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Type | Label | Description |
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Statement | ||
Theorem | gneispace 43801* | The predicate that 𝐹 is a (generic) Seifert and Threlfall neighborhood space. (Contributed by RP, 14-Apr-2021.) |
⊢ 𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓∀𝑛 ∈ (𝑓‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝))))} ⇒ ⊢ (𝐹 ∈ 𝑉 → (𝐹 ∈ 𝐴 ↔ (Fun 𝐹 ∧ ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹 ∧ ∀𝑝 ∈ dom 𝐹((𝐹‘𝑝) ≠ ∅ ∧ ∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝))))))) | ||
Theorem | gneispacef 43802* | A generic neighborhood space is a function with a range that is a subset of the powerset of the powerset of its domain. (Contributed by RP, 15-Apr-2021.) |
⊢ 𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓∀𝑛 ∈ (𝑓‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝))))} ⇒ ⊢ (𝐹 ∈ 𝐴 → 𝐹:dom 𝐹⟶(𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅})) | ||
Theorem | gneispacef2 43803* | A generic neighborhood space is a function with a range that is a subset of the powerset of the powerset of its domain. (Contributed by RP, 15-Apr-2021.) |
⊢ 𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓∀𝑛 ∈ (𝑓‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝))))} ⇒ ⊢ (𝐹 ∈ 𝐴 → 𝐹:dom 𝐹⟶𝒫 𝒫 dom 𝐹) | ||
Theorem | gneispacefun 43804* | A generic neighborhood space is a function. (Contributed by RP, 15-Apr-2021.) |
⊢ 𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓∀𝑛 ∈ (𝑓‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝))))} ⇒ ⊢ (𝐹 ∈ 𝐴 → Fun 𝐹) | ||
Theorem | gneispacern 43805* | A generic neighborhood space has a range that is a subset of the powerset of the powerset of its domain. (Contributed by RP, 15-Apr-2021.) |
⊢ 𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓∀𝑛 ∈ (𝑓‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝))))} ⇒ ⊢ (𝐹 ∈ 𝐴 → ran 𝐹 ⊆ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅})) | ||
Theorem | gneispacern2 43806* | A generic neighborhood space has a range that is a subset of the powerset of the powerset of its domain. (Contributed by RP, 15-Apr-2021.) |
⊢ 𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓∀𝑛 ∈ (𝑓‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝))))} ⇒ ⊢ (𝐹 ∈ 𝐴 → ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹) | ||
Theorem | gneispace0nelrn 43807* | A generic neighborhood space has a nonempty set of neighborhoods for every point in its domain. (Contributed by RP, 15-Apr-2021.) |
⊢ 𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓∀𝑛 ∈ (𝑓‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝))))} ⇒ ⊢ (𝐹 ∈ 𝐴 → ∀𝑝 ∈ dom 𝐹(𝐹‘𝑝) ≠ ∅) | ||
Theorem | gneispace0nelrn2 43808* | A generic neighborhood space has a nonempty set of neighborhoods for every point in its domain. (Contributed by RP, 15-Apr-2021.) |
⊢ 𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓∀𝑛 ∈ (𝑓‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝))))} ⇒ ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑃 ∈ dom 𝐹) → (𝐹‘𝑃) ≠ ∅) | ||
Theorem | gneispace0nelrn3 43809* | A generic neighborhood space has a nonempty set of neighborhoods for every point in its domain. (Contributed by RP, 15-Apr-2021.) |
⊢ 𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓∀𝑛 ∈ (𝑓‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝))))} ⇒ ⊢ (𝐹 ∈ 𝐴 → ¬ ∅ ∈ ran 𝐹) | ||
Theorem | gneispaceel 43810* | Every neighborhood of a point in a generic neighborhood space contains that point. (Contributed by RP, 15-Apr-2021.) |
