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Theorem List for Metamath Proof Explorer - 39301-39400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlcmineqlem10 39301* Induction step of lcmineqlem13 39304 (deduction form). (Contributed by metakunt, 12-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑀 < 𝑁)       (𝜑 → ∫(0[,]1)((𝑥↑((𝑀 + 1) − 1)) · ((1 − 𝑥)↑(𝑁 − (𝑀 + 1)))) d𝑥 = ((𝑀 / (𝑁𝑀)) · ∫(0[,]1)((𝑥↑(𝑀 − 1)) · ((1 − 𝑥)↑(𝑁𝑀))) d𝑥))

Theoremlcmineqlem11 39302 Induction step, continuation for binomial coefficients. (Contributed by metakunt, 12-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑀 < 𝑁)       (𝜑 → (1 / ((𝑀 + 1) · (𝑁C(𝑀 + 1)))) = ((𝑀 / (𝑁𝑀)) · (1 / (𝑀 · (𝑁C𝑀)))))

Theoremlcmineqlem12 39303* Base case for induction. (Contributed by metakunt, 12-May-2024.)
(𝜑𝑁 ∈ ℕ)       (𝜑 → ∫(0[,]1)((𝑡↑(1 − 1)) · ((1 − 𝑡)↑(𝑁 − 1))) d𝑡 = (1 / (1 · (𝑁C1))))

Theoremlcmineqlem13 39304* Induction proof for lcm integral. (Contributed by metakunt, 12-May-2024.)
𝐹 = ∫(0[,]1)((𝑥↑(𝑀 − 1)) · ((1 − 𝑥)↑(𝑁𝑀))) d𝑥    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑀𝑁)       (𝜑𝐹 = (1 / (𝑀 · (𝑁C𝑀))))

Theoremlcmineqlem14 39305 Technical lemma for inequality estimate. (Contributed by metakunt, 12-May-2024.)
(𝜑𝐴 ∈ ℕ)    &   (𝜑𝐵 ∈ ℕ)    &   (𝜑𝐶 ∈ ℕ)    &   (𝜑𝐷 ∈ ℕ)    &   (𝜑𝐸 ∈ ℕ)    &   (𝜑 → (𝐴 · 𝐶) ∥ 𝐷)    &   (𝜑 → (𝐵 · 𝐶) ∥ 𝐸)    &   (𝜑𝐷𝐸)    &   (𝜑 → (𝐴 gcd 𝐵) = 1)       (𝜑 → ((𝐴 · 𝐵) · 𝐶) ∥ 𝐸)

Theoremlcmineqlem15 39306* F times the least common multiple of 1 to n is a natural number. (Contributed by metakunt, 10-May-2024.)
𝐹 = ∫(0[,]1)((𝑥↑(𝑀 − 1)) · ((1 − 𝑥)↑(𝑁𝑀))) d𝑥    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑀𝑁)       (𝜑 → ((lcm‘(1...𝑁)) · 𝐹) ∈ ℕ)

Theoremlcmineqlem16 39307 Technical divisibility lemma. (Contributed by metakunt, 12-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑀𝑁)       (𝜑 → (𝑀 · (𝑁C𝑀)) ∥ (lcm‘(1...𝑁)))

Theoremlcmineqlem17 39308 Inequality of 2^{2n}. (Contributed by metakunt, 29-Apr-2024.)
(𝜑𝑁 ∈ ℕ0)       (𝜑 → (2↑(2 · 𝑁)) ≤ (((2 · 𝑁) + 1) · ((2 · 𝑁)C𝑁)))

Theoremlcmineqlem18 39309 Technical lemma to shift factors in binomial coefficient. (Contributed by metakunt, 12-May-2024.)
(𝜑𝑁 ∈ ℕ)       (𝜑 → ((𝑁 + 1) · (((2 · 𝑁) + 1)C(𝑁 + 1))) = (((2 · 𝑁) + 1) · ((2 · 𝑁)C𝑁)))

Theoremlcmineqlem19 39310 Dividing implies inequality for lcm inequality lemma. (Contributed by metakunt, 12-May-2024.)
(𝜑𝑁 ∈ ℕ)       (𝜑 → ((𝑁 · ((2 · 𝑁) + 1)) · ((2 · 𝑁)C𝑁)) ∥ (lcm‘(1...((2 · 𝑁) + 1))))

Theoremlcmineqlem20 39311 Inequality for lcm lemma. (Contributed by metakunt, 12-May-2024.)
(𝜑𝑁 ∈ ℕ)       (𝜑 → (𝑁 · (2↑(2 · 𝑁))) ≤ (lcm‘(1...((2 · 𝑁) + 1))))

Theoremlcmineqlem21 39312 The lcm inequality lemma without base cases 7 and 8. (Contributed by metakunt, 12-May-2024.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑 → 4 ≤ 𝑁)       (𝜑 → (2↑((2 · 𝑁) + 2)) ≤ (lcm‘(1...((2 · 𝑁) + 1))))

Theoremlcmineqlem22 39313 The lcm inequality lemma without base cases 7 and 8. (Contributed by metakunt, 12-May-2024.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑 → 4 ≤ 𝑁)       (𝜑 → ((2↑((2 · 𝑁) + 1)) ≤ (lcm‘(1...((2 · 𝑁) + 1))) ∧ (2↑((2 · 𝑁) + 2)) ≤ (lcm‘(1...((2 · 𝑁) + 2)))))

