P. 330 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 32901-33000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	rexdiv 32901	The extended real division operation when both arguments are real. (Contributed by Thierry Arnoux, 18-Dec-2016.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) → (𝐴 /𝑒 𝐵) = (𝐴 / 𝐵))
Theorem	xdivrec 32902	Relationship between division and reciprocal. (Contributed by Thierry Arnoux, 5-Jul-2017.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) → (𝐴 /𝑒 𝐵) = (𝐴 ·e (1 /𝑒 𝐵)))
Theorem	xdivid 32903	A number divided by itself is one. (Contributed by Thierry Arnoux, 18-Dec-2016.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → (𝐴 /𝑒 𝐴) = 1)
Theorem	xdiv0 32904	Division into zero is zero. (Contributed by Thierry Arnoux, 18-Dec-2016.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → (0 /𝑒 𝐴) = 0)
Theorem	xdiv0rp 32905	Division into zero is zero. (Contributed by Thierry Arnoux, 18-Dec-2016.)
⊢ (𝐴 ∈ ℝ+ → (0 /𝑒 𝐴) = 0)
Theorem	eliccioo 32906	Membership in a closed interval of extended reals versus the same open interval. (Contributed by Thierry Arnoux, 18-Dec-2016.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 = 𝐴 ∨ 𝐶 ∈ (𝐴(,)𝐵) ∨ 𝐶 = 𝐵)))
Theorem	elxrge02 32907	Elementhood in the set of nonnegative extended reals. (Contributed by Thierry Arnoux, 18-Dec-2016.)
⊢ (𝐴 ∈ (0[,]+∞) ↔ (𝐴 = 0 ∨ 𝐴 ∈ ℝ+ ∨ 𝐴 = +∞))
Theorem	xdivpnfrp 32908	Plus infinity divided by a positive real number is plus infinity. (Contributed by Thierry Arnoux, 18-Dec-2016.)
⊢ (𝐴 ∈ ℝ+ → (+∞ /𝑒 𝐴) = +∞)
Theorem	rpxdivcld 32909	Closure law for extended division of positive reals. (Contributed by Thierry Arnoux, 18-Dec-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (𝐴 /𝑒 𝐵) ∈ ℝ+)
Theorem	xrpxdivcld 32910	Closure law for extended division of positive extended reals. (Contributed by Thierry Arnoux, 18-Dec-2016.)
⊢ (𝜑 → 𝐴 ∈ (0[,]+∞))    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (𝐴 /𝑒 𝐵) ∈ (0[,]+∞))
21.3.8  Words over a set - misc additions
Theorem	wrdres 32911	Condition for the restriction of a word to be a word itself. (Contributed by Thierry Arnoux, 5-Oct-2018.)
⊢ ((𝑊 ∈ Word 𝑆 ∧ 𝑁 ∈ (0...(♯‘𝑊))) → (𝑊 ↾ (0..^𝑁)) ∈ Word 𝑆)
Theorem	wrdsplex 32912*	Existence of a split of a word at a given index. (Contributed by Thierry Arnoux, 11-Oct-2018.) (Proof shortened by AV, 3-Nov-2022.)
⊢ ((𝑊 ∈ Word 𝑆 ∧ 𝑁 ∈ (0...(♯‘𝑊))) → ∃𝑣 ∈ Word 𝑆𝑊 = ((𝑊 ↾ (0..^𝑁)) ++ 𝑣))
Theorem	wrdfsupp 32913	A word has finite support. (Contributed by Thierry Arnoux, 27-May-2025.)
⊢ (𝜑 → 𝑍 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑊 ∈ Word 𝑆)    ⇒   ⊢ (𝜑 → 𝑊 finSupp 𝑍)
Theorem	wrdpmcl 32914	Closure of a word with permuted symbols. (Contributed by Thierry Arnoux, 27-May-2025.)
⊢ 𝐽 = (0..^(♯‘𝑊))    &   ⊢ (𝜑 → 𝐸:𝐽–1-1-onto→𝐽)    &   ⊢ (𝜑 → 𝑊 ∈ Word 𝑆)    ⇒   ⊢ (𝜑 → (𝑊 ∘ 𝐸) ∈ Word 𝑆)
Theorem	pfx1s2 32915	The prefix of length 1 of a length 2 word. (Contributed by Thierry Arnoux, 19-Sep-2023.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (⟨“𝐴𝐵”⟩ prefix 1) = ⟨“𝐴”⟩)
Theorem	pfxrn2 32916	The range of a prefix of a word is a subset of the range of that word. Stronger version of pfxrn 14590. (Contributed by Thierry Arnoux, 12-Dec-2023.)
⊢ ((𝑊 ∈ Word 𝑆 ∧ 𝐿 ∈ (0...(♯‘𝑊))) → ran (𝑊 prefix 𝐿) ⊆ ran 𝑊)
Theorem	pfxrn3 32917	Express the range of a prefix of a word. Stronger version of pfxrn2 32916. (Contributed by Thierry Arnoux, 13-Dec-2023.)
⊢ ((𝑊 ∈ Word 𝑆 ∧ 𝐿 ∈ (0...(♯‘𝑊))) → ran (𝑊 prefix 𝐿) = (𝑊 “ (0..^𝐿)))
Theorem	pfxf1 32918	Condition for a prefix to be injective. (Contributed by Thierry Arnoux, 13-Dec-2023.)
⊢ (𝜑 → 𝑊 ∈ Word 𝑆)    &   ⊢ (𝜑 → 𝑊:dom 𝑊–1-1→𝑆)    &   ⊢ (𝜑 → 𝐿 ∈ (0...(♯‘𝑊)))    ⇒   ⊢ (𝜑 → (𝑊 prefix 𝐿):dom (𝑊 prefix 𝐿)–1-1→𝑆)
Theorem	s1f1 32919	Conditions for a length 1 string to be a one-to-one function. (Contributed by Thierry Arnoux, 11-Dec-2023.)