⊢ 𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓∀𝑛 ∈ (𝑓‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝))))} ⇒ ⊢ (𝐹 ∈ 𝐴 → ∀𝑝 ∈ dom 𝐹∀𝑛 ∈ (𝐹‘𝑝)𝑝 ∈ 𝑛) | ||
Theorem | gneispaceel2 43811* | Every neighborhood of a point in a generic neighborhood space contains that point. (Contributed by RP, 15-Apr-2021.) |
⊢ 𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓∀𝑛 ∈ (𝑓‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝))))} ⇒ ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑃 ∈ dom 𝐹 ∧ 𝑁 ∈ (𝐹‘𝑃)) → 𝑃 ∈ 𝑁) | ||
Theorem | gneispacess 43812* | All supersets of a neighborhood of a point (limited to the domain of the neighborhood space) are also neighborhoods of that point. (Contributed by RP, 15-Apr-2021.) |
⊢ 𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓∀𝑛 ∈ (𝑓‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝))))} ⇒ ⊢ (𝐹 ∈ 𝐴 → ∀𝑝 ∈ dom 𝐹∀𝑛 ∈ (𝐹‘𝑝)∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝))) | ||
Theorem | gneispacess2 43813* | All supersets of a neighborhood of a point (limited to the domain of the neighborhood space) are also neighborhoods of that point. (Contributed by RP, 15-Apr-2021.) |
⊢ 𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓∀𝑛 ∈ (𝑓‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝))))} ⇒ ⊢ (((𝐹 ∈ 𝐴 ∧ 𝑃 ∈ dom 𝐹) ∧ (𝑁 ∈ (𝐹‘𝑃) ∧ 𝑆 ∈ 𝒫 dom 𝐹 ∧ 𝑁 ⊆ 𝑆)) → 𝑆 ∈ (𝐹‘𝑃)) | ||
See https://kerodon.net/ for a work in progress by Jacob Lurie. | ||
See https://kerodon.net/tag/0004 for introduction to the topological simplex of dimension 𝑁. | ||
Theorem | k0004lem1 43814 | Application of ssin 4232 to range of a function. (Contributed by RP, 1-Apr-2021.) |
⊢ (𝐷 = (𝐵 ∩ 𝐶) → ((𝐹:𝐴⟶𝐵 ∧ (𝐹 “ 𝐴) ⊆ 𝐶) ↔ 𝐹:𝐴⟶𝐷)) | ||
Theorem | k0004lem2 43815 | A mapping with a particular restricted range is also a mapping to that range. (Contributed by RP, 1-Apr-2021.) |
⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ⊆ 𝐵) → ((𝐹 ∈ (𝐵 ↑m 𝐴) ∧ (𝐹 “ 𝐴) ⊆ 𝐶) ↔ 𝐹 ∈ (𝐶 ↑m 𝐴))) | ||
Theorem | k0004lem3 43816 | When the value of a mapping on a singleton is known, the mapping is a completely known singleton. (Contributed by RP, 2-Apr-2021.) |
⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐵) → ((𝐹 ∈ (𝐵 ↑m {𝐴}) ∧ (𝐹‘𝐴) = 𝐶) ↔ 𝐹 = {〈𝐴, 𝐶〉})) | ||
Theorem | k0004val 43817* | The topological simplex of dimension 𝑁 is the set of real vectors where the components are nonnegative and sum to 1. (Contributed by RP, 29-Mar-2021.) |
⊢ 𝐴 = (𝑛 ∈ ℕ0 ↦ {𝑡 ∈ ((0[,]1) ↑m (1...(𝑛 + 1))) ∣ Σ𝑘 ∈ (1...(𝑛 + 1))(𝑡‘𝑘) = 1}) ⇒ ⊢ (𝑁 ∈ ℕ0 → (𝐴‘𝑁) = {𝑡 ∈ ((0[,]1) ↑m (1...(𝑁 + 1))) ∣ Σ𝑘 ∈ (1...(𝑁 + 1))(𝑡‘𝑘) = 1}) | ||
Theorem | k0004ss1 43818* | The topological simplex of dimension 𝑁 is a subset of the real vectors of dimension (𝑁 + 1). (Contributed by RP, 29-Mar-2021.) |
⊢ 𝐴 = (𝑛 ∈ ℕ0 ↦ {𝑡 ∈ ((0[,]1) ↑m (1...(𝑛 + 1))) ∣ Σ𝑘 ∈ (1...(𝑛 + 1))(𝑡‘𝑘) = 1}) ⇒ ⊢ (𝑁 ∈ ℕ0 → (𝐴‘𝑁) ⊆ (ℝ ↑m (1...(𝑁 + 1)))) | ||
Theorem | k0004ss2 43819* | The topological simplex of dimension 𝑁 is a subset of the base set of a real vector space of dimension (𝑁 + 1). (Contributed by RP, 29-Mar-2021.) |
⊢ 𝐴 = (𝑛 ∈ ℕ0 ↦ {𝑡 ∈ ((0[,]1) ↑m (1...(𝑛 + 1))) ∣ Σ𝑘 ∈ (1...(𝑛 + 1))(𝑡‘𝑘) = 1}) ⇒ ⊢ (𝑁 ∈ ℕ0 → (𝐴‘𝑁) ⊆ (Base‘(ℝ^‘(1...(𝑁 + 1))))) | ||
Theorem | k0004ss3 43820* | The topological simplex of dimension 𝑁 is a subset of the base set of Euclidean space of dimension (𝑁 + 1). (Contributed by RP, 29-Mar-2021.) |
⊢ 𝐴 = (𝑛 ∈ ℕ0 ↦ {𝑡 ∈ ((0[,]1) ↑m (1...(𝑛 + 1))) ∣ Σ𝑘 ∈ (1...(𝑛 + 1))(𝑡‘𝑘) = 1}) ⇒ ⊢ (𝑁 ∈ ℕ0 → (𝐴‘𝑁) ⊆ (Base‘(𝔼hil‘(𝑁 + 1)))) | ||
