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Theorem List for Metamath Proof Explorer - 30601-30700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdpmul10 30601 Multiply by 10 a decimal expansion. (Contributed by Thierry Arnoux, 25-Dec-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℝ       ((𝐴.𝐵) · 10) = 𝐴𝐵

Theoremdecdiv10 30602 Divide a decimal number by 10. (Contributed by Thierry Arnoux, 25-Dec-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℝ       (𝐴𝐵 / 10) = (𝐴.𝐵)

Theoremdpmul100 30603 Multiply by 100 a decimal expansion. (Contributed by Thierry Arnoux, 25-Dec-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℝ       ((𝐴.𝐵𝐶) · 100) = 𝐴𝐵𝐶

Theoremdp3mul10 30604 Multiply by 10 a decimal expansion with 3 digits. (Contributed by Thierry Arnoux, 25-Dec-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℝ       ((𝐴.𝐵𝐶) · 10) = (𝐴𝐵.𝐶)

Theoremdpmul1000 30605 Multiply by 1000 a decimal expansion. (Contributed by Thierry Arnoux, 25-Dec-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℕ0    &   𝐷 ∈ ℝ       ((𝐴.𝐵𝐶𝐷) · 1000) = 𝐴𝐵𝐶𝐷

Theoremdpval3rp 30606 Value of the decimal point construct. (Contributed by Thierry Arnoux, 16-Dec-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℝ+       (𝐴.𝐵) = 𝐴𝐵

Theoremdp0u 30607 Add a zero in the tenths place. (Contributed by Thierry Arnoux, 16-Dec-2021.)
𝐴 ∈ ℕ0       (𝐴.0) = 𝐴

Theoremdp0h 30608 Remove a zero in the units places. (Contributed by Thierry Arnoux, 16-Dec-2021.)
𝐴 ∈ ℝ+       (0.𝐴) = (𝐴 / 10)

Theoremrpdpcl 30609 Closure of the decimal point in the positive real numbers. (Contributed by Thierry Arnoux, 16-Dec-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℝ+       (𝐴.𝐵) ∈ ℝ+

Theoremdplt 30610 Comparing two decimal expansions (equal higher places). (Contributed by Thierry Arnoux, 16-Dec-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℝ+    &   𝐶 ∈ ℝ+    &   𝐵 < 𝐶       (𝐴.𝐵) < (𝐴.𝐶)

Theoremdplti 30611 Comparing a decimal expansions with the next higher integer. (Contributed by Thierry Arnoux, 16-Dec-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℝ+    &   𝐶 ∈ ℕ0    &   𝐵 < 10    &   (𝐴 + 1) = 𝐶       (𝐴.𝐵) < 𝐶

Theoremdpgti 30612 Comparing a decimal expansions with the next lower integer. (Contributed by Thierry Arnoux, 16-Dec-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℝ+       𝐴 < (𝐴.𝐵)

Theoremdpltc 30613 Comparing two decimal integers (unequal higher places). (Contributed by Thierry Arnoux, 16-Dec-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℝ+    &   𝐶 ∈ ℕ0    &   𝐷 ∈ ℝ+    &   𝐴 < 𝐶    &   𝐵 < 10       (𝐴.𝐵) < (𝐶.𝐷)

Theoremdpexpp1 30614 Add one zero to the mantisse, and a one to the exponent in a scientific notation. (Contributed by Thierry Arnoux, 16-Dec-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℝ+    &   (𝑃 + 1) = 𝑄    &   𝑃 ∈ ℤ    &   𝑄 ∈ ℤ       ((𝐴.𝐵) · (10↑𝑃)) = ((0.𝐴𝐵) · (10↑𝑄))

Theorem0dp2dp 30615 Multiply by 10 a decimal expansion which starts with a zero. (Contributed by Thierry Arnoux, 16-Dec-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℝ+       ((0.𝐴𝐵) · 10) = (𝐴.𝐵)