⊢ (𝜑 → 𝐼 ∈ 𝐷)    ⇒   ⊢ (𝜑 → ⟨“𝐼”⟩:dom ⟨“𝐼”⟩–1-1→𝐷)
Theorem	s2rnOLD 32920	Obsolete version of s2rn 14867 as of 1-Aug-2025. Range of a length 2 string. (Contributed by Thierry Arnoux, 19-Sep-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝐼 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐽 ∈ 𝐷)    ⇒   ⊢ (𝜑 → ran ⟨“𝐼𝐽”⟩ = {𝐼, 𝐽})
Theorem	s2f1 32921	Conditions for a length 2 string to be a one-to-one function. (Contributed by Thierry Arnoux, 19-Sep-2023.)
⊢ (𝜑 → 𝐼 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐽 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐼 ≠ 𝐽)    ⇒   ⊢ (𝜑 → ⟨“𝐼𝐽”⟩:dom ⟨“𝐼𝐽”⟩–1-1→𝐷)
Theorem	s3rnOLD 32922	Obsolete version of s2rn 14867 as of 1-Aug-2025. Range of a length 3 string. (Contributed by Thierry Arnoux, 19-Sep-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝐼 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐽 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐾 ∈ 𝐷)    ⇒   ⊢ (𝜑 → ran ⟨“𝐼𝐽𝐾”⟩ = {𝐼, 𝐽, 𝐾})
Theorem	s3f1 32923	Conditions for a length 3 string to be a one-to-one function. (Contributed by Thierry Arnoux, 19-Sep-2023.)
⊢ (𝜑 → 𝐼 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐽 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐾 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐼 ≠ 𝐽)    &   ⊢ (𝜑 → 𝐽 ≠ 𝐾)    &   ⊢ (𝜑 → 𝐾 ≠ 𝐼)    ⇒   ⊢ (𝜑 → ⟨“𝐼𝐽𝐾”⟩:dom ⟨“𝐼𝐽𝐾”⟩–1-1→𝐷)
Theorem	s3clhash 32924	Closure of the words of length 3 in a preimage using the hash function. (Contributed by Thierry Arnoux, 27-Sep-2023.)
⊢ ⟨“𝐼𝐽𝐾”⟩ ∈ (◡♯ “ {3})
Theorem	ccatf1 32925	Conditions for a concatenation to be injective. (Contributed by Thierry Arnoux, 11-Dec-2023.)
⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 ∈ Word 𝑆)    &   ⊢ (𝜑 → 𝐵 ∈ Word 𝑆)    &   ⊢ (𝜑 → 𝐴:dom 𝐴–1-1→𝑆)    &   ⊢ (𝜑 → 𝐵:dom 𝐵–1-1→𝑆)    &   ⊢ (𝜑 → (ran 𝐴 ∩ ran 𝐵) = ∅)    ⇒   ⊢ (𝜑 → (𝐴 ++ 𝐵):dom (𝐴 ++ 𝐵)–1-1→𝑆)
Theorem	pfxlsw2ccat 32926	Reconstruct a word from its prefix and its last two symbols. (Contributed by Thierry Arnoux, 26-Sep-2023.)
⊢ 𝑁 = (♯‘𝑊)    ⇒   ⊢ ((𝑊 ∈ Word 𝑉 ∧ 2 ≤ 𝑁) → 𝑊 = ((𝑊 prefix (𝑁 − 2)) ++ ⟨“(𝑊‘(𝑁 − 2))(𝑊‘(𝑁 − 1))”⟩))
Theorem	ccatws1f1o 32927	Conditions for the concatenation of a word and a singleton word to be bijective. (Contributed by Thierry Arnoux, 27-May-2025.)
⊢ 𝑁 = (♯‘𝑇)    &   ⊢ 𝐽 = (0..^(𝑁 + 1))    &   ⊢ (𝜑 → 𝑇:(0..^𝑁)–1-1-onto→(0..^𝑁))    ⇒   ⊢ (𝜑 → (𝑇 ++ ⟨“𝑁”⟩):𝐽–1-1-onto→𝐽)
Theorem	ccatws1f1olast 32928	Two ways to reorder symbols in a word 𝑊 according to permutation 𝑇, and add a last symbol 𝑋. (Contributed by Thierry Arnoux, 27-May-2025.)
⊢ 𝑁 = (♯‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ Word 𝑆)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑇:(0..^𝑁)–1-1-onto→(0..^𝑁))    ⇒   ⊢ (𝜑 → ((𝑊 ++ ⟨“𝑋”⟩) ∘ (𝑇 ++ ⟨“𝑁”⟩)) = ((𝑊 ∘ 𝑇) ++ ⟨“𝑋”⟩))
Theorem	wrdt2ind 32929*	Perform an induction over the structure of a word of even length. (Contributed by Thierry Arnoux, 26-Sep-2023.)
⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = (𝑦 ++ ⟨“𝑖𝑗”⟩) → (𝜑 ↔ 𝜃))    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))    &   ⊢ 𝜓    &   ⊢ ((𝑦 ∈ Word 𝐵 ∧ 𝑖 ∈ 𝐵 ∧ 𝑗 ∈ 𝐵) → (𝜒 → 𝜃))    ⇒   ⊢ ((𝐴 ∈ Word 𝐵 ∧ 2 ∥ (♯‘𝐴)) → 𝜏)
Theorem	swrdrn2 32930	The range of a subword is a subset of the range of that word. Stronger version of swrdrn 14557. (Contributed by Thierry Arnoux, 12-Dec-2023.)
⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑊))) → ran (𝑊 substr ⟨𝑀, 𝑁⟩) ⊆ ran 𝑊)
Theorem	swrdrn3 32931	Express the range of a subword. Stronger version of swrdrn2 32930. (Contributed by Thierry Arnoux, 13-Dec-2023.)
⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑊))) → ran (𝑊 substr ⟨𝑀, 𝑁⟩) = (𝑊 “ (𝑀..^𝑁)))
Theorem	swrdf1 32932	Condition for a subword to be injective. (Contributed by Thierry Arnoux, 12-Dec-2023.)
⊢ (𝜑 → 𝑊 ∈ Word 𝐷)    &   ⊢ (𝜑 → 𝑀 ∈ (0...𝑁))    &   ⊢ (𝜑 → 𝑁 ∈ (0...(♯‘𝑊)))    &   ⊢ (𝜑 → 𝑊:dom 𝑊–1-1→𝐷)    ⇒   ⊢ (𝜑 → (𝑊 substr ⟨𝑀, 𝑁⟩):dom (𝑊 substr ⟨𝑀, 𝑁⟩)–1-1→𝐷)
Theorem	swrdrndisj 32933	Condition for the range of two subwords of an injective word to be disjoint. (Contributed by Thierry Arnoux, 13-Dec-2023.)
⊢ (𝜑 → 𝑊 ∈ Word 𝐷)    &   ⊢ (𝜑 → 𝑀 ∈ (0...𝑁))    &   ⊢ (𝜑 → 𝑁 ∈ (0...(♯‘𝑊)))    &   ⊢ (𝜑 → 𝑊:dom 𝑊–1-1→𝐷)    &   ⊢ (𝜑 → 𝑂 ∈ (𝑁...𝑃))    &   ⊢ (𝜑 → 𝑃 ∈ (𝑁...(♯‘𝑊)))    ⇒   ⊢ (𝜑 → (ran (𝑊 substr ⟨𝑀, 𝑁⟩) ∩ ran (𝑊 substr ⟨𝑂, 𝑃⟩)) = ∅)
21.3.8.1  Splicing words (substring replacement)
Theorem	splfv3 32934	Symbols to the right of a splice are unaffected. (Contributed by Thierry Arnoux, 14-Dec-2023.)
⊢ (𝜑 → 𝑆 ∈ Word 𝐴)    &   ⊢ (𝜑 → 𝐹 ∈ (0...𝑇))    &   ⊢ (𝜑 → 𝑇 ∈ (0...(♯‘𝑆)))    &   ⊢ (𝜑 → 𝑅 ∈ Word 𝐴)    &   ⊢ (𝜑 → 𝑋 ∈ (0..^((♯‘𝑆) − 𝑇)))    &   ⊢ (𝜑 → 𝐾 = (𝐹 + (♯‘𝑅)))    ⇒   ⊢ (𝜑 → ((𝑆 splice ⟨𝐹, 𝑇, 𝑅⟩)‘(𝑋 + 𝐾)) = (𝑆‘(𝑋 + 𝑇)))
21.3.8.2  Cyclic shift of words
Theorem	1cshid 32935	Cyclically shifting a single letter word keeps it unchanged. (Contributed by Thierry Arnoux, 21-Nov-2023.)
⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℤ ∧ (♯‘𝑊) = 1) → (𝑊 cyclShift 𝑁) = 𝑊)
Theorem	cshw1s2 32936	Cyclically shifting a length 2 word swaps its symbols. (Contributed by Thierry Arnoux, 19-Sep-2023.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (⟨“𝐴𝐵”⟩ cyclShift 1) = ⟨“𝐵𝐴”⟩)
Theorem	cshwrnid 32937	Cyclically shifting a word preserves its range. (Contributed by Thierry Arnoux, 19-Sep-2023.)
⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℤ) → ran (𝑊 cyclShift 𝑁) = ran 𝑊)
Theorem	cshf1o 32938	Condition for the cyclic shift to be a bijection. (Contributed by Thierry Arnoux, 4-Oct-2023.)
⊢ ((𝑊 ∈ Word 𝐷 ∧ 𝑊:dom 𝑊–1-1→𝐷 ∧ 𝑁 ∈ ℤ) → (𝑊 cyclShift 𝑁):dom 𝑊–1-1-onto→ran 𝑊)
21.3.9  Extensible Structures
21.3.9.1  Structure restriction operator
Theorem	ressplusf 32939	The group operation function +𝑓 of a structure's restriction is the operation function's restriction to the new base. (Contributed by Thierry Arnoux, 26-Mar-2017.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝐻 = (𝐺 ↾s 𝐴)    &   ⊢ ⨣ = (+g‘𝐺)    &   ⊢ ⨣ Fn (𝐵 × 𝐵)    &   ⊢ 𝐴 ⊆ 𝐵    ⇒   ⊢ (+𝑓‘𝐻) = ( ⨣ ↾ (𝐴 × 𝐴))
Theorem	ressnm 32940	The norm in a restricted structure. (Contributed by Thierry Arnoux, 8-Oct-2017.)
⊢ 𝐻 = (𝐺 ↾s 𝐴)    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ 𝑁 = (norm‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Mnd ∧ 0 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) → (𝑁 ↾ 𝐴) = (norm‘𝐻))
Theorem	abvpropd2 32941	Weaker version of abvpropd 20748. (Contributed by Thierry Arnoux, 8-Nov-2017.)
⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿))    &   ⊢ (𝜑 → (+g‘𝐾) = (+g‘𝐿))    &   ⊢ (𝜑 → (.r‘𝐾) = (.r‘𝐿))    ⇒   ⊢ (𝜑 → (AbsVal‘𝐾) = (AbsVal‘𝐿))
21.3.9.2  Posets
Theorem	ressprs 32942	The restriction of a proset is a proset. (Contributed by Thierry Arnoux, 11-Sep-2015.)
