HomeTheorem List (p. 432 of 500)
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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | dvdsacongtr 43101 | Alternating congruence passes from a base to a dividing base. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
| ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ) ∧ (𝐷 ∥ 𝐴 ∧ (𝐴 ∥ (𝐵 − 𝐶) ∨ 𝐴 ∥ (𝐵 − -𝐶)))) → (𝐷 ∥ (𝐵 − 𝐶) ∨ 𝐷 ∥ (𝐵 − -𝐶))) | ||
| Theorem | coprmdvdsb 43102 | Multiplication by a coprime number does not affect divisibility. (Contributed by Stefan O'Rear, 23-Sep-2014.) |
| ⊢ ((𝐾 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑀 ∈ ℤ ∧ (𝐾 gcd 𝑀) = 1)) → (𝐾 ∥ 𝑁 ↔ 𝐾 ∥ (𝑀 · 𝑁))) | ||
| Theorem | modabsdifz 43103 | Divisibility in terms of modular reduction by the absolute value of the base. (Contributed by Stefan O'Rear, 26-Sep-2014.) |
| ⊢ ((𝑁 ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ 𝑀 ≠ 0) → ((𝑁 − (𝑁 mod (abs‘𝑀))) / 𝑀) ∈ ℤ) | ||
| Theorem | dvdsabsmod0 43104 | Divisibility in terms of modular reduction by the absolute value of the base. (Contributed by Stefan O'Rear, 24-Sep-2014.) (Proof shortened by OpenAI, 3-Jul-2020.) |
| ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≠ 0) → (𝑀 ∥ 𝑁 ↔ (𝑁 mod (abs‘𝑀)) = 0)) | ||
| Theorem | jm2.18 43105 | Theorem 2.18 of [JonesMatijasevic] p. 696. Direct relationship of the exponential function to X and Y sequences. (Contributed by Stefan O'Rear, 14-Oct-2014.) |
| ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝐾 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((((2 · 𝐴) · 𝐾) − (𝐾↑2)) − 1) ∥ (((𝐴 Xrm 𝑁) − ((𝐴 − 𝐾) · (𝐴 Yrm 𝑁))) − (𝐾↑𝑁))) | ||
| Theorem | jm2.19lem1 43106 | Lemma for jm2.19 43110. X and Y values are coprime. (Contributed by Stefan O'Rear, 23-Sep-2014.) |
| ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ) → ((𝐴 Xrm 𝑀) gcd (𝐴 Yrm 𝑀)) = 1) | ||
| Theorem | jm2.19lem2 43107 | Lemma for jm2.19 43110. (Contributed by Stefan O'Rear, 23-Sep-2014.) |
| ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐴 Yrm 𝑀) ∥ (𝐴 Yrm 𝑁) ↔ (𝐴 Yrm 𝑀) ∥ (𝐴 Yrm (𝑁 + 𝑀)))) | ||
| Theorem | jm2.19lem3 43108 | Lemma for jm2.19 43110. (Contributed by Stefan O'Rear, 26-Sep-2014.) |
| ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐼 ∈ ℕ0) → ((𝐴 Yrm 𝑀) ∥ (𝐴 Yrm 𝑁) ↔ (𝐴 Yrm 𝑀) ∥ (𝐴 Yrm (𝑁 + (𝐼 · 𝑀))))) | ||
| Theorem | jm2.19lem4 43109 | Lemma for jm2.19 43110. Extend to ZZ by symmetry. TODO: use zindbi 43063. (Contributed by Stefan O'Rear, 26-Sep-2014.) |
| ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐼 ∈ ℤ) → ((𝐴 Yrm 𝑀) ∥ (𝐴 Yrm 𝑁) ↔ (𝐴 Yrm 𝑀) ∥ (𝐴 Yrm (𝑁 + (𝐼 · 𝑀))))) | ||
| Theorem | jm2.19 43110 | Lemma 2.19 of [JonesMatijasevic] p. 696. Transfer divisibility constraints between Y-values and their indices. (Contributed by Stefan O'Rear, 24-Sep-2014.) |
| ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ 𝑁 ↔ (𝐴 Yrm 𝑀) ∥ (𝐴 Yrm 𝑁))) | ||
| Theorem | jm2.21 43111 | Lemma for jm2.20nn 43114. Express X and Y values as a binomial. (Contributed by Stefan O'Rear, 26-Sep-2014.) |
| ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ ∧ 𝐽 ∈ ℤ) → ((𝐴 Xrm (𝑁 · 𝐽)) + ((√‘((𝐴↑2) − 1)) · (𝐴 Yrm (𝑁 · 𝐽)))) = (((𝐴 Xrm 𝑁) + ((√‘((𝐴↑2) − 1)) · (𝐴 Yrm 𝑁)))↑𝐽)) | ||
| Theorem | jm2.22 43112* | Lemma for jm2.20nn 43114. Applying binomial theorem and taking irrational part. (Contributed by Stefan O'Rear, 26-Sep-2014.) (Revised by Stefan O'Rear, 6-May-2015.) |
| ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ ∧ 𝐽 ∈ ℕ0) → (𝐴 Yrm (𝑁 · 𝐽)) = Σ𝑖 ∈ {𝑥 ∈ (0...𝐽) ∣ ¬ 2 ∥ 𝑥} ((𝐽C𝑖) · (((𝐴 Xrm 𝑁)↑(𝐽 − 𝑖)) · (((𝐴 Yrm 𝑁)↑𝑖) · (((𝐴↑2) − 1)↑((𝑖 − 1) / 2)))))) | ||
| Theorem | jm2.23 43113 | Lemma for jm2.20nn 43114. Truncate binomial expansion p-adicly. (Contributed by Stefan O'Rear, 26-Sep-2014.) |
| ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ ∧ 𝐽 ∈ ℕ) → ((𝐴 Yrm 𝑁)↑3) ∥ ((𝐴 Yrm (𝑁 · 𝐽)) − (𝐽 · (((𝐴 Xrm 𝑁)↑(𝐽 − 1)) · (𝐴 Yrm 𝑁))))) | ||
| Theorem | jm2.20nn 43114 | Lemma 2.20 of [JonesMatijasevic] p. 696, the "first step down lemma". (Contributed by Stefan O'Rear, 27-Sep-2014.) |
| ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (((𝐴 Yrm 𝑁)↑2) ∥ (𝐴 Yrm 𝑀) ↔ (𝑁 · (𝐴 Yrm 𝑁)) ∥ 𝑀)) | ||
| Theorem | jm2.25lem1 43115 | Lemma for jm2.26 43119. (Contributed by Stefan O'Rear, 2-Oct-2014.) |
| ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ) ∧ (𝐴 ∥ (𝐶 − 𝐷) ∨ 𝐴 ∥ (𝐶 − -𝐷))) → ((𝐴 ∥ (𝐷 − 𝐵) ∨ 𝐴 ∥ (𝐷 − -𝐵)) ↔ (𝐴 ∥ (𝐶 − 𝐵) ∨ 𝐴 ∥ (𝐶 − -𝐵)))) | ||
| Theorem | jm2.25 43116 | Lemma for jm2.26 43119. Remainders mod X(2n) are negaperiodic mod 2n. (Contributed by Stefan O'Rear, 2-Oct-2014.) |
| ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐼 ∈ ℤ) → ((𝐴 Xrm 𝑁) ∥ ((𝐴 Yrm (𝑀 + (𝐼 · (2 · 𝑁)))) − (𝐴 Yrm 𝑀)) ∨ (𝐴 Xrm 𝑁) ∥ ((𝐴 Yrm (𝑀 + (𝐼 · (2 · 𝑁)))) − -(𝐴 Yrm 𝑀)))) | ||
| Theorem | jm2.26a 43117 | Lemma for jm2.26 43119. Reverse direction is required to prove forward direction, so do it separately. Induction on difference between K and M, together with the addition formula fact that adding 2N only inverts sign. (Contributed by Stefan O'Rear, 2-Oct-2014.) |
| ⊢ (((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → (((2 · 𝑁) ∥ (𝐾 − 𝑀) ∨ (2 · 𝑁) ∥ (𝐾 − -𝑀)) → ((𝐴 Xrm 𝑁) ∥ ((𝐴 Yrm 𝐾) − (𝐴 Yrm 𝑀)) ∨ (𝐴 Xrm 𝑁) ∥ ((𝐴 Yrm 𝐾) − -(𝐴 Yrm 𝑀))))) | ||
| Theorem | jm2.26lem3 43118 | Lemma for jm2.26 43119. Use acongrep 43097 to find K', M' ~ K, M in [ 0,N ]. Thus Y(K') ~ Y(M') and both are small; K' = M' on pain of contradicting 2.24, so K ~ M. (Contributed by Stefan O'Rear, 3-Oct-2014.) |
| ⊢ (((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ) ∧ (𝐾 ∈ (0...𝑁) ∧ 𝑀 ∈ (0...𝑁)) ∧ ((𝐴 Xrm 𝑁) ∥ ((𝐴 Yrm 𝐾) − (𝐴 Yrm 𝑀)) ∨ (𝐴 Xrm 𝑁) ∥ ((𝐴 Yrm 𝐾) − -(𝐴 Yrm 𝑀)))) → 𝐾 = 𝑀) | ||
| Theorem | jm2.26 43119 | Lemma 2.26 of [JonesMatijasevic] p. 697, the "second step down lemma". (Contributed by Stefan O'Rear, 2-Oct-2014.) |
| ⊢ (((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ) ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → (((𝐴 Xrm 𝑁) ∥ ((𝐴 Yrm 𝐾) − (𝐴 Yrm 𝑀)) ∨ (𝐴 Xrm 𝑁) ∥ ((𝐴 Yrm 𝐾) − -(𝐴 Yrm 𝑀))) ↔ ((2 · 𝑁) ∥ (𝐾 − 𝑀) ∨ (2 · 𝑁) ∥ (𝐾 − -𝑀)))) | ||
| Theorem | jm2.15nn0 43120 | Lemma 2.15 of [JonesMatijasevic] p. 695. Yrm is a polynomial for fixed N, so has the expected congruence property. (Contributed by Stefan O'Rear, 1-Oct-2014.) |
| ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → (𝐴 − 𝐵) ∥ ((𝐴 Yrm 𝑁) − (𝐵 Yrm 𝑁))) | ||
| Theorem | jm2.16nn0 43121 | Lemma 2.16 of [JonesMatijasevic] p. 695. This may be regarded as a special case of jm2.15nn0 43120 if Yrm is redefined as described in rmyluc 43054. (Contributed by Stefan O'Rear, 1-Oct-2014.) |
| ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → (𝐴 − 1) ∥ ((𝐴 Yrm 𝑁) − 𝑁)) | ||
| Theorem | jm2.27a 43122 | Lemma for jm2.27 43125. Reverse direction after existential quantifiers are expanded. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
| ⊢ (𝜑 → 𝐴 ∈ (ℤ≥‘2)) & ⊢ (𝜑 → 𝐵 ∈ ℕ) & ⊢ (𝜑 → 𝐶 ∈ ℕ) & ⊢ (𝜑 → 𝐷 ∈ ℕ0) & ⊢ (𝜑 → 𝐸 ∈ ℕ0) & ⊢ (𝜑 → 𝐹 ∈ ℕ0) & ⊢ (𝜑 → 𝐺 ∈ ℕ0) & ⊢ (𝜑 → 𝐻 ∈ ℕ0) & ⊢ (𝜑 → 𝐼 ∈ ℕ0) & ⊢ (𝜑 → 𝐽 ∈ ℕ0) & ⊢ (𝜑 → ((𝐷↑2) − (((𝐴↑2) − 1) · (𝐶↑2))) = 1) & ⊢ (𝜑 → ((𝐹↑2) − (((𝐴↑2) − 1) · (𝐸↑2))) = 1) & ⊢ (𝜑 → 𝐺 ∈ (ℤ≥‘2)) & ⊢ (𝜑 → ((𝐼↑2) − (((𝐺↑2) − 1) · (𝐻↑2))) = 1) & ⊢ (𝜑 → 𝐸 = ((𝐽 + 1) · (2 · (𝐶↑2)))) & ⊢ (𝜑 → 𝐹 ∥ (𝐺 − 𝐴)) & ⊢ (𝜑 → (2 · 𝐶) ∥ (𝐺 − 1)) & ⊢ (𝜑 → 𝐹 ∥ (𝐻 − 𝐶)) & ⊢ (𝜑 → (2 · 𝐶) ∥ (𝐻 − 𝐵)) & ⊢ (𝜑 → 𝐵 ≤ 𝐶) & ⊢ (𝜑 → 𝑃 ∈ ℤ) & ⊢ (𝜑 → 𝐷 = (𝐴 Xrm 𝑃)) & ⊢ (𝜑 → 𝐶 = (𝐴 Yrm 𝑃)) & ⊢ (𝜑 → 𝑄 ∈ ℤ) & ⊢ (𝜑 → 𝐹 = (𝐴 Xrm 𝑄)) & ⊢ (𝜑 → 𝐸 = (𝐴 Yrm 𝑄)) & ⊢ (𝜑 → 𝑅 ∈ ℤ) & ⊢ (𝜑 → 𝐼 = (𝐺 Xrm 𝑅)) & ⊢ (𝜑 → 𝐻 = (𝐺 Yrm 𝑅)) ⇒ ⊢ (𝜑 → 𝐶 = (𝐴 Yrm 𝐵)) | ||
| Theorem | jm2.27b 43123 | Lemma for jm2.27 43125. Expand existential quantifiers for reverse direction. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
| ⊢ (𝜑 → 𝐴 ∈ (ℤ≥‘2)) & ⊢ (𝜑 → 𝐵 ∈ ℕ) & ⊢ (𝜑 → 𝐶 ∈ ℕ) & ⊢ (𝜑 → 𝐷 ∈ ℕ0) & ⊢ (𝜑 → 𝐸 ∈ ℕ0) & ⊢ (𝜑 → 𝐹 ∈ ℕ0) & ⊢ (𝜑 → 𝐺 ∈ ℕ0) & ⊢ (𝜑 → 𝐻 ∈ ℕ0) & ⊢ (𝜑 → 𝐼 ∈ ℕ0) & ⊢ (𝜑 → 𝐽 ∈ ℕ0) & ⊢ (𝜑 → ((𝐷↑2) − (((𝐴↑2) − 1) · (𝐶↑2))) = 1) & ⊢ (𝜑 → ((𝐹↑2) − (((𝐴↑2) − 1) · (𝐸↑2))) = 1) & ⊢ (𝜑 → 𝐺 ∈ (ℤ≥‘2)) & ⊢ (𝜑 → ((𝐼↑2) − (((𝐺↑2) − 1) · (𝐻↑2))) = 1) & ⊢ (𝜑 → 𝐸 = ((𝐽 + 1) · (2 · (𝐶↑2)))) & ⊢ (𝜑 → 𝐹 ∥ (𝐺 − 𝐴)) & ⊢ (𝜑 → (2 · 𝐶) ∥ (𝐺 − 1)) & ⊢ (𝜑 → 𝐹 ∥ (𝐻 − 𝐶)) & ⊢ (𝜑 → (2 · 𝐶) ∥ (𝐻 − 𝐵)) & ⊢ (𝜑 → 𝐵 ≤ 𝐶) ⇒ ⊢ (𝜑 → 𝐶 = (𝐴 Yrm 𝐵)) | ||
| Theorem | jm2.27c 43124 | Lemma for jm2.27 43125. Forward direction with substitutions. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
