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Type | Label | Description |
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Statement | ||
Theorem | pm11.71 41101* | Theorem *11.71 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.) |
⊢ ((∃𝑥𝜑 ∧ ∃𝑦𝜒) → ((∀𝑥(𝜑 → 𝜓) ∧ ∀𝑦(𝜒 → 𝜃)) ↔ ∀𝑥∀𝑦((𝜑 ∧ 𝜒) → (𝜓 ∧ 𝜃)))) | ||
Theorem | sbeqal1 41102* | If 𝑥 = 𝑦 always implies 𝑥 = 𝑧, then 𝑦 = 𝑧. (Contributed by Andrew Salmon, 2-Jun-2011.) |
⊢ (∀𝑥(𝑥 = 𝑦 → 𝑥 = 𝑧) → 𝑦 = 𝑧) | ||
Theorem | sbeqal1i 41103* | Suppose you know 𝑥 = 𝑦 implies 𝑥 = 𝑧, assuming 𝑥 and 𝑧 are distinct. Then, 𝑦 = 𝑧. (Contributed by Andrew Salmon, 3-Jun-2011.) |
⊢ (𝑥 = 𝑦 → 𝑥 = 𝑧) ⇒ ⊢ 𝑦 = 𝑧 | ||
Theorem | sbeqal2i 41104* | If 𝑥 = 𝑦 implies 𝑥 = 𝑧, then we can infer 𝑧 = 𝑦. (Contributed by Andrew Salmon, 3-Jun-2011.) |
⊢ (𝑥 = 𝑦 → 𝑥 = 𝑧) ⇒ ⊢ 𝑧 = 𝑦 | ||
Theorem | axc5c4c711 41105 | Proof of a theorem that can act as a sole axiom for pure predicate calculus with ax-gen 1797 as the inference rule. This proof extends the idea of axc5c711 36214 and related theorems. (Contributed by Andrew Salmon, 14-Jul-2011.) |
⊢ ((∀𝑥∀𝑦 ¬ ∀𝑥∀𝑦(∀𝑦𝜑 → 𝜓) → (𝜑 → ∀𝑦(∀𝑦𝜑 → 𝜓))) → (∀𝑦𝜑 → ∀𝑦𝜓)) | ||
Theorem | axc5c4c711toc5 41106 | Rederivation of sp 2180 from axc5c4c711 41105. Note that ax6 2391 is used for the rederivation. (Contributed by Andrew Salmon, 14-Jul-2011.) Revised to use ax6v 1971 instead of ax6 2391, so that this rederivation requires only ax6v 1971 and propositional calculus. (Revised by BJ, 14-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∀𝑥𝜑 → 𝜑) | ||
Theorem | axc5c4c711toc4 41107 | Rederivation of axc4 2329 from axc5c4c711 41105. Note that only propositional calculus is required for the rederivation. (Contributed by Andrew Salmon, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∀𝑥(∀𝑥𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓)) | ||
Theorem | axc5c4c711toc7 41108 | Rederivation of axc7 2325 from axc5c4c711 41105. Note that neither axc7 2325 nor ax-11 2158 are required for the rederivation. (Contributed by Andrew Salmon, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑) | ||
Theorem | axc5c4c711to11 41109 | Rederivation of ax-11 2158 from axc5c4c711 41105. Note that ax-11 2158 is not required for the rederivation. (Contributed by Andrew Salmon, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑) | ||
Theorem | axc11next 41110* | This theorem shows that, given axextb 2773, we can derive a version of axc11n 2437. However, it is weaker than axc11n 2437 because it has a distinct variable requirement. (Contributed by Andrew Salmon, 16-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∀𝑥 𝑥 = 𝑧 → ∀𝑧 𝑧 = 𝑥) | ||
Theorem | pm13.13a 41111 | One result of theorem *13.13 in [WhiteheadRussell] p. 178. A note on the section - to make the theorems more usable, and because inequality is notation for set theory (it is not defined in the predicate calculus section), this section will use classes instead of sets. (Contributed by Andrew Salmon, 3-Jun-2011.) |