⊢ 𝐵 = (Base‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (𝐾 ↾s 𝐴) ∈ Proset )
Theorem	posrasymb 32943	A poset ordering is asymetric. (Contributed by Thierry Arnoux, 13-Sep-2018.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵))    ⇒   ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋) ↔ 𝑋 = 𝑌))
Theorem	odutos 32944	Being a toset is a self-dual property. (Contributed by Thierry Arnoux, 13-Sep-2018.)
⊢ 𝐷 = (ODual‘𝐾)    ⇒   ⊢ (𝐾 ∈ Toset → 𝐷 ∈ Toset)
Theorem	tlt2 32945	In a Toset, two elements must compare. (Contributed by Thierry Arnoux, 13-Apr-2018.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ < = (lt‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 ∨ 𝑌 < 𝑋))
Theorem	tlt3 32946	In a Toset, two elements must compare strictly, or be equal. (Contributed by Thierry Arnoux, 13-Apr-2018.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ < = (lt‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 = 𝑌 ∨ 𝑋 < 𝑌 ∨ 𝑌 < 𝑋))
Theorem	trleile 32947	In a Toset, two elements must compare. (Contributed by Thierry Arnoux, 12-Sep-2018.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵))    ⇒   ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 ∨ 𝑌 ≤ 𝑋))
Theorem	toslublem 32948*	Lemma for toslub 32949 and xrsclat 32987. (Contributed by Thierry Arnoux, 17-Feb-2018.) (Revised by NM, 15-Sep-2018.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ < = (lt‘𝐾)    &   ⊢ (𝜑 → 𝐾 ∈ Toset)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ ≤ = (le‘𝐾)    ⇒   ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((∀𝑏 ∈ 𝐴 𝑏 ≤ 𝑎 ∧ ∀𝑐 ∈ 𝐵 (∀𝑏 ∈ 𝐴 𝑏 ≤ 𝑐 → 𝑎 ≤ 𝑐)) ↔ (∀𝑏 ∈ 𝐴 ¬ 𝑎 < 𝑏 ∧ ∀𝑏 ∈ 𝐵 (𝑏 < 𝑎 → ∃𝑑 ∈ 𝐴 𝑏 < 𝑑))))
Theorem	toslub 32949	In a toset, the lowest upper bound lub, defined for partial orders is the supremum, sup(𝐴, 𝐵, < ), defined for total orders. (these are the set.mm definitions: lowest upper bound and supremum are normally synonymous). Note that those two values are also equal if such a supremum does not exist: in that case, both are equal to the empty set. (Contributed by Thierry Arnoux, 15-Feb-2018.) (Revised by Thierry Arnoux, 24-Sep-2018.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ < = (lt‘𝐾)    &   ⊢ (𝜑 → 𝐾 ∈ Toset)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → ((lub‘𝐾)‘𝐴) = sup(𝐴, 𝐵, < ))
Theorem	tosglblem 32950*	Lemma for tosglb 32951 and xrsclat 32987. (Contributed by Thierry Arnoux, 17-Feb-2018.) (Revised by NM, 15-Sep-2018.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ < = (lt‘𝐾)    &   ⊢ (𝜑 → 𝐾 ∈ Toset)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ ≤ = (le‘𝐾)    ⇒   ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((∀𝑏 ∈ 𝐴 𝑎 ≤ 𝑏 ∧ ∀𝑐 ∈ 𝐵 (∀𝑏 ∈ 𝐴 𝑐 ≤ 𝑏 → 𝑐 ≤ 𝑎)) ↔ (∀𝑏 ∈ 𝐴 ¬ 𝑎◡ < 𝑏 ∧ ∀𝑏 ∈ 𝐵 (𝑏◡ < 𝑎 → ∃𝑑 ∈ 𝐴 𝑏◡ < 𝑑))))
Theorem	tosglb 32951	Same theorem as toslub 32949, for infinimum. (Contributed by Thierry Arnoux, 17-Feb-2018.) (Revised by AV, 28-Sep-2020.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ < = (lt‘𝐾)    &   ⊢ (𝜑 → 𝐾 ∈ Toset)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → ((glb‘𝐾)‘𝐴) = inf(𝐴, 𝐵, < ))
21.3.9.3  Complete lattices
Theorem	clatp0cl 32952	The poset zero of a complete lattice belongs to its base. (Contributed by Thierry Arnoux, 17-Feb-2018.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 0 = (0.‘𝑊)    ⇒   ⊢ (𝑊 ∈ CLat → 0 ∈ 𝐵)
Theorem	clatp1cl 32953	The poset one of a complete lattice belongs to its base. (Contributed by Thierry Arnoux, 17-Feb-2018.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 1 = (1.‘𝑊)    ⇒   ⊢ (𝑊 ∈ CLat → 1 ∈ 𝐵)
21.3.9.4  Order Theory
Syntax	cmnt 32954	Extend class notation with monotone functions.
class Monot
Syntax	cmgc 32955	Extend class notation with the monotone Galois connection.
class MGalConn
Definition	df-mnt 32956*	Define a monotone function between two ordered sets. (Contributed by Thierry Arnoux, 20-Apr-2024.)
⊢ Monot = (𝑣 ∈ V, 𝑤 ∈ V ↦ ⦋(Base‘𝑣) / 𝑎⦌{𝑓 ∈ ((Base‘𝑤) ↑m 𝑎) ∣ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑎 (𝑥(le‘𝑣)𝑦 → (𝑓‘𝑥)(le‘𝑤)(𝑓‘𝑦))})
Definition	df-mgc 32957*	Define monotone Galois connections. See mgcval 32963 for an expanded version. (Contributed by Thierry Arnoux, 20-Apr-2024.)