Theoremdpadd2 30616 Addition with one decimal, no carry. (Contributed by Thierry Arnoux, 29-Dec-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℝ+    &   𝐶 ∈ ℕ0    &   𝐷 ∈ ℝ+    &   𝐸 ∈ ℕ0    &   𝐹 ∈ ℝ+    &   𝐺 ∈ ℕ0    &   𝐻 ∈ ℕ0    &   (𝐺 + 𝐻) = 𝐼    &   ((𝐴.𝐵) + (𝐶.𝐷)) = (𝐸.𝐹)       ((𝐺.𝐴𝐵) + (𝐻.𝐶𝐷)) = (𝐼.𝐸𝐹)

𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℕ0    &   𝐷 ∈ ℕ0    &   𝐸 ∈ ℕ0    &   𝐹 ∈ ℕ0    &   (𝐴𝐵 + 𝐶𝐷) = 𝐸𝐹       ((𝐴.𝐵) + (𝐶.𝐷)) = (𝐸.𝐹)

𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℕ0    &   𝐷 ∈ ℕ0    &   𝐸 ∈ ℕ0    &   𝐺 ∈ ℕ0    &   𝐹 ∈ ℕ0    &   𝐻 ∈ ℕ0    &   𝐼 ∈ ℕ0    &   (𝐴𝐵𝐶 + 𝐷𝐸𝐹) = 𝐺𝐻𝐼       ((𝐴.𝐵𝐶) + (𝐷.𝐸𝐹)) = (𝐺.𝐻𝐼)

Theoremdpmul 30619 Multiplication with one decimal point. (Contributed by Thierry Arnoux, 26-Dec-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℕ0    &   𝐷 ∈ ℕ0    &   𝐸 ∈ ℕ0    &   𝐺 ∈ ℕ0    &   𝐽 ∈ ℕ0    &   𝐾 ∈ ℕ0    &   (𝐴 · 𝐶) = 𝐹    &   (𝐴 · 𝐷) = 𝑀    &   (𝐵 · 𝐶) = 𝐿    &   (𝐵 · 𝐷) = 𝐸𝐾    &   ((𝐿 + 𝑀) + 𝐸) = 𝐺𝐽    &   (𝐹 + 𝐺) = 𝐼       ((𝐴.𝐵) · (𝐶.𝐷)) = (𝐼.𝐽𝐾)

Theoremdpmul4 30620 An upper bound to multiplication of decimal numbers with 4 digits. (Contributed by Thierry Arnoux, 25-Dec-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℕ0    &   𝐷 ∈ ℕ0    &   𝐸 ∈ ℕ0    &   𝐺 ∈ ℕ0    &   𝐽 ∈ ℕ0    &   𝐾 ∈ ℕ0    &   𝐹 ∈ ℕ0    &   𝐻 ∈ ℕ0    &   𝐼 ∈ ℕ0    &   𝐿 ∈ ℕ0    &   𝑀 ∈ ℕ0    &   𝑁 ∈ ℕ0    &   𝑂 ∈ ℕ0    &   𝑃 ∈ ℕ0    &   𝑄 ∈ ℕ0    &   𝑅 ∈ ℕ0    &   𝑆 ∈ ℕ0    &   𝑇 ∈ ℕ0    &   𝑈 ∈ ℕ0    &   𝑊 ∈ ℕ0    &   𝑋 ∈ ℕ0    &   𝑌 ∈ ℕ0    &   𝑍 ∈ ℕ0    &   𝑈 < 10    &   𝑃 < 10    &   𝑄 < 10    &   (𝐿𝑀𝑁 + 𝑂) = 𝑅𝑆𝑇𝑈    &   ((𝐴.𝐵) · (𝐸.𝐹)) = (𝐼.𝐽𝐾)    &   ((𝐶.𝐷) · (𝐺.𝐻)) = (𝑂.𝑃𝑄)    &   (𝐼𝐽𝐾1 + 𝑅𝑆𝑇) = 𝑊𝑋𝑌𝑍    &   (((𝐴.𝐵) + (𝐶.𝐷)) · ((𝐸.𝐹) + (𝐺.𝐻))) = (((𝐼.𝐽𝐾) + (𝐿.𝑀𝑁)) + (𝑂.𝑃𝑄))       ((𝐴.𝐵𝐶𝐷) · (𝐸.𝐹𝐺𝐻)) < (𝑊.𝑋𝑌𝑍)

Theoremthreehalves 30621 Example theorem demonstrating decimal expansions. (Contributed by Thierry Arnoux, 27-Dec-2021.)
(3 / 2) = (1.5)

