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	xdivval 32901*	Value of division: the (unique) element 𝑥 such that (𝐵 · 𝑥) = 𝐴. This is meaningful only when 𝐵 is nonzero. (Contributed by Thierry Arnoux, 17-Dec-2016.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) → (𝐴 /𝑒 𝐵) = (℩𝑥 ∈ ℝ* (𝐵 ·e 𝑥) = 𝐴))
Theorem	xrecex 32902*	Existence of reciprocal of nonzero real number. (Contributed by Thierry Arnoux, 17-Dec-2016.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → ∃𝑥 ∈ ℝ (𝐴 ·e 𝑥) = 1)
Theorem	xmulcand 32903	Cancellation law for extended multiplication. (Contributed by Thierry Arnoux, 17-Dec-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ≠ 0)    ⇒   ⊢ (𝜑 → ((𝐶 ·e 𝐴) = (𝐶 ·e 𝐵) ↔ 𝐴 = 𝐵))
Theorem	xreceu 32904*	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 32905	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 32906	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 32907	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 32908	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 32909	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 32910	A number divided by itself is one. (Contributed by Thierry Arnoux, 18-Dec-2016.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → (𝐴 /𝑒 𝐴) = 1)
Theorem	xdiv0 32911	Division into zero is zero. (Contributed by Thierry Arnoux, 18-Dec-2016.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → (0 /𝑒 𝐴) = 0)
Theorem	xdiv0rp 32912	Division into zero is zero. (Contributed by Thierry Arnoux, 18-Dec-2016.)
⊢ (𝐴 ∈ ℝ+ → (0 /𝑒 𝐴) = 0)
Theorem	eliccioo 32913	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 32914	Elementhood in the set of nonnegative extended reals. (Contributed by Thierry Arnoux, 18-Dec-2016.)
⊢ (𝐴 ∈ (0[,]+∞) ↔ (𝐴 = 0 ∨ 𝐴 ∈ ℝ+ ∨ 𝐴 = +∞))
Theorem	xdivpnfrp 32915	Plus infinity divided by a positive real number is plus infinity. (Contributed by Thierry Arnoux, 18-Dec-2016.)
⊢ (𝐴 ∈ ℝ+ → (+∞ /𝑒 𝐴) = +∞)
Theorem	rpxdivcld 32916	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 32917	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	wrdfd 32918	A word is a zero-based sequence with a recoverable upper limit, deduction version. (Contributed by Thierry Arnoux, 22-Dec-2021.)
⊢ (𝜑 → 𝑁 = (♯‘𝑊))    &   ⊢ (𝜑 → 𝑊 ∈ Word 𝑆)    ⇒   ⊢ (𝜑 → 𝑊:(0..^𝑁)⟶𝑆)
Theorem	wrdres 32919	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 32920*	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 32921	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 32922	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 32923	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 32924	The range of a prefix of a word is a subset of the range of that word. Stronger version of pfxrn 14723. (Contributed by Thierry Arnoux, 12-Dec-2023.)
⊢ ((𝑊 ∈ Word 𝑆 ∧ 𝐿 ∈ (0...(♯‘𝑊))) → ran (𝑊 prefix 𝐿) ⊆ ran 𝑊)
Theorem	pfxrn3 32925	Express the range of a prefix of a word. Stronger version of pfxrn2 32924. (Contributed by Thierry Arnoux, 13-Dec-2023.)
⊢ ((𝑊 ∈ Word 𝑆 ∧ 𝐿 ∈ (0...(♯‘𝑊))) → ran (𝑊 prefix 𝐿) = (𝑊 “ (0..^𝐿)))
Theorem	pfxf1 32926	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 32927	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 32928	Obsolete version of s2rn 15002 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 32929	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 32930	Obsolete version of s2rn 15002 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 32931	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 32932	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 32933	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	ccatdmss 32934	The domain of a concatenated word is a superset of the domain of the first word. (Contributed by Thierry Arnoux, 19-Jun-2025.)
⊢ (𝜑 → 𝐴 ∈ Word 𝑆)    &   ⊢ (𝜑 → 𝐵 ∈ Word 𝑆)    ⇒   ⊢ (𝜑 → dom 𝐴 ⊆ dom (𝐴 ++ 𝐵))
Theorem	pfxlsw2ccat 32935	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 32936	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 32937	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 32938*	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 32939	The range of a subword is a subset of the range of that word. Stronger version of swrdrn 14690. (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 32940	Express the range of a subword. Stronger version of swrdrn2 32939. (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 32941	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 32942	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 32943	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 32944	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 32945	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 32946	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 32947	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 32948	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 32949	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 32950	Weaker version of abvpropd 20836. (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  The opposite group
Theorem	oppgle 32951	less-than relation of an opposite group. (Contributed by Thierry Arnoux, 13-Apr-2018.)
⊢ 𝑂 = (oppg‘𝑅)    &   ⊢ ≤ = (le‘𝑅)    ⇒   ⊢ ≤ = (le‘𝑂)
Theorem	oppgleOLD 32952	Obsolete version of oppgle 32951 as of 27-Oct-2024. less-than relation of an opposite group. (Contributed by Thierry Arnoux, 13-Apr-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝑂 = (oppg‘𝑅)    &   ⊢ ≤ = (le‘𝑅)    ⇒   ⊢ ≤ = (le‘𝑂)
Theorem	oppglt 32953	less-than relation of an opposite group. (Contributed by Thierry Arnoux, 13-Apr-2018.)