⊢ MGalConn = (𝑣 ∈ V, 𝑤 ∈ V ↦ ⦋(Base‘𝑣) / 𝑎⦌⦋(Base‘𝑤) / 𝑏⦌{⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑏 ↑m 𝑎) ∧ 𝑔 ∈ (𝑎 ↑m 𝑏)) ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ((𝑓‘𝑥)(le‘𝑤)𝑦 ↔ 𝑥(le‘𝑣)(𝑔‘𝑦)))})
Theorem	mntoval 32958*	Operation value of the monotone function. (Contributed by Thierry Arnoux, 23-Apr-2024.)
⊢ 𝐴 = (Base‘𝑉)    &   ⊢ 𝐵 = (Base‘𝑊)    &   ⊢ ≤ = (le‘𝑉)    &   ⊢ ≲ = (le‘𝑊)    ⇒   ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑉Monot𝑊) = {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝑓‘𝑥) ≲ (𝑓‘𝑦))})
Theorem	ismnt 32959*	Express the statement "𝐹 is monotone". (Contributed by Thierry Arnoux, 23-Apr-2024.)
⊢ 𝐴 = (Base‘𝑉)    &   ⊢ 𝐵 = (Base‘𝑊)    &   ⊢ ≤ = (le‘𝑉)    &   ⊢ ≲ = (le‘𝑊)    ⇒   ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝐹 ∈ (𝑉Monot𝑊) ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)))))
Theorem	ismntd 32960	Property of being a monotone increasing function, deduction version. (Contributed by Thierry Arnoux, 24-Apr-2024.)
⊢ 𝐴 = (Base‘𝑉)    &   ⊢ 𝐵 = (Base‘𝑊)    &   ⊢ ≤ = (le‘𝑉)    &   ⊢ ≲ = (le‘𝑊)    &   ⊢ (𝜑 → 𝑉 ∈ 𝐶)    &   ⊢ (𝜑 → 𝑊 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑉Monot𝑊))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑋 ≤ 𝑌)    ⇒   ⊢ (𝜑 → (𝐹‘𝑋) ≲ (𝐹‘𝑌))
Theorem	mntf 32961	A monotone function is a function. (Contributed by Thierry Arnoux, 24-Apr-2024.)
⊢ 𝐴 = (Base‘𝑉)    &   ⊢ 𝐵 = (Base‘𝑊)    ⇒   ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌 ∧ 𝐹 ∈ (𝑉Monot𝑊)) → 𝐹:𝐴⟶𝐵)
Theorem	mgcoval 32962*	Operation value of the monotone Galois connection. (Contributed by Thierry Arnoux, 23-Apr-2024.)
⊢ 𝐴 = (Base‘𝑉)    &   ⊢ 𝐵 = (Base‘𝑊)    &   ⊢ ≤ = (le‘𝑉)    &   ⊢ ≲ = (le‘𝑊)    ⇒   ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑉MGalConn𝑊) = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐵 ↑m 𝐴) ∧ 𝑔 ∈ (𝐴 ↑m 𝐵)) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝑓‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝑔‘𝑦)))})
Theorem	mgcval 32963*	Monotone Galois connection between two functions 𝐹 and 𝐺. If this relation is satisfied, 𝐹 is called the lower adjoint of 𝐺, and 𝐺 is called the upper adjoint of 𝐹. Technically, this is implemented as an operation taking a pair of structures 𝑉 and 𝑊, expected to be posets, which gives a relation between pairs of functions 𝐹 and 𝐺. If such a relation exists, it can be proven to be unique. Galois connections generalize the fundamental theorem of Galois theory about the correspondence between subgroups and subfields. (Contributed by Thierry Arnoux, 23-Apr-2024.)
⊢ 𝐴 = (Base‘𝑉)    &   ⊢ 𝐵 = (Base‘𝑊)    &   ⊢ ≤ = (le‘𝑉)    &   ⊢ ≲ = (le‘𝑊)    &   ⊢ 𝐻 = (𝑉MGalConn𝑊)    &   ⊢ (𝜑 → 𝑉 ∈ Proset )    &   ⊢ (𝜑 → 𝑊 ∈ Proset )    ⇒   ⊢ (𝜑 → (𝐹𝐻𝐺 ↔ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)))))
Theorem	mgcf1 32964	The lower adjoint 𝐹 of a Galois connection is a function. (Contributed by Thierry Arnoux, 24-Apr-2024.)
⊢ 𝐴 = (Base‘𝑉)    &   ⊢ 𝐵 = (Base‘𝑊)    &   ⊢ ≤ = (le‘𝑉)    &   ⊢ ≲ = (le‘𝑊)    &   ⊢ 𝐻 = (𝑉MGalConn𝑊)    &   ⊢ (𝜑 → 𝑉 ∈ Proset )    &   ⊢ (𝜑 → 𝑊 ∈ Proset )    &   ⊢ (𝜑 → 𝐹𝐻𝐺)    ⇒   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
Theorem	mgcf2 32965	The upper adjoint 𝐺 of a Galois connection is a function. (Contributed by Thierry Arnoux, 24-Apr-2024.)
⊢ 𝐴 = (Base‘𝑉)    &   ⊢ 𝐵 = (Base‘𝑊)    &   ⊢ ≤ = (le‘𝑉)    &   ⊢ ≲ = (le‘𝑊)    &   ⊢ 𝐻 = (𝑉MGalConn𝑊)    &   ⊢ (𝜑 → 𝑉 ∈ Proset )    &   ⊢ (𝜑 → 𝑊 ∈ Proset )    &   ⊢ (𝜑 → 𝐹𝐻𝐺)    ⇒   ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)
Theorem	mgccole1 32966	An inequality for the kernel operator 𝐺 ∘ 𝐹. (Contributed by Thierry Arnoux, 26-Apr-2024.)