Theorem1mhdrd 30622 Example theorem demonstrating decimal expansions. (Contributed by Thierry Arnoux, 27-Dec-2021.)
((0.99) + (0.01)) = 1

20.3.6.2  Division in the extended real number system

Syntaxcxdiv 30623 Extend class notation to include division of extended reals.
class /𝑒

Definitiondf-xdiv 30624* Define division over extended real numbers. (Contributed by Thierry Arnoux, 17-Dec-2016.)
/𝑒 = (𝑥 ∈ ℝ*, 𝑦 ∈ (ℝ ∖ {0}) ↦ (𝑧 ∈ ℝ* (𝑦 ·e 𝑧) = 𝑥))

Theoremxdivval 30625* 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))

Theoremxrecex 30626* Existence of reciprocal of nonzero real number. (Contributed by Thierry Arnoux, 17-Dec-2016.)
((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → ∃𝑥 ∈ ℝ (𝐴 ·e 𝑥) = 1)

Theoremxmulcand 30627 Cancellation law for extended multiplication. (Contributed by Thierry Arnoux, 17-Dec-2016.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐶 ≠ 0)       (𝜑 → ((𝐶 ·e 𝐴) = (𝐶 ·e 𝐵) ↔ 𝐴 = 𝐵))

Theoremxreceu 30628* 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)

Theoremxdivcld 30629 Closure law for the extended division. (Contributed by Thierry Arnoux, 15-Mar-2017.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐵 ≠ 0)       (𝜑 → (𝐴 /𝑒 𝐵) ∈ ℝ*)

Theoremxdivcl 30630 Closure law for the extended division. (Contributed by Thierry Arnoux, 15-Mar-2017.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) → (𝐴 /𝑒 𝐵) ∈ ℝ*)

Theoremxdivmul 30631 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))

Theoremrexdiv 30632 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))

Theoremxdivrec 30633 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)))

Theoremxdivid 30634 A number divided by itself is one. (Contributed by Thierry Arnoux, 18-Dec-2016.)
((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → (𝐴 /𝑒 𝐴) = 1)

Theoremxdiv0 30635 Division into zero is zero. (Contributed by Thierry Arnoux, 18-Dec-2016.)
((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → (0 /𝑒 𝐴) = 0)

Theoremxdiv0rp 30636 Division into zero is zero. (Contributed by Thierry Arnoux, 18-Dec-2016.)
(𝐴 ∈ ℝ+ → (0 /𝑒 𝐴) = 0)

Theoremeliccioo 30637 Membership in a closed interval of extended reals versus the same open interval. (Contributed by Thierry Arnoux, 18-Dec-2016.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴𝐵) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 = 𝐴𝐶 ∈ (𝐴(,)𝐵) ∨ 𝐶 = 𝐵)))

Theoremelxrge02 30638 Elementhood in the set of nonnegative extended reals. (Contributed by Thierry Arnoux, 18-Dec-2016.)
(𝐴 ∈ (0[,]+∞) ↔ (𝐴 = 0 ∨ 𝐴 ∈ ℝ+𝐴 = +∞))

Theoremxdivpnfrp 30639 Plus infinity divided by a positive real number is plus infinity. (Contributed by Thierry Arnoux, 18-Dec-2016.)
(𝐴 ∈ ℝ+ → (+∞ /𝑒 𝐴) = +∞)

Theoremrpxdivcld 30640 Closure law for extended division of positive reals. (Contributed by Thierry Arnoux, 18-Dec-2016.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)       (𝜑 → (𝐴 /𝑒 𝐵) ∈ ℝ+)

Theoremxrpxdivcld 30641 Closure law for extended division of positive extended reals. (Contributed by Thierry Arnoux, 18-Dec-2016.)
(𝜑𝐴 ∈ (0[,]+∞))    &   (𝜑𝐵 ∈ ℝ+)       (𝜑 → (𝐴 /𝑒 𝐵) ∈ (0[,]+∞))

20.3.7  Words over a set - misc additions

Theoremwrdfd 30642 A word is a zero-based sequence with a recoverable upper limit, deduction version. (Contributed by Thierry Arnoux, 22-Dec-2021.)
(𝜑𝑁 = (♯‘𝑊))    &   (𝜑𝑊 ∈ Word 𝑆)       (𝜑𝑊:(0..^𝑁)⟶𝑆)