Theoremlcmineqlem23 39314 Penultimate step to the lcm inequality lemma. (Contributed by metakunt, 12-May-2024.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑 → 9 ≤ 𝑁)       (𝜑 → (2↑𝑁) ≤ (lcm‘(1...𝑁)))

Theoremlcmineqlem 39315 The least common multiple inequality lemma, a central result for future use. Theorem 3.1 from https://www3.nd.edu/%7eandyp/notes/AKS.pdf (Contributed by metakunt, 16-May-2024.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑 → 7 ≤ 𝑁)       (𝜑 → (2↑𝑁) ≤ (lcm‘(1...𝑁)))

20.25.4  Logarithm inequalities

Theorem3lexlogpow5ineq1 39316 First inequality in inequality chain, proposed by Mario Carneiro (Contributed by metakunt, 22-May-2024.)
7 < ((3 / 2)↑5)

Theorem3lexlogpow5ineq2 39317 Second inequality in inequality chain, proposed by Mario Carneiro. (Contributed by metakunt, 22-May-2024.)
(𝜑𝑋 ∈ ℝ)    &   (𝜑 → 3 ≤ 𝑋)       (𝜑 → ((3 / 2)↑5) ≤ ((2 logb 𝑋)↑5))

Theorem3lexlogpow5ineq3 39318 Combined inequality chain for a specific power of the binary logarithm, proposed by Mario Carneiro. (Contributed by metakunt, 22-May-2024.)
(𝜑𝑋 ∈ ℝ)    &   (𝜑 → 3 ≤ 𝑋)       (𝜑 → 7 < ((2 logb 𝑋)↑5))

20.25.5  Miscellaneous results for AKS formalisation

Theorem5bc2eq10 39319 The value of 5 choose 2. (Contributed by metakunt, 8-Jun-2024.)
(5C2) = 10

Theoremfacp2 39320 The factorial of a successor's successor. (Contributed by metakunt, 19-Apr-2024.)
(𝑁 ∈ ℕ0 → (!‘(𝑁 + 2)) = ((!‘𝑁) · ((𝑁 + 1) · (𝑁 + 2))))

Theorem2np3bcnp1 39321 Part of induction step for 2ap1caineq 39322. (Contributed by metakunt, 8-Jun-2024.)
(𝜑𝑁 ∈ ℕ0)       (𝜑 → (((2 · (𝑁 + 1)) + 1)C(𝑁 + 1)) = ((((2 · 𝑁) + 1)C𝑁) · (2 · (((2 · 𝑁) + 3) / (𝑁 + 2)))))

Theorem2ap1caineq 39322 Inequality for Theorem 6.6 for AKS. (Contributed by metakunt, 8-Jun-2024.)
(𝜑𝑁 ∈ ℤ)    &   (𝜑 → 2 ≤ 𝑁)       (𝜑 → (2↑(𝑁 + 1)) < (((2 · 𝑁) + 1)C𝑁))

20.25.6  Permutation results

Theoremmetakunt1 39323* A is an endomapping. (Contributed by metakunt, 23-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))))       (𝜑𝐴:(1...𝑀)⟶(1...𝑀))

Theoremmetakunt2 39324* A is an endomapping. (Contributed by metakunt, 23-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝑀, 𝐼, if(𝑥 < 𝐼, 𝑥, (𝑥 + 1))))       (𝜑𝐴:(1...𝑀)⟶(1...𝑀))

Theoremmetakunt3 39325* Value of A. (Contributed by metakunt, 23-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))))    &   (𝜑𝑋 ∈ (1...𝑀))       (𝜑 → (𝐴𝑋) = if(𝑋 = 𝐼, 𝑀, if(𝑋 < 𝐼, 𝑋, (𝑋 − 1))))

Theoremmetakunt4 39326* Value of A. (Contributed by metakunt, 23-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝑀, 𝐼, if(𝑥 < 𝐼, 𝑥, (𝑥 + 1))))    &   (𝜑𝑋 ∈ (1...𝑀))       (𝜑 → (𝐴𝑋) = if(𝑋 = 𝑀, 𝐼, if(𝑋 < 𝐼, 𝑋, (𝑋 + 1))))

Theoremmetakunt5 39327* C is the left inverse for A. (Contributed by metakunt, 24-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))))    &   𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))))    &   (𝜑𝑋 ∈ (1...𝑀))       ((𝜑𝑋 = 𝐼) → (𝐶‘(𝐴𝑋)) = 𝑋)

Theoremmetakunt6 39328* C is the left inverse for A. (Contributed by metakunt, 24-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))))    &   𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))))    &   (𝜑𝑋 ∈ (1...𝑀))       ((𝜑𝑋 < 𝐼) → (𝐶‘(𝐴𝑋)) = 𝑋)

Theoremmetakunt7 39329* C is the left inverse for A. (Contributed by metakunt, 24-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))))    &   𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))))    &   (𝜑𝑋 ∈ (1...𝑀))       ((𝜑𝐼 < 𝑋) → ((𝐴𝑋) = (𝑋 − 1) ∧ ¬ (𝐴𝑋) = 𝑀 ∧ ¬ (𝐴𝑋) < 𝐼))