Theorem | k0004val0 43821* | The topological simplex of dimension 0 is a singleton. (Contributed by RP, 2-Apr-2021.) |
⊢ 𝐴 = (𝑛 ∈ ℕ0 ↦ {𝑡 ∈ ((0[,]1) ↑m (1...(𝑛 + 1))) ∣ Σ𝑘 ∈ (1...(𝑛 + 1))(𝑡‘𝑘) = 1}) ⇒ ⊢ (𝐴‘0) = {{〈1, 1〉}} | ||
Theorem | inductionexd 43822 | Simple induction example. (Contributed by Stanislas Polu, 9-Mar-2020.) |
⊢ (𝑁 ∈ ℕ → 3 ∥ ((4↑𝑁) + 5)) | ||
Theorem | wwlemuld 43823 | Natural deduction form of lemul2d 13114. (Contributed by Stanislas Polu, 9-Mar-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → (𝐶 · 𝐴) ≤ (𝐶 · 𝐵)) & ⊢ (𝜑 → 0 < 𝐶) ⇒ ⊢ (𝜑 → 𝐴 ≤ 𝐵) | ||
Theorem | leeq1d 43824 | Specialization of breq1d 5163 to reals and less than. (Contributed by Stanislas Polu, 9-Mar-2020.) |
⊢ (𝜑 → 𝐴 ≤ 𝐶) & ⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) ⇒ ⊢ (𝜑 → 𝐵 ≤ 𝐶) | ||
Theorem | leeq2d 43825 | Specialization of breq2d 5165 to reals and less than. (Contributed by Stanislas Polu, 9-Mar-2020.) |
⊢ (𝜑 → 𝐴 ≤ 𝐶) & ⊢ (𝜑 → 𝐶 = 𝐷) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) ⇒ ⊢ (𝜑 → 𝐴 ≤ 𝐷) | ||
Theorem | absmulrposd 43826 | Specialization of absmuld with absidd 15427. (Contributed by Stanislas Polu, 9-Mar-2020.) |
⊢ (𝜑 → 0 ≤ 𝐴) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (abs‘(𝐴 · 𝐵)) = (𝐴 · (abs‘𝐵))) | ||
Theorem | imadisjld 43827 | Natural dduction form of one side of imadisj 6089. (Contributed by Stanislas Polu, 9-Mar-2020.) |
⊢ (𝜑 → (dom 𝐴 ∩ 𝐵) = ∅) ⇒ ⊢ (𝜑 → (𝐴 “ 𝐵) = ∅) | ||
Theorem | wnefimgd 43828 | The image of a mapping from A is nonempty if A is nonempty. (Contributed by Stanislas Polu, 9-Mar-2020.) |
⊢ (𝜑 → 𝐴 ≠ ∅) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) ⇒ ⊢ (𝜑 → (𝐹 “ 𝐴) ≠ ∅) | ||
Theorem | fco2d 43829 | Natural deduction form of fco2 6755. (Contributed by Stanislas Polu, 9-Mar-2020.) |
⊢ (𝜑 → 𝐺:𝐴⟶𝐵) & ⊢ (𝜑 → (𝐹 ↾ 𝐵):𝐵⟶𝐶) ⇒ ⊢ (𝜑 → (𝐹 ∘ 𝐺):𝐴⟶𝐶) | ||
Theorem | wfximgfd 43830 | The value of a function on its domain is in the image of the function. (Contributed by Stanislas Polu, 9-Mar-2020.) |
⊢ (𝜑 → 𝐶 ∈ 𝐴) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) ⇒ ⊢ (𝜑 → (𝐹‘𝐶) ∈ (𝐹 “ 𝐴)) | ||
Theorem | extoimad 43831* | If |f(x)| <= C for all x then it applies to all x in the image of |f(x)| (Contributed by Stanislas Polu, 9-Mar-2020.) |
⊢ (𝜑 → 𝐹:ℝ⟶ℝ) & ⊢ (𝜑 → ∀𝑦 ∈ ℝ (abs‘(𝐹‘𝑦)) ≤ 𝐶) ⇒ ⊢ (𝜑 → ∀𝑥 ∈ (abs “ (𝐹 “ ℝ))𝑥 ≤ 𝐶) | ||
Theorem | imo72b2lem0 43832* | Lemma for imo72b2 43839. (Contributed by Stanislas Polu, 9-Mar-2020.) |
⊢ (𝜑 → 𝐹:ℝ⟶ℝ) & ⊢ (𝜑 → 𝐺:ℝ⟶ℝ) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → ((𝐹‘(𝐴 + 𝐵)) + (𝐹‘(𝐴 − 𝐵))) = (2 · ((𝐹‘𝐴) · (𝐺‘𝐵)))) & ⊢ (𝜑 → ∀𝑦 ∈ ℝ (abs‘(𝐹‘𝑦)) ≤ 1) ⇒ ⊢ (𝜑 → ((abs‘(𝐹‘𝐴)) · (abs‘(𝐺‘𝐵))) ≤ sup((abs “ (𝐹 “ ℝ)), ℝ, < )) | ||
Theorem | suprleubrd 43833* | Natural deduction form of specialized suprleub 12232. (Contributed by Stanislas Polu, 9-Mar-2020.) |
⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → 𝐴 ≠ ∅) & ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝐵) ⇒ ⊢ (𝜑 → sup(𝐴, ℝ, < ) ≤ 𝐵) | ||
Theorem | imo72b2lem2 43834* | Lemma for imo72b2 43839. (Contributed by Stanislas Polu, 9-Mar-2020.) |
⊢ (𝜑 → 𝐹:ℝ⟶ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → ∀𝑧 ∈ ℝ (abs‘(𝐹‘𝑧)) ≤ 𝐶) ⇒ ⊢ (𝜑 → sup((abs “ (𝐹 “ ℝ)), ℝ, < ) ≤ 𝐶) | ||
Theorem | suprlubrd 43835* | Natural deduction form of specialized suprlub 12230. (Contributed by Stanislas Polu, 9-Mar-2020.) |
⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → 𝐴 ≠ ∅) & ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → ∃𝑧 ∈ 𝐴 𝐵 < 𝑧) ⇒ ⊢ (𝜑 → 𝐵 < sup(𝐴, ℝ, < )) | ||