| ⊢ (𝜑 → 𝐴 ∈ (ℤ≥‘2)) & ⊢ (𝜑 → 𝐵 ∈ ℕ) & ⊢ (𝜑 → 𝐶 ∈ ℕ) & ⊢ (𝜑 → 𝐶 = (𝐴 Yrm 𝐵)) & ⊢ 𝐷 = (𝐴 Xrm 𝐵) & ⊢ 𝑄 = (𝐵 · (𝐴 Yrm 𝐵)) & ⊢ 𝐸 = (𝐴 Yrm (2 · 𝑄)) & ⊢ 𝐹 = (𝐴 Xrm (2 · 𝑄)) & ⊢ 𝐺 = (𝐴 + ((𝐹↑2) · ((𝐹↑2) − 𝐴))) & ⊢ 𝐻 = (𝐺 Yrm 𝐵) & ⊢ 𝐼 = (𝐺 Xrm 𝐵) & ⊢ 𝐽 = ((𝐸 / (2 · (𝐶↑2))) − 1) ⇒ ⊢ (𝜑 → (((𝐷 ∈ ℕ0 ∧ 𝐸 ∈ ℕ0 ∧ 𝐹 ∈ ℕ0) ∧ (𝐺 ∈ ℕ0 ∧ 𝐻 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ (𝐽 ∈ ℕ0 ∧ (((((𝐷↑2) − (((𝐴↑2) − 1) · (𝐶↑2))) = 1 ∧ ((𝐹↑2) − (((𝐴↑2) − 1) · (𝐸↑2))) = 1 ∧ 𝐺 ∈ (ℤ≥‘2)) ∧ (((𝐼↑2) − (((𝐺↑2) − 1) · (𝐻↑2))) = 1 ∧ 𝐸 = ((𝐽 + 1) · (2 · (𝐶↑2))) ∧ 𝐹 ∥ (𝐺 − 𝐴))) ∧ (((2 · 𝐶) ∥ (𝐺 − 1) ∧ 𝐹 ∥ (𝐻 − 𝐶)) ∧ ((2 · 𝐶) ∥ (𝐻 − 𝐵) ∧ 𝐵 ≤ 𝐶)))))) | ||
| Theorem | jm2.27 43125* | Lemma 2.27 of [JonesMatijasevic] p. 697; rmY is a diophantine relation. 0 was excluded from the range of B and the lower limit of G was imposed because the source proof does not seem to work otherwise; quite possible I'm just missing something. The source proof uses both i and I; i has been changed to j to avoid collision. This theorem is basically nothing but substitution instances, all the work is done in jm2.27a 43122 and jm2.27c 43124. Once Diophantine relations have been defined, the content of the theorem is "rmY is Diophantine". (Contributed by Stefan O'Rear, 4-Oct-2014.) |
| ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 = (𝐴 Yrm 𝐵) ↔ ∃𝑑 ∈ ℕ0 ∃𝑒 ∈ ℕ0 ∃𝑓 ∈ ℕ0 ∃𝑔 ∈ ℕ0 ∃ℎ ∈ ℕ0 ∃𝑖 ∈ ℕ0 ∃𝑗 ∈ ℕ0 (((((𝑑↑2) − (((𝐴↑2) − 1) · (𝐶↑2))) = 1 ∧ ((𝑓↑2) − (((𝐴↑2) − 1) · (𝑒↑2))) = 1 ∧ 𝑔 ∈ (ℤ≥‘2)) ∧ (((𝑖↑2) − (((𝑔↑2) − 1) · (ℎ↑2))) = 1 ∧ 𝑒 = ((𝑗 + 1) · (2 · (𝐶↑2))) ∧ 𝑓 ∥ (𝑔 − 𝐴))) ∧ (((2 · 𝐶) ∥ (𝑔 − 1) ∧ 𝑓 ∥ (ℎ − 𝐶)) ∧ ((2 · 𝐶) ∥ (ℎ − 𝐵) ∧ 𝐵 ≤ 𝐶))))) | ||
| Theorem | jm2.27dlem1 43126* | Lemma for rmydioph 43131. Substitution of a tuple restriction into a projection that doesn't care. (Contributed by Stefan O'Rear, 11-Oct-2014.) |
| ⊢ 𝐴 ∈ (1...𝐵) ⇒ ⊢ (𝑎 = (𝑏 ↾ (1...𝐵)) → (𝑎‘𝐴) = (𝑏‘𝐴)) | ||
| Theorem | jm2.27dlem2 43127 | Lemma for rmydioph 43131. This theorem is used along with the next three to efficiently infer steps like 7 ∈ (1...;10). (Contributed by Stefan O'Rear, 11-Oct-2014.) |
| ⊢ 𝐴 ∈ (1...𝐵) & ⊢ 𝐶 = (𝐵 + 1) & ⊢ 𝐵 ∈ ℕ ⇒ ⊢ 𝐴 ∈ (1...𝐶) | ||
| Theorem | jm2.27dlem3 43128 | Lemma for rmydioph 43131. Infer membership of the endpoint of a range. (Contributed by Stefan O'Rear, 11-Oct-2014.) |
| ⊢ 𝐴 ∈ ℕ ⇒ ⊢ 𝐴 ∈ (1...𝐴) | ||
| Theorem | jm2.27dlem4 43129 | Lemma for rmydioph 43131. Infer ℕ-hood of large numbers. (Contributed by Stefan O'Rear, 11-Oct-2014.) |
| ⊢ 𝐴 ∈ ℕ & ⊢ 𝐵 = (𝐴 + 1) ⇒ ⊢ 𝐵 ∈ ℕ | ||
| Theorem | jm2.27dlem5 43130 | Lemma for rmydioph 43131. Used with sselii 3927 to infer membership of midpoints of range; jm2.27dlem2 43127 is deprecated. (Contributed by Stefan O'Rear, 11-Oct-2014.) |
| ⊢ 𝐵 = (𝐴 + 1) & ⊢ (1...𝐵) ⊆ (1...𝐶) ⇒ ⊢ (1...𝐴) ⊆ (1...𝐶) | ||
| Theorem | rmydioph 43131 | jm2.27 43125 restated in terms of Diophantine sets. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.) |
| ⊢ {𝑎 ∈ (ℕ0 ↑m (1...3)) ∣ ((𝑎‘1) ∈ (ℤ≥‘2) ∧ (𝑎‘3) = ((𝑎‘1) Yrm (𝑎‘2)))} ∈ (Dioph‘3) | ||
| Theorem | rmxdiophlem 43132* | X can be expressed in terms of Y, so it is also Diophantine. (Contributed by Stefan O'Rear, 15-Oct-2014.) |
| ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ ℕ0) → (𝑋 = (𝐴 Xrm 𝑁) ↔ ∃𝑦 ∈ ℕ0 (𝑦 = (𝐴 Yrm 𝑁) ∧ ((𝑋↑2) − (((𝐴↑2) − 1) · (𝑦↑2))) = 1))) | ||
| Theorem | rmxdioph 43133 | X is a Diophantine function. (Contributed by Stefan O'Rear, 17-Oct-2014.) |
| ⊢ {𝑎 ∈ (ℕ0 ↑m (1...3)) ∣ ((𝑎‘1) ∈ (ℤ≥‘2) ∧ (𝑎‘3) = ((𝑎‘1) Xrm (𝑎‘2)))} ∈ (Dioph‘3) | ||
| Theorem | jm3.1lem1 43134 | Lemma for jm3.1 43137. (Contributed by Stefan O'Rear, 16-Oct-2014.) |
| ⊢ (𝜑 → 𝐴 ∈ (ℤ≥‘2)) & ⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘2)) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → (𝐾 Yrm (𝑁 + 1)) ≤ 𝐴) ⇒ ⊢ (𝜑 → (𝐾↑𝑁) < 𝐴) | ||
| Theorem | jm3.1lem2 43135 | Lemma for jm3.1 43137. (Contributed by Stefan O'Rear, 16-Oct-2014.) |
| ⊢ (𝜑 → 𝐴 ∈ (ℤ≥‘2)) & ⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘2)) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → (𝐾 Yrm (𝑁 + 1)) ≤ 𝐴) ⇒ ⊢ (𝜑 → (𝐾↑𝑁) < ((((2 · 𝐴) · 𝐾) − (𝐾↑2)) − 1)) | ||
| Theorem | jm3.1lem3 43136 | Lemma for jm3.1 43137. (Contributed by Stefan O'Rear, 17-Oct-2014.) |
| ⊢ (𝜑 → 𝐴 ∈ (ℤ≥‘2)) & ⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘2)) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → (𝐾 Yrm (𝑁 + 1)) ≤ 𝐴) ⇒ ⊢ (𝜑 → ((((2 · 𝐴) · 𝐾) − (𝐾↑2)) − 1) ∈ ℕ) | ||
| Theorem | jm3.1 43137 | Diophantine expression for exponentiation. Lemma 3.1 of [JonesMatijasevic] p. 698. (Contributed by Stefan O'Rear, 16-Oct-2014.) |
