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Theorem List for Metamath Proof Explorer - 33201-33300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremaxc4i-o 33201 Inference version of ax-c4 33187. (Contributed by NM, 3-Jan-1993.) (New usage is discouraged.)
(∀𝑥𝜑𝜓)       (∀𝑥𝜑 → ∀𝑥𝜓)

Theoremequid1 33202 Proof of equid 1926 from our older axioms. This is often an axiom of equality in textbook systems, but we don't need it as an axiom since it can be proved from our other axioms (although the proof, as you can see below, is not as obvious as you might think). This proof uses only axioms without distinct variable conditions and requires no dummy variables. A simpler proof, similar to Tarski's, is possible if we make use of ax-5 1827; see the proof of equid 1926. See equid1ALT 33228 for an alternate proof. (Contributed by NM, 10-Jan-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥 = 𝑥

Theoremequcomi1 33203 Proof of equcomi 1931 from equid1 33202, avoiding use of ax-5 1827 (the only use of ax-5 1827 is via ax7 1930, so using ax-7 1922 instead would remove dependency on ax-5 1827). (Contributed by BJ, 8-Jul-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = 𝑦𝑦 = 𝑥)

Theoremaecom-o 33204 Commutation law for identical variable specifiers. The antecedent and consequent are true when 𝑥 and 𝑦 are substituted with the same variable. Lemma L12 in [Megill] p. 445 (p. 12 of the preprint). Version of aecom 2299 using ax-c11 33190. Unlike axc11nfromc11 33229, this version does not require ax-5 1827 (see comment of equcomi1 33203). (Contributed by NM, 10-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)

Theoremaecoms-o 33205 A commutation rule for identical variable specifiers. Version of aecoms 2300 using ax-c11 33190. (Contributed by NM, 10-May-1993.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦𝜑)       (∀𝑦 𝑦 = 𝑥𝜑)

Theoremhbae-o 33206 All variables are effectively bound in an identical variable specifier. Version of hbae 2303 using ax-c11 33190. (Contributed by NM, 13-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑧𝑥 𝑥 = 𝑦)

Theoremdral1-o 33207 Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). Version of dral1 2313 using ax-c11 33190. (Contributed by NM, 24-Nov-1994.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓))

Theoremax12fromc15 33208 Rederivation of axiom ax-12 2034 from ax-c15 33192, ax-c11 33190 (used through dral1-o 33207), and other older axioms. See theorem axc15 2291 for the derivation of ax-c15 33192 from ax-12 2034.

An open problem is whether we can prove this using ax-c11n 33191 instead of ax-c11 33190.

This proof uses newer axioms ax-4 1728 and ax-6 1875, but since these are proved from the older axioms above, this is acceptable and lets us avoid having to reprove several earlier theorems to use ax-c4 33187 and ax-c10 33189. (Contributed by NM, 22-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

(𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))

Theoremax13fromc9 33209 Derive ax-13 2234 from ax-c9 33193 and other older axioms.

This proof uses newer axioms ax-4 1728 and ax-6 1875, but since these are proved from the older axioms above, this is acceptable and lets us avoid having to reprove several earlier theorems to use ax-c4 33187 and ax-c10 33189. (Contributed by NM, 21-Dec-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))

21.22.3  Legacy theorems using obsolete axioms

These theorems were mostly intended to study properties of the older axiom schemes and are not useful outside of this section. They should not be used outside of this section. They may be deleted when they are deemed to no longer be of interest.

Theoremax5ALT 33210* Axiom to quantify a variable over a formula in which it does not occur. Axiom C5 in [Megill] p. 444 (p. 11 of the preprint). Also appears as Axiom B6 (p. 75) of system S2 of [Tarski] p. 77 and Axiom C5-1 of [Monk2] p. 113.

(This theorem simply repeats ax-5 1827 so that we can include the following note, which applies only to the obsolete axiomatization.)