Theorem | imo72b2lem1 43836* | Lemma for imo72b2 43839. (Contributed by Stanislas Polu, 9-Mar-2020.) |
⊢ (𝜑 → 𝐹:ℝ⟶ℝ) & ⊢ (𝜑 → ∃𝑥 ∈ ℝ (𝐹‘𝑥) ≠ 0) & ⊢ (𝜑 → ∀𝑦 ∈ ℝ (abs‘(𝐹‘𝑦)) ≤ 1) ⇒ ⊢ (𝜑 → 0 < sup((abs “ (𝐹 “ ℝ)), ℝ, < )) | ||
Theorem | lemuldiv3d 43837 | 'Less than or equal to' relationship between division and multiplication. (Contributed by Stanislas Polu, 9-Mar-2020.) |
⊢ (𝜑 → (𝐵 · 𝐴) ≤ 𝐶) & ⊢ (𝜑 → 0 < 𝐴) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) ⇒ ⊢ (𝜑 → 𝐵 ≤ (𝐶 / 𝐴)) | ||
Theorem | lemuldiv4d 43838 | 'Less than or equal to' relationship between division and multiplication. (Contributed by Stanislas Polu, 9-Mar-2020.) |
⊢ (𝜑 → 𝐵 ≤ (𝐶 / 𝐴)) & ⊢ (𝜑 → 0 < 𝐴) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐵 · 𝐴) ≤ 𝐶) | ||
Theorem | imo72b2 43839* | IMO 1972 B2. (14th International Mathematical Olympiad in Poland, problem B2). (Contributed by Stanislas Polu, 9-Mar-2020.) |
⊢ (𝜑 → 𝐹:ℝ⟶ℝ) & ⊢ (𝜑 → 𝐺:ℝ⟶ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → ∀𝑢 ∈ ℝ ∀𝑣 ∈ ℝ ((𝐹‘(𝑢 + 𝑣)) + (𝐹‘(𝑢 − 𝑣))) = (2 · ((𝐹‘𝑢) · (𝐺‘𝑣)))) & ⊢ (𝜑 → ∀𝑦 ∈ ℝ (abs‘(𝐹‘𝑦)) ≤ 1) & ⊢ (𝜑 → ∃𝑥 ∈ ℝ (𝐹‘𝑥) ≠ 0) ⇒ ⊢ (𝜑 → (abs‘(𝐺‘𝐵)) ≤ 1) | ||
This section formalizes theorems necessary to reproduce the equality and inequality generator described in "Neural Theorem Proving on Inequality Problems" http://aitp-conference.org/2020/abstract/paper_18.pdf. Other theorems required: 0red 11267 1red 11265 readdcld 11293 remulcld 11294 eqcomd 2732. | ||
Theorem | int-addcomd 43840 | AdditionCommutativity generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐵 + 𝐶) = (𝐶 + 𝐴)) | ||
Theorem | int-addassocd 43841 | AdditionAssociativity generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐷 ∈ ℝ) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐵 + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + 𝐷)) | ||
Theorem | int-addsimpd 43842 | AdditionSimplification generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → 0 = (𝐴 − 𝐵)) | ||
Theorem | int-mulcomd 43843 | MultiplicationCommutativity generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐵 · 𝐶) = (𝐶 · 𝐴)) | ||
Theorem | int-mulassocd 43844 | MultiplicationAssociativity generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐷 ∈ ℝ) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐵 · (𝐶 · 𝐷)) = ((𝐴 · 𝐶) · 𝐷)) | ||
Theorem | int-mulsimpd 43845 | MultiplicationSimplification generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐵 ≠ 0) ⇒ ⊢ (𝜑 → 1 = (𝐴 / 𝐵)) | ||
Theorem | int-leftdistd 43846 | AdditionMultiplicationLeftDistribution generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐷 ∈ ℝ) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → ((𝐶 + 𝐷) · 𝐵) = ((𝐶 · 𝐴) + (𝐷 · 𝐴))) | ||
Theorem | int-rightdistd 43847 | AdditionMultiplicationRightDistribution generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐷 ∈ ℝ) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐵 · (𝐶 + 𝐷)) = ((𝐴 · 𝐶) + (𝐴 · 𝐷))) | ||
Theorem | int-sqdefd 43848 | SquareDefinition generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐴 · 𝐵) = (𝐴↑2)) | ||
Theorem | int-mul11d 43849 | First MultiplicationOne generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐴 · 1) = 𝐵) | ||
Theorem | int-mul12d 43850 | Second MultiplicationOne generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (1 · 𝐴) = 𝐵) | ||
Theorem | int-add01d 43851 | First AdditionZero generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐴 + 0) = 𝐵) | ||
