ∈ ℕ → 1 ≤ 𝐴)
Theorem	nngt1ne1 11381	A positive integer is greater than one iff it is not equal to one. (Contributed by NM, 7-Oct-2004.)
⊢ (𝐴 ∈ ℕ → (1 < 𝐴 ↔ 𝐴 ≠ 1))
Theorem	nnle1eq1 11382	A positive integer is less than or equal to one iff it is equal to one. (Contributed by NM, 3-Apr-2005.)
⊢ (𝐴 ∈ ℕ → (𝐴 ≤ 1 ↔ 𝐴 = 1))
Theorem	nngt0 11383	A positive integer is positive. (Contributed by NM, 26-Sep-1999.)
⊢ (𝐴 ∈ ℕ → 0 < 𝐴)
Theorem	nnnlt1 11384	A positive integer is not less than one. (Contributed by NM, 18-Jan-2004.) (Revised by Mario Carneiro, 27-May-2016.)
⊢ (𝐴 ∈ ℕ → ¬ 𝐴 < 1)
Theorem	nnnle0 11385	A positive integer is not less than or equal to zero . (Contributed by AV, 13-May-2020.)
⊢ (𝐴 ∈ ℕ → ¬ 𝐴 ≤ 0)
Theorem	nnne0 11386	A positive integer is nonzero. See nnne0ALT 11389 for a shorter proof using ax-pre-mulgt0 10329. This proof avoids 0lt1 10874, and thus ax-pre-mulgt0 10329, by splitting ax-1ne0 10321 into the two separate cases 0 < 1 and 1 < 0. (Contributed by NM, 27-Sep-1999.) Remove dependency on ax-pre-mulgt0 10329. (Revised by Steven Nguyen, 30-Jan-2023.)
⊢ (𝐴 ∈ ℕ → 𝐴 ≠ 0)
Theorem	0nnn 11387	Zero is not a positive integer. (Contributed by NM, 25-Aug-1999.) Remove dependency on ax-pre-mulgt0 10329. (Revised by Steven Nguyen, 30-Jan-2023.)
⊢ ¬ 0 ∈ ℕ
Theorem	0nnnALT 11388	Alternate proof of 0nnn 11387, which requires ax-pre-mulgt0 10329 but is not based on nnne0 11386 (and which can therefore be used in nnne0ALT 11389). (Contributed by NM, 25-Aug-1999.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ ¬ 0 ∈ ℕ
Theorem	nnne0ALT 11389	Alternate version of nnne0 11386. A positive integer is nonzero. (Contributed by NM, 27-Sep-1999.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ ℕ → 𝐴 ≠ 0)
Theorem	nngt0i 11390	A positive integer is positive (inference version). (Contributed by NM, 17-Sep-1999.)
⊢ 𝐴 ∈ ℕ    ⇒   ⊢ 0 < 𝐴
Theorem	nnne0i 11391	A positive integer is nonzero (inference version). (Contributed by NM, 25-Aug-1999.)
⊢ 𝐴 ∈ ℕ    ⇒   ⊢ 𝐴 ≠ 0
Theorem	nndivre 11392	The quotient of a real and a positive integer is real. (Contributed by NM, 28-Nov-2008.)
⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ) → (𝐴 / 𝑁) ∈ ℝ)
Theorem	nnrecre 11393	The reciprocal of a positive integer is real. (Contributed by NM, 8-Feb-2008.)
⊢ (𝑁 ∈ ℕ → (1 / 𝑁) ∈ ℝ)
Theorem	nnrecgt0 11394	The reciprocal of a positive integer is positive. (Contributed by NM, 25-Aug-1999.)
⊢ (𝐴 ∈ ℕ → 0 < (1 / 𝐴))
Theorem	nnsub 11395	Subtraction of positive integers. (Contributed by NM, 20-Aug-2001.) (Revised by Mario Carneiro, 16-May-2014.)
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 < 𝐵 ↔ (𝐵 − 𝐴) ∈ ℕ))
Theorem	nnsubi 11396	Subtraction of positive integers. (Contributed by NM, 19-Aug-2001.)
⊢ 𝐴 ∈ ℕ    &   ⊢ 𝐵 ∈ ℕ    ⇒   ⊢ (𝐴 < 𝐵 ↔ (𝐵 − 𝐴) ∈ ℕ)
Theorem	nndiv 11397*	Two ways to express "𝐴 divides 𝐵 " for positive integers. (Contributed by NM, 3-Feb-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (∃𝑥 ∈ ℕ (𝐴 · 𝑥) = 𝐵 ↔ (𝐵 / 𝐴) ∈ ℕ))
Theorem	nndivtr 11398	Transitive property of divisibility: if 𝐴 divides 𝐵 and 𝐵 divides 𝐶, then 𝐴 divides 𝐶. Typically, 𝐶 would be an integer, although the theorem holds for complex 𝐶. (Contributed by NM, 3-May-2005.)
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℂ) ∧ ((𝐵 / 𝐴) ∈ ℕ ∧ (𝐶 / 𝐵) ∈ ℕ)) → (𝐶 / 𝐴) ∈ ℕ)
Theorem	nnge1d 11399	A positive integer is one or greater. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    ⇒   ⊢ (𝜑 → 1 ≤ 𝐴)
Theorem	nngt0d 11400	A positive integer is positive. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    ⇒   ⊢ (𝜑 → 0 < 𝐴)