Theoremwrdres 30643 Condition for the restriction of a word to be a word itself. (Contributed by Thierry Arnoux, 5-Oct-2018.)
((𝑊 ∈ Word 𝑆𝑁 ∈ (0...(♯‘𝑊))) → (𝑊 ↾ (0..^𝑁)) ∈ Word 𝑆)

Theoremwrdsplex 30644* 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 𝑆𝑁 ∈ (0...(♯‘𝑊))) → ∃𝑣 ∈ Word 𝑆𝑊 = ((𝑊 ↾ (0..^𝑁)) ++ 𝑣))

Theorempfx1s2 30645 The prefix of length 1 of a length 2 word. (Contributed by Thierry Arnoux, 19-Sep-2023.)
((𝐴𝑉𝐵𝑉) → (⟨“𝐴𝐵”⟩ prefix 1) = ⟨“𝐴”⟩)

Theorempfxrn2 30646 The range of a prefix of a word is a subset of the range of that word. Stronger version of pfxrn 14042. (Contributed by Thierry Arnoux, 12-Dec-2023.)
((𝑊 ∈ Word 𝑆𝐿 ∈ (0...(♯‘𝑊))) → ran (𝑊 prefix 𝐿) ⊆ ran 𝑊)

Theorempfxrn3 30647 Express the range of a prefix of a word. Stronger version of pfxrn2 30646. (Contributed by Thierry Arnoux, 13-Dec-2023.)
((𝑊 ∈ Word 𝑆𝐿 ∈ (0...(♯‘𝑊))) → ran (𝑊 prefix 𝐿) = (𝑊 “ (0..^𝐿)))

Theorempfxf1 30648 Condition for a prefix to be injective. (Contributed by Thierry Arnoux, 13-Dec-2023.)
(𝜑𝑊 ∈ Word 𝑆)    &   (𝜑𝑊:dom 𝑊1-1𝑆)    &   (𝜑𝐿 ∈ (0...(♯‘𝑊)))       (𝜑 → (𝑊 prefix 𝐿):dom (𝑊 prefix 𝐿)–1-1𝑆)

Theorems1f1 30649 Conditions for a length 1 string to be a one-to-one function. (Contributed by Thierry Arnoux, 11-Dec-2023.)
(𝜑𝐼𝐷)       (𝜑 → ⟨“𝐼”⟩:dom ⟨“𝐼”⟩–1-1𝐷)

Theorems2rn 30650 Range of a length 2 string. (Contributed by Thierry Arnoux, 19-Sep-2023.)
(𝜑𝐼𝐷)    &   (𝜑𝐽𝐷)       (𝜑 → ran ⟨“𝐼𝐽”⟩ = {𝐼, 𝐽})

Theorems2f1 30651 Conditions for a length 2 string to be a one-to-one function. (Contributed by Thierry Arnoux, 19-Sep-2023.)
(𝜑𝐼𝐷)    &   (𝜑𝐽𝐷)    &   (𝜑𝐼𝐽)       (𝜑 → ⟨“𝐼𝐽”⟩:dom ⟨“𝐼𝐽”⟩–1-1𝐷)

Theorems3rn 30652 Range of a length 3 string. (Contributed by Thierry Arnoux, 19-Sep-2023.)
(𝜑𝐼𝐷)    &   (𝜑𝐽𝐷)    &   (𝜑𝐾𝐷)       (𝜑 → ran ⟨“𝐼𝐽𝐾”⟩ = {𝐼, 𝐽, 𝐾})

Theorems3f1 30653 Conditions for a length 3 string to be a one-to-one function. (Contributed by Thierry Arnoux, 19-Sep-2023.)
(𝜑𝐼𝐷)    &   (𝜑𝐽𝐷)    &   (𝜑𝐾𝐷)    &   (𝜑𝐼𝐽)    &   (𝜑𝐽𝐾)    &   (𝜑𝐾𝐼)       (𝜑 → ⟨“𝐼𝐽𝐾”⟩:dom ⟨“𝐼𝐽𝐾”⟩–1-1𝐷)

