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Type | Label | Description |
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Statement | ||
Theorem | axc11next 44401* | This theorem shows that, given axextb 2708, we can derive a version of axc11n 2428. However, it is weaker than axc11n 2428 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 44402 | 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 44403 | 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 44404 | Theorem *13.14 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.) |
⊢ (([𝐴 / 𝑥]𝜑 ∧ ¬ 𝜑) → 𝑥 ≠ 𝐴) | ||
Theorem | pm13.192 44405* | 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 44406 | 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 44407 | 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 44408* | Theorem *13.195 in [WhiteheadRussell] p. 179. This theorem is very similar to sbc5 3818. (Contributed by Andrew Salmon, 3-Jun-2011.) (Revised by NM, 4-Jan-2017.) |
⊢ (∃𝑦(𝑦 = 𝐴 ∧ 𝜑) ↔ [𝐴 / 𝑦]𝜑) | ||
Theorem | pm13.196a 44409* | 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 44410* | 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 44411* | 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 44412 | Equivalence between two different forms of ℩. (Contributed by Andrew Salmon, 15-Jul-2011.) |
⊢ (∃!𝑥𝜑 → ∩ {𝑥 ∣ 𝜑} = (℩𝑥𝜑)) | ||
Theorem | iotaexeu 44413 | The iota class exists. This theorem does not require ax-nul 5311 for its proof. (Contributed by Andrew Salmon, 11-Jul-2011.) |
⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V) | ||
Theorem | iotasbc 44414* | 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 44415* | 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 44416* | 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 44417* | 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 44418* | 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 44419* | 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 44420* | 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 44421* | 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 44422* | 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 44423 | Theorem *14.18 in [WhiteheadRussell] p. 189. (Contributed by Andrew Salmon, 11-Jul-2011.) |
⊢ (∃!𝑥𝜑 → (∀𝑥𝜓 → [(℩𝑥𝜑) / 𝑥]𝜓)) | ||
Theorem | iotaequ 44424* | Theorem *14.2 in [WhiteheadRussell] p. 189. (Contributed by Andrew Salmon, 11-Jul-2011.) |
⊢ (℩𝑥𝑥 = 𝑦) = 𝑦 | ||
Theorem | iotavalb 44425* | Theorem *14.202 in [WhiteheadRussell] p. 189. A biconditional version of iotaval 6533. (Contributed by Andrew Salmon, 11-Jul-2011.) |
⊢ (∃!𝑥𝜑 → (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ↔ (℩𝑥𝜑) = 𝑦)) | ||
Theorem | iotasbc5 44426* | 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 44427* | Theorem *14.24 in [WhiteheadRussell] p. 191. (Contributed by Andrew Salmon, 12-Jul-2011.) |
⊢ (∃!𝑥𝜑 → ∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = (℩𝑥𝜑))) | ||
Theorem | iotavalsb 44428* | 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 44429 | Theorem *14.25 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 12-Jul-2011.) |
⊢ (∃!𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) ↔ [(℩𝑥𝜑) / 𝑥]𝜓)) | ||
Theorem | sbaniota 44430 | Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 12-Jul-2011.) |
⊢ (∃!𝑥𝜑 → (∃𝑥(𝜑 ∧ 𝜓) ↔ [(℩𝑥𝜑) / 𝑥]𝜓)) | ||
Theorem | eubiOLD 44431 | Obsolete proof of eubi 2581 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 44432 | 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 44433* | 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 44434 | 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 44435* | Equality between two ways of saying "the complement of 𝐴". (Contributed by Andrew Salmon, 15-Jul-2011.) |
⊢ (V ∖ 𝐴) = {𝑥 ∣ ¬ 𝑥 ∈ 𝐴} | ||
Theorem | compne 44436 | 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 44437 | 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 44438 | Contrapositive law for subsets. (Contributed by Andrew Salmon, 15-Jul-2011.) |
⊢ (𝐴 ⊆ (V ∖ 𝐵) ↔ 𝐵 ⊆ (V ∖ 𝐴)) | ||
Theorem | conss1 44439 | Contrapositive law for subsets. (Contributed by Andrew Salmon, 15-Jul-2011.) |
⊢ ((V ∖ 𝐴) ⊆ 𝐵 ↔ (V ∖ 𝐵) ⊆ 𝐴) | ||