⊢ ((𝜑 ∧ 𝑥 = 𝐴) → [𝐴 / 𝑥]𝜑) | ||
Theorem | pm13.13b 41112 | Theorem *13.13 in [WhiteheadRussell] p. 178 with different variable substitution. (Contributed by Andrew Salmon, 3-Jun-2011.) |
⊢ (([𝐴 / 𝑥]𝜑 ∧ 𝑥 = 𝐴) → 𝜑) | ||
Theorem | pm13.14 41113 | Theorem *13.14 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.) |
⊢ (([𝐴 / 𝑥]𝜑 ∧ ¬ 𝜑) → 𝑥 ≠ 𝐴) | ||
Theorem | pm13.192 41114* | Theorem *13.192 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.) (Revised by NM, 4-Jan-2017.) |
⊢ (∃𝑦(∀𝑥(𝑥 = 𝐴 ↔ 𝑥 = 𝑦) ∧ 𝜑) ↔ [𝐴 / 𝑦]𝜑) | ||
Theorem | pm13.193 41115 | Theorem *13.193 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.) |
⊢ ((𝜑 ∧ 𝑥 = 𝑦) ↔ ([𝑦 / 𝑥]𝜑 ∧ 𝑥 = 𝑦)) | ||
Theorem | pm13.194 41116 | Theorem *13.194 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.) |
⊢ ((𝜑 ∧ 𝑥 = 𝑦) ↔ ([𝑦 / 𝑥]𝜑 ∧ 𝜑 ∧ 𝑥 = 𝑦)) | ||
Theorem | pm13.195 41117* | Theorem *13.195 in [WhiteheadRussell] p. 179. This theorem is very similar to sbc5 3748. (Contributed by Andrew Salmon, 3-Jun-2011.) (Revised by NM, 4-Jan-2017.) |
⊢ (∃𝑦(𝑦 = 𝐴 ∧ 𝜑) ↔ [𝐴 / 𝑦]𝜑) | ||
Theorem | pm13.196a 41118* | Theorem *13.196 in [WhiteheadRussell] p. 179. The only difference is the position of the substituted variable. (Contributed by Andrew Salmon, 3-Jun-2011.) |
⊢ (¬ 𝜑 ↔ ∀𝑦([𝑦 / 𝑥]𝜑 → 𝑦 ≠ 𝑥)) | ||
Theorem | 2sbc6g 41119* | Theorem *13.21 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.) |
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (∀𝑧∀𝑤((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) → 𝜑) ↔ [𝐴 / 𝑧][𝐵 / 𝑤]𝜑)) | ||
Theorem | 2sbc5g 41120* | Theorem *13.22 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.) |
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (∃𝑧∃𝑤((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) ∧ 𝜑) ↔ [𝐴 / 𝑧][𝐵 / 𝑤]𝜑)) | ||
Theorem | iotain 41121 | Equivalence between two different forms of ℩. (Contributed by Andrew Salmon, 15-Jul-2011.) |
⊢ (∃!𝑥𝜑 → ∩ {𝑥 ∣ 𝜑} = (℩𝑥𝜑)) | ||
Theorem | iotaexeu 41122 | The iota class exists. This theorem does not require ax-nul 5174 for its proof. (Contributed by Andrew Salmon, 11-Jul-2011.) |
⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V) | ||
Theorem | iotasbc 41123* | Definition *14.01 in [WhiteheadRussell] p. 184. In Principia Mathematica, Russell and Whitehead define ℩ in terms of a function of (℩𝑥𝜑). Their definition differs in that a function of (℩𝑥𝜑) evaluates to "false" when there isn't a single 𝑥 that satisfies 𝜑. (Contributed by Andrew Salmon, 11-Jul-2011.) |
⊢ (∃!𝑥𝜑 → ([(℩𝑥𝜑) / 𝑦]𝜓 ↔ ∃𝑦(∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ∧ 𝜓))) | ||
Theorem | iotasbc2 41124* | Theorem *14.111 in [WhiteheadRussell] p. 184. (Contributed by Andrew Salmon, 11-Jul-2011.) |
⊢ ((∃!𝑥𝜑 ∧ ∃!𝑥𝜓) → ([(℩𝑥𝜑) / 𝑦][(℩𝑥𝜓) / 𝑧]𝜒 ↔ ∃𝑦∃𝑧(∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ∧ ∀𝑥(𝜓 ↔ 𝑥 = 𝑧) ∧ 𝜒))) | ||
Theorem | pm14.12 41125* | Theorem *14.12 in [WhiteheadRussell] p. 184. (Contributed by Andrew Salmon, 11-Jul-2011.) |