Theoremmetakunt8 39330* C is the left inverse for A. (Contributed by metakunt, 24-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))))    &   𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))))    &   (𝜑𝑋 ∈ (1...𝑀))       ((𝜑𝐼 < 𝑋) → (𝐶‘(𝐴𝑋)) = 𝑋)

Theoremmetakunt9 39331* C is the left inverse for A. (Contributed by metakunt, 24-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))))    &   𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))))    &   (𝜑𝑋 ∈ (1...𝑀))       (𝜑 → (𝐶‘(𝐴𝑋)) = 𝑋)

Theoremmetakunt10 39332* C is the right inverse for A. (Contributed by metakunt, 24-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))))    &   𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))))    &   (𝜑𝑋 ∈ (1...𝑀))       ((𝜑𝑋 = 𝑀) → (𝐴‘(𝐶𝑋)) = 𝑋)

Theoremmetakunt11 39333* C is the right inverse for A. (Contributed by metakunt, 24-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))))    &   𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))))    &   (𝜑𝑋 ∈ (1...𝑀))       ((𝜑𝑋 < 𝐼) → (𝐴‘(𝐶𝑋)) = 𝑋)

Theoremmetakunt12 39334* C is the right inverse for A. (Contributed by metakunt, 25-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))))    &   𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))))    &   (𝜑𝑋 ∈ (1...𝑀))       ((𝜑 ∧ ¬ (𝑋 = 𝑀𝑋 < 𝐼)) → (𝐴‘(𝐶𝑋)) = 𝑋)

Theoremmetakunt13 39335* C is the right inverse for A. (Contributed by metakunt, 25-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))))    &   𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))))    &   (𝜑𝑋 ∈ (1...𝑀))       (𝜑 → (𝐴‘(𝐶𝑋)) = 𝑋)

Theoremmetakunt14 39336* A is a primitive permutation that moves the I-th element to the end and C is its inverse that moves the last element back to the I-th position. (Contributed by metakunt, 25-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))))    &   𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))))       (𝜑 → (𝐴:(1...𝑀)–1-1-onto→(1...𝑀) ∧ 𝐴 = 𝐶))

Theoremmetakunt15 39337* Construction of another permutation. (Contributed by metakunt, 25-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐹 = (𝑥 ∈ (1...(𝐼 − 1)) ↦ (𝑥 + (𝑀𝐼)))       (𝜑𝐹:(1...(𝐼 − 1))–1-1-onto→(((𝑀𝐼) + 1)...(𝑀 − 1)))

Theoremmetakunt16 39338* Construction of another permutation. (Contributed by metakunt, 25-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐹 = (𝑥 ∈ (𝐼...(𝑀 − 1)) ↦ (𝑥 + (1 − 𝐼)))       (𝜑𝐹:(𝐼...(𝑀 − 1))–1-1-onto→(1...(𝑀𝐼)))

Theoremmetakunt17 39339 The union of three disjoint bijections is a bijection. (Contributed by metakunt, 28-May-2024.)
(𝜑𝐺:𝐴1-1-onto𝑋)    &   (𝜑𝐻:𝐵1-1-onto𝑌)    &   (𝜑𝐼:𝐶1-1-onto𝑍)    &   (𝜑 → (𝐴𝐵) = ∅)    &   (𝜑 → (𝐴𝐶) = ∅)    &   (𝜑 → (𝐵𝐶) = ∅)    &   (𝜑 → (𝑋𝑌) = ∅)    &   (𝜑 → (𝑋𝑍) = ∅)    &   (𝜑 → (𝑌𝑍) = ∅)    &   (𝜑𝐹 = ((𝐺𝐻) ∪ 𝐼))    &   (𝜑𝐷 = ((𝐴𝐵) ∪ 𝐶))    &   (𝜑𝑊 = ((𝑋𝑌) ∪ 𝑍))       (𝜑𝐹:𝐷1-1-onto𝑊)

Theoremmetakunt18 39340 Disjoint domains and codomains. (Contributed by metakunt, 28-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)       (𝜑 → ((((1...(𝐼 − 1)) ∩ (𝐼...(𝑀 − 1))) = ∅ ∧ ((1...(𝐼 − 1)) ∩ {𝑀}) = ∅ ∧ ((𝐼...(𝑀 − 1)) ∩ {𝑀}) = ∅) ∧ (((((𝑀𝐼) + 1)...(𝑀 − 1)) ∩ (1...(𝑀𝐼))) = ∅ ∧ ((((𝑀𝐼) + 1)...(𝑀 − 1)) ∩ {𝑀}) = ∅ ∧ ((1...(𝑀𝐼)) ∩ {𝑀}) = ∅)))

Theoremmetakunt19 39341* Domains on restrictions of functions. (Contributed by metakunt, 28-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐵 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝑀, 𝑀, if(𝑥 < 𝐼, (𝑥 + (𝑀𝐼)), (𝑥 + (1 − 𝐼)))))    &   𝐶 = (𝑥 ∈ (1...(𝐼 − 1)) ↦ (𝑥 + (𝑀𝐼)))    &   𝐷 = (𝑥 ∈ (𝐼...(𝑀 − 1)) ↦ (𝑥 + (1 − 𝐼)))       (𝜑 → ((𝐶 Fn (1...(𝐼 − 1)) ∧ 𝐷 Fn (𝐼...(𝑀 − 1)) ∧ (𝐶𝐷) Fn ((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1)))) ∧ {⟨𝑀, 𝑀⟩} Fn {𝑀}))