Theorems3clhash 30654 Closure of the words of length 3 in a preimage using the hash function. (Contributed by Thierry Arnoux, 27-Sep-2023.)
⟨“𝐼𝐽𝐾”⟩ ∈ (♯ “ {3})

Theoremccatf1 30655 Conditions for a concatenation to be injective. (Contributed by Thierry Arnoux, 11-Dec-2023.)
(𝜑𝑆𝑉)    &   (𝜑𝐴 ∈ Word 𝑆)    &   (𝜑𝐵 ∈ Word 𝑆)    &   (𝜑𝐴:dom 𝐴1-1𝑆)    &   (𝜑𝐵:dom 𝐵1-1𝑆)    &   (𝜑 → (ran 𝐴 ∩ ran 𝐵) = ∅)       (𝜑 → (𝐴 ++ 𝐵):dom (𝐴 ++ 𝐵)–1-1𝑆)

Theorempfxlsw2ccat 30656 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))”⟩))

Theoremwrdt2ind 30657* Perform an induction over the structure of a word of even length. (Contributed by Thierry Arnoux, 26-Sep-2023.)
(𝑥 = ∅ → (𝜑𝜓))    &   (𝑥 = 𝑦 → (𝜑𝜒))    &   (𝑥 = (𝑦 ++ ⟨“𝑖𝑗”⟩) → (𝜑𝜃))    &   (𝑥 = 𝐴 → (𝜑𝜏))    &   𝜓    &   ((𝑦 ∈ Word 𝐵𝑖𝐵𝑗𝐵) → (𝜒𝜃))       ((𝐴 ∈ Word 𝐵 ∧ 2 ∥ (♯‘𝐴)) → 𝜏)

Theoremswrdrn2 30658 The range of a subword is a subset of the range of that word. Stronger version of swrdrn 14009. (Contributed by Thierry Arnoux, 12-Dec-2023.)
((𝑊 ∈ Word 𝑉𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑊))) → ran (𝑊 substr ⟨𝑀, 𝑁⟩) ⊆ ran 𝑊)

Theoremswrdrn3 30659 Express the range of a subword. Stronger version of swrdrn2 30658. (Contributed by Thierry Arnoux, 13-Dec-2023.)
((𝑊 ∈ Word 𝑉𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑊))) → ran (𝑊 substr ⟨𝑀, 𝑁⟩) = (𝑊 “ (𝑀..^𝑁)))

Theoremswrdf1 30660 Condition for a subword to be injective. (Contributed by Thierry Arnoux, 12-Dec-2023.)
(𝜑𝑊 ∈ Word 𝐷)    &   (𝜑𝑀 ∈ (0...𝑁))    &   (𝜑𝑁 ∈ (0...(♯‘𝑊)))    &   (𝜑𝑊:dom 𝑊1-1𝐷)       (𝜑 → (𝑊 substr ⟨𝑀, 𝑁⟩):dom (𝑊 substr ⟨𝑀, 𝑁⟩)–1-1𝐷)

Theoremswrdrndisj 30661 Condition for the range of two subwords of an injective word to be disjoint. (Contributed by Thierry Arnoux, 13-Dec-2023.)
(𝜑𝑊 ∈ Word 𝐷)    &   (𝜑𝑀 ∈ (0...𝑁))    &   (𝜑𝑁 ∈ (0...(♯‘𝑊)))    &   (𝜑𝑊:dom 𝑊1-1𝐷)    &   (𝜑𝑂 ∈ (𝑁...𝑃))    &   (𝜑𝑃 ∈ (𝑁...(♯‘𝑊)))       (𝜑 → (ran (𝑊 substr ⟨𝑀, 𝑁⟩) ∩ ran (𝑊 substr ⟨𝑂, 𝑃⟩)) = ∅)

20.3.7.1  Splicing words (substring replacement)

Theoremsplfv3 30662 Symbols to the right of a splice are unaffected. (Contributed by Thierry Arnoux, 14-Dec-2023.)
(𝜑𝑆 ∈ Word 𝐴)    &   (𝜑𝐹 ∈ (0...𝑇))    &   (𝜑𝑇 ∈ (0...(♯‘𝑆)))    &   (𝜑𝑅 ∈ Word 𝐴)    &   (𝜑𝑋 ∈ (0..^((♯‘𝑆) − 𝑇)))    &   (𝜑𝐾 = (𝐹 + (♯‘𝑅)))       (𝜑 → ((𝑆 splice ⟨𝐹, 𝑇, 𝑅⟩)‘(𝑋 + 𝐾)) = (𝑆‘(𝑋 + 𝑇)))