This axiom is logically redundant in the (logically complete) predicate calculus axiom system consisting of ax-gen 1713, ax-c4 33187, ax-c5 33186, ax-11 2021, ax-c7 33188, ax-7 1922, ax-c9 33193, ax-c10 33189, ax-c11 33190, ax-8 1979, ax-9 1986, ax-c14 33194, ax-c15 33192, and ax-c16 33195: in that system, we can derive any instance of ax-5 1827 not containing wff variables by induction on formula length, using ax5eq 33235 and ax5el 33240 for the basis together with hbn 2131, hbal 2023, and hbim 2112. However, if we omit this axiom, our development would be quite inconvenient since we could work only with specific instances of wffs containing no wff variables - this axiom introduces the concept of a setvar variable not occurring in a wff (as opposed to just two setvar variables being distinct). (Contributed by NM, 19-Aug-2017.) (New usage is discouraged.) (Proof modification is discouraged.)

(𝜑 → ∀𝑥𝜑)

Theoremsps-o 33211 Generalization of antecedent. (Contributed by NM, 5-Jan-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)       (∀𝑥𝜑𝜓)

Theoremhbequid 33212 Bound-variable hypothesis builder for 𝑥 = 𝑥. This theorem tells us that any variable, including 𝑥, is effectively not free in 𝑥 = 𝑥, even though 𝑥 is technically free according to the traditional definition of free variable. (The proof does not use ax-c10 33189.) (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 23-Mar-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥)

Theoremnfequid-o 33213 Bound-variable hypothesis builder for 𝑥 = 𝑥. This theorem tells us that any variable, including 𝑥, is effectively not free in 𝑥 = 𝑥, even though 𝑥 is technically free according to the traditional definition of free variable. (The proof uses only ax-4 1728, ax-7 1922, ax-c9 33193, and ax-gen 1713. This shows that this can be proved without ax6 2239, even though the theorem equid 1926 cannot be. A shorter proof using ax6 2239 is obtainable from equid 1926 and hbth 1720.) Remark added 2-Dec-2015 NM: This proof does implicitly use ax6v 1876, which is used for the derivation of axc9 2290, unless we consider ax-c9 33193 the starting axiom rather than ax-13 2234. (Contributed by NM, 13-Jan-2011.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑦 𝑥 = 𝑥

Theoremaxc5c7 33214 Proof of a single axiom that can replace ax-c5 33186 and ax-c7 33188. See axc5c7toc5 33215 and axc5c7toc7 33216 for the rederivation of those axioms. (Contributed by Scott Fenton, 12-Sep-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
((∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥𝜑) → 𝜑)

Theoremaxc5c7toc5 33215 Rederivation of ax-c5 33186 from axc5c7 33214. Only propositional calculus is used for the rederivation. (Contributed by Scott Fenton, 12-Sep-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥𝜑𝜑)

Theoremaxc5c7toc7 33216 Rederivation of ax-c7 33188 from axc5c7 33214. Only propositional calculus is used for the rederivation. (Contributed by Scott Fenton, 12-Sep-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 ¬ ∀𝑥𝜑𝜑)

Theoremaxc711 33217 Proof of a single axiom that can replace both ax-c7 33188 and ax-11 2021. See axc711toc7 33219 and axc711to11 33220 for the rederivation of those axioms. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 ¬ ∀𝑦𝑥𝜑 → ∀𝑦𝜑)

Theoremnfa1-o 33218 𝑥 is not free in 𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝑥𝜑

Theoremaxc711toc7 33219 Rederivation of ax-c7 33188 from axc711 33217. Note that ax-c7 33188 and ax-11 2021 are not used by the rederivation. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 ¬ ∀𝑥𝜑𝜑)

Theoremaxc711to11 33220 Rederivation of ax-11 2021 from axc711 33217. Note that ax-c7 33188 and ax-11 2021 are not used by the rederivation. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)