Theoremmetakunt20 39342* Show that B coincides on the union of bijections of functions. (Contributed by metakunt, 28-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐵 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝑀, 𝑀, if(𝑥 < 𝐼, (𝑥 + (𝑀𝐼)), (𝑥 + (1 − 𝐼)))))    &   𝐶 = (𝑥 ∈ (1...(𝐼 − 1)) ↦ (𝑥 + (𝑀𝐼)))    &   𝐷 = (𝑥 ∈ (𝐼...(𝑀 − 1)) ↦ (𝑥 + (1 − 𝐼)))    &   (𝜑𝑋 ∈ (1...𝑀))    &   (𝜑𝑋 = 𝑀)       (𝜑 → (𝐵𝑋) = (((𝐶𝐷) ∪ {⟨𝑀, 𝑀⟩})‘𝑋))

Theoremmetakunt21 39343* Show that B coincides on the union of bijections of functions. (Contributed by metakunt, 28-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐵 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝑀, 𝑀, if(𝑥 < 𝐼, (𝑥 + (𝑀𝐼)), (𝑥 + (1 − 𝐼)))))    &   𝐶 = (𝑥 ∈ (1...(𝐼 − 1)) ↦ (𝑥 + (𝑀𝐼)))    &   𝐷 = (𝑥 ∈ (𝐼...(𝑀 − 1)) ↦ (𝑥 + (1 − 𝐼)))    &   (𝜑𝑋 ∈ (1...𝑀))    &   (𝜑 → ¬ 𝑋 = 𝑀)    &   (𝜑𝑋 < 𝐼)       (𝜑 → (𝐵𝑋) = (((𝐶𝐷) ∪ {⟨𝑀, 𝑀⟩})‘𝑋))

Theoremmetakunt22 39344* Show that B coincides on the union of bijections of functions. (Contributed by metakunt, 28-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐵 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝑀, 𝑀, if(𝑥 < 𝐼, (𝑥 + (𝑀𝐼)), (𝑥 + (1 − 𝐼)))))    &   𝐶 = (𝑥 ∈ (1...(𝐼 − 1)) ↦ (𝑥 + (𝑀𝐼)))    &   𝐷 = (𝑥 ∈ (𝐼...(𝑀 − 1)) ↦ (𝑥 + (1 − 𝐼)))    &   (𝜑𝑋 ∈ (1...𝑀))    &   (𝜑 → ¬ 𝑋 = 𝑀)    &   (𝜑 → ¬ 𝑋 < 𝐼)       (𝜑 → (𝐵𝑋) = (((𝐶𝐷) ∪ {⟨𝑀, 𝑀⟩})‘𝑋))

Theoremmetakunt23 39345* B coincides on the union of bijections of functions. (Contributed by metakunt, 28-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐵 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝑀, 𝑀, if(𝑥 < 𝐼, (𝑥 + (𝑀𝐼)), (𝑥 + (1 − 𝐼)))))    &   𝐶 = (𝑥 ∈ (1...(𝐼 − 1)) ↦ (𝑥 + (𝑀𝐼)))    &   𝐷 = (𝑥 ∈ (𝐼...(𝑀 − 1)) ↦ (𝑥 + (1 − 𝐼)))    &   (𝜑𝑋 ∈ (1...𝑀))       (𝜑 → (𝐵𝑋) = (((𝐶𝐷) ∪ {⟨𝑀, 𝑀⟩})‘𝑋))

Theoremmetakunt24 39346 Technical condition such that metakunt17 39339 holds (Contributed by metakunt, 28-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)       (𝜑 → ((((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∩ {𝑀}) = ∅ ∧ (1...𝑀) = (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∪ {𝑀}) ∧ (1...𝑀) = (((((𝑀𝐼) + 1)...(𝑀 − 1)) ∪ (1...(𝑀𝐼))) ∪ {𝑀})))

Theoremmetakunt25 39347* B is a permutation. (Contributed by metakunt, 28-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐵 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝑀, 𝑀, if(𝑥 < 𝐼, (𝑥 + (𝑀𝐼)), (𝑥 + (1 − 𝐼)))))       (𝜑𝐵:(1...𝑀)–1-1-onto→(1...𝑀))

20.25.7  Unused lemmas scheduled for deletion

Theoremandiff 39348 Adding biconditional when antecedents are conjuncted. (Contributed by metakunt, 16-Apr-2024.)
(𝜑 → (𝜒𝜃))    &   (𝜓 → (𝜃𝜒))       ((𝜑𝜓) → (𝜒𝜃))

Theoremfac2xp3 39349 Factorial of 2x+3, sublemma for sublemma for AKS. (Contributed by metakunt, 19-Apr-2024.)
(𝑥 ∈ ℕ0 → (!‘((2 · 𝑥) + 3)) = ((!‘((2 · 𝑥) + 1)) · (((2 · 𝑥) + 2) · ((2 · 𝑥) + 3))))