⊢ 𝐴 = (Base‘𝑉)    &   ⊢ 𝐵 = (Base‘𝑊)    &   ⊢ ≤ = (le‘𝑉)    &   ⊢ ≲ = (le‘𝑊)    &   ⊢ 𝐻 = (𝑉MGalConn𝑊)    &   ⊢ (𝜑 → 𝑉 ∈ Proset )    &   ⊢ (𝜑 → 𝑊 ∈ Proset )    &   ⊢ (𝜑 → 𝐹𝐻𝐺)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    ⇒   ⊢ (𝜑 → 𝑋 ≤ (𝐺‘(𝐹‘𝑋)))
Theorem	mgccole2 32967	Inequality for the closure operator (𝐹 ∘ 𝐺) of the Galois connection 𝐻. (Contributed by Thierry Arnoux, 26-Apr-2024.)
⊢ 𝐴 = (Base‘𝑉)    &   ⊢ 𝐵 = (Base‘𝑊)    &   ⊢ ≤ = (le‘𝑉)    &   ⊢ ≲ = (le‘𝑊)    &   ⊢ 𝐻 = (𝑉MGalConn𝑊)    &   ⊢ (𝜑 → 𝑉 ∈ Proset )    &   ⊢ (𝜑 → 𝑊 ∈ Proset )    &   ⊢ (𝜑 → 𝐹𝐻𝐺)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐹‘(𝐺‘𝑌)) ≲ 𝑌)
Theorem	mgcmnt1 32968	The lower adjoint 𝐹 of a Galois connection is monotonically increasing. (Contributed by Thierry Arnoux, 26-Apr-2024.)
⊢ 𝐴 = (Base‘𝑉)    &   ⊢ 𝐵 = (Base‘𝑊)    &   ⊢ ≤ = (le‘𝑉)    &   ⊢ ≲ = (le‘𝑊)    &   ⊢ 𝐻 = (𝑉MGalConn𝑊)    &   ⊢ (𝜑 → 𝑉 ∈ Proset )    &   ⊢ (𝜑 → 𝑊 ∈ Proset )    &   ⊢ (𝜑 → 𝐹𝐻𝐺)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑋 ≤ 𝑌)    ⇒   ⊢ (𝜑 → (𝐹‘𝑋) ≲ (𝐹‘𝑌))
Theorem	mgcmnt2 32969	The upper adjoint 𝐺 of a Galois connection is monotonically increasing. (Contributed by Thierry Arnoux, 26-Apr-2024.)
⊢ 𝐴 = (Base‘𝑉)    &   ⊢ 𝐵 = (Base‘𝑊)    &   ⊢ ≤ = (le‘𝑉)    &   ⊢ ≲ = (le‘𝑊)    &   ⊢ 𝐻 = (𝑉MGalConn𝑊)    &   ⊢ (𝜑 → 𝑉 ∈ Proset )    &   ⊢ (𝜑 → 𝑊 ∈ Proset )    &   ⊢ (𝜑 → 𝐹𝐻𝐺)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑋 ≲ 𝑌)    ⇒   ⊢ (𝜑 → (𝐺‘𝑋) ≤ (𝐺‘𝑌))
Theorem	mgcmntco 32970*	A Galois connection like statement, for two functions with same range. (Contributed by Thierry Arnoux, 26-Apr-2024.)
⊢ 𝐴 = (Base‘𝑉)    &   ⊢ 𝐵 = (Base‘𝑊)    &   ⊢ ≤ = (le‘𝑉)    &   ⊢ ≲ = (le‘𝑊)    &   ⊢ 𝐻 = (𝑉MGalConn𝑊)    &   ⊢ (𝜑 → 𝑉 ∈ Proset )    &   ⊢ (𝜑 → 𝑊 ∈ Proset )    &   ⊢ (𝜑 → 𝐹𝐻𝐺)    &   ⊢ 𝐶 = (Base‘𝑋)    &   ⊢ < = (le‘𝑋)    &   ⊢ (𝜑 → 𝑋 ∈ Proset )    &   ⊢ (𝜑 → 𝐾 ∈ (𝑉Monot𝑋))    &   ⊢ (𝜑 → 𝐿 ∈ (𝑊Monot𝑋))    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 (𝐾‘𝑥) < (𝐿‘(𝐹‘𝑥)) ↔ ∀𝑦 ∈ 𝐵 (𝐾‘(𝐺‘𝑦)) < (𝐿‘𝑦)))
Theorem	dfmgc2lem 32971*	Lemma for dfmgc2, backwards direction. (Contributed by Thierry Arnoux, 26-Apr-2024.)
⊢ 𝐴 = (Base‘𝑉)    &   ⊢ 𝐵 = (Base‘𝑊)    &   ⊢ ≤ = (le‘𝑉)    &   ⊢ ≲ = (le‘𝑊)    &   ⊢ 𝐻 = (𝑉MGalConn𝑊)    &   ⊢ (𝜑 → 𝑉 ∈ Proset )    &   ⊢ (𝜑 → 𝑊 ∈ Proset )    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)))    &   ⊢ (𝜑 → ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢 ≲ 𝑣 → (𝐺‘𝑢) ≤ (𝐺‘𝑣)))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ≤ (𝐺‘(𝐹‘𝑥)))    &   ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐵) → (𝐹‘(𝐺‘𝑢)) ≲ 𝑢)    ⇒   ⊢ (𝜑 → 𝐹𝐻𝐺)
Theorem	dfmgc2 32972*	Alternate definition of the monotone Galois connection. (Contributed by Thierry Arnoux, 26-Apr-2024.)