Theoremaxc5c711 33221 Proof of a single axiom that can replace ax-c5 33186, ax-c7 33188, and ax-11 2021 in a subsystem that includes these axioms plus ax-c4 33187 and ax-gen 1713 (and propositional calculus). See axc5c711toc5 33222, axc5c711toc7 33223, and axc5c711to11 33224 for the rederivation of those axioms. This theorem extends the idea in Scott Fenton's axc5c7 33214. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
((∀𝑥𝑦 ¬ ∀𝑥𝑦𝜑 → ∀𝑥𝜑) → 𝜑)

Theoremaxc5c711toc5 33222 Rederivation of ax-c5 33186 from axc5c711 33221. Only propositional calculus is used by the rederivation. (Contributed by NM, 19-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥𝜑𝜑)

Theoremaxc5c711toc7 33223 Rederivation of ax-c7 33188 from axc5c711 33221. Note that ax-c7 33188 and ax-11 2021 are not used by the rederivation. The use of alimi 1730 (which uses ax-c5 33186) is allowed since we have already proved axc5c711toc5 33222. (Contributed by NM, 19-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 ¬ ∀𝑥𝜑𝜑)

Theoremaxc5c711to11 33224 Rederivation of ax-11 2021 from axc5c711 33221. Note that ax-c7 33188 and ax-11 2021 are not used by the rederivation. The use of alimi 1730 (which uses ax-c5 33186) is allowed since we have already proved axc5c711toc5 33222. (Contributed by NM, 19-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)

Theoremequidqe 33225 equid 1926 with existential quantifier without using ax-c5 33186 or ax-5 1827. (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 27-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
¬ ∀𝑦 ¬ 𝑥 = 𝑥

Theoremaxc5sp1 33226 A special case of ax-c5 33186 without using ax-c5 33186 or ax-5 1827. (Contributed by NM, 13-Jan-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑦 ¬ 𝑥 = 𝑥 → ¬ 𝑥 = 𝑥)

Theoremequidq 33227 equid 1926 with universal quantifier without using ax-c5 33186 or ax-5 1827. (Contributed by NM, 13-Jan-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑦 𝑥 = 𝑥

Theoremequid1ALT 33228 Alternate proof of equid 1926 and equid1 33202 from older axioms ax-c7 33188, ax-c10 33189 and ax-c9 33193. (Contributed by NM, 10-Jan-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥 = 𝑥

Theoremaxc11nfromc11 33229 Rederivation of ax-c11n 33191 from original version ax-c11 33190. See theorem axc11 2302 for the derivation of ax-c11 33190 from ax-c11n 33191.

This theorem should not be referenced in any proof. Instead, use ax-c11n 33191 above so that uses of ax-c11n 33191 can be more easily identified, or use aecom-o 33204 when this form is needed for studies involving ax-c11 33190 and omitting ax-5 1827. (Contributed by NM, 16-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

(∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)

Theoremnaecoms-o 33230 A commutation rule for distinct variable specifiers. Version of naecoms 2301 using ax-c11 33190. (Contributed by NM, 2-Jan-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦𝜑)       (¬ ∀𝑦 𝑦 = 𝑥𝜑)

Theoremhbnae-o 33231 All variables are effectively bound in a distinct variable specifier. Lemma L19 in [Megill] p. 446 (p. 14 of the preprint). Version of hbnae 2305 using ax-c11 33190. (Contributed by NM, 13-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦)

Theoremdvelimf-o 33232 Proof of dvelimh 2324 that uses ax-c11 33190 but not ax-c15 33192, ax-c11n 33191, or ax-12 2034. Version of dvelimh 2324 using ax-c11 33190 instead of axc11 2302. (Contributed by NM, 12-Nov-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → ∀𝑥𝜑)    &   (𝜓 → ∀𝑧𝜓)    &   (𝑧 = 𝑦 → (𝜑𝜓))       (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))

Theoremdral2-o 33233 Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). Version of dral2 2312 using ax-c11 33190. (Contributed by NM, 27-Feb-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥 𝑥 = 𝑦 → (∀𝑧𝜑 ↔ ∀𝑧𝜓))

Theoremaev-o 33234* A "distinctor elimination" lemma with no restrictions on variables in the consequent, proved without using ax-c16 33195. Version of aev 1970 using ax-c11 33190. (Contributed by NM, 8-Nov-2006.) (Proof shortened by Andrew Salmon, 21-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑤 = 𝑣)

Theoremax5eq 33235* Theorem to add distinct quantifier to atomic formula. (This theorem demonstrates the induction basis for ax-5 1827 considered as a metatheorem. Do not use it for later proofs - use ax-5 1827 instead, to avoid reference to the redundant axiom ax-c16 33195.) (Contributed by NM, 10-Jan-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)

Theoremdveeq2-o 33236* Quantifier introduction when one pair of variables is distinct. Version of dveeq2 2286 using ax-c15 33192. (Contributed by NM, 2-Jan-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))

Theoremaxc16g-o 33237* A generalization of axiom ax-c16 33195. Version of axc16g 2119 using ax-c11 33190. (Contributed by NM, 15-May-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))

Theoremdveeq1-o 33238* Quantifier introduction when one pair of variables is distinct. Version of dveeq1 2288 using ax-c11 . (Contributed by NM, 2-Jan-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))

Theoremdveeq1-o16 33239* Version of dveeq1 2288 using ax-c16 33195 instead of ax-5 1827. (Contributed by NM, 29-Apr-2008.) TODO: Recover proof from older set.mm to remove use of ax-5 1827. (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))

Theoremax5el 33240* Theorem to add distinct quantifier to atomic formula. This theorem demonstrates the induction basis for ax-5 1827 considered as a metatheorem.) (Contributed by NM, 22-Jun-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥𝑦 → ∀𝑧 𝑥𝑦)

Theoremaxc11n-16 33241* This theorem shows that, given ax-c16 33195, we can derive a version of ax-c11n 33191. However, it is weaker than ax-c11n 33191 because it has a distinct variable requirement. (Contributed by Andrew Salmon, 27-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑧 → ∀𝑧 𝑧 = 𝑥)

Theoremdveel2ALT 33242* Alternate proof of dveel2 2359 using ax-c16 33195 instead of ax-5 1827. (Contributed by NM, 10-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑧𝑦 → ∀𝑥 𝑧𝑦))

Theoremax12f 33243 Basis step for constructing a substitution instance of ax-c15 33192 without using ax-c15 33192. We can start with any formula 𝜑 in which 𝑥 is not free. (Contributed by NM, 21-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → ∀𝑥𝜑)       (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))

Theoremax12eq 33244 Basis step for constructing a substitution instance of ax-c15 33192 without using ax-c15 33192. Atomic formula for equality predicate. (Contributed by NM, 22-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝑧 = 𝑤 → ∀𝑥(𝑥 = 𝑦𝑧 = 𝑤))))

Theoremax12el 33245 Basis step for constructing a substitution instance of ax-c15 33192 without using ax-c15 33192. Atomic formula for membership predicate. (Contributed by NM, 22-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝑧𝑤 → ∀𝑥(𝑥 = 𝑦𝑧𝑤))))

Theoremax12indn 33246 Induction step for constructing a substitution instance of ax-c15 33192 without using ax-c15 33192. Negation case. (Contributed by NM, 21-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))       (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (¬ 𝜑 → ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑))))

Theoremax12indi 33247 Induction step for constructing a substitution instance of ax-c15 33192 without using ax-c15 33192. Implication case. (Contributed by NM, 21-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))    &   (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜓 → ∀𝑥(𝑥 = 𝑦𝜓))))       (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → ((𝜑𝜓) → ∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)))))

Theoremax12indalem 33248 Lemma for ax12inda2 33250 and ax12inda 33251. (Contributed by NM, 24-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))       (¬ ∀𝑦 𝑦 = 𝑧 → (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑)))))

Theoremax12inda2ALT 33249* Alternate proof of ax12inda2 33250, slightly more direct and not requiring ax-c16 33195. (Contributed by NM, 4-May-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))       (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑))))

Theoremax12inda2 33250* Induction step for constructing a substitution instance of ax-c15 33192 without using ax-c15 33192. Quantification case. When 𝑧 and 𝑦 are distinct, this theorem avoids the dummy variables needed by the more general ax12inda 33251. (Contributed by NM, 24-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))       (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑))))