| ⊢ (((𝐴 ∈ (ℤ≥‘2) ∧ 𝐾 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ) ∧ (𝐾 Yrm (𝑁 + 1)) ≤ 𝐴) → (𝐾↑𝑁) = (((𝐴 Xrm 𝑁) − ((𝐴 − 𝐾) · (𝐴 Yrm 𝑁))) mod ((((2 · 𝐴) · 𝐾) − (𝐾↑2)) − 1))) | ||
| Theorem | expdiophlem1 43138* | Lemma for expdioph 43140. Fully expanded expression for exponential. (Contributed by Stefan O'Rear, 17-Oct-2014.) |
| ⊢ (𝐶 ∈ ℕ0 → (((𝐴 ∈ (ℤ≥‘2) ∧ 𝐵 ∈ ℕ) ∧ 𝐶 = (𝐴↑𝐵)) ↔ ∃𝑑 ∈ ℕ0 ∃𝑒 ∈ ℕ0 ∃𝑓 ∈ ℕ0 ((𝐴 ∈ (ℤ≥‘2) ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑑 = (𝐴 Yrm (𝐵 + 1))) ∧ ((𝑑 ∈ (ℤ≥‘2) ∧ 𝑒 = (𝑑 Yrm 𝐵)) ∧ ((𝑑 ∈ (ℤ≥‘2) ∧ 𝑓 = (𝑑 Xrm 𝐵)) ∧ (𝐶 < ((((2 · 𝑑) · 𝐴) − (𝐴↑2)) − 1) ∧ ((((2 · 𝑑) · 𝐴) − (𝐴↑2)) − 1) ∥ ((𝑓 − ((𝑑 − 𝐴) · 𝑒)) − 𝐶)))))))) | ||
| Theorem | expdiophlem2 43139 | Lemma for expdioph 43140. Exponentiation on a restricted domain is Diophantine. (Contributed by Stefan O'Rear, 17-Oct-2014.) |
| ⊢ {𝑎 ∈ (ℕ0 ↑m (1...3)) ∣ (((𝑎‘1) ∈ (ℤ≥‘2) ∧ (𝑎‘2) ∈ ℕ) ∧ (𝑎‘3) = ((𝑎‘1)↑(𝑎‘2)))} ∈ (Dioph‘3) | ||
| Theorem | expdioph 43140 | The exponential function is Diophantine. This result completes and encapsulates our development using Pell equation solution sequences and is sometimes regarded as Matiyasevich's theorem properly. (Contributed by Stefan O'Rear, 17-Oct-2014.) |
| ⊢ {𝑎 ∈ (ℕ0 ↑m (1...3)) ∣ (𝑎‘3) = ((𝑎‘1)↑(𝑎‘2))} ∈ (Dioph‘3) | ||
| Theorem | setindtr 43141* | Set induction for sets contained in a transitive set. If we are allowed to assume Infinity, then all sets have a transitive closure and this reduces to setind 9644; however, this version is useful without Infinity. (Contributed by Stefan O'Rear, 28-Oct-2014.) |
| ⊢ (∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) → (∃𝑦(Tr 𝑦 ∧ 𝐵 ∈ 𝑦) → 𝐵 ∈ 𝐴)) | ||
| Theorem | setindtrs 43142* | Set induction scheme without Infinity. See comments at setindtr 43141. (Contributed by Stefan O'Rear, 28-Oct-2014.) |
| ⊢ (∀𝑦 ∈ 𝑥 𝜓 → 𝜑) & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒)) ⇒ ⊢ (∃𝑧(Tr 𝑧 ∧ 𝐵 ∈ 𝑧) → 𝜒) | ||
| Theorem | dford3lem1 43143* | Lemma for dford3 43145. (Contributed by Stefan O'Rear, 28-Oct-2014.) |
| ⊢ ((Tr 𝑁 ∧ ∀𝑦 ∈ 𝑁 Tr 𝑦) → ∀𝑏 ∈ 𝑁 (Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦)) | ||
| Theorem | dford3lem2 43144* | Lemma for dford3 43145. (Contributed by Stefan O'Rear, 28-Oct-2014.) |
| ⊢ ((Tr 𝑥 ∧ ∀𝑦 ∈ 𝑥 Tr 𝑦) → 𝑥 ∈ On) | ||
| Theorem | dford3 43145* | Ordinals are precisely the hereditarily transitive classes. Definition 1.2 of [Schloeder] p. 1. (Contributed by Stefan O'Rear, 28-Oct-2014.) |
| ⊢ (Ord 𝑁 ↔ (Tr 𝑁 ∧ ∀𝑥 ∈ 𝑁 Tr 𝑥)) | ||
| Theorem | dford4 43146* | dford3 43145 expressed in primitives to demonstrate shortness. (Contributed by Stefan O'Rear, 28-Oct-2014.) |
| ⊢ (Ord 𝑁 ↔ ∀𝑎∀𝑏∀𝑐((𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑎) → (𝑏 ∈ 𝑁 ∧ (𝑐 ∈ 𝑏 → 𝑐 ∈ 𝑎)))) | ||
| Theorem | wopprc 43147 | Unrelated: Wiener pairs treat proper classes symmetrically. (Contributed by Stefan O'Rear, 19-Sep-2014.) |
| ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ ¬ 1o ∈ {{{𝐴}, ∅}, {{𝐵}}}) | ||
| Theorem | rpnnen3lem 43148* | Lemma for rpnnen3 43149. (Contributed by Stefan O'Rear, 18-Jan-2015.) |
| ⊢ (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) ∧ 𝑎 < 𝑏) → {𝑐 ∈ ℚ ∣ 𝑐 < 𝑎} ≠ {𝑐 ∈ ℚ ∣ 𝑐 < 𝑏}) | ||
| Theorem | rpnnen3 43149 | Dedekind cut injection of ℝ into 𝒫 ℚ. (Contributed by Stefan O'Rear, 18-Jan-2015.) |
| ⊢ ℝ ≼ 𝒫 ℚ | ||
| Theorem | axac10 43150 | Characterization of choice similar to dffin1-5 10286. (Contributed by Stefan O'Rear, 6-Jan-2015.) |
| ⊢ ( ≈ “ On) = V | ||
| Theorem | harinf 43151 | The Hartogs number of an infinite set is at least ω. MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.) |
| ⊢ ((𝑆 ∈ 𝑉 ∧ ¬ 𝑆 ∈ Fin) → ω ⊆ (har‘𝑆)) | ||
| Theorem | wdom2d2 43152* | Deduction for weak dominance by a Cartesian product. MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.) |
| ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ (𝜑 → 𝐶 ∈ 𝑋) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑥 = 𝑋) ⇒ ⊢ (𝜑 → 𝐴 ≼* (𝐵 × 𝐶)) | ||
| Theorem | ttac 43153 | Tarski's theorem about choice: infxpidm 10460 is equivalent to ax-ac 10357. (Contributed by Stefan O'Rear, 4-Nov-2014.) (Proof shortened by Stefan O'Rear, 10-Jul-2015.) |
| ⊢ (CHOICE ↔ ∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐)) | ||