⊢ 𝐴 = (Base‘𝑉)    &   ⊢ 𝐵 = (Base‘𝑊)    &   ⊢ ≤ = (le‘𝑉)    &   ⊢ ≲ = (le‘𝑊)    &   ⊢ 𝐻 = (𝑉MGalConn𝑊)    &   ⊢ (𝜑 → 𝑉 ∈ Proset )    &   ⊢ (𝜑 → 𝑊 ∈ Proset )    ⇒   ⊢ (𝜑 → (𝐹𝐻𝐺 ↔ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)) ∧ ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢 ≲ 𝑣 → (𝐺‘𝑢) ≤ (𝐺‘𝑣))) ∧ (∀𝑢 ∈ 𝐵 (𝐹‘(𝐺‘𝑢)) ≲ 𝑢 ∧ ∀𝑥 ∈ 𝐴 𝑥 ≤ (𝐺‘(𝐹‘𝑥)))))))
Theorem	mgcmnt1d 32973	Galois connection implies monotonicity of the left adjoint. (Contributed by Thierry Arnoux, 21-Jul-2024.)
⊢ 𝐻 = (𝑉MGalConn𝑊)    &   ⊢ (𝜑 → 𝑉 ∈ Proset )    &   ⊢ (𝜑 → 𝑊 ∈ Proset )    &   ⊢ (𝜑 → 𝐹𝐻𝐺)    ⇒   ⊢ (𝜑 → 𝐹 ∈ (𝑉Monot𝑊))
Theorem	mgcmnt2d 32974	Galois connection implies monotonicity of the right adjoint. (Contributed by Thierry Arnoux, 21-Jul-2024.)
⊢ 𝐻 = (𝑉MGalConn𝑊)    &   ⊢ (𝜑 → 𝑉 ∈ Proset )    &   ⊢ (𝜑 → 𝑊 ∈ Proset )    &   ⊢ (𝜑 → 𝐹𝐻𝐺)    ⇒   ⊢ (𝜑 → 𝐺 ∈ (𝑊Monot𝑉))
Theorem	mgccnv 32975	The inverse Galois connection is the Galois connection of the dual orders. (Contributed by Thierry Arnoux, 26-Apr-2024.)
⊢ 𝐻 = (𝑉MGalConn𝑊)    &   ⊢ 𝑀 = ((ODual‘𝑊)MGalConn(ODual‘𝑉))    ⇒   ⊢ ((𝑉 ∈ Proset ∧ 𝑊 ∈ Proset ) → (𝐹𝐻𝐺 ↔ 𝐺𝑀𝐹))
Theorem	pwrssmgc 32976*	Given a function 𝐹, exhibit a Galois connection between subsets of its domain and subsets of its range. (Contributed by Thierry Arnoux, 26-Apr-2024.)
⊢ 𝐺 = (𝑛 ∈ 𝒫 𝑌 ↦ (◡𝐹 “ 𝑛))    &   ⊢ 𝐻 = (𝑚 ∈ 𝒫 𝑋 ↦ {𝑦 ∈ 𝑌 ∣ (◡𝐹 “ {𝑦}) ⊆ 𝑚})    &   ⊢ 𝑉 = (toInc‘𝒫 𝑌)    &   ⊢ 𝑊 = (toInc‘𝒫 𝑋)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹:𝑋⟶𝑌)    ⇒   ⊢ (𝜑 → 𝐺(𝑉MGalConn𝑊)𝐻)
Theorem	mgcf1olem1 32977	Property of a Galois connection, lemma for mgcf1o 32979. (Contributed by Thierry Arnoux, 26-Jul-2024.)
⊢ 𝐻 = (𝑉MGalConn𝑊)    &   ⊢ 𝐴 = (Base‘𝑉)    &   ⊢ 𝐵 = (Base‘𝑊)    &   ⊢ ≤ = (le‘𝑉)    &   ⊢ ≲ = (le‘𝑊)    &   ⊢ (𝜑 → 𝑉 ∈ Poset)    &   ⊢ (𝜑 → 𝑊 ∈ Poset)    &   ⊢ (𝜑 → 𝐹𝐻𝐺)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (𝐹‘(𝐺‘(𝐹‘𝑋))) = (𝐹‘𝑋))
Theorem	mgcf1olem2 32978	Property of a Galois connection, lemma for mgcf1o 32979. (Contributed by Thierry Arnoux, 26-Jul-2024.)
⊢ 𝐻 = (𝑉MGalConn𝑊)    &   ⊢ 𝐴 = (Base‘𝑉)    &   ⊢ 𝐵 = (Base‘𝑊)    &   ⊢ ≤ = (le‘𝑉)    &   ⊢ ≲ = (le‘𝑊)    &   ⊢ (𝜑 → 𝑉 ∈ Poset)    &   ⊢ (𝜑 → 𝑊 ∈ Poset)    &   ⊢ (𝜑 → 𝐹𝐻𝐺)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐺‘(𝐹‘(𝐺‘𝑌))) = (𝐺‘𝑌))
Theorem	mgcf1o 32979	Given a Galois connection, exhibit an order isomorphism. (Contributed by Thierry Arnoux, 26-Jul-2024.)
⊢ 𝐻 = (𝑉MGalConn𝑊)    &   ⊢ 𝐴 = (Base‘𝑉)    &   ⊢ 𝐵 = (Base‘𝑊)    &   ⊢ ≤ = (le‘𝑉)    &   ⊢ ≲ = (le‘𝑊)    &   ⊢ (𝜑 → 𝑉 ∈ Poset)    &   ⊢ (𝜑 → 𝑊 ∈ Poset)    &   ⊢ (𝜑 → 𝐹𝐻𝐺)    ⇒   ⊢ (𝜑 → (𝐹 ↾ ran 𝐺) Isom ≤ , ≲ (ran 𝐺, ran 𝐹))
21.3.9.5  Extended reals Structure - misc additions
Axiom	ax-xrssca 32980	Assume the scalar component of the extended real structure is the field of the real numbers (this has to be defined in the main body of set.mm). (Contributed by Thierry Arnoux, 22-Oct-2017.)
⊢ ℝfld = (Scalar‘ℝ*𝑠)
Axiom	ax-xrsvsca 32981	Assume the scalar product of the extended real structure is the extended real number multiplication operation (this has to be defined in the main body of set.mm). (Contributed by Thierry Arnoux, 22-Oct-2017.)