⊢ (∃!𝑥𝜑 → ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) | ||
Theorem | pm14.122a 41126* | Theorem *14.122 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.) |
⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝜑 ↔ 𝑥 = 𝐴) ↔ (∀𝑥(𝜑 → 𝑥 = 𝐴) ∧ [𝐴 / 𝑥]𝜑))) | ||
Theorem | pm14.122b 41127* | Theorem *14.122 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.) |
⊢ (𝐴 ∈ 𝑉 → ((∀𝑥(𝜑 → 𝑥 = 𝐴) ∧ [𝐴 / 𝑥]𝜑) ↔ (∀𝑥(𝜑 → 𝑥 = 𝐴) ∧ ∃𝑥𝜑))) | ||
Theorem | pm14.122c 41128* | Theorem *14.122 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.) |
⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝜑 ↔ 𝑥 = 𝐴) ↔ (∀𝑥(𝜑 → 𝑥 = 𝐴) ∧ ∃𝑥𝜑))) | ||
Theorem | pm14.123a 41129* | Theorem *14.123 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑧∀𝑤(𝜑 ↔ (𝑧 = 𝐴 ∧ 𝑤 = 𝐵)) ↔ (∀𝑧∀𝑤(𝜑 → (𝑧 = 𝐴 ∧ 𝑤 = 𝐵)) ∧ [𝐴 / 𝑧][𝐵 / 𝑤]𝜑))) | ||
Theorem | pm14.123b 41130* | Theorem *14.123 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((∀𝑧∀𝑤(𝜑 → (𝑧 = 𝐴 ∧ 𝑤 = 𝐵)) ∧ [𝐴 / 𝑧][𝐵 / 𝑤]𝜑) ↔ (∀𝑧∀𝑤(𝜑 → (𝑧 = 𝐴 ∧ 𝑤 = 𝐵)) ∧ ∃𝑧∃𝑤𝜑))) | ||
Theorem | pm14.123c 41131* | Theorem *14.123 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑧∀𝑤(𝜑 ↔ (𝑧 = 𝐴 ∧ 𝑤 = 𝐵)) ↔ (∀𝑧∀𝑤(𝜑 → (𝑧 = 𝐴 ∧ 𝑤 = 𝐵)) ∧ ∃𝑧∃𝑤𝜑))) | ||
Theorem | pm14.18 41132 | Theorem *14.18 in [WhiteheadRussell] p. 189. (Contributed by Andrew Salmon, 11-Jul-2011.) |
⊢ (∃!𝑥𝜑 → (∀𝑥𝜓 → [(℩𝑥𝜑) / 𝑥]𝜓)) | ||
Theorem | iotaequ 41133* | Theorem *14.2 in [WhiteheadRussell] p. 189. (Contributed by Andrew Salmon, 11-Jul-2011.) |
⊢ (℩𝑥𝑥 = 𝑦) = 𝑦 | ||
Theorem | iotavalb 41134* | Theorem *14.202 in [WhiteheadRussell] p. 189. A biconditional version of iotaval 6298. (Contributed by Andrew Salmon, 11-Jul-2011.) |
⊢ (∃!𝑥𝜑 → (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ↔ (℩𝑥𝜑) = 𝑦)) | ||
Theorem | iotasbc5 41135* | Theorem *14.205 in [WhiteheadRussell] p. 190. (Contributed by Andrew Salmon, 11-Jul-2011.) |
⊢ (∃!𝑥𝜑 → ([(℩𝑥𝜑) / 𝑦]𝜓 ↔ ∃𝑦(𝑦 = (℩𝑥𝜑) ∧ 𝜓))) | ||
Theorem | pm14.24 41136* | Theorem *14.24 in [WhiteheadRussell] p. 191. (Contributed by Andrew Salmon, 12-Jul-2011.) |
⊢ (∃!𝑥𝜑 → ∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = (℩𝑥𝜑))) | ||
Theorem | iotavalsb 41137* | Theorem *14.242 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.) |
⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ([𝑦 / 𝑧]𝜓 ↔ [(℩𝑥𝜑) / 𝑧]𝜓)) | ||
Theorem | sbiota1 41138 | Theorem *14.25 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 12-Jul-2011.) |
⊢ (∃!𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) ↔ [(℩𝑥𝜑) / 𝑥]𝜓)) | ||
Theorem | sbaniota 41139 | Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 12-Jul-2011.) |
⊢ (∃!𝑥𝜑 → (∃𝑥(𝜑 ∧ 𝜓) ↔ [(℩𝑥𝜑) / 𝑥]𝜓)) | ||
Theorem | eubiOLD 41140 | Obsolete proof of eubi 2644 as of 7-Oct-2022. (Contributed by Andrew Salmon, 11-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∃!𝑥𝜑 ↔ ∃!𝑥𝜓)) | ||
Theorem | iotasbcq 41141 | Theorem *14.272 in [WhiteheadRussell] p. 193. (Contributed by Andrew Salmon, 11-Jul-2011.) |