Theoremax12inda 33251* Induction step for constructing a substitution instance of ax-c15 33192 without using ax-c15 33192. Quantification case. (When 𝑧 and 𝑦 are distinct, ax12inda2 33250 may be used instead to avoid the dummy variable 𝑤 in the proof.) (Contributed by NM, 24-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑤 → (𝑥 = 𝑤 → (𝜑 → ∀𝑥(𝑥 = 𝑤𝜑))))       (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑))))

Theoremax12v2-o 33252* Rederivation of ax-c15 33192 from ax12v 2035 (without using ax-c15 33192 or the full ax-12 2034). Thus, the hypothesis (ax12v 2035) provides an alternate axiom that can be used in place of ax-c15 33192. See also axc15 2291. (Contributed by NM, 2-Feb-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧𝜑)))       (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))

Theoremax12a2-o 33253* Derive ax-c15 33192 from a hypothesis in the form of ax-12 2034, without using ax-12 2034 or ax-c15 33192. The hypothesis is weaker than ax-12 2034, with 𝑧 both distinct from 𝑥 and not occurring in 𝜑. Thus, the hypothesis provides an alternate axiom that can be used in place of ax-12 2034, if we also have ax-c11 33190, which this proof uses. As theorem ax12 2292 shows, the distinct variable conditions are optional. An open problem is whether we can derive this with ax-c11n 33191 instead of ax-c11 33190. (Contributed by NM, 2-Feb-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = 𝑧 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑧𝜑)))       (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))

Theoremaxc11-o 33254 Show that ax-c11 33190 can be derived from ax-c11n 33191 and ax-12 2034. An open problem is whether this theorem can be derived from ax-c11n 33191 and the others when ax-12 2034 is replaced with ax-c15 33192 or ax12v 2035. See theorem axc11nfromc11 33229 for the rederivation of ax-c11n 33191 from axc11 2302.

Normally, axc11 2302 should be used rather than ax-c11 33190 or axc11-o 33254, except by theorems specifically studying the latter's properties. (Contributed by NM, 16-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

(∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))

Theoremfsumshftd 33255* Index shift of a finite sum with a weaker "implicit substitution" hypothesis than fsumshft 14354. The proof demonstrates how this can be derived starting from from fsumshft 14354. (Contributed by NM, 1-Nov-2019.)
(𝜑𝐾 ∈ ℤ)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℤ)    &   ((𝜑𝑗 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)    &   ((𝜑𝑗 = (𝑘𝐾)) → 𝐴 = 𝐵)       (𝜑 → Σ𝑗 ∈ (𝑀...𝑁)𝐴 = Σ𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾))𝐵)

TheoremfsumshftdOLD 33256* Obsolete version of fsumshftd 33255 as of 1-Nov-2019. (Contributed by NM, 1-Nov-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝜑𝐾 ∈ ℤ)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℤ)    &   ((𝜑𝑗 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)    &   ((𝜑𝑗 = (𝑘𝐾)) → 𝐴 = 𝐵)       (𝜑 → Σ𝑗 ∈ (𝑀...𝑁)𝐴 = Σ𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾))𝐵)

Axiomax-riotaBAD 33257 Define restricted description binder. In case it doesn't exist, we return a set which is not a member of the domain of discourse 𝐴. See also comments for df-iota 5768. (Contributed by NM, 15-Sep-2011.) (Revised by Mario Carneiro, 15-Oct-2016.) WARNING: THIS "AXIOM", WHICH IS THE OLD df-riota 6511, CONFLICTS WITH (THE NEW) df-riota 6511 AND MAKES THE SYSTEM IN set.mm INCONSISTENT. IT IS TEMPORARY AND WILL BE DELETED AFTER ALL USES ARE ELIMINATED.
(𝑥𝐴 𝜑) = if(∃!𝑥𝐴 𝜑, (℩𝑥(𝑥𝐴𝜑)), (Undef‘{𝑥𝑥𝐴}))