⊢ ·e = ( ·𝑠 ‘ℝ*𝑠)
Theorem	xrs0 32982	The zero of the extended real numbers. The extended real is not a group, as its addition is not associative. (cf. xaddass 13145 and df-xrs 17403), however it has a zero. (Contributed by Thierry Arnoux, 13-Jun-2017.)
⊢ 0 = (0g‘ℝ*𝑠)
Theorem	xrslt 32983	The "strictly less than" relation for the extended real structure. (Contributed by Thierry Arnoux, 30-Jan-2018.)
⊢ < = (lt‘ℝ*𝑠)
Theorem	xrsinvgval 32984	The inversion operation in the extended real numbers. The extended real is not a group, as its addition is not associative. (cf. xaddass 13145 and df-xrs 17403), however it has an inversion operation. (Contributed by Thierry Arnoux, 13-Jun-2017.)
⊢ (𝐵 ∈ ℝ* → ((invg‘ℝ*𝑠)‘𝐵) = -𝑒𝐵)
Theorem	xrsmulgzz 32985	The "multiple" function in the extended real numbers structure. (Contributed by Thierry Arnoux, 14-Jun-2017.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℝ*) → (𝐴(.g‘ℝ*𝑠)𝐵) = (𝐴 ·e 𝐵))
Theorem	xrstos 32986	The extended real numbers form a toset. (Contributed by Thierry Arnoux, 15-Feb-2018.)
⊢ ℝ*𝑠 ∈ Toset
Theorem	xrsclat 32987	The extended real numbers form a complete lattice. (Contributed by Thierry Arnoux, 15-Feb-2018.)
⊢ ℝ*𝑠 ∈ CLat
Theorem	xrsp0 32988	The poset 0 of the extended real numbers is minus infinity. (Contributed by Thierry Arnoux, 18-Feb-2018.) (Proof shortened by AV, 28-Sep-2020.)
⊢ -∞ = (0.‘ℝ*𝑠)
Theorem	xrsp1 32989	The poset 1 of the extended real numbers is plus infinity. (Contributed by Thierry Arnoux, 18-Feb-2018.)
⊢ +∞ = (1.‘ℝ*𝑠)
21.3.9.6  The extended nonnegative real numbers commutative monoid
Theorem	xrge00 32990	The zero of the extended nonnegative real numbers monoid. (Contributed by Thierry Arnoux, 30-Jan-2017.)
⊢ 0 = (0g‘(ℝ*𝑠 ↾s (0[,]+∞)))
Theorem	xrge0mulgnn0 32991	The group multiple function in the extended nonnegative real numbers. (Contributed by Thierry Arnoux, 14-Jun-2017.)
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ (0[,]+∞)) → (𝐴(.g‘(ℝ*𝑠 ↾s (0[,]+∞)))𝐵) = (𝐴 ·e 𝐵))
Theorem	xrge0addass 32992	Associativity of extended nonnegative real addition. (Contributed by Thierry Arnoux, 8-Jun-2017.)
⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) → ((𝐴 +𝑒 𝐵) +𝑒 𝐶) = (𝐴 +𝑒 (𝐵 +𝑒 𝐶)))
Theorem	xrge0addgt0 32993	The sum of nonnegative and positive numbers is positive. See addgtge0 11602. (Contributed by Thierry Arnoux, 6-Jul-2017.)
⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞)) ∧ 0 < 𝐴) → 0 < (𝐴 +𝑒 𝐵))
Theorem	xrge0adddir 32994	Right-distributivity of extended nonnegative real multiplication over addition. (Contributed by Thierry Arnoux, 30-Jun-2017.)
⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) → ((𝐴 +𝑒 𝐵) ·e 𝐶) = ((𝐴 ·e 𝐶) +𝑒 (𝐵 ·e 𝐶)))
Theorem	xrge0adddi 32995	Left-distributivity of extended nonnegative real multiplication over addition. (Contributed by Thierry Arnoux, 6-Sep-2018.)
⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) → (𝐶 ·e (𝐴 +𝑒 𝐵)) = ((𝐶 ·e 𝐴) +𝑒 (𝐶 ·e 𝐵)))
Theorem	xrge0npcan 32996	Extended nonnegative real version of npcan 11366. (Contributed by Thierry Arnoux, 9-Jun-2017.)
⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐵 ≤ 𝐴) → ((𝐴 +𝑒 -𝑒𝐵) +𝑒 𝐵) = 𝐴)
Theorem	fsumrp0cl 32997*	Closure of a finite sum of nonnegative reals. (Contributed by Thierry Arnoux, 25-Jun-2017.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,)+∞))    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ (0[,)+∞))
21.3.10  Algebra
21.3.10.1  Monoids
Theorem	mndcld 32998	Closure of the operation of a monoid. (Contributed by Thierry Arnoux, 3-Aug-2025.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Mnd)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐵)
Theorem	mndassd 32999	A monoid operation is associative. (Contributed by Thierry Arnoux, 3-Aug-2025.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Mnd)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍)))
Theorem	mndlrinv 33000	In a monoid, if an element 𝑋 has both a left inverse 𝑀 and a right inverse 𝑁, they are equal. (Contributed by Thierry Arnoux, 3-Aug-2025.)
⊢ 𝐵 = (Base‘𝐸)    &   ⊢ 0 = (0g‘𝐸)    &   ⊢ + = (+g‘𝐸)    &   ⊢ (𝜑 → 𝐸 ∈ Mnd)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑀 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑁 ∈ 𝐵)    &   ⊢ (𝜑 → (𝑀 + 𝑋) = 0 )    &   ⊢ (𝜑 → (𝑋 + 𝑁) = 0 )    ⇒   ⊢ (𝜑 → 𝑀 = 𝑁)