⊢ (∀𝑥(𝜑 ↔ 𝜓) → ([(℩𝑥𝜑) / 𝑦]𝜒 ↔ [(℩𝑥𝜓) / 𝑦]𝜒)) | ||
Theorem | elnev 41142* | Any set that contains one element less than the universe is not equal to it. (Contributed by Andrew Salmon, 16-Jun-2011.) |
⊢ (𝐴 ∈ V ↔ {𝑥 ∣ ¬ 𝑥 = 𝐴} ≠ V) | ||
Theorem | rusbcALT 41143 | A version of Russell's paradox which is proven using proper substitution. (Contributed by Andrew Salmon, 18-Jun-2011.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V | ||
Theorem | compeq 41144* | Equality between two ways of saying "the complement of 𝐴". (Contributed by Andrew Salmon, 15-Jul-2011.) |
⊢ (V ∖ 𝐴) = {𝑥 ∣ ¬ 𝑥 ∈ 𝐴} | ||
Theorem | compne 41145 | The complement of 𝐴 is not equal to 𝐴. (Contributed by Andrew Salmon, 15-Jul-2011.) (Proof shortened by BJ, 11-Nov-2021.) |
⊢ (V ∖ 𝐴) ≠ 𝐴 | ||
Theorem | compab 41146 | Two ways of saying "the complement of a class abstraction". (Contributed by Andrew Salmon, 15-Jul-2011.) (Proof shortened by Mario Carneiro, 11-Dec-2016.) |
⊢ (V ∖ {𝑧 ∣ 𝜑}) = {𝑧 ∣ ¬ 𝜑} | ||
Theorem | conss2 41147 | Contrapositive law for subsets. (Contributed by Andrew Salmon, 15-Jul-2011.) |
⊢ (𝐴 ⊆ (V ∖ 𝐵) ↔ 𝐵 ⊆ (V ∖ 𝐴)) | ||
Theorem | conss1 41148 | Contrapositive law for subsets. (Contributed by Andrew Salmon, 15-Jul-2011.) |
⊢ ((V ∖ 𝐴) ⊆ 𝐵 ↔ (V ∖ 𝐵) ⊆ 𝐴) | ||
Theorem | ralbidar 41149 | More general form of ralbida 3194. (Contributed by Andrew Salmon, 25-Jul-2011.) |
⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜑) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜒)) | ||
Theorem | rexbidar 41150 | More general form of rexbida 3277. (Contributed by Andrew Salmon, 25-Jul-2011.) |
⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜑) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜒)) | ||
Theorem | dropab1 41151 | Theorem to aid use of the distinctor reduction theorem with ordered pair class abstraction. (Contributed by Andrew Salmon, 25-Jul-2011.) |
⊢ (∀𝑥 𝑥 = 𝑦 → {〈𝑥, 𝑧〉 ∣ 𝜑} = {〈𝑦, 𝑧〉 ∣ 𝜑}) | ||
Theorem | dropab2 41152 | Theorem to aid use of the distinctor reduction theorem with ordered pair class abstraction. (Contributed by Andrew Salmon, 25-Jul-2011.) |
⊢ (∀𝑥 𝑥 = 𝑦 → {〈𝑧, 𝑥〉 ∣ 𝜑} = {〈𝑧, 𝑦〉 ∣ 𝜑}) | ||
Theorem | ipo0 41153 | If the identity relation partially orders any class, then that class is the null class. (Contributed by Andrew Salmon, 25-Jul-2011.) |
⊢ ( I Po 𝐴 ↔ 𝐴 = ∅) | ||
Theorem | ifr0 41154 | A class that is founded by the identity relation is null. (Contributed by Andrew Salmon, 25-Jul-2011.) |
⊢ ( I Fr 𝐴 ↔ 𝐴 = ∅) | ||
Theorem | ordpss 41155 | ordelpss 6187 with an antecedent removed. (Contributed by Andrew Salmon, 25-Jul-2011.) |
⊢ (Ord 𝐵 → (𝐴 ∈ 𝐵 → 𝐴 ⊊ 𝐵)) | ||
Theorem | fvsb 41156* | Explicit substitution of a value of a function into a wff. (Contributed by Andrew Salmon, 1-Aug-2011.) |
⊢ (∃!𝑦 𝐴𝐹𝑦 → ([(𝐹‘𝐴) / 𝑥]𝜑 ↔ ∃𝑥(∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑥) ∧ 𝜑))) | ||
Theorem | fveqsb 41157* | Implicit substitution of a value of a function into a wff. (Contributed by Andrew Salmon, 1-Aug-2011.) |