TheoremriotaclbgBAD 33258* Closure of restricted iota. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 24-Dec-2016.)
(𝐴𝑉 → (∃!𝑥𝐴 𝜑 ↔ (𝑥𝐴 𝜑) ∈ 𝐴))

TheoremriotaclbBAD 33259* Closure of restricted iota. (Contributed by NM, 15-Sep-2011.)
𝐴 ∈ V       (∃!𝑥𝐴 𝜑 ↔ (𝑥𝐴 𝜑) ∈ 𝐴)

Theoremriotasvd 33260* Deduction version of riotasv 33263. (Contributed by NM, 4-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
(𝜑𝐷 = (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)))    &   (𝜑𝐷𝐴)       ((𝜑𝐴𝑉) → ((𝑦𝐵𝜓) → 𝐷 = 𝐶))

Theoremriotasv2d 33261* Value of description binder 𝐷 for a single-valued class expression 𝐶(𝑦) (as in e.g. reusv2 4800). Special case of riota2f 6532. (Contributed by NM, 2-Mar-2013.)
𝑦𝜑    &   (𝜑𝑦𝐹)    &   (𝜑 → Ⅎ𝑦𝜒)    &   (𝜑𝐷 = (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)))    &   ((𝜑𝑦 = 𝐸) → (𝜓𝜒))    &   ((𝜑𝑦 = 𝐸) → 𝐶 = 𝐹)    &   (𝜑𝐷𝐴)    &   (𝜑𝐸𝐵)    &   (𝜑𝜒)       ((𝜑𝐴𝑉) → 𝐷 = 𝐹)

Theoremriotasv2s 33262* The value of description binder 𝐷 for a single-valued class expression 𝐶(𝑦) (as in e.g. reusv2 4800) in the form of a substitution instance. Special case of riota2f 6532. (Contributed by NM, 3-Mar-2013.) (Proof shortened by Mario Carneiro, 6-Dec-2016.)
𝐷 = (𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶))       ((𝐴𝑉𝐷𝐴 ∧ (𝐸𝐵[𝐸 / 𝑦]𝜑)) → 𝐷 = 𝐸 / 𝑦𝐶)

Theoremriotasv 33263* Value of description binder 𝐷 for a single-valued class expression 𝐶(𝑦) (as in e.g. reusv2 4800). Special case of riota2f 6532. (Contributed by NM, 26-Jan-2013.) (Proof shortened by Mario Carneiro, 6-Dec-2016.)
𝐴 ∈ V    &   𝐷 = (𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶))       ((𝐷𝐴𝑦𝐵𝜑) → 𝐷 = 𝐶)

Theoremriotasv3d 33264* A property 𝜒 holding for a representative of a single-valued class expression 𝐶(𝑦) (see e.g. reusv2 4800) also holds for its description binder 𝐷 (in the form of property 𝜃). (Contributed by NM, 5-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑦𝜃)    &   (𝜑𝐷 = (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)))    &   ((𝜑𝐶 = 𝐷) → (𝜒𝜃))    &   (𝜑 → ((𝑦𝐵𝜓) → 𝜒))    &   (𝜑𝐷𝐴)    &   (𝜑 → ∃𝑦𝐵 𝜓)       ((𝜑𝐴𝑉) → 𝜃)

21.22.4  Experiments with weak deduction theorem

Theoremelimhyps 33265 A version of elimhyp 4096 using explicit substitution. (Contributed by NM, 15-Jun-2019.)
[𝐵 / 𝑥]𝜑       [if(𝜑, 𝑥, 𝐵) / 𝑥]𝜑

Theoremdedths 33266 A version of weak deduction theorem dedth 4089 using explicit substitution. (Contributed by NM, 15-Jun-2019.)
[if(𝜑, 𝑥, 𝐵) / 𝑥]𝜓       (𝜑𝜓)