| Theorem | pw2f1ocnv 43154* | Define a bijection between characteristic functions and subsets. EDITORIAL: extracted from pw2en 9004, which can be easily reproved in terms of this. (Contributed by Stefan O'Rear, 18-Jan-2015.) (Revised by Stefan O'Rear, 9-Jul-2015.) |
| ⊢ 𝐹 = (𝑥 ∈ (2o ↑m 𝐴) ↦ (◡𝑥 “ {1o})) ⇒ ⊢ (𝐴 ∈ 𝑉 → (𝐹:(2o ↑m 𝐴)–1-1-onto→𝒫 𝐴 ∧ ◡𝐹 = (𝑦 ∈ 𝒫 𝐴 ↦ (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1o, ∅))))) | ||
| Theorem | pw2f1o2 43155* | Define a bijection between characteristic functions and subsets. EDITORIAL: extracted from pw2en 9004, which can be easily reproved in terms of this. (Contributed by Stefan O'Rear, 18-Jan-2015.) |
| ⊢ 𝐹 = (𝑥 ∈ (2o ↑m 𝐴) ↦ (◡𝑥 “ {1o})) ⇒ ⊢ (𝐴 ∈ 𝑉 → 𝐹:(2o ↑m 𝐴)–1-1-onto→𝒫 𝐴) | ||
| Theorem | pw2f1o2val 43156* | Function value of the pw2f1o2 43155 bijection. (Contributed by Stefan O'Rear, 18-Jan-2015.) (Revised by Stefan O'Rear, 6-May-2015.) |
| ⊢ 𝐹 = (𝑥 ∈ (2o ↑m 𝐴) ↦ (◡𝑥 “ {1o})) ⇒ ⊢ (𝑋 ∈ (2o ↑m 𝐴) → (𝐹‘𝑋) = (◡𝑋 “ {1o})) | ||
| Theorem | pw2f1o2val2 43157* | Membership in a mapped set under the pw2f1o2 43155 bijection. (Contributed by Stefan O'Rear, 18-Jan-2015.) (Revised by Stefan O'Rear, 6-May-2015.) |
| ⊢ 𝐹 = (𝑥 ∈ (2o ↑m 𝐴) ↦ (◡𝑥 “ {1o})) ⇒ ⊢ ((𝑋 ∈ (2o ↑m 𝐴) ∧ 𝑌 ∈ 𝐴) → (𝑌 ∈ (𝐹‘𝑋) ↔ (𝑋‘𝑌) = 1o)) | ||
| Theorem | limsuc2 43158 | Limit ordinals in the sense inclusive of zero contain all successors of their members. (Contributed by Stefan O'Rear, 20-Jan-2015.) |
| ⊢ ((Ord 𝐴 ∧ 𝐴 = ∪ 𝐴) → (𝐵 ∈ 𝐴 ↔ suc 𝐵 ∈ 𝐴)) | ||
| Theorem | wepwsolem 43159* | Transfer an ordering on characteristic functions by isomorphism to the power set. (Contributed by Stefan O'Rear, 18-Jan-2015.) |
| ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 ((𝑧 ∈ 𝑦 ∧ ¬ 𝑧 ∈ 𝑥) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑤 ∈ 𝑥 ↔ 𝑤 ∈ 𝑦)))} & ⊢ 𝑈 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧) E (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))} & ⊢ 𝐹 = (𝑎 ∈ (2o ↑m 𝐴) ↦ (◡𝑎 “ {1o})) ⇒ ⊢ (𝐴 ∈ V → 𝐹 Isom 𝑈, 𝑇 ((2o ↑m 𝐴), 𝒫 𝐴)) | ||
| Theorem | wepwso 43160* | A well-ordering induces a strict ordering on the power set. EDITORIAL: when well-orderings are set like, this can be strengthened to remove 𝐴 ∈ 𝑉. (Contributed by Stefan O'Rear, 18-Jan-2015.) |
| ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 ((𝑧 ∈ 𝑦 ∧ ¬ 𝑧 ∈ 𝑥) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑤 ∈ 𝑥 ↔ 𝑤 ∈ 𝑦)))} ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 We 𝐴) → 𝑇 Or 𝒫 𝐴) | ||
| Theorem | dnnumch1 43161* | Define an enumeration of a set from a choice function; second part, it restricts to a bijection. EDITORIAL: overlaps dfac8a 9928. (Contributed by Stefan O'Rear, 18-Jan-2015.) |
| ⊢ 𝐹 = recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧)))) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺‘𝑦) ∈ 𝑦)) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ On (𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴) | ||
| Theorem | dnnumch2 43162* | Define an enumeration (weak dominance version) of a set from a choice function. (Contributed by Stefan O'Rear, 18-Jan-2015.) |
| ⊢ 𝐹 = recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧)))) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺‘𝑦) ∈ 𝑦)) ⇒ ⊢ (𝜑 → 𝐴 ⊆ ran 𝐹) | ||
| Theorem | dnnumch3lem 43163* | Value of the ordinal injection function. (Contributed by Stefan O'Rear, 18-Jan-2015.) |
| ⊢ 𝐹 = recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧)))) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺‘𝑦) ∈ 𝑦)) ⇒ ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤) = ∩ (◡𝐹 “ {𝑤})) | ||
| Theorem | dnnumch3 43164* | Define an injection from a set into the ordinals using a choice function. (Contributed by Stefan O'Rear, 18-Jan-2015.) |
| ⊢ 𝐹 = recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧)))) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺‘𝑦) ∈ 𝑦)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥})):𝐴–1-1→On) | ||
| Theorem | dnwech 43165* | Define a well-ordering from a choice function. (Contributed by Stefan O'Rear, 18-Jan-2015.) |
| ⊢ 𝐹 = recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧)))) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺‘𝑦) ∈ 𝑦)) & ⊢ 𝐻 = {⟨𝑣, 𝑤⟩ ∣ ∩ (◡𝐹 “ {𝑣}) ∈ ∩ (◡𝐹 “ {𝑤})} ⇒ ⊢ (𝜑 → 𝐻 We 𝐴) | ||
| Theorem | fnwe2val 43166* | Lemma for fnwe2 43170. Substitute variables. (Contributed by Stefan O'Rear, 19-Jan-2015.) |
