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| 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[,]+∞)) | ||
| 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〉)) = ∅) | ||
| 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))) | ||
| 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) | ||
| 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)) | ||
| 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)) | ||
| 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, < )) | ||
| 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) | ||
| 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)) | ||
| 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)) | ||
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