Theorem | int-add02d 43852 | Second AdditionZero generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (0 + 𝐴) = 𝐵) | ||
Theorem | int-sqgeq0d 43853 | SquareGEQZero generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → 0 ≤ (𝐴 · 𝐵)) | ||
Theorem | int-eqprincd 43854 | PrincipleOfEquality generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → (𝐴 + 𝐶) = (𝐵 + 𝐷)) | ||
Theorem | int-eqtransd 43855 | EqualityTransitivity generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → 𝐴 = 𝐶) | ||
Theorem | int-eqmvtd 43856 | EquMoveTerm generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐷 ∈ ℝ) & ⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐴 = (𝐶 + 𝐷)) ⇒ ⊢ (𝜑 → 𝐶 = (𝐵 − 𝐷)) | ||
Theorem | int-eqineqd 43857 | EquivalenceImpliesDoubleInequality generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → 𝐵 ≤ 𝐴) | ||
Theorem | int-ineqmvtd 43858 | IneqMoveTerm generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐷 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ≤ 𝐴) & ⊢ (𝜑 → 𝐴 = (𝐶 + 𝐷)) ⇒ ⊢ (𝜑 → (𝐵 − 𝐷) ≤ 𝐶) | ||
Theorem | int-ineq1stprincd 43859 | FirstPrincipleOfInequality generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐷 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ≤ 𝐴) & ⊢ (𝜑 → 𝐷 ≤ 𝐶) ⇒ ⊢ (𝜑 → (𝐵 + 𝐷) ≤ (𝐴 + 𝐶)) | ||
Theorem | int-ineq2ndprincd 43860 | SecondPrincipleOfInequality generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ≤ 𝐴) & ⊢ (𝜑 → 0 ≤ 𝐶) ⇒ ⊢ (𝜑 → (𝐵 · 𝐶) ≤ (𝐴 · 𝐶)) | ||
Theorem | int-ineqtransd 43861 | InequalityTransitivity generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ≤ 𝐴) & ⊢ (𝜑 → 𝐶 ≤ 𝐵) ⇒ ⊢ (𝜑 → 𝐶 ≤ 𝐴) | ||
This section formalizes theorems used in an n-digit addition proof generator. Other theorems required: deccl 12744 addcomli 11456 00id 11439 addridi 11451 addlidi 11452 eqid 2726 dec0h 12751 decadd 12783 decaddc 12784. | ||
Theorem | unitadd 43862 | Theorem used in conjunction with decaddc 12784 to absorb carry when generating n-digit addition synthetic proofs. (Contributed by Stanislas Polu, 7-Apr-2020.) |
⊢ (𝐴 + 𝐵) = 𝐹 & ⊢ (𝐶 + 1) = 𝐵 & ⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐶 ∈ ℕ0 ⇒ ⊢ ((𝐴 + 𝐶) + 1) = 𝐹 | ||
Theorem | gsumws3 43863 | Valuation of a length 3 word in a monoid. (Contributed by Stanislas Polu, 9-Sep-2020.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Mnd ∧ (𝑆 ∈ 𝐵 ∧ (𝑇 ∈ 𝐵 ∧ 𝑈 ∈ 𝐵))) → (𝐺 Σg 〈“𝑆𝑇𝑈”〉) = (𝑆 + (𝑇 + 𝑈))) | ||
Theorem | gsumws4 43864 | Valuation of a length 4 word in a monoid. (Contributed by Stanislas Polu, 10-Sep-2020.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Mnd ∧ (𝑆 ∈ 𝐵 ∧ (𝑇 ∈ 𝐵 ∧ (𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵)))) → (𝐺 Σg 〈“𝑆𝑇𝑈𝑉”〉) = (𝑆 + (𝑇 + (𝑈 + 𝑉)))) | ||
Theorem | amgm2d 43865 | Arithmetic-geometric mean inequality for 𝑛 = 2, derived from amgmlem 27018. (Contributed by Stanislas Polu, 8-Sep-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) ⇒ ⊢ (𝜑 → ((𝐴 · 𝐵)↑𝑐(1 / 2)) ≤ ((𝐴 + 𝐵) / 2)) | ||
Theorem | amgm3d 43866 | Arithmetic-geometric mean inequality for 𝑛 = 3. (Contributed by Stanislas Polu, 11-Sep-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) & ⊢ (𝜑 → 𝐶 ∈ ℝ+) ⇒ ⊢ (𝜑 → ((𝐴 · (𝐵 · 𝐶))↑𝑐(1 / 3)) ≤ ((𝐴 + (𝐵 + 𝐶)) / 3)) | ||
Theorem | amgm4d 43867 | Arithmetic-geometric mean inequality for 𝑛 = 4. (Contributed by Stanislas Polu, 11-Sep-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) & ⊢ (𝜑 → 𝐶 ∈ ℝ+) & ⊢ (𝜑 → 𝐷 ∈ ℝ+) ⇒ ⊢ (𝜑 → ((𝐴 · (𝐵 · (𝐶 · 𝐷)))↑𝑐(1 / 4)) ≤ ((𝐴 + (𝐵 + (𝐶 + 𝐷))) / 4)) | ||