⊢ (𝑥 = (𝐹‘𝐴) → (𝜑 ↔ 𝜓)) & ⊢ Ⅎ𝑥𝜓 ⇒ ⊢ (∃!𝑦 𝐴𝐹𝑦 → (𝜓 ↔ ∃𝑥(∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑥) ∧ 𝜑))) | ||
Theorem | xpexb 41158 | A Cartesian product exists iff its converse does. Corollary 6.9(1) in [TakeutiZaring] p. 26. (Contributed by Andrew Salmon, 13-Nov-2011.) |
⊢ ((𝐴 × 𝐵) ∈ V ↔ (𝐵 × 𝐴) ∈ V) | ||
Theorem | trelpss 41159 | An element of a transitive set is a proper subset of it. Theorem 7.2 in [TakeutiZaring] p. 35. Unlike tz7.2 5503, ax-reg 9040 is required for its proof. (Contributed by Andrew Salmon, 13-Nov-2011.) |
⊢ ((Tr 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ⊊ 𝐴) | ||
Theorem | addcomgi 41160 | Generalization of commutative law for addition. Simplifies proofs dealing with vectors. However, it is dependent on our particular definition of ordered pair. (Contributed by Andrew Salmon, 28-Jan-2012.) (Revised by Mario Carneiro, 6-May-2015.) |
⊢ (𝐴 + 𝐵) = (𝐵 + 𝐴) | ||
Syntax | cplusr 41161 | Introduce the operation of vector addition. |
class +𝑟 | ||
Syntax | cminusr 41162 | Introduce the operation of vector subtraction. |
class -𝑟 | ||
Syntax | ctimesr 41163 | Introduce the operation of scalar multiplication. |
class .𝑣 | ||
Syntax | cptdfc 41164 | PtDf is a predicate that is crucial for the definition of lines as well as proving a number of important theorems. |
class PtDf(𝐴, 𝐵) | ||
Syntax | crr3c 41165 | RR3 is a class. |
class RR3 | ||
Syntax | cline3 41166 | line3 is a class. |
class line3 | ||
Definition | df-addr 41167* | Define the operation of vector addition. (Contributed by Andrew Salmon, 27-Jan-2012.) |
⊢ +𝑟 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑣 ∈ ℝ ↦ ((𝑥‘𝑣) + (𝑦‘𝑣)))) | ||
Definition | df-subr 41168* | Define the operation of vector subtraction. (Contributed by Andrew Salmon, 27-Jan-2012.) |
⊢ -𝑟 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑣 ∈ ℝ ↦ ((𝑥‘𝑣) − (𝑦‘𝑣)))) | ||
Definition | df-mulv 41169* | Define the operation of scalar multiplication. (Contributed by Andrew Salmon, 27-Jan-2012.) |
⊢ .𝑣 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑣 ∈ ℝ ↦ (𝑥 · (𝑦‘𝑣)))) | ||
Theorem | addrval 41170* | Value of the operation of vector addition. (Contributed by Andrew Salmon, 27-Jan-2012.) |
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴+𝑟𝐵) = (𝑣 ∈ ℝ ↦ ((𝐴‘𝑣) + (𝐵‘𝑣)))) | ||
Theorem | subrval 41171* | Value of the operation of vector subtraction. (Contributed by Andrew Salmon, 27-Jan-2012.) |
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴-𝑟𝐵) = (𝑣 ∈ ℝ ↦ ((𝐴‘𝑣) − (𝐵‘𝑣)))) | ||
Theorem | mulvval 41172* | Value of the operation of scalar multiplication. (Contributed by Andrew Salmon, 27-Jan-2012.) |
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴.𝑣𝐵) = (𝑣 ∈ ℝ ↦ (𝐴 · (𝐵‘𝑣)))) | ||
Theorem | addrfv 41173 | Vector addition at a value. The operation takes each vector 𝐴 and 𝐵 and forms a new vector whose values are the sum of each of the values of 𝐴 and 𝐵. (Contributed by Andrew Salmon, 27-Jan-2012.) |