Theoremprodsplit 39350* Product split into two factors, original by Steven Nguyen. (Contributed by metakunt, 21-Apr-2024.)
(𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℤ)    &   (𝜑𝑀𝑁)    &   (𝜑𝐾 ∈ ℕ0)    &   ((𝜑𝑘 ∈ (𝑀...(𝑁 + 𝐾))) → 𝐴 ∈ ℂ)       (𝜑 → ∏𝑘 ∈ (𝑀...(𝑁 + 𝐾))𝐴 = (∏𝑘 ∈ (𝑀...𝑁)𝐴 · ∏𝑘 ∈ ((𝑁 + 1)...(𝑁 + 𝐾))𝐴))

Theorem2xp3dxp2ge1d 39351 2x+3 is greater than or equal to x+2 for x >= -1, a deduction version (Contributed by metakunt, 21-Apr-2024.)
(𝜑𝑋 ∈ (-1[,)+∞))       (𝜑 → 1 ≤ (((2 · 𝑋) + 3) / (𝑋 + 2)))

Theoremfactwoffsmonot 39352 A factorial with offset is monotonely increasing. (Contributed by metakunt, 20-Apr-2024.)
(((𝑋 ∈ ℕ0𝑌 ∈ ℕ0𝑋𝑌) ∧ 𝑁 ∈ ℕ0) → (!‘(𝑋 + 𝑁)) ≤ (!‘(𝑌 + 𝑁)))

20.26  Mathbox for Steven Nguyen

20.26.1  Utility theorems

Theoremioin9i8 39353 Miscellaneous inference creating a biconditional from an implied converse implication. (Contributed by Steven Nguyen, 17-Jul-2022.)
(𝜑 → (𝜓𝜒))    &   (𝜒 → ¬ 𝜃)    &   (𝜓𝜃)       (𝜑 → (𝜓𝜃))

Theoremjaodd 39354 Double deduction form of jaoi 854. (Contributed by Steven Nguyen, 17-Jul-2022.)
(𝜑 → (𝜓 → (𝜒𝜃)))    &   (𝜑 → (𝜓 → (𝜏𝜃)))       (𝜑 → (𝜓 → ((𝜒𝜏) → 𝜃)))

Theoremnsb 39355 Generalization rule for negated wff. (Contributed by Steven Nguyen, 3-May-2023.)
¬ 𝜑        ¬ [𝑥 / 𝑦]𝜑

Theoremsbtd 39356* A true statement is true upon substitution (deduction). A similar proof is possible for icht 43922. (Contributed by SN, 4-May-2024.)
(𝜑𝜓)       (𝜑 → [𝑡 / 𝑥]𝜓)

Theoremsbn1 39357 One direction of sbn 2289, using fewer axioms. Compare 19.2 1982. (Contributed by Steven Nguyen, 18-Aug-2023.)
([𝑡 / 𝑥] ¬ 𝜑 → ¬ [𝑡 / 𝑥]𝜑)

Theoremsbor2 39358 One direction of sbor 2318, using fewer axioms. Compare 19.33 1886. (Contributed by Steven Nguyen, 18-Aug-2023.)
(([𝑡 / 𝑥]𝜑 ∨ [𝑡 / 𝑥]𝜓) → [𝑡 / 𝑥](𝜑𝜓))

Theorem19.9dev 39359* 19.9d 2205 in the case of an existential quantifier, avoiding the ax-10 2146 from nfex 2345 that would be used for the hypothesis of 19.9d 2205, at the cost of an additional DV condition on 𝑦, 𝜑. (Contributed by SN, 26-May-2024.)
(𝜑 → Ⅎ𝑥𝜓)       (𝜑 → (∃𝑥𝑦𝜓 ↔ ∃𝑦𝜓))

Theorem3rspcedvd 39360* Triple application of rspcedvd 3612. (Contributed by Steven Nguyen, 27-Feb-2023.)
(𝜑𝐴𝐷)    &   (𝜑𝐵𝐷)    &   (𝜑𝐶𝐷)    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))    &   ((𝜑𝑦 = 𝐵) → (𝜒𝜃))    &   ((𝜑𝑧 = 𝐶) → (𝜃𝜏))    &   (𝜑𝜏)       (𝜑 → ∃𝑥𝐷𝑦𝐷𝑧𝐷 𝜓)

Theoremrabeqcda 39361* When 𝜓 is always true in a context, a restricted class abstraction is equal to the restricting class. Deduction form of rabeqc 3664. (Contributed by Steven Nguyen, 7-Jun-2023.)
((𝜑𝑥𝐴) → 𝜓)       (𝜑 → {𝑥𝐴𝜓} = 𝐴)

Theoremrabdif 39362* Move difference in and out of a restricted class abstraction. (Contributed by Steven Nguyen, 6-Jun-2023.)
({𝑥𝐴𝜑} ∖ 𝐵) = {𝑥 ∈ (𝐴𝐵) ∣ 𝜑}

Theoremsn-axrep5v 39363* A condensed form of axrep5 5183. (Contributed by SN, 21-Sep-2023.)
(∀𝑤𝑥 ∃*𝑧𝜑 → ∃𝑦𝑧(𝑧𝑦 ↔ ∃𝑤𝑥 𝜑))

Theoremsn-axprlem3 39364* axprlem3 5314 using only Tarski's FOL axiom schemes and ax-rep 5177. (Contributed by SN, 22-Sep-2023.)
𝑦𝑧(𝑧𝑦 ↔ ∃𝑤𝑥 if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏))