Theorem | ralbidar 44440 | More general form of ralbida 3267. (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 44441 | More general form of rexbida 3269. (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 44442 | 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 44443 | 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 44444 | 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 44445 | A class that is founded by the identity relation is null. (Contributed by Andrew Salmon, 25-Jul-2011.) |
⊢ ( I Fr 𝐴 ↔ 𝐴 = ∅) | ||
Theorem | ordpss 44446 | ordelpss 6413 with an antecedent removed. (Contributed by Andrew Salmon, 25-Jul-2011.) |
⊢ (Ord 𝐵 → (𝐴 ∈ 𝐵 → 𝐴 ⊊ 𝐵)) | ||
Theorem | fvsb 44447* | 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 44448* | 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 44449 | 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 44450 | An element of a transitive set is a proper subset of it. Theorem 7.2 in [TakeutiZaring] p. 35. Unlike tz7.2 5671, ax-reg 9629 is required for its proof. (Contributed by Andrew Salmon, 13-Nov-2011.) |
⊢ ((Tr 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ⊊ 𝐴) | ||
Theorem | addcomgi 44451 | 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 44452 | Introduce the operation of vector addition. |
class +𝑟 | ||
Syntax | cminusr 44453 | Introduce the operation of vector subtraction. |
class -𝑟 | ||
Syntax | ctimesr 44454 | Introduce the operation of scalar multiplication. |
class .𝑣 | ||
Syntax | cptdfc 44455 | 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 44456 | RR3 is a class. |
class RR3 | ||
Syntax | cline3 44457 | line3 is a class. |
class line3 | ||
Definition | df-addr 44458* | 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 44459* | 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 44460* | 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 44461* | 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 44462* | 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 44463* | 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 44464 | 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 44465 | 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 44466 | 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 44467 | Vector addition produces a function. (Contributed by Andrew Salmon, 27-Jan-2012.) |
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴+𝑟𝐵) Fn ℝ) | ||
Theorem | subrfn 44468 | Vector subtraction produces a function. (Contributed by Andrew Salmon, 27-Jan-2012.) |
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴-𝑟𝐵) Fn ℝ) | ||
Theorem | mulvfn 44469 | Scalar multiplication producees a function. (Contributed by Andrew Salmon, 27-Jan-2012.) |
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴.𝑣𝐵) Fn ℝ) | ||
Theorem | addrcom 44470 | Vector addition is commutative. (Contributed by Andrew Salmon, 28-Jan-2012.) |
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴+𝑟𝐵) = (𝐵+𝑟𝐴)) | ||
Definition | df-ptdf 44471* | 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 44472 | 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 44473* | 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 8422 in 2008. He developed a tool called "completeusersproof" that assists developing proofs using his "virtual deduction" method: https://us.metamath.org/other.html#completeusersproof 8422. His virtual deduction method is explained in the comment for wvd1 44566. 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 44474 | 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 44475 | Exportation implication also converting the consequent from a biconditional to an implication. Derived automatically from exbirVD 44850. (Contributed by Alan Sare, 31-Dec-2011.) |
⊢ (((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)) → (𝜑 → (𝜓 → (𝜃 → 𝜒)))) | ||
Theorem | 3impexpbicom 44476 | Version of 3impexp 1357 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 44477 | Inference associated with 3impexpbicom 44476. Derived automatically from 3impexpbicomiVD 44855. (Contributed by Alan Sare, 31-Dec-2011.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜃 ↔ 𝜏)) ⇒ ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜏 ↔ 𝜃)))) | ||
Theorem | bi1imp 44478 | Importation inference similar to imp 406, 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 44479 | Importation inference similar to imp 406, except both implications of the hypothesis are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.) |