20.3.7.2  Cyclic shift of words

Theorem1cshid 30663 Cyclically shifting a single letter word keeps it unchanged. (Contributed by Thierry Arnoux, 21-Nov-2023.)
((𝑊 ∈ Word 𝑉𝑁 ∈ ℤ ∧ (♯‘𝑊) = 1) → (𝑊 cyclShift 𝑁) = 𝑊)

Theoremcshw1s2 30664 Cyclically shifting a length 2 word swaps its symbols. (Contributed by Thierry Arnoux, 19-Sep-2023.)
((𝐴𝑉𝐵𝑉) → (⟨“𝐴𝐵”⟩ cyclShift 1) = ⟨“𝐵𝐴”⟩)

Theoremcshwrnid 30665 Cyclically shifting a word preserves its range. (Contributed by Thierry Arnoux, 19-Sep-2023.)
((𝑊 ∈ Word 𝑉𝑁 ∈ ℤ) → ran (𝑊 cyclShift 𝑁) = ran 𝑊)

Theoremcshf1o 30666 Condition for the cyclic shift to be a bijection. (Contributed by Thierry Arnoux, 4-Oct-2023.)
((𝑊 ∈ Word 𝐷𝑊:dom 𝑊1-1𝐷𝑁 ∈ ℤ) → (𝑊 cyclShift 𝑁):dom 𝑊1-1-onto→ran 𝑊)

20.3.8  Extensible Structures

20.3.8.1  Structure restriction operator

Theoremressplusf 30667 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 = (𝐺s 𝐴)    &    = (+g𝐺)    &    Fn (𝐵 × 𝐵)    &   𝐴𝐵       (+𝑓𝐻) = ( ↾ (𝐴 × 𝐴))

Theoremressnm 30668 The norm in a restricted structure. (Contributed by Thierry Arnoux, 8-Oct-2017.)
𝐻 = (𝐺s 𝐴)    &   𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   𝑁 = (norm‘𝐺)       ((𝐺 ∈ Mnd ∧ 0𝐴𝐴𝐵) → (𝑁𝐴) = (norm‘𝐻))

Theoremabvpropd2 30669 Weaker version of abvpropd 19610. (Contributed by Thierry Arnoux, 8-Nov-2017.)
(𝜑 → (Base‘𝐾) = (Base‘𝐿))    &   (𝜑 → (+g𝐾) = (+g𝐿))    &   (𝜑 → (.r𝐾) = (.r𝐿))       (𝜑 → (AbsVal‘𝐾) = (AbsVal‘𝐿))

20.3.8.2  The opposite group

Theoremoppgle 30670 less-than relation of an opposite group. (Contributed by Thierry Arnoux, 13-Apr-2018.)
𝑂 = (oppg𝑅)    &    = (le‘𝑅)        = (le‘𝑂)

Theoremoppglt 30671 less-than relation of an opposite group. (Contributed by Thierry Arnoux, 13-Apr-2018.)
𝑂 = (oppg𝑅)    &    < = (lt‘𝑅)       (𝑅𝑉< = (lt‘𝑂))

20.3.8.3  Posets

Theoremressprs 30672 The restriction of a proset is a proset. (Contributed by Thierry Arnoux, 11-Sep-2015.)
𝐵 = (Base‘𝐾)       ((𝐾 ∈ Proset ∧ 𝐴𝐵) → (𝐾s 𝐴) ∈ Proset )

Theoremoduprs 30673 Being a proset is a self-dual property. (Contributed by Thierry Arnoux, 13-Sep-2018.)
𝐷 = (ODual‘𝐾)       (𝐾 ∈ Proset → 𝐷 ∈ Proset )

Theoremposrasymb 30674 A poset ordering is asymetric. (Contributed by Thierry Arnoux, 13-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = ((le‘𝐾) ∩ (𝐵 × 𝐵))       ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌𝑌 𝑋) ↔ 𝑋 = 𝑌))