⊢ ((𝐴 ∈ 𝐸 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ ℝ) → ((𝐴+𝑟𝐵)‘𝐶) = ((𝐴‘𝐶) + (𝐵‘𝐶))) | ||
Theorem | subrfv 41174 | Vector subtraction at a value. (Contributed by Andrew Salmon, 27-Jan-2012.) |
⊢ ((𝐴 ∈ 𝐸 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ ℝ) → ((𝐴-𝑟𝐵)‘𝐶) = ((𝐴‘𝐶) − (𝐵‘𝐶))) | ||
Theorem | mulvfv 41175 | Scalar multiplication at a value. (Contributed by Andrew Salmon, 27-Jan-2012.) |
⊢ ((𝐴 ∈ 𝐸 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ ℝ) → ((𝐴.𝑣𝐵)‘𝐶) = (𝐴 · (𝐵‘𝐶))) | ||
Theorem | addrfn 41176 | Vector addition produces a function. (Contributed by Andrew Salmon, 27-Jan-2012.) |
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴+𝑟𝐵) Fn ℝ) | ||
Theorem | subrfn 41177 | Vector subtraction produces a function. (Contributed by Andrew Salmon, 27-Jan-2012.) |
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴-𝑟𝐵) Fn ℝ) | ||
Theorem | mulvfn 41178 | Scalar multiplication producees a function. (Contributed by Andrew Salmon, 27-Jan-2012.) |
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴.𝑣𝐵) Fn ℝ) | ||
Theorem | addrcom 41179 | Vector addition is commutative. (Contributed by Andrew Salmon, 28-Jan-2012.) |
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴+𝑟𝐵) = (𝐵+𝑟𝐴)) | ||
Definition | df-ptdf 41180* | Define the predicate PtDf, which is a utility definition used to shorten definitions and simplify proofs. (Contributed by Andrew Salmon, 15-Jul-2012.) |
⊢ PtDf(𝐴, 𝐵) = (𝑥 ∈ ℝ ↦ (((𝑥.𝑣(𝐵-𝑟𝐴)) +𝑣 𝐴) “ {1, 2, 3})) | ||
Definition | df-rr3 41181 | Define the set of all points RR3. We define each point 𝐴 as a function to allow the use of vector addition and subtraction as well as scalar multiplication in our proofs. (Contributed by Andrew Salmon, 15-Jul-2012.) |
⊢ RR3 = (ℝ ↑m {1, 2, 3}) | ||
Definition | df-line3 41182* | Define the set of all lines. A line is an infinite subset of RR3 that satisfies a PtDf property. (Contributed by Andrew Salmon, 15-Jul-2012.) |
⊢ line3 = {𝑥 ∈ 𝒫 RR3 ∣ (2o ≼ 𝑥 ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑧 ≠ 𝑦 → ran PtDf(𝑦, 𝑧) = 𝑥))} | ||
We are sad to report the passing of long-time contributor Alan Sare (Nov. 9, 1954 - Mar. 23, 2019). Alan's first contribution to Metamath was a shorter proof for tfrlem8 8003 in 2008. He developed a tool called "completeusersproof" that assists developing proofs using his "virtual deduction" method: https://us.metamath.org/other.html#completeusersproof 8003. His virtual deduction method is explained in the comment for wvd1 41275. Below are some excerpts from his first emails to NM in 2007: ...I have been interested in proving set theory theorems for many years for mental exercise. I enjoy it. I have used a book by Martin Zuckerman. It is informal. I am interested in completely and perfectly proving theorems. Mr. Zuckerman leaves out most of the steps of a proof, of course, like most authors do, as you have noted. A complete