Theorem | spALT 43868 | sp 2172 can be proven from the other classic axioms. (Contributed by Rohan Ridenour, 3-Nov-2023.) (Proof modification is discouraged.) Use sp 2172 instead. (New usage is discouraged.) |
⊢ (∀𝑥𝜑 → 𝜑) | ||
Theorem | elnelneqd 43869 | Two classes are not equal if there is an element of one which is not an element of the other. (Contributed by Rohan Ridenour, 11-Aug-2023.) |
⊢ (𝜑 → 𝐶 ∈ 𝐴) & ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐵) ⇒ ⊢ (𝜑 → ¬ 𝐴 = 𝐵) | ||
Theorem | elnelneq2d 43870 | Two classes are not equal if one but not the other is an element of a given class. (Contributed by Rohan Ridenour, 12-Aug-2023.) |
⊢ (𝜑 → 𝐴 ∈ 𝐶) & ⊢ (𝜑 → ¬ 𝐵 ∈ 𝐶) ⇒ ⊢ (𝜑 → ¬ 𝐴 = 𝐵) | ||
Theorem | rr-spce 43871* | Prove an existential. (Contributed by Rohan Ridenour, 12-Aug-2023.) |
⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝜓) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) ⇒ ⊢ (𝜑 → ∃𝑥𝜓) | ||
Theorem | rexlimdvaacbv 43872* | Unpack a restricted existential antecedent while changing the variable with implicit substitution. The equivalent of this theorem without the bound variable change is rexlimdvaa 3146. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜃)) & ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝜃)) → 𝜒) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒)) | ||
Theorem | rexlimddvcbvw 43873* | Unpack a restricted existential assumption while changing the variable with implicit substitution. Similar to rexlimdvaacbv 43872. The equivalent of this theorem without the bound variable change is rexlimddv 3151. Version of rexlimddvcbv 43874 with a disjoint variable condition, which does not require ax-13 2366. (Contributed by Rohan Ridenour, 3-Aug-2023.) (Revised by GG, 2-Apr-2024.) |
⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜃) & ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝜒)) → 𝜓) & ⊢ (𝑥 = 𝑦 → (𝜃 ↔ 𝜒)) ⇒ ⊢ (𝜑 → 𝜓) | ||
Theorem | rexlimddvcbv 43874* | Unpack a restricted existential assumption while changing the variable with implicit substitution. Similar to rexlimdvaacbv 43872. The equivalent of this theorem without the bound variable change is rexlimddv 3151. Usage of this theorem is discouraged because it depends on ax-13 2366, see rexlimddvcbvw 43873 for a weaker version that does not require it. (Contributed by Rohan Ridenour, 3-Aug-2023.) (New usage is discouraged.) |
⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜃) & ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝜒)) → 𝜓) & ⊢ (𝑥 = 𝑦 → (𝜃 ↔ 𝜒)) ⇒ ⊢ (𝜑 → 𝜓) | ||
Theorem | rr-elrnmpt3d 43875* | Elementhood in an image set. (Contributed by Rohan Ridenour, 11-Aug-2023.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) & ⊢ (𝜑 → 𝐶 ∈ 𝐴) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝐵 = 𝐷) ⇒ ⊢ (𝜑 → 𝐷 ∈ ran 𝐹) | ||
Theorem | finnzfsuppd 43876* | If a function is zero outside of a finite set, it has finite support. (Contributed by Rohan Ridenour, 13-May-2024.) |
⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ (𝜑 → 𝐹 Fn 𝐷) & ⊢ (𝜑 → 𝑍 ∈ 𝑈) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑥 ∈ 𝐴 ∨ (𝐹‘𝑥) = 𝑍)) ⇒ ⊢ (𝜑 → 𝐹 finSupp 𝑍) | ||
Theorem | rr-phpd 43877 | Equivalent of php 9244 without negation. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
⊢ (𝜑 → 𝐴 ∈ ω) & ⊢ (𝜑 → 𝐵 ⊆ 𝐴) & ⊢ (𝜑 → 𝐴 ≈ 𝐵) ⇒ ⊢ (𝜑 → 𝐴 = 𝐵) | ||
Theorem | suceqd 43878 | Deduction associated with suceq 6442. (Contributed by Rohan Ridenour, 8-Aug-2023.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → suc 𝐴 = suc 𝐵) | ||