⊢ 𝑂 = (oppg‘𝑅)    &   ⊢ < = (lt‘𝑅)    ⇒   ⊢ (𝑅 ∈ 𝑉 → < = (lt‘𝑂))
21.3.9.3  Posets
Theorem	ressprs 32954	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 32955	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	resspos 32956	The restriction of a Poset is a Poset. (Contributed by Thierry Arnoux, 20-Jan-2018.)
⊢ ((𝐹 ∈ Poset ∧ 𝐴 ∈ 𝑉) → (𝐹 ↾s 𝐴) ∈ Poset)
Theorem	resstos 32957	The restriction of a Toset is a Toset. (Contributed by Thierry Arnoux, 20-Jan-2018.)
⊢ ((𝐹 ∈ Toset ∧ 𝐴 ∈ 𝑉) → (𝐹 ↾s 𝐴) ∈ Toset)
Theorem	odutos 32958	Being a toset is a self-dual property. (Contributed by Thierry Arnoux, 13-Sep-2018.)
⊢ 𝐷 = (ODual‘𝐾)    ⇒   ⊢ (𝐾 ∈ Toset → 𝐷 ∈ Toset)
Theorem	tlt2 32959	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 32960	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 32961	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 32962*	Lemma for toslub 32963 and xrsclat 33013. (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 32963	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 32964*	Lemma for tosglb 32965 and xrsclat 33013. (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 32965	Same theorem as toslub 32963, 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, < ))
21.3.9.4  Complete lattices
Theorem	clatp0cl 32966	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 32967	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)
21.3.9.5  Order Theory
Syntax	cmnt 32968	Extend class notation with monotone functions.
class Monot
Syntax	cmgc 32969	Extend class notation with the monotone Galois connection.
class MGalConn
Definition	df-mnt 32970*	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 32971*	Define monotone Galois connections. See mgcval 32977 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 32972*	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 32973*	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 32974	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 32975	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 32976*	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 32977*	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 32978	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 32979	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 32980	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 32981	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 32982	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 32983	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 32984*	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 32985*	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 32986*	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 32987	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 32988	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 32989	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 32990*	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 32991	Property of a Galois connection, lemma for mgcf1o 32993. (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 32992	Property of a Galois connection, lemma for mgcf1o 32993. (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 32993	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))
21.3.9.6  Chains
Syntax	cchn 32994	Extend class notation with the class of (finite) chains.
class ( < Chain𝐴)
Definition	df-chn 32995*	Define the class of (finite) chains. A chain is defined to be a sequence of objects, where each object is less than the next one in the sequence. The term "chain" is usually used in order theory. In the context of algebra, chains are often called "towers", for example for fields, or "series", for example for subgroup or subnormal series. (Contributed by Thierry Arnoux, 19-Jun-2025.)
⊢ ( < Chain𝐴) = {𝑐 ∈ Word 𝐴 ∣ ∀𝑛 ∈ (dom 𝑐 ∖ {0})(𝑐‘(𝑛 − 1)) < (𝑐‘𝑛)}
Theorem	ischn 32996*	Property of being a chain. (Contributed by Thierry Arnoux, 19-Jun-2025.)
⊢ (𝐶 ∈ ( < Chain𝐴) ↔ (𝐶 ∈ Word 𝐴 ∧ ∀𝑛 ∈ (dom 𝐶 ∖ {0})(𝐶‘(𝑛 − 1)) < (𝐶‘𝑛)))
Theorem	chnwrd 32997	A chain is an ordered sequence, i.e. a word. (Contributed by Thierry Arnoux, 19-Jun-2025.)
⊢ (𝜑 → 𝐶 ∈ ( < Chain𝐴))    ⇒   ⊢ (𝜑 → 𝐶 ∈ Word 𝐴)
Theorem	chnltm1 32998	Basic property of a chain. (Contributed by Thierry Arnoux, 19-Jun-2025.)
⊢ (𝜑 → 𝐶 ∈ ( < Chain𝐴))    &   ⊢ (𝜑 → 𝑁 ∈ (dom 𝐶 ∖ {0}))    ⇒   ⊢ (𝜑 → (𝐶‘(𝑁 − 1)) < (𝐶‘𝑁))
Theorem	pfxchn 32999	A prefix of a chain is still a chain. (Contributed by Thierry Arnoux, 19-Jun-2025.)
⊢ (𝜑 → 𝐶 ∈ ( < Chain𝐴))    &   ⊢ (𝜑 → 𝐿 ∈ (0...(♯‘𝐶)))    ⇒   ⊢ (𝜑 → (𝐶 prefix 𝐿) ∈ ( < Chain𝐴))
Theorem	s1chn 33000	A singleton word is always a chain. (Contributed by Thierry Arnoux, 19-Oct-2025.)
⊢ (𝜑 → 𝑋 ∈ 𝐴)    ⇒   ⊢ (𝜑 → ⟨“𝑋”⟩ ∈ ( < Chain𝐴))