Theoremsn-el 39365* A version of el 5258 with an inner existential quantifier on 𝑥, which avoids ax-7 2016 and ax-8 2117. (Contributed by SN, 18-Sep-2023.)
𝑦𝑥 𝑥𝑦

Theoremsn-dtru 39366* dtru 5259 without ax-8 2117 or ax-12 2179. (Contributed by SN, 21-Sep-2023.)
¬ ∀𝑥 𝑥 = 𝑦

Theorempssexg 39367 The proper subset of a set is also a set. (Contributed by Steven Nguyen, 17-Jul-2022.)
((𝐴𝐵𝐵𝐶) → 𝐴 ∈ V)

Theorempssn0 39368 A proper superset is nonempty. (Contributed by Steven Nguyen, 17-Jul-2022.)
(𝐴𝐵𝐵 ≠ ∅)

Theorempsspwb 39369 Classes are proper subclasses if and only if their power classes are proper subclasses. (Contributed by Steven Nguyen, 17-Jul-2022.)
(𝐴𝐵 ↔ 𝒫 𝐴 ⊊ 𝒫 𝐵)

Theoremxppss12 39370 Proper subset theorem for Cartesian product. (Contributed by Steven Nguyen, 17-Jul-2022.)
((𝐴𝐵𝐶𝐷) → (𝐴 × 𝐶) ⊊ (𝐵 × 𝐷))

Theoremelpwbi 39371 Membership in a power set, biconditional. (Contributed by Steven Nguyen, 17-Jul-2022.) (Proof shortened by Steven Nguyen, 16-Sep-2022.)
𝐵 ∈ V       (𝐴𝐵𝐴 ∈ 𝒫 𝐵)

Theoremopelxpii 39372 Ordered pair membership in a Cartesian product (implication). (Contributed by Steven Nguyen, 17-Jul-2022.)
𝐴𝐶    &   𝐵𝐷       𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷)

Theoremiunsn 39373* Indexed union of a singleton. Compare dfiun2 4944 and rnmpt 5815. (Contributed by Steven Nguyen, 7-Jun-2023.)
𝑥𝐴 {𝐵} = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}

Theoremimaopab 39374* The image of a class of ordered pairs. (Contributed by Steven Nguyen, 6-Jun-2023.)
({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴) = {𝑦 ∣ ∃𝑥𝐴 𝜑}

Theoremfnsnbt 39375 A function's domain is a singleton iff the function is a singleton. (Contributed by Steven Nguyen, 18-Aug-2023.)
(𝐴 ∈ V → (𝐹 Fn {𝐴} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩}))

Theoremfnimasnd 39376 The image of a function by a singleton whose element is in the domain of the function. (Contributed by Steven Nguyen, 7-Jun-2023.)
(𝜑𝐹 Fn 𝐴)    &   (𝜑𝑆𝐴)       (𝜑 → (𝐹 “ {𝑆}) = {(𝐹𝑆)})

Theoremdfqs2 39377* Alternate definition of quotient set. (Contributed by Steven Nguyen, 7-Jun-2023.)
(𝐴 / 𝑅) = ran (𝑥𝐴 ↦ [𝑥]𝑅)

Theoremdfqs3 39378* Alternate definition of quotient set. (Contributed by Steven Nguyen, 7-Jun-2023.)
(𝐴 / 𝑅) = 𝑥𝐴 {[𝑥]𝑅}

Theoremqseq12d 39379 Equality theorem for quotient set, deduction form. (Contributed by Steven Nguyen, 30-Apr-2023.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴 / 𝐶) = (𝐵 / 𝐷))

Theoremqsalrel 39380* The quotient set is equal to the singleton of 𝐴 when all elements are related and 𝐴 is nonempty. (Contributed by SN, 8-Jun-2023.)
((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → 𝑥 𝑦)    &   (𝜑 Er 𝐴)    &   (𝜑𝑁𝐴)       (𝜑 → (𝐴 / ) = {𝐴})

Theoremfzosumm1 39381* Separate out the last term in a finite sum. (Contributed by Steven Nguyen, 22-Aug-2023.)
(𝜑 → (𝑁 − 1) ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (𝑀..^𝑁)) → 𝐴 ∈ ℂ)    &   (𝑘 = (𝑁 − 1) → 𝐴 = 𝐵)    &   (𝜑𝑁 ∈ ℤ)       (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)𝐴 = (Σ𝑘 ∈ (𝑀..^(𝑁 − 1))𝐴 + 𝐵))

Theoremccatcan2d 39382 Cancellation law for concatenation. (Contributed by SN, 6-Sep-2023.)
(𝜑𝐴 ∈ Word 𝑉)    &   (𝜑𝐵 ∈ Word 𝑉)    &   (𝜑𝐶 ∈ Word 𝑉)       (𝜑 → ((𝐴 ++ 𝐶) = (𝐵 ++ 𝐶) ↔ 𝐴 = 𝐵))

Theoremnelsubginvcld 39383 The inverse of a non-subgroup-member is a non-subgroup-member. (Contributed by Steven Nguyen, 15-Apr-2023.)
(𝜑𝐺 ∈ Grp)    &   (𝜑𝑆 ∈ (SubGrp‘𝐺))    &   (𝜑𝑋 ∈ (𝐵𝑆))    &   𝐵 = (Base‘𝐺)    &   𝑁 = (invg𝐺)       (𝜑 → (𝑁𝑋) ∈ (𝐵𝑆))