Theorem | tfindsd 43879* | Deduction associated with tfinds 7870. (Contributed by Rohan Ridenour, 8-Aug-2023.) |
⊢ (𝑥 = ∅ → (𝜓 ↔ 𝜒)) & ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜃)) & ⊢ (𝑥 = suc 𝑦 → (𝜓 ↔ 𝜏)) & ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜂)) & ⊢ (𝜑 → 𝜒) & ⊢ ((𝜑 ∧ 𝑦 ∈ On ∧ 𝜃) → 𝜏) & ⊢ ((𝜑 ∧ Lim 𝑥 ∧ ∀𝑦 ∈ 𝑥 𝜃) → 𝜓) & ⊢ (𝜑 → 𝐴 ∈ On) ⇒ ⊢ (𝜑 → 𝜂) | ||
Syntax | cmnring 43880 | Extend class notation with the monoid ring function. |
class MndRing | ||
Definition | df-mnring 43881* | Define the monoid ring function. This takes a monoid 𝑀 and a ring 𝑅 and produces a free left module over 𝑅 with a product extending the monoid function on 𝑀. (Contributed by Rohan Ridenour, 13-May-2024.) |
⊢ MndRing = (𝑟 ∈ V, 𝑚 ∈ V ↦ ⦋(𝑟 freeLMod (Base‘𝑚)) / 𝑣⦌(𝑣 sSet 〈(.r‘ndx), (𝑥 ∈ (Base‘𝑣), 𝑦 ∈ (Base‘𝑣) ↦ (𝑣 Σg (𝑎 ∈ (Base‘𝑚), 𝑏 ∈ (Base‘𝑚) ↦ (𝑖 ∈ (Base‘𝑚) ↦ if(𝑖 = (𝑎(+g‘𝑚)𝑏), ((𝑥‘𝑎)(.r‘𝑟)(𝑦‘𝑏)), (0g‘𝑟))))))〉)) | ||
Theorem | mnringvald 43882* | Value of the monoid ring function. (Contributed by Rohan Ridenour, 14-May-2024.) |
⊢ 𝐹 = (𝑅 MndRing 𝑀) & ⊢ · = (.r‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐴 = (Base‘𝑀) & ⊢ + = (+g‘𝑀) & ⊢ 𝑉 = (𝑅 freeLMod 𝐴) & ⊢ 𝐵 = (Base‘𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑈) & ⊢ (𝜑 → 𝑀 ∈ 𝑊) ⇒ ⊢ (𝜑 → 𝐹 = (𝑉 sSet 〈(.r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑉 Σg (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎 + 𝑏), ((𝑥‘𝑎) · (𝑦‘𝑏)), 0 )))))〉)) | ||
Theorem | mnringnmulrd 43883 | Components of a monoid ring other than its ring product match its underlying free module. (Contributed by Rohan Ridenour, 14-May-2024.) (Revised by AV, 1-Nov-2024.) |
⊢ 𝐹 = (𝑅 MndRing 𝑀) & ⊢ 𝐸 = Slot (𝐸‘ndx) & ⊢ (𝐸‘ndx) ≠ (.r‘ndx) & ⊢ 𝐴 = (Base‘𝑀) & ⊢ 𝑉 = (𝑅 freeLMod 𝐴) & ⊢ (𝜑 → 𝑅 ∈ 𝑈) & ⊢ (𝜑 → 𝑀 ∈ 𝑊) ⇒ ⊢ (𝜑 → (𝐸‘𝑉) = (𝐸‘𝐹)) | ||
Theorem | mnringnmulrdOLD 43884 | Obsolete version of mnringnmulrd 43883 as of 1-Nov-2024. Components of a monoid ring other than its ring product match its underlying free module. (Contributed by Rohan Ridenour, 14-May-2024.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ 𝐹 = (𝑅 MndRing 𝑀) & ⊢ 𝐸 = Slot 𝑁 & ⊢ 𝑁 ∈ ℕ & ⊢ 𝑁 ≠ (.r‘ndx) & ⊢ 𝐴 = (Base‘𝑀) & ⊢ 𝑉 = (𝑅 freeLMod 𝐴) & ⊢ (𝜑 → 𝑅 ∈ 𝑈) & ⊢ (𝜑 → 𝑀 ∈ 𝑊) ⇒ ⊢ (𝜑 → (𝐸‘𝑉) = (𝐸‘𝐹)) | ||
Theorem | mnringbased 43885 | The base set of a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024.) (Proof shortened by AV, 1-Nov-2024.) |
⊢ 𝐹 = (𝑅 MndRing 𝑀) & ⊢ 𝐴 = (Base‘𝑀) & ⊢ 𝑉 = (𝑅 freeLMod 𝐴) & ⊢ 𝐵 = (Base‘𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑈) & ⊢ (𝜑 → 𝑀 ∈ 𝑊) ⇒ ⊢ (𝜑 → 𝐵 = (Base‘𝐹)) | ||
Theorem | mnringbasedOLD 43886 | Obsolete version of mnringnmulrd 43883 as of 1-Nov-2024. The base set of a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ 𝐹 = (𝑅 MndRing 𝑀) & ⊢ 𝐴 = (Base‘𝑀) & ⊢ 𝑉 = (𝑅 freeLMod 𝐴) & ⊢ 𝐵 = (Base‘𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑈) & ⊢ (𝜑 → 𝑀 ∈ 𝑊) ⇒ ⊢ (𝜑 → 𝐵 = (Base‘𝐹)) | ||
Theorem | mnringbaserd 43887 | The base set of a monoid ring. Converse of mnringbased 43885. (Contributed by Rohan Ridenour, 14-May-2024.) |
⊢ 𝐹 = (𝑅 MndRing 𝑀) & ⊢ 𝐵 = (Base‘𝐹) & ⊢ 𝐴 = (Base‘𝑀) & ⊢ 𝑉 = (𝑅 freeLMod 𝐴) & ⊢ (𝜑 → 𝑅 ∈ 𝑈) & ⊢ (𝜑 → 𝑀 ∈ 𝑊) ⇒ ⊢ (𝜑 → 𝐵 = (Base‘𝑉)) | ||
Theorem | mnringelbased 43888 | Membership in the base set of a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024.) |
⊢ 𝐹 = (𝑅 MndRing 𝑀) & ⊢ 𝐵 = (Base‘𝐹) & ⊢ 𝐴 = (Base‘𝑀) & ⊢ 𝐶 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ 𝑈) & ⊢ (𝜑 → 𝑀 ∈ 𝑊) ⇒ ⊢ (𝜑 → (𝑋 ∈ 𝐵 ↔ (𝑋 ∈ (𝐶 ↑m 𝐴) ∧ 𝑋 finSupp 0 ))) | ||