Theoremtospos 30675 A Toset is a Poset. (Contributed by Thierry Arnoux, 20-Jan-2018.)
(𝐹 ∈ Toset → 𝐹 ∈ Poset)

Theoremresspos 30676 The restriction of a Poset is a Poset. (Contributed by Thierry Arnoux, 20-Jan-2018.)
((𝐹 ∈ Poset ∧ 𝐴𝑉) → (𝐹s 𝐴) ∈ Poset)

Theoremresstos 30677 The restriction of a Toset is a Toset. (Contributed by Thierry Arnoux, 20-Jan-2018.)
((𝐹 ∈ Toset ∧ 𝐴𝑉) → (𝐹s 𝐴) ∈ Toset)

Theoremtleile 30678 In a Toset, two elements must compare. (Contributed by Thierry Arnoux, 11-Feb-2018.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)       ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌𝑌 𝑋))

Theoremtltnle 30679 In a Toset, less-than is equivalent to not inverse "less than or equal to" see pltnle 17572. (Contributed by Thierry Arnoux, 11-Feb-2018.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    < = (lt‘𝐾)       ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 ↔ ¬ 𝑌 𝑋))

Theoremodutos 30680 Being a toset is a self-dual property. (Contributed by Thierry Arnoux, 13-Sep-2018.)
𝐷 = (ODual‘𝐾)       (𝐾 ∈ Toset → 𝐷 ∈ Toset)

Theoremtlt2 30681 In a Toset, two elements must compare. (Contributed by Thierry Arnoux, 13-Apr-2018.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    < = (lt‘𝐾)       ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌𝑌 < 𝑋))

Theoremtlt3 30682 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 ∧ 𝑋𝐵𝑌𝐵) → (𝑋 = 𝑌𝑋 < 𝑌𝑌 < 𝑋))

Theoremtrleile 30683 In a Toset, two elements must compare. (Contributed by Thierry Arnoux, 12-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = ((le‘𝐾) ∩ (𝐵 × 𝐵))       ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌𝑌 𝑋))

Theoremtoslublem 30684* Lemma for toslub 30685 and xrsclat 30718. (Contributed by Thierry Arnoux, 17-Feb-2018.) (Revised by NM, 15-Sep-2018.)
𝐵 = (Base‘𝐾)    &    < = (lt‘𝐾)    &   (𝜑𝐾 ∈ Toset)    &   (𝜑𝐴𝐵)    &    = (le‘𝐾)       ((𝜑𝑎𝐵) → ((∀𝑏𝐴 𝑏 𝑎 ∧ ∀𝑐𝐵 (∀𝑏𝐴 𝑏 𝑐𝑎 𝑐)) ↔ (∀𝑏𝐴 ¬ 𝑎 < 𝑏 ∧ ∀𝑏𝐵 (𝑏 < 𝑎 → ∃𝑑𝐴 𝑏 < 𝑑))))

Theoremtoslub 30685 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)    &   (𝜑𝐾 ∈ Toset)    &   (𝜑𝐴𝐵)       (𝜑 → ((lub‘𝐾)‘𝐴) = sup(𝐴, 𝐵, < ))

Theoremtosglblem 30686* Lemma for tosglb 30687 and xrsclat 30718. (Contributed by Thierry Arnoux, 17-Feb-2018.) (Revised by NM, 15-Sep-2018.)
𝐵 = (Base‘𝐾)    &    < = (lt‘𝐾)    &   (𝜑𝐾 ∈ Toset)    &   (𝜑𝐴𝐵)    &    = (le‘𝐾)       ((𝜑𝑎𝐵) → ((∀𝑏𝐴 𝑎 𝑏 ∧ ∀𝑐𝐵 (∀𝑏𝐴 𝑐 𝑏𝑐 𝑎)) ↔ (∀𝑏𝐴 ¬ 𝑎 < 𝑏 ∧ ∀𝑏𝐵 (𝑏 < 𝑎 → ∃𝑑𝐴 𝑏 < 𝑑))))

Theoremtosglb 30687 Same theorem as toslub 30685, for infinimum. (Contributed by Thierry Arnoux, 17-Feb-2018.) (Revised by AV, 28-Sep-2020.)
𝐵 = (Base‘𝐾)    &    < = (lt‘𝐾)    &   (𝜑𝐾 ∈ Toset)    &   (𝜑𝐴𝐵)       (𝜑 → ((glb‘𝐾)‘𝐴) = inf(𝐴, 𝐵, < ))