proof for higher theorems would require a volume of writing similar to the Metamath documents. So I am frustrated when I am not capable of constructing a proof and Zuckerman leaves out steps I do not understand. I could search for the steps in other texts, but I don't do that too much. Metamath may be the answer for me.... ...If we go beyond mathematics, I believe that it is possible to write down all human knowledge in a way similar to the way you have explicated large areas of mathematics. Of course, that would be a much, much more difficult job. For example, it is possible to take a hard science like physics, construct axioms based on experimental results, and to cast all of physics into a collection of axioms and theorems. Maybe this has already been attempted, although I am not familiar with it. When one then moves on to the soft sciences such as social science, this job gets much more difficult. The key is: All human thought consists of logical operations on abstract objects. Usually, these logical operations are done informally. There is no reason why one cannot take any subject and explicate it and take it down to the indivisible postulates in a formal rigorous way.... ...When I read a math book or an engineering book I come across something I don't understand and I am compelled to understand it. But, often it is hopeless. I don't have the time. Or, I would have to read the same thing by multiple authors in the hope that different authors would give parts of the working proof that others have omitted. It is very inefficient. Because I have always been inclined to "get to the bottom" for a 100% fully understood proof.... | ||
Theorem | idiALT 41183 | Placeholder for idi 1. Though unnecessary, this theorem is sometimes used in proofs in this mathbox for pedagogical purposes. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝜑 ⇒ ⊢ 𝜑 | ||
Theorem | exbir 41184 | Exportation implication also converting the consequent from a biconditional to an implication. Derived automatically from exbirVD 41559. (Contributed by Alan Sare, 31-Dec-2011.) |
⊢ (((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)) → (𝜑 → (𝜓 → (𝜃 → 𝜒)))) | ||
Theorem | 3impexpbicom 41185 | Version of 3impexp 1355 where in addition the consequent is commuted. (Contributed by Alan Sare, 31-Dec-2011.) |
⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜃 ↔ 𝜏)) ↔ (𝜑 → (𝜓 → (𝜒 → (𝜏 ↔ 𝜃))))) | ||
Theorem | 3impexpbicomi 41186 | Inference associated with 3impexpbicom 41185. Derived automatically from 3impexpbicomiVD 41564. (Contributed by Alan Sare, 31-Dec-2011.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜃 ↔ 𝜏)) ⇒ ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜏 ↔ 𝜃)))) | ||
Theorem | bi1imp 41187 | Importation inference similar to imp 410, except the outermost implication of the hypothesis is a biconditional. (Contributed by Alan Sare, 6-Nov-2017.) |
⊢ (𝜑 ↔ (𝜓 → 𝜒)) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝜒) | ||
Theorem | bi2imp 41188 | Importation inference similar to imp 410, except both implications of the hypothesis are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.) |
⊢ (𝜑 ↔ (𝜓 ↔ 𝜒)) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝜒) | ||
Theorem | bi3impb 41189 | Similar to 3impb 1112 with implication in hypothesis replaced by biconditional. (Contributed by Alan Sare, 6-Nov-2017.) |
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) ↔ 𝜃) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) | ||
Theorem | bi3impa 41190 | Similar to 3impa 1107 with implication in hypothesis replaced by biconditional. (Contributed by Alan Sare, 6-Nov-2017.) |
⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ 𝜃) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) | ||
Theorem | bi23impib 41191 | 3impib 1113 with the inner implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.) |
⊢ (𝜑 → ((𝜓 ∧ 𝜒) ↔ 𝜃)) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) | ||
Theorem | bi13impib 41192 | 3impib 1113 with the outer implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.) |
⊢ (𝜑 ↔ ((𝜓 ∧ 𝜒) → 𝜃)) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) | ||
Theorem | bi123impib 41193 | 3impib 1113 with the implications of the hypothesis biconditionals. (Contributed by Alan Sare, 6-Nov-2017.) |
⊢ (𝜑 ↔ ((𝜓 ∧ 𝜒) ↔ 𝜃)) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) | ||
Theorem | bi13impia 41194 | 3impia 1114 with the outer implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.) |
⊢ ((𝜑 ∧ 𝜓) ↔ (𝜒 → 𝜃)) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) | ||
Theorem | bi123impia 41195 | 3impia 1114 with the implications of the hypothesis biconditionals. (Contributed by Alan Sare, 6-Nov-2017.) |
⊢ ((𝜑 ∧ 𝜓) ↔ (𝜒 ↔ 𝜃)) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) | ||
Theorem | bi33imp12 41196 | 3imp 1108 with innermost implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.) |
⊢ (𝜑 → (𝜓 → (𝜒 ↔ 𝜃))) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) | ||
Theorem | bi23imp13 41197 | 3imp 1108 with middle implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.) |
⊢ (𝜑 → (𝜓 ↔ (𝜒 → 𝜃))) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) | ||
Theorem | bi13imp23 41198 | 3imp 1108 with outermost implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.) |
⊢ (𝜑 ↔ (𝜓 → (𝜒 → 𝜃))) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) | ||
Theorem | bi13imp2 41199 | Similar to 3imp 1108 except the outermost and innermost implications are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.) |
⊢ (𝜑 ↔ (𝜓 → (𝜒 ↔ 𝜃))) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) | ||
Theorem | bi12imp3 41200 | Similar to 3imp 1108 except all but innermost implication are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.) |
⊢ (𝜑 ↔ (𝜓 ↔ (𝜒 → 𝜃))) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) |
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