| ⊢ (𝑧 = (𝐹‘𝑥) → 𝑆 = 𝑈) & ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝐹‘𝑥)𝑅(𝐹‘𝑦) ∨ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑈𝑦))} ⇒ ⊢ (𝑎𝑇𝑏 ↔ ((𝐹‘𝑎)𝑅(𝐹‘𝑏) ∨ ((𝐹‘𝑎) = (𝐹‘𝑏) ∧ 𝑎⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑏))) | ||
| Theorem | fnwe2lem1 43167* | Lemma for fnwe2 43170. Substitution in well-ordering hypothesis. (Contributed by Stefan O'Rear, 19-Jan-2015.) |
| ⊢ (𝑧 = (𝐹‘𝑥) → 𝑆 = 𝑈) & ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝐹‘𝑥)𝑅(𝐹‘𝑦) ∨ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑈𝑦))} & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑈 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑥)}) ⇒ ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ⦋(𝐹‘𝑎) / 𝑧⦌𝑆 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑎)}) | ||
| Theorem | fnwe2lem2 43168* | Lemma for fnwe2 43170. An element which is in a minimal fiber and minimal within its fiber is minimal globally; thus 𝑇 is well-founded. (Contributed by Stefan O'Rear, 19-Jan-2015.) |
| ⊢ (𝑧 = (𝐹‘𝑥) → 𝑆 = 𝑈) & ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝐹‘𝑥)𝑅(𝐹‘𝑦) ∨ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑈𝑦))} & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑈 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑥)}) & ⊢ (𝜑 → (𝐹 ↾ 𝐴):𝐴⟶𝐵) & ⊢ (𝜑 → 𝑅 We 𝐵) & ⊢ (𝜑 → 𝑎 ⊆ 𝐴) & ⊢ (𝜑 → 𝑎 ≠ ∅) ⇒ ⊢ (𝜑 → ∃𝑏 ∈ 𝑎 ∀𝑐 ∈ 𝑎 ¬ 𝑐𝑇𝑏) | ||
| Theorem | fnwe2lem3 43169* | Lemma for fnwe2 43170. Trichotomy. (Contributed by Stefan O'Rear, 19-Jan-2015.) |
| ⊢ (𝑧 = (𝐹‘𝑥) → 𝑆 = 𝑈) & ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝐹‘𝑥)𝑅(𝐹‘𝑦) ∨ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑈𝑦))} & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑈 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑥)}) & ⊢ (𝜑 → (𝐹 ↾ 𝐴):𝐴⟶𝐵) & ⊢ (𝜑 → 𝑅 We 𝐵) & ⊢ (𝜑 → 𝑎 ∈ 𝐴) & ⊢ (𝜑 → 𝑏 ∈ 𝐴) ⇒ ⊢ (𝜑 → (𝑎𝑇𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑇𝑎)) | ||
| Theorem | fnwe2 43170* | A well-ordering can be constructed on a partitioned set by patching together well-orderings on each partition using a well-ordering on the partitions themselves. Similar to fnwe 8068 but does not require the within-partition ordering to be globally well. (Contributed by Stefan O'Rear, 19-Jan-2015.) |
| ⊢ (𝑧 = (𝐹‘𝑥) → 𝑆 = 𝑈) & ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝐹‘𝑥)𝑅(𝐹‘𝑦) ∨ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑈𝑦))} & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑈 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑥)}) & ⊢ (𝜑 → (𝐹 ↾ 𝐴):𝐴⟶𝐵) & ⊢ (𝜑 → 𝑅 We 𝐵) ⇒ ⊢ (𝜑 → 𝑇 We 𝐴) | ||
| Theorem | aomclem1 43171* |
Lemma for dfac11 43179. This is the beginning of the proof that
multiple
choice is equivalent to choice. Our goal is to construct, by
transfinite recursion, a well-ordering of (𝑅1‘𝐴). In what
follows, 𝐴 is the index of the rank we wish to
well-order, 𝑧 is
the collection of well-orderings constructed so far, dom 𝑧 is
the
set of ordinal indices of constructed ranks i.e. the next rank to
construct, and 𝑦 is a postulated multiple-choice
function.
Successor case 1, define a simple ordering from the well-ordered predecessor. (Contributed by Stefan O'Rear, 18-Jan-2015.) |
| ⊢ 𝐵 = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ (𝑅1‘∪ dom 𝑧)((𝑐 ∈ 𝑏 ∧ ¬ 𝑐 ∈ 𝑎) ∧ ∀𝑑 ∈ (𝑅1‘∪ dom 𝑧)(𝑑(𝑧‘∪ dom 𝑧)𝑐 → (𝑑 ∈ 𝑎 ↔ 𝑑 ∈ 𝑏)))} & ⊢ (𝜑 → dom 𝑧 ∈ On) & ⊢ (𝜑 → dom 𝑧 = suc ∪ dom 𝑧) & ⊢ (𝜑 → ∀𝑎 ∈ dom 𝑧(𝑧‘𝑎) We (𝑅1‘𝑎)) ⇒ ⊢ (𝜑 → 𝐵 Or (𝑅1‘dom 𝑧)) | ||
| Theorem | aomclem2 43172* | Lemma for dfac11 43179. Successor case 2, a choice function for subsets of (𝑅1‘dom 𝑧). (Contributed by Stefan O'Rear, 18-Jan-2015.) |
| ⊢ 𝐵 = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ (𝑅1‘∪ dom 𝑧)((𝑐 ∈ 𝑏 ∧ ¬ 𝑐 ∈ 𝑎) ∧ ∀𝑑 ∈ (𝑅1‘∪ dom 𝑧)(𝑑(𝑧‘∪ dom 𝑧)𝑐 → (𝑑 ∈ 𝑎 ↔ 𝑑 ∈ 𝑏)))} & ⊢ 𝐶 = (𝑎 ∈ V ↦ sup((𝑦‘𝑎), (𝑅1‘dom 𝑧), 𝐵)) & ⊢ (𝜑 → dom 𝑧 ∈ On) & ⊢ (𝜑 → dom 𝑧 = suc ∪ dom 𝑧) & ⊢ (𝜑 → ∀𝑎 ∈ dom 𝑧(𝑧‘𝑎) We (𝑅1‘𝑎)) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → dom 𝑧 ⊆ 𝐴) & ⊢ (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1‘𝐴)(𝑎 ≠ ∅ → (𝑦‘𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅}))) ⇒ ⊢ (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1‘dom 𝑧)(𝑎 ≠ ∅ → (𝐶‘𝑎) ∈ 𝑎)) | ||