Theoremnelsubgcld 39384 A non-subgroup-member plus a subgroup member is a non-subgroup-member. (Contributed by Steven Nguyen, 15-Apr-2023.)
(𝜑𝐺 ∈ Grp)    &   (𝜑𝑆 ∈ (SubGrp‘𝐺))    &   (𝜑𝑋 ∈ (𝐵𝑆))    &   𝐵 = (Base‘𝐺)    &   (𝜑𝑌𝑆)    &    + = (+g𝐺)       (𝜑 → (𝑋 + 𝑌) ∈ (𝐵𝑆))

Theoremnelsubgsubcld 39385 A non-subgroup-member minus a subgroup member is a non-subgroup-member. (Contributed by Steven Nguyen, 15-Apr-2023.)
(𝜑𝐺 ∈ Grp)    &   (𝜑𝑆 ∈ (SubGrp‘𝐺))    &   (𝜑𝑋 ∈ (𝐵𝑆))    &   𝐵 = (Base‘𝐺)    &   (𝜑𝑌𝑆)    &    = (-g𝐺)       (𝜑 → (𝑋 𝑌) ∈ (𝐵𝑆))

Theoremrnasclg 39386 The set of injected scalars is also interpretable as the span of the identity. (Contributed by Mario Carneiro, 9-Mar-2015.)
𝐴 = (algSc‘𝑊)    &    1 = (1r𝑊)    &   𝑁 = (LSpan‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑊 ∈ Ring) → ran 𝐴 = (𝑁‘{ 1 }))

Theoremselvval2lem1 39387 𝑇 is an associative algebra. For simplicity, 𝐼 stands for (𝐼𝐽) and we have 𝐽𝑊 instead of 𝐽𝐼. (Contributed by SN, 15-Dec-2023.)
𝑈 = (𝐼 mPoly 𝑅)    &   𝑇 = (𝐽 mPoly 𝑈)    &   (𝜑𝐼𝑉)    &   (𝜑𝐽𝑊)    &   (𝜑𝑅 ∈ CRing)       (𝜑𝑇 ∈ AssAlg)

Theoremselvval2lem2 39388 𝐷 is a ring homomorphism. (Contributed by SN, 15-Dec-2023.)
𝑈 = (𝐼 mPoly 𝑅)    &   𝑇 = (𝐽 mPoly 𝑈)    &   𝐶 = (algSc‘𝑇)    &   𝐷 = (𝐶 ∘ (algSc‘𝑈))    &   (𝜑𝐼𝑉)    &   (𝜑𝐽𝑊)    &   (𝜑𝑅 ∈ CRing)       (𝜑𝐷 ∈ (𝑅 RingHom 𝑇))

Theoremselvval2lem3 39389 The third argument passed to evalSub is in the domain. (Contributed by SN, 15-Dec-2023.)
𝑈 = (𝐼 mPoly 𝑅)    &   𝑇 = (𝐽 mPoly 𝑈)    &   𝐶 = (algSc‘𝑇)    &   𝐷 = (𝐶 ∘ (algSc‘𝑈))    &   (𝜑𝐼𝑉)    &   (𝜑𝐽𝑊)    &   (𝜑𝑅 ∈ CRing)       (𝜑 → ran 𝐷 ∈ (SubRing‘𝑇))

Theoremselvval2lemn 39390 A lemma to illustrate the purpose of selvval2lem3 39389 and the value of 𝑄. Will be renamed in the future when this section is moved to main. (Contributed by SN, 5-Nov-2023.)
𝑈 = ((𝐼𝐽) mPoly 𝑅)    &   𝑇 = (𝐽 mPoly 𝑈)    &   𝐶 = (algSc‘𝑇)    &   𝐷 = (𝐶 ∘ (algSc‘𝑈))    &   𝑄 = ((𝐼 evalSub 𝑇)‘ran 𝐷)    &   𝑊 = (𝐼 mPoly 𝑆)    &   𝑆 = (𝑇s ran 𝐷)    &   𝑋 = (𝑇s (𝐵m 𝐼))    &   𝐵 = (Base‘𝑇)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝐽𝐼)       (𝜑𝑄 ∈ (𝑊 RingHom 𝑋))

Theoremselvval2lem4 39391 The fourth argument passed to evalSub is in the domain (a polynomial in (𝐼 mPoly (𝐽 mPoly ((𝐼𝐽) mPoly 𝑅)))). (Contributed by SN, 5-Nov-2023.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝐵 = (Base‘𝑃)    &   𝑈 = ((𝐼𝐽) mPoly 𝑅)    &   𝑇 = (𝐽 mPoly 𝑈)    &   𝐶 = (algSc‘𝑇)    &   𝐷 = (𝐶 ∘ (algSc‘𝑈))    &   𝑆 = (𝑇s ran 𝐷)    &   𝑊 = (𝐼 mPoly 𝑆)    &   𝑋 = (Base‘𝑊)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝐽𝐼)    &   (𝜑𝐹𝐵)       (𝜑 → (𝐷𝐹) ∈ 𝑋)