TheoremrenegclALT 33267 Closure law for negative of reals. Demonstrates use of weak deduction theorem with explicit substitution. The proof is much longer than that of renegcl 10223. (Contributed by NM, 15-Jun-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴 ∈ ℝ → -𝐴 ∈ ℝ)

Theoremelimhyps2 33268 Generalization of elimhyps 33265 that is not useful unless we can separately prove 𝐴 ∈ V. (Contributed by NM, 13-Jun-2019.)
[𝐵 / 𝑥]𝜑       [if([𝐴 / 𝑥]𝜑, 𝐴, 𝐵) / 𝑥]𝜑

Theoremdedths2 33269 Generalization of dedths 33266 that is not useful unless we can separately prove 𝐴 ∈ V. (Contributed by NM, 13-Jun-2019.)
[if([𝐴 / 𝑥]𝜑, 𝐴, 𝐵) / 𝑥]𝜓       ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)

Theorem19.9alt 33270 Version of 19.9t 2059 for universal quantifier. (Contributed by NM, 9-Nov-2020.)
(Ⅎ𝑥𝜑 → (∀𝑥𝜑𝜑))

Theoremnfcxfrdf 33271 A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by NM, 19-Nov-2020.)
𝑥𝜑    &   (𝜑𝐴 = 𝐵)    &   (𝜑𝑥𝐵)       (𝜑𝑥𝐴)

Theoremnfded 33272 A deduction theorem that converts a not-free inference directly to deduction form. The first hypothesis is the hypothesis of the deduction form. The second is an equality deduction (e.g. (𝑥𝐴 {𝑦 ∣ ∀𝑥𝑦𝐴} = 𝐴)) that starts from abidnf 3342. The last is assigned to the inference form (e.g. 𝑥 {𝑦 ∣ ∀𝑥𝑦𝐴}) whose hypothesis is satisfied using nfaba1 2756. (Contributed by NM, 19-Nov-2020.)
(𝜑𝑥𝐴)    &   (𝑥𝐴𝐵 = 𝐶)    &   𝑥𝐵       (𝜑𝑥𝐶)

Theoremnfded2 33273 A deduction theorem that converts a not-free inference directly to deduction form. The first 2 hypotheses are the hypotheses of the deduction form. The third is an equality deduction (e.g. ((𝑥𝐴𝑥𝐵) → ⟨{𝑦 ∣ ∀𝑥𝑦𝐴}, {𝑦 ∣ ∀𝑥𝑦𝐵}⟩ = ⟨𝐴, 𝐵⟩) for nfopd 4357) that starts from abidnf 3342. The last is assigned to the inference form (e.g. 𝑥⟨{𝑦 ∣ ∀𝑥𝑦𝐴}, {𝑦 ∣ ∀𝑥𝑦𝐵}⟩ for nfop 4356) whose hypotheses are satisfied using nfaba1 2756. (Contributed by NM, 19-Nov-2020.)
(𝜑𝑥𝐴)    &   (𝜑𝑥𝐵)    &   ((𝑥𝐴𝑥𝐵) → 𝐶 = 𝐷)    &   𝑥𝐶       (𝜑𝑥𝐷)

TheoremnfunidALT2 33274 Deduction version of nfuni 4378. (Contributed by NM, 19-Nov-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝑥𝐴)       (𝜑𝑥 𝐴)

TheoremnfunidALT 33275 Deduction version of nfuni 4378. (Contributed by NM, 19-Nov-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝑥𝐴)       (𝜑𝑥 𝐴)

TheoremnfopdALT 33276 Deduction version of bound-variable hypothesis builder nfop 4356. This shows how the deduction version of a not-free theorem such as nfop 4356 can be created from the corresponding not-free inference theorem. (Contributed by NM, 19-Nov-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝑥𝐴)    &   (𝜑𝑥𝐵)       (𝜑𝑥𝐴, 𝐵⟩)

21.22.5  Miscellanea

Theoremcnaddcom 33277 Recover the commutative law of addition for complex numbers from the Abelian group structure. (Contributed by NM, 17-Mar-2013.) (Proof modification is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))