Theorem | mnringbasefd 43889 | Elements of a monoid ring are functions. (Contributed by Rohan Ridenour, 14-May-2024.) |
⊢ 𝐹 = (𝑅 MndRing 𝑀) & ⊢ 𝐵 = (Base‘𝐹) & ⊢ 𝐴 = (Base‘𝑀) & ⊢ 𝐶 = (Base‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ 𝑈) & ⊢ (𝜑 → 𝑀 ∈ 𝑊) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → 𝑋:𝐴⟶𝐶) | ||
Theorem | mnringbasefsuppd 43890 | Elements of a monoid ring are finitely supported. (Contributed by Rohan Ridenour, 14-May-2024.) |
⊢ 𝐹 = (𝑅 MndRing 𝑀) & ⊢ 𝐵 = (Base‘𝐹) & ⊢ 0 = (0g‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ 𝑈) & ⊢ (𝜑 → 𝑀 ∈ 𝑊) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → 𝑋 finSupp 0 ) | ||
Theorem | mnringaddgd 43891 | The additive operation of a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024.) (Proof shortened by AV, 1-Nov-2024.) |
⊢ 𝐹 = (𝑅 MndRing 𝑀) & ⊢ 𝐴 = (Base‘𝑀) & ⊢ 𝑉 = (𝑅 freeLMod 𝐴) & ⊢ (𝜑 → 𝑅 ∈ 𝑈) & ⊢ (𝜑 → 𝑀 ∈ 𝑊) ⇒ ⊢ (𝜑 → (+g‘𝑉) = (+g‘𝐹)) | ||
Theorem | mnringaddgdOLD 43892 | Obsolete version of mnringaddgd 43891 as of 1-Nov-2024. The additive operation of a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ 𝐹 = (𝑅 MndRing 𝑀) & ⊢ 𝐴 = (Base‘𝑀) & ⊢ 𝑉 = (𝑅 freeLMod 𝐴) & ⊢ (𝜑 → 𝑅 ∈ 𝑈) & ⊢ (𝜑 → 𝑀 ∈ 𝑊) ⇒ ⊢ (𝜑 → (+g‘𝑉) = (+g‘𝐹)) | ||
Theorem | mnring0gd 43893 | The additive identity of a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024.) |
⊢ 𝐹 = (𝑅 MndRing 𝑀) & ⊢ 𝐴 = (Base‘𝑀) & ⊢ 𝑉 = (𝑅 freeLMod 𝐴) & ⊢ (𝜑 → 𝑅 ∈ 𝑈) & ⊢ (𝜑 → 𝑀 ∈ 𝑊) ⇒ ⊢ (𝜑 → (0g‘𝑉) = (0g‘𝐹)) | ||
Theorem | mnring0g2d 43894 | The additive identity of a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024.) |
⊢ 𝐹 = (𝑅 MndRing 𝑀) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐴 = (Base‘𝑀) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑀 ∈ 𝑊) ⇒ ⊢ (𝜑 → (𝐴 × { 0 }) = (0g‘𝐹)) | ||
Theorem | mnringmulrd 43895* | The ring product of a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024.) |
⊢ 𝐹 = (𝑅 MndRing 𝑀) & ⊢ 𝐵 = (Base‘𝐹) & ⊢ · = (.r‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐴 = (Base‘𝑀) & ⊢ + = (+g‘𝑀) & ⊢ (𝜑 → 𝑅 ∈ 𝑈) & ⊢ (𝜑 → 𝑀 ∈ 𝑊) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝐹 Σg (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎 + 𝑏), ((𝑥‘𝑎) · (𝑦‘𝑏)), 0 ))))) = (.r‘𝐹)) | ||
Theorem | mnringscad 43896 | The scalar ring of a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024.) (Proof shortened by AV, 1-Nov-2024.) |
⊢ 𝐹 = (𝑅 MndRing 𝑀) & ⊢ (𝜑 → 𝑅 ∈ 𝑈) & ⊢ (𝜑 → 𝑀 ∈ 𝑊) ⇒ ⊢ (𝜑 → 𝑅 = (Scalar‘𝐹)) | ||
Theorem | mnringscadOLD 43897 | Obsolete version of mnringscad 43896 as of 1-Nov-2024. The scalar ring of a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ 𝐹 = (𝑅 MndRing 𝑀) & ⊢ (𝜑 → 𝑅 ∈ 𝑈) & ⊢ (𝜑 → 𝑀 ∈ 𝑊) ⇒ ⊢ (𝜑 → 𝑅 = (Scalar‘𝐹)) | ||
Theorem | mnringvscad 43898 | The scalar product of a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024.) (Proof shortened by AV, 1-Nov-2024.) |
⊢ 𝐹 = (𝑅 MndRing 𝑀) & ⊢ 𝐵 = (Base‘𝑀) & ⊢ 𝑉 = (𝑅 freeLMod 𝐵) & ⊢ (𝜑 → 𝑅 ∈ 𝑈) & ⊢ (𝜑 → 𝑀 ∈ 𝑊) ⇒ ⊢ (𝜑 → ( ·𝑠 ‘𝑉) = ( ·𝑠 ‘𝐹)) | ||
Theorem | mnringvscadOLD 43899 | Obsolete version of mnringvscad 43898 as of 1-Nov-2024. The scalar product of a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ 𝐹 = (𝑅 MndRing 𝑀) & ⊢ 𝐵 = (Base‘𝑀) & ⊢ 𝑉 = (𝑅 freeLMod 𝐵) & ⊢ (𝜑 → 𝑅 ∈ 𝑈) & ⊢ (𝜑 → 𝑀 ∈ 𝑊) ⇒ ⊢ (𝜑 → ( ·𝑠 ‘𝑉) = ( ·𝑠 ‘𝐹)) | ||
Theorem | mnringlmodd 43900 | Monoid rings are left modules. (Contributed by Rohan Ridenour, 14-May-2024.) |
⊢ 𝐹 = (𝑅 MndRing 𝑀) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑀 ∈ 𝑈) ⇒ ⊢ (𝜑 → 𝐹 ∈ LMod) |
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