Theoremselvval2lem5 39392* The fifth argument passed to evalSub is in the domain (a function 𝐼𝐸). (Contributed by SN, 22-Feb-2024.)
𝑈 = ((𝐼𝐽) mPoly 𝑅)    &   𝑇 = (𝐽 mPoly 𝑈)    &   𝐶 = (algSc‘𝑇)    &   𝐸 = (Base‘𝑇)    &   𝐹 = (𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝐶‘(((𝐼𝐽) mVar 𝑅)‘𝑥))))    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝐽𝐼)       (𝜑𝐹 ∈ (𝐸m 𝐼))

Theoremselvcl 39393 Closure of the "variable selection" function. (Contributed by SN, 22-Feb-2024.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝐵 = (Base‘𝑃)    &   𝑈 = ((𝐼𝐽) mPoly 𝑅)    &   𝑇 = (𝐽 mPoly 𝑈)    &   𝐸 = (Base‘𝑇)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝐽𝐼)    &   (𝜑𝐹𝐵)       (𝜑 → (((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) ∈ 𝐸)

Theoremfrlmfielbas 39394 The vectors of a finite free module are the functions from 𝐼 to 𝑁. (Contributed by SN, 31-Aug-2023.)
𝐹 = (𝑅 freeLMod 𝐼)    &   𝑁 = (Base‘𝑅)    &   𝐵 = (Base‘𝐹)       ((𝑅𝑉𝐼 ∈ Fin) → (𝑋𝐵𝑋:𝐼𝑁))

Theoremfrlmfzwrd 39395 A vector of a module with indices from 0 to 𝑁 is a word over the scalars of the module. (Contributed by SN, 31-Aug-2023.)
𝑊 = (𝐾 freeLMod (0...𝑁))    &   𝐵 = (Base‘𝑊)    &   𝑆 = (Base‘𝐾)       (𝑋𝐵𝑋 ∈ Word 𝑆)

Theoremfrlmfzowrd 39396 A vector of a module with indices from 0 to 𝑁 − 1 is a word over the scalars of the module. (Contributed by SN, 31-Aug-2023.)
𝑊 = (𝐾 freeLMod (0..^𝑁))    &   𝐵 = (Base‘𝑊)    &   𝑆 = (Base‘𝐾)       (𝑋𝐵𝑋 ∈ Word 𝑆)

Theoremfrlmfzolen 39397 The dimension of a vector of a module with indices from 0 to 𝑁 − 1. (Contributed by SN, 1-Sep-2023.)
𝑊 = (𝐾 freeLMod (0..^𝑁))    &   𝐵 = (Base‘𝑊)    &   𝑆 = (Base‘𝐾)       ((𝑁 ∈ ℕ0𝑋𝐵) → (♯‘𝑋) = 𝑁)

Theoremfrlmfzowrdb 39398 The vectors of a module with indices 0 to 𝑁 − 1 are the length- 𝑁 words over the scalars of the module. (Contributed by SN, 1-Sep-2023.)
𝑊 = (𝐾 freeLMod (0..^𝑁))    &   𝐵 = (Base‘𝑊)    &   𝑆 = (Base‘𝐾)       ((𝐾𝑉𝑁 ∈ ℕ0) → (𝑋𝐵 ↔ (𝑋 ∈ Word 𝑆 ∧ (♯‘𝑋) = 𝑁)))

Theoremfrlmfzoccat 39399 The concatenation of two vectors of dimension 𝑁 and 𝑀 forms a vector of dimension 𝑁 + 𝑀. (Contributed by SN, 31-Aug-2023.)
𝑊 = (𝐾 freeLMod (0..^𝐿))    &   𝑋 = (𝐾 freeLMod (0..^𝑀))    &   𝑌 = (𝐾 freeLMod (0..^𝑁))    &   𝐵 = (Base‘𝑊)    &   𝐶 = (Base‘𝑋)    &   𝐷 = (Base‘𝑌)    &   (𝜑𝐾 ∈ Ring)    &   (𝜑 → (𝑀 + 𝑁) = 𝐿)    &   (𝜑𝑀 ∈ ℕ0)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝑈𝐶)    &   (𝜑𝑉𝐷)       (𝜑 → (𝑈 ++ 𝑉) ∈ 𝐵)

Theoremfrlmvscadiccat 39400 Scalar multiplication distributes over concatenation. (Contributed by SN, 6-Sep-2023.)
𝑊 = (𝐾 freeLMod (0..^𝐿))    &   𝑋 = (𝐾 freeLMod (0..^𝑀))    &   𝑌 = (𝐾 freeLMod (0..^𝑁))    &   𝐵 = (Base‘𝑊)    &   𝐶 = (Base‘𝑋)    &   𝐷 = (Base‘𝑌)    &   (𝜑𝐾 ∈ Ring)    &   (𝜑 → (𝑀 + 𝑁) = 𝐿)    &   (𝜑𝑀 ∈ ℕ0)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝑈𝐶)    &   (𝜑𝑉𝐷)    &   𝑂 = ( ·𝑠𝑊)    &    = ( ·𝑠𝑋)    &    · = ( ·𝑠𝑌)    &   𝑆 = (Base‘𝐾)    &   (𝜑𝐴𝑆)       (𝜑 → (𝐴𝑂(𝑈 ++ 𝑉)) = ((𝐴 𝑈) ++ (𝐴 · 𝑉)))

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