⊢ (𝜑 ↔ (𝜓 ↔ 𝜒)) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝜒) | ||
Theorem | bi3impb 44480 | Similar to 3impb 1114 with implication in hypothesis replaced by biconditional. (Contributed by Alan Sare, 6-Nov-2017.) |
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) ↔ 𝜃) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) | ||
Theorem | bi3impa 44481 | Similar to 3impa 1109 with implication in hypothesis replaced by biconditional. (Contributed by Alan Sare, 6-Nov-2017.) |
⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ 𝜃) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) | ||
Theorem | bi23impib 44482 | 3impib 1115 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 44483 | 3impib 1115 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 44484 | 3impib 1115 with the implications of the hypothesis biconditionals. (Contributed by Alan Sare, 6-Nov-2017.) |
⊢ (𝜑 ↔ ((𝜓 ∧ 𝜒) ↔ 𝜃)) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) | ||
Theorem | bi13impia 44485 | 3impia 1116 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 44486 | 3impia 1116 with the implications of the hypothesis biconditionals. (Contributed by Alan Sare, 6-Nov-2017.) |
⊢ ((𝜑 ∧ 𝜓) ↔ (𝜒 ↔ 𝜃)) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) | ||
Theorem | bi33imp12 44487 | 3imp 1110 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 44488 | 3imp 1110 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 44489 | 3imp 1110 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 44490 | Similar to 3imp 1110 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 44491 | Similar to 3imp 1110 except all but innermost implication are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.) |
⊢ (𝜑 ↔ (𝜓 ↔ (𝜒 → 𝜃))) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) | ||
Theorem | bi23imp1 44492 | Similar to 3imp 1110 except all but outermost implication are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.) |
⊢ (𝜑 → (𝜓 ↔ (𝜒 ↔ 𝜃))) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) | ||
Theorem | bi123imp0 44493 | Similar to 3imp 1110 except all implications are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.) |
⊢ (𝜑 ↔ (𝜓 ↔ (𝜒 ↔ 𝜃))) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) | ||
Theorem | 4animp1 44494 | A single hypothesis unification deduction with an assertion which is an implication with a 4-right-nested conjunction antecedent. (Contributed by Alan Sare, 30-May-2018.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜏 ↔ 𝜃)) ⇒ ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏) | ||
Theorem | 4an31 44495 | A rearrangement of conjuncts for a 4-right-nested conjunction. (Contributed by Alan Sare, 30-May-2018.) |
⊢ ((((𝜒 ∧ 𝜓) ∧ 𝜑) ∧ 𝜃) → 𝜏) ⇒ ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏) | ||
Theorem | 4an4132 44496 | A rearrangement of conjuncts for a 4-right-nested conjunction. (Contributed by Alan Sare, 30-May-2018.) |
⊢ ((((𝜃 ∧ 𝜒) ∧ 𝜓) ∧ 𝜑) → 𝜏) ⇒ ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏) | ||
Theorem | expcomdg 44497 | Biconditional form of expcomd 416. (Contributed by Alan Sare, 22-Jul-2012.) (New usage is discouraged.) |
⊢ ((𝜑 → ((𝜓 ∧ 𝜒) → 𝜃)) ↔ (𝜑 → (𝜒 → (𝜓 → 𝜃)))) | ||
Theorem | iidn3 44498 | idn3 44612 without virtual deduction connectives. Special theorem needed for the Virtual Deduction translation tool. (Contributed by Alan Sare, 23-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → (𝜓 → (𝜒 → 𝜒))) | ||
Theorem | ee222 44499 | e222 44633 without virtual deduction connectives. Special theorem needed for the Virtual Deduction translation tool. (Contributed by Alan Sare, 7-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → (𝜓 → 𝜒)) & ⊢ (𝜑 → (𝜓 → 𝜃)) & ⊢ (𝜑 → (𝜓 → 𝜏)) & ⊢ (𝜒 → (𝜃 → (𝜏 → 𝜂))) ⇒ ⊢ (𝜑 → (𝜓 → 𝜂)) | ||
Theorem | ee3bir 44500 | Right-biconditional form of e3 44734 without virtual deduction connectives. Special theorem needed for the Virtual Deduction translation tool. (Contributed by Alan Sare, 22-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃))) & ⊢ (𝜏 ↔ 𝜃) ⇒ ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜏))) |
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