20.3.8.4  Complete lattices

Theoremclatp0cl 30688 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 → 0𝐵)

Theoremclatp1cl 30689 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 → 1𝐵)

20.3.8.5  Order Theory

Syntaxcmnt 30690 Extend class notation with monotone functions.
class Monot

Syntaxcmgc 30691 Extend class notation with the monotone Galois connection.
class MGalConn

Definitiondf-mnt 30692* 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) ∣ ∀𝑥𝑎𝑦𝑎 (𝑥(le‘𝑣)𝑦 → (𝑓𝑥)(le‘𝑤)(𝑓𝑦))})

Definitiondf-mgc 30693* Define monotone Galois connections. See mgcval 30699 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 ∈ (𝑏m 𝑎) ∧ 𝑔 ∈ (𝑎m 𝑏)) ∧ ∀𝑥𝑎𝑦𝑏 ((𝑓𝑥)(le‘𝑤)𝑦𝑥(le‘𝑣)(𝑔𝑦)))})

Theoremmntoval 30694* Operation value of the monotone function. (Contributed by Thierry Arnoux, 23-Apr-2024.)
𝐴 = (Base‘𝑉)    &   𝐵 = (Base‘𝑊)    &    = (le‘𝑉)    &    = (le‘𝑊)       ((𝑉𝑋𝑊𝑌) → (𝑉Monot𝑊) = {𝑓 ∈ (𝐵m 𝐴) ∣ ∀𝑥𝐴𝑦𝐴 (𝑥 𝑦 → (𝑓𝑥) (𝑓𝑦))})

Theoremismnt 30695* Express the statement "𝐹 is monotone". (Contributed by Thierry Arnoux, 23-Apr-2024.)
𝐴 = (Base‘𝑉)    &   𝐵 = (Base‘𝑊)    &    = (le‘𝑉)    &    = (le‘𝑊)       ((𝑉𝑋𝑊𝑌) → (𝐹 ∈ (𝑉Monot𝑊) ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 𝑦 → (𝐹𝑥) (𝐹𝑦)))))

Theoremismntd 30696 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)    &   (𝜑𝑉𝐶)    &   (𝜑𝑊𝐷)    &   (𝜑𝐹 ∈ (𝑉Monot𝑊))    &   (𝜑𝑋𝐴)    &   (𝜑𝑌𝐴)    &   (𝜑𝑋 𝑌)       (𝜑 → (𝐹𝑋) (𝐹𝑌))

Theoremmntf 30697 A monotone function is a function. (Contributed by Thierry Arnoux, 24-Apr-2024.)
𝐴 = (Base‘𝑉)    &   𝐵 = (Base‘𝑊)       ((𝑉𝑋𝑊𝑌𝐹 ∈ (𝑉Monot𝑊)) → 𝐹:𝐴𝐵)

Theoremmgcoval 30698* Operation value of the monotone Galois connection. (Contributed by Thierry Arnoux, 23-Apr-2024.)
𝐴 = (Base‘𝑉)    &   𝐵 = (Base‘𝑊)    &    = (le‘𝑉)    &    = (le‘𝑊)       ((𝑉𝑋𝑊𝑌) → (𝑉MGalConn𝑊) = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐵m 𝐴) ∧ 𝑔 ∈ (𝐴m 𝐵)) ∧ ∀𝑥𝐴𝑦𝐵 ((𝑓𝑥) 𝑦𝑥 (𝑔𝑦)))})

Theoremmgcval 30699* 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)    &   (𝜑𝑉 ∈ Proset )    &   (𝜑𝑊 ∈ Proset )       (𝜑 → (𝐹𝐻𝐺 ↔ ((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ∀𝑥𝐴𝑦𝐵 ((𝐹𝑥) 𝑦𝑥 (𝐺𝑦)))))

Theoremmgcf1 30700 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)    &   (𝜑𝑉 ∈ Proset )    &   (𝜑𝑊 ∈ Proset )    &   (𝜑𝐹𝐻𝐺)       (𝜑𝐹:𝐴𝐵)

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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 454 45301-45326
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