| Theorem | aomclem3 43173* | Lemma for dfac11 43179. Successor case 3, our required well-ordering. (Contributed by Stefan O'Rear, 19-Jan-2015.) |
| ⊢ 𝐵 = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ (𝑅1‘∪ dom 𝑧)((𝑐 ∈ 𝑏 ∧ ¬ 𝑐 ∈ 𝑎) ∧ ∀𝑑 ∈ (𝑅1‘∪ dom 𝑧)(𝑑(𝑧‘∪ dom 𝑧)𝑐 → (𝑑 ∈ 𝑎 ↔ 𝑑 ∈ 𝑏)))} & ⊢ 𝐶 = (𝑎 ∈ V ↦ sup((𝑦‘𝑎), (𝑅1‘dom 𝑧), 𝐵)) & ⊢ 𝐷 = recs((𝑎 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑎)))) & ⊢ 𝐸 = {⟨𝑎, 𝑏⟩ ∣ ∩ (◡𝐷 “ {𝑎}) ∈ ∩ (◡𝐷 “ {𝑏})} & ⊢ (𝜑 → dom 𝑧 ∈ On) & ⊢ (𝜑 → dom 𝑧 = suc ∪ dom 𝑧) & ⊢ (𝜑 → ∀𝑎 ∈ dom 𝑧(𝑧‘𝑎) We (𝑅1‘𝑎)) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → dom 𝑧 ⊆ 𝐴) & ⊢ (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1‘𝐴)(𝑎 ≠ ∅ → (𝑦‘𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅}))) ⇒ ⊢ (𝜑 → 𝐸 We (𝑅1‘dom 𝑧)) | ||
| Theorem | aomclem4 43174* | Lemma for dfac11 43179. Limit case. Patch together well-orderings constructed so far using fnwe2 43170 to cover the limit rank. (Contributed by Stefan O'Rear, 20-Jan-2015.) |
| ⊢ 𝐹 = {⟨𝑎, 𝑏⟩ ∣ ((rank‘𝑎) E (rank‘𝑏) ∨ ((rank‘𝑎) = (rank‘𝑏) ∧ 𝑎(𝑧‘suc (rank‘𝑎))𝑏))} & ⊢ (𝜑 → dom 𝑧 ∈ On) & ⊢ (𝜑 → dom 𝑧 = ∪ dom 𝑧) & ⊢ (𝜑 → ∀𝑎 ∈ dom 𝑧(𝑧‘𝑎) We (𝑅1‘𝑎)) ⇒ ⊢ (𝜑 → 𝐹 We (𝑅1‘dom 𝑧)) | ||
| Theorem | aomclem5 43175* | Lemma for dfac11 43179. Combine the successor case with the limit case. (Contributed by Stefan O'Rear, 20-Jan-2015.) |
| ⊢ 𝐵 = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ (𝑅1‘∪ dom 𝑧)((𝑐 ∈ 𝑏 ∧ ¬ 𝑐 ∈ 𝑎) ∧ ∀𝑑 ∈ (𝑅1‘∪ dom 𝑧)(𝑑(𝑧‘∪ dom 𝑧)𝑐 → (𝑑 ∈ 𝑎 ↔ 𝑑 ∈ 𝑏)))} & ⊢ 𝐶 = (𝑎 ∈ V ↦ sup((𝑦‘𝑎), (𝑅1‘dom 𝑧), 𝐵)) & ⊢ 𝐷 = recs((𝑎 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑎)))) & ⊢ 𝐸 = {⟨𝑎, 𝑏⟩ ∣ ∩ (◡𝐷 “ {𝑎}) ∈ ∩ (◡𝐷 “ {𝑏})} & ⊢ 𝐹 = {⟨𝑎, 𝑏⟩ ∣ ((rank‘𝑎) E (rank‘𝑏) ∨ ((rank‘𝑎) = (rank‘𝑏) ∧ 𝑎(𝑧‘suc (rank‘𝑎))𝑏))} & ⊢ 𝐺 = (if(dom 𝑧 = ∪ dom 𝑧, 𝐹, 𝐸) ∩ ((𝑅1‘dom 𝑧) × (𝑅1‘dom 𝑧))) & ⊢ (𝜑 → dom 𝑧 ∈ On) & ⊢ (𝜑 → ∀𝑎 ∈ dom 𝑧(𝑧‘𝑎) We (𝑅1‘𝑎)) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → dom 𝑧 ⊆ 𝐴) & ⊢ (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1‘𝐴)(𝑎 ≠ ∅ → (𝑦‘𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅}))) ⇒ ⊢ (𝜑 → 𝐺 We (𝑅1‘dom 𝑧)) | ||
| Theorem | aomclem6 43176* | Lemma for dfac11 43179. Transfinite induction, close over 𝑧. (Contributed by Stefan O'Rear, 20-Jan-2015.) |
| ⊢ 𝐵 = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ (𝑅1‘∪ dom 𝑧)((𝑐 ∈ 𝑏 ∧ ¬ 𝑐 ∈ 𝑎) ∧ ∀𝑑 ∈ (𝑅1‘∪ dom 𝑧)(𝑑(𝑧‘∪ dom 𝑧)𝑐 → (𝑑 ∈ 𝑎 ↔ 𝑑 ∈ 𝑏)))} & ⊢ 𝐶 = (𝑎 ∈ V ↦ sup((𝑦‘𝑎), (𝑅1‘dom 𝑧), 𝐵)) & ⊢ 𝐷 = recs((𝑎 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑎)))) & ⊢ 𝐸 = {⟨𝑎, 𝑏⟩ ∣ ∩ (◡𝐷 “ {𝑎}) ∈ ∩ (◡𝐷 “ {𝑏})} & ⊢ 𝐹 = {⟨𝑎, 𝑏⟩ ∣ ((rank‘𝑎) E (rank‘𝑏) ∨ ((rank‘𝑎) = (rank‘𝑏) ∧ 𝑎(𝑧‘suc (rank‘𝑎))𝑏))} & ⊢ 𝐺 = (if(dom 𝑧 = ∪ dom 𝑧, 𝐹, 𝐸) ∩ ((𝑅1‘dom 𝑧) × (𝑅1‘dom 𝑧))) & ⊢ 𝐻 = recs((𝑧 ∈ V ↦ 𝐺)) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1‘𝐴)(𝑎 ≠ ∅ → (𝑦‘𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅}))) ⇒ ⊢ (𝜑 → (𝐻‘𝐴) We (𝑅1‘𝐴)) | ||
| Theorem | aomclem7 43177* | Lemma for dfac11 43179. (𝑅1‘𝐴) is well-orderable. (Contributed by Stefan O'Rear, 20-Jan-2015.) |
| ⊢ 𝐵 = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ (𝑅1‘∪ dom 𝑧)((𝑐 ∈ 𝑏 ∧ ¬ 𝑐 ∈ 𝑎) ∧ ∀𝑑 ∈ (𝑅1‘∪ dom 𝑧)(𝑑(𝑧‘∪ dom 𝑧)𝑐 → (𝑑 ∈ 𝑎 ↔ 𝑑 ∈ 𝑏)))} & ⊢ 𝐶 = (𝑎 ∈ V ↦ sup((𝑦‘𝑎), (𝑅1‘dom 𝑧), 𝐵)) & ⊢ 𝐷 = recs((𝑎 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑎)))) & ⊢ 𝐸 = {⟨𝑎, 𝑏⟩ ∣ ∩ (◡𝐷 “ {𝑎}) ∈ ∩ (◡𝐷 “ {𝑏})} & ⊢ 𝐹 = {⟨𝑎, 𝑏⟩ ∣ ((rank‘𝑎) E (rank‘𝑏) ∨ ((rank‘𝑎) = (rank‘𝑏) ∧ 𝑎(𝑧‘suc (rank‘𝑎))𝑏))} & ⊢ 𝐺 = (if(dom 𝑧 = ∪ dom 𝑧, 𝐹, 𝐸) ∩ ((𝑅1‘dom 𝑧) × (𝑅1‘dom 𝑧))) & ⊢ 𝐻 = recs((𝑧 ∈ V ↦ 𝐺)) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1‘𝐴)(𝑎 ≠ ∅ → (𝑦‘𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅}))) ⇒ ⊢ (𝜑 → ∃𝑏 𝑏 We (𝑅1‘𝐴)) | ||
| Theorem | aomclem8 43178* | Lemma for dfac11 43179. Perform variable substitutions. This is the most we can say without invoking regularity. (Contributed by Stefan O'Rear, 20-Jan-2015.) |
| ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1‘𝐴)(𝑎 ≠ ∅ → (𝑦‘𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅}))) ⇒ ⊢ (𝜑 → ∃𝑏 𝑏 We (𝑅1‘𝐴)) | ||
| Theorem | dfac11 43179* |
The right-hand side of this theorem (compare with ac4 10373),
sometimes
known as the "axiom of multiple choice", is a choice
equivalent.
Curiously, this statement cannot be proved without ax-reg 9485, despite
not mentioning the cumulative hierarchy in any way as most consequences
of regularity do.
