P. 331 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 33001-33100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	xreceu 33001*	Existential uniqueness of reciprocals. Theorem I.8 of [Apostol] p. 18. (Contributed by Thierry Arnoux, 17-Dec-2016.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) → ∃!𝑥 ∈ ℝ* (𝐵 ·e 𝑥) = 𝐴)
Theorem	xdivcld 33002	Closure law for the extended division. (Contributed by Thierry Arnoux, 15-Mar-2017.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ≠ 0)    ⇒   ⊢ (𝜑 → (𝐴 /𝑒 𝐵) ∈ ℝ*)
Theorem	xdivcl 33003	Closure law for the extended division. (Contributed by Thierry Arnoux, 15-Mar-2017.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) → (𝐴 /𝑒 𝐵) ∈ ℝ*)
Theorem	xdivmul 33004	Relationship between division and multiplication. (Contributed by Thierry Arnoux, 24-Dec-2016.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ (𝐶 ∈ ℝ ∧ 𝐶 ≠ 0)) → ((𝐴 /𝑒 𝐶) = 𝐵 ↔ (𝐶 ·e 𝐵) = 𝐴))
Theorem	rexdiv 33005	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 33006	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 33007	A number divided by itself is one. (Contributed by Thierry Arnoux, 18-Dec-2016.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → (𝐴 /𝑒 𝐴) = 1)
Theorem	xdiv0 33008	Division into zero is zero. (Contributed by Thierry Arnoux, 18-Dec-2016.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → (0 /𝑒 𝐴) = 0)
Theorem	xdiv0rp 33009	Division into zero is zero. (Contributed by Thierry Arnoux, 18-Dec-2016.)
⊢ (𝐴 ∈ ℝ+ → (0 /𝑒 𝐴) = 0)
Theorem	eliccioo 33010	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 33011	Elementhood in the set of nonnegative extended reals. (Contributed by Thierry Arnoux, 18-Dec-2016.)
⊢ (𝐴 ∈ (0[,]+∞) ↔ (𝐴 = 0 ∨ 𝐴 ∈ ℝ+ ∨ 𝐴 = +∞))
Theorem	xdivpnfrp 33012	Plus infinity divided by a positive real number is plus infinity. (Contributed by Thierry Arnoux, 18-Dec-2016.)
⊢ (𝐴 ∈ ℝ+ → (+∞ /𝑒 𝐴) = +∞)
Theorem	rpxdivcld 33013	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 33014	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 33015	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 33016*	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 33017	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 33018	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 33019	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 33020	The range of a prefix of a word is a subset of the range of that word. Stronger version of pfxrn 14637. (Contributed by Thierry Arnoux, 12-Dec-2023.)
⊢ ((𝑊 ∈ Word 𝑆 ∧ 𝐿 ∈ (0...(♯‘𝑊))) → ran (𝑊 prefix 𝐿) ⊆ ran 𝑊)
Theorem	pfxrn3 33021	Express the range of a prefix of a word. Stronger version of pfxrn2 33020. (Contributed by Thierry Arnoux, 13-Dec-2023.)
⊢ ((𝑊 ∈ Word 𝑆 ∧ 𝐿 ∈ (0...(♯‘𝑊))) → ran (𝑊 prefix 𝐿) = (𝑊 “ (0..^𝐿)))
Theorem	pfxf1 33022	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 33023	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 33024	Obsolete version of s2rn 14914 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 33025	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 33026	Obsolete version of s2rn 14914 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 33027	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 33028	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 33029	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 33030	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 33031	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 33032	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 33033*	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 33034	The range of a subword is a subset of the range of that word. Stronger version of swrdrn 14604. (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 33035	Express the range of a subword. Stronger version of swrdrn2 33034. (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 33036	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 33037	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 33038	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 33039	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 33040	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 33041	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 33042	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 33043	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 33044	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 33045	Weaker version of abvpropd 20801. (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 33046	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 33047	A poset ordering is asymmetric. (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 33048	Being a toset is a self-dual property. (Contributed by Thierry Arnoux, 13-Sep-2018.)
⊢ 𝐷 = (ODual‘𝐾)    ⇒   ⊢ (𝐾 ∈ Toset → 𝐷 ∈ Toset)
Theorem	tlt2 33049	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 33050	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 33051	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 33052*	Lemma for toslub 33053 and xrsclat 33091. (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 33053	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 33054*	Lemma for tosglb 33055 and xrsclat 33091. (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 33055	Same theorem as toslub 33053, 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 33056	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 33057	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 33058	Extend class notation with monotone functions.
class Monot
Syntax	cmgc 33059	Extend class notation with the monotone Galois connection.
class MGalConn
Definition	df-mnt 33060*	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 33061*	Define monotone Galois connections. See mgcval 33067 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 33062*	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 33063*	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 33064	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 33065	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 33066*	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 33067*	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 33068	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 33069	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 33070	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 33071	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 33072	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 33073	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 33074*	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 33075*	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 33076*	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 33077	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 33078	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 33079	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 33080*	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 33081	Property of a Galois connection, lemma for mgcf1o 33083. (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 33082	Property of a Galois connection, lemma for mgcf1o 33083. (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 33083	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 33084	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 33085	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 33086	The zero of the extended real numbers. The extended real is not a group, as its addition is not associative. (cf. xaddass 13190 and df-xrs 17455), however it has a zero. (Contributed by Thierry Arnoux, 13-Jun-2017.)
⊢ 0 = (0g‘ℝ*𝑠)
Theorem	xrslt 33087	The "strictly less than" relation for the extended real structure. (Contributed by Thierry Arnoux, 30-Jan-2018.)
⊢ < = (lt‘ℝ*𝑠)
Theorem	xrsinvgval 33088	The inversion operation in the extended real numbers. The extended real is not a group, as its addition is not associative. (cf. xaddass 13190 and df-xrs 17455), however it has an inversion operation. (Contributed by Thierry Arnoux, 13-Jun-2017.)
⊢ (𝐵 ∈ ℝ* → ((invg‘ℝ*𝑠)‘𝐵) = -𝑒𝐵)
Theorem	xrsmulgzz 33089	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 33090	The extended real numbers form a toset. (Contributed by Thierry Arnoux, 15-Feb-2018.)
⊢ ℝ*𝑠 ∈ Toset
Theorem	xrsclat 33091	The extended real numbers form a complete lattice. (Contributed by Thierry Arnoux, 15-Feb-2018.)
⊢ ℝ*𝑠 ∈ CLat
Theorem	xrsp0 33092	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 33093	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 33094	The zero of the extended nonnegative real numbers monoid. (Contributed by Thierry Arnoux, 30-Jan-2017.)
⊢ 0 = (0g‘(ℝ*𝑠 ↾s (0[,]+∞)))
Theorem	xrge0mulgnn0 33095	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 33096	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 33097	The sum of nonnegative and positive numbers is positive. See addgtge0 11627. (Contributed by Thierry Arnoux, 6-Jul-2017.)
⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞)) ∧ 0 < 𝐴) → 0 < (𝐴 +𝑒 𝐵))
Theorem	xrge0adddir 33098	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 33099	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 33100	Extended nonnegative real version of npcan 11391. (Contributed by Thierry Arnoux, 9-Jun-2017.)
⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐵 ≤ 𝐴) → ((𝐴 +𝑒 -𝑒𝐵) +𝑒 𝐵) = 𝐴)