This is definition (MC) of [Schechter] p. 141. EDITORIAL: the proof is not original with me of course but I lost my reference sometime after writing it. A multiple choice function allows any total order to be extended to a choice function, which in turn defines a well-ordering. Since a well-ordering on a set defines a simple ordering of the power set, this allows the trivial well-ordering of the empty set to be transfinitely bootstrapped up the cumulative hierarchy to any desired level. (Contributed by Stefan O'Rear, 20-Jan-2015.) (Revised by Stefan O'Rear, 1-Jun-2015.) |
| ⊢ (CHOICE ↔ ∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅}))) | ||
| Theorem | kelac1 43180* | Kelley's choice, basic form: if a collection of sets can be cast as closed sets in the factors of a topology, and there is a definable element in each topology (which need not be in the closed set - if it were this would be trivial), then compactness (via finite intersection) guarantees that the final product is nonempty. (Contributed by Stefan O'Rear, 22-Feb-2015.) |
| ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑆 ≠ ∅) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐽 ∈ Top) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐶 ∈ (Clsd‘𝐽)) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐵:𝑆–1-1-onto→𝐶) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑈 ∈ ∪ 𝐽) & ⊢ (𝜑 → (∏t‘(𝑥 ∈ 𝐼 ↦ 𝐽)) ∈ Comp) ⇒ ⊢ (𝜑 → X𝑥 ∈ 𝐼 𝑆 ≠ ∅) | ||
| Theorem | kelac2lem 43181 | Lemma for kelac2 43182 and dfac21 43183: knob topologies are compact. (Contributed by Stefan O'Rear, 22-Feb-2015.) |
| ⊢ (𝑆 ∈ 𝑉 → (topGen‘{𝑆, {𝒫 ∪ 𝑆}}) ∈ Comp) | ||
| Theorem | kelac2 43182* | Kelley's choice, most common form: compactness of a product of knob topologies recovers choice. (Contributed by Stefan O'Rear, 22-Feb-2015.) |
| ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑆 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑆 ≠ ∅) & ⊢ (𝜑 → (∏t‘(𝑥 ∈ 𝐼 ↦ (topGen‘{𝑆, {𝒫 ∪ 𝑆}}))) ∈ Comp) ⇒ ⊢ (𝜑 → X𝑥 ∈ 𝐼 𝑆 ≠ ∅) | ||
| Theorem | dfac21 43183 | Tychonoff's theorem is a choice equivalent. Definition AC21 of Schechter p. 461. (Contributed by Stefan O'Rear, 22-Feb-2015.) (Revised by Mario Carneiro, 27-Aug-2015.) |
| ⊢ (CHOICE ↔ ∀𝑓(𝑓:dom 𝑓⟶Comp → (∏t‘𝑓) ∈ Comp)) | ||
| Syntax | clfig 43184 | Extend class notation with the class of finitely generated left modules. |
| class LFinGen | ||
| Definition | df-lfig 43185 | Define the class of finitely generated left modules. Finite generation of subspaces can be interpreted using ↾s. (Contributed by Stefan O'Rear, 1-Jan-2015.) |
| ⊢ LFinGen = {𝑤 ∈ LMod ∣ (Base‘𝑤) ∈ ((LSpan‘𝑤) “ (𝒫 (Base‘𝑤) ∩ Fin))} | ||
| Theorem | islmodfg 43186* | Property of a finitely generated left module. (Contributed by Stefan O'Rear, 1-Jan-2015.) |
| ⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) ⇒ ⊢ (𝑊 ∈ LMod → (𝑊 ∈ LFinGen ↔ ∃𝑏 ∈ 𝒫 𝐵(𝑏 ∈ Fin ∧ (𝑁‘𝑏) = 𝐵))) | ||
| Theorem | islssfg 43187* | Property of a finitely generated left (sub)module. (Contributed by Stefan O'Rear, 1-Jan-2015.) |
| ⊢ 𝑋 = (𝑊 ↾s 𝑈) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → (𝑋 ∈ LFinGen ↔ ∃𝑏 ∈ 𝒫 𝑈(𝑏 ∈ Fin ∧ (𝑁‘𝑏) = 𝑈))) | ||
| Theorem | islssfg2 43188* | Property of a finitely generated left (sub)module, with a relaxed constraint on the spanning vectors. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
| ⊢ 𝑋 = (𝑊 ↾s 𝑈) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ 𝐵 = (Base‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → (𝑋 ∈ LFinGen ↔ ∃𝑏 ∈ (𝒫 𝐵 ∩ Fin)(𝑁‘𝑏) = 𝑈)) | ||
| Theorem | islssfgi 43189 | Finitely spanned subspaces are finitely generated. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
| ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑋 = (𝑊 ↾s (𝑁‘𝐵)) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ⊆ 𝑉 ∧ 𝐵 ∈ Fin) → 𝑋 ∈ LFinGen) | ||
| Theorem | fglmod 43190 | Finitely generated left modules are left modules. (Contributed by Stefan O'Rear, 1-Jan-2015.) |
| ⊢ (𝑀 ∈ LFinGen → 𝑀 ∈ LMod) | ||
| Theorem | lsmfgcl 43191 | The sum of two finitely generated submodules is finitely generated. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
| ⊢ 𝑈 = (LSubSp‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝐷 = (𝑊 ↾s 𝐴) & ⊢ 𝐸 = (𝑊 ↾s 𝐵) & ⊢ 𝐹 = (𝑊 ↾s (𝐴 ⊕ 𝐵)) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → 𝐵 ∈ 𝑈) & ⊢ (𝜑 → 𝐷 ∈ LFinGen) & ⊢ (𝜑 → 𝐸 ∈ LFinGen) ⇒ ⊢ (𝜑 → 𝐹 ∈ LFinGen) | ||
| Syntax | clnm 43192 | Extend class notation with the class of Noetherian left modules. |
| class LNoeM | ||
| Definition | df-lnm 43193* | A left-module is Noetherian iff it is hereditarily finitely generated. (Contributed by Stefan O'Rear, 12-Dec-2014.) |
| ⊢ LNoeM = {𝑤 ∈ LMod ∣ ∀𝑖 ∈ (LSubSp‘𝑤)(𝑤 ↾s 𝑖) ∈ LFinGen} | ||
| Theorem | islnm 43194* | Property of being a Noetherian left module. (Contributed by Stefan O'Rear, 12-Dec-2014.) |
| ⊢ 𝑆 = (LSubSp‘𝑀) ⇒ ⊢ (𝑀 ∈ LNoeM ↔ (𝑀 ∈ LMod ∧ ∀𝑖 ∈ 𝑆 (𝑀 ↾s 𝑖) ∈ LFinGen)) | ||
| Theorem | islnm2 43195* | Property of being a Noetherian left module with finite generation expanded in terms of spans. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ 𝑆 = (LSubSp‘𝑀) & ⊢ 𝑁 = (LSpan‘𝑀) ⇒ ⊢ (𝑀 ∈ LNoeM ↔ (𝑀 ∈ LMod ∧ ∀𝑖 ∈ 𝑆 ∃𝑔 ∈ (𝒫 𝐵 ∩ Fin)𝑖 = (𝑁‘𝑔))) | ||
| Theorem | lnmlmod 43196 | A Noetherian left module is a left module. (Contributed by Stefan O'Rear, 12-Dec-2014.) |
| ⊢ (𝑀 ∈ LNoeM → 𝑀 ∈ LMod) | ||
| Theorem | lnmlssfg 43197 | A submodule of Noetherian module is finitely generated. (Contributed by Stefan O'Rear, 1-Jan-2015.) |
| ⊢ 𝑆 = (LSubSp‘𝑀) & ⊢ 𝑅 = (𝑀 ↾s 𝑈) ⇒ ⊢ ((𝑀 ∈ LNoeM ∧ 𝑈 ∈ 𝑆) → 𝑅 ∈ LFinGen) | ||
| Theorem | lnmlsslnm 43198 | All submodules of a Noetherian module are Noetherian. (Contributed by Stefan O'Rear, 1-Jan-2015.) |
| ⊢ 𝑆 = (LSubSp‘𝑀) & ⊢ 𝑅 = (𝑀 ↾s 𝑈) ⇒ ⊢ ((𝑀 ∈ LNoeM ∧ 𝑈 ∈ 𝑆) → 𝑅 ∈ LNoeM) | ||
| Theorem | lnmfg 43199 | A Noetherian left module is finitely generated. (Contributed by Stefan O'Rear, 12-Dec-2014.) |
| ⊢ (𝑀 ∈ LNoeM → 𝑀 ∈ LFinGen) | ||
| Theorem | kercvrlsm 43200 | The domain of a linear function is the subspace sum of the kernel and any subspace which covers the range. (Contributed by Stefan O'Rear, 24-Jan-2015.) (Revised by Stefan O'Rear, 6-May-2015.) |
| ⊢ 𝑈 = (LSubSp‘𝑆) & ⊢ ⊕ = (LSSum‘𝑆) & ⊢ 0 = (0g‘𝑇) & ⊢ 𝐾 = (◡𝐹 “ { 0 }) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ (𝜑 → 𝐹 ∈ (𝑆 LMHom 𝑇)) & ⊢ (𝜑 → 𝐷 ∈ 𝑈) & ⊢ (𝜑 → (𝐹 “ 𝐷) = ran 𝐹) ⇒ ⊢ (𝜑 → (𝐾 ⊕ 𝐷) = 𝐵) | ||
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