Theoremtoycom 33278* Show the commutative law for an operation 𝑂 on a toy structure class 𝐶 of commuatitive operations on . This illustrates how a structure class can be partially specialized. In practice, we would ordinarily define a new constant such as "CAbel" in place of 𝐶. (Contributed by NM, 17-Mar-2013.) (Proof modification is discouraged.)
𝐶 = {𝑔 ∈ Abel ∣ (Base‘𝑔) = ℂ}    &    + = (+g𝐾)       ((𝐾𝐶𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))

21.22.6  Atoms, hyperplanes, and covering in a left vector space (or module)

Syntaxclsa 33279 Extend class notation with all 1-dim subspaces (atoms) of a left module or left vector space.
class LSAtoms

Syntaxclsh 33280 Extend class notation with all subspaces of a left module or left vector space that are hyperplanes.
class LSHyp

Definitiondf-lsatoms 33281* Define the set of all 1-dim subspaces (atoms) of a left module or left vector space. (Contributed by NM, 9-Apr-2014.)
LSAtoms = (𝑤 ∈ V ↦ ran (𝑣 ∈ ((Base‘𝑤) ∖ {(0g𝑤)}) ↦ ((LSpan‘𝑤)‘{𝑣})))

Definitiondf-lshyp 33282* Define the set of all hyperplanes of a left module or left vector space. Also called co-atoms, these are subspaces that are one dimension less that the full space. (Contributed by NM, 29-Jun-2014.)
LSHyp = (𝑤 ∈ V ↦ {𝑠 ∈ (LSubSp‘𝑤) ∣ (𝑠 ≠ (Base‘𝑤) ∧ ∃𝑣 ∈ (Base‘𝑤)((LSpan‘𝑤)‘(𝑠 ∪ {𝑣})) = (Base‘𝑤))})

Theoremlshpset 33283* The set of all hyperplanes of a left module or left vector space. The vector 𝑣 is called a generating vector for the hyperplane. (Contributed by NM, 29-Jun-2014.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   𝐻 = (LSHyp‘𝑊)       (𝑊𝑋𝐻 = {𝑠𝑆 ∣ (𝑠𝑉 ∧ ∃𝑣𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉)})

Theoremislshp 33284* The predicate "is a hyperplane" (of a left module or left vector space). (Contributed by NM, 29-Jun-2014.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   𝐻 = (LSHyp‘𝑊)       (𝑊𝑋 → (𝑈𝐻 ↔ (𝑈𝑆𝑈𝑉 ∧ ∃𝑣𝑉 (𝑁‘(𝑈 ∪ {𝑣})) = 𝑉)))

Theoremislshpsm 33285* Hyperplane properties expressed with subspace sum. (Contributed by NM, 3-Jul-2014.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   𝑆 = (LSubSp‘𝑊)    &    = (LSSum‘𝑊)    &   𝐻 = (LSHyp‘𝑊)    &   (𝜑𝑊 ∈ LMod)       (𝜑 → (𝑈𝐻 ↔ (𝑈𝑆𝑈𝑉 ∧ ∃𝑣𝑉 (𝑈 (𝑁‘{𝑣})) = 𝑉)))

Theoremlshplss 33286 A hyperplane is a subspace. (Contributed by NM, 3-Jul-2014.)
𝑆 = (LSubSp‘𝑊)    &   𝐻 = (LSHyp‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑈𝐻)       (𝜑𝑈𝑆)

Theoremlshpne 33287 A hyperplane is not equal to the vector space. (Contributed by NM, 4-Jul-2014.)
𝑉 = (Base‘𝑊)    &   𝐻 = (LSHyp‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑈𝐻)       (𝜑𝑈𝑉)

Theoremlshpnel 33288 A hyperplane's generating vector does not belong to the hyperplane. (Contributed by NM, 3-Jul-2014.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &    = (LSSum‘𝑊)    &   𝐻 = (LSHyp‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑈𝐻)    &   (𝜑𝑋𝑉)    &   (𝜑 → (𝑈 (𝑁‘{𝑋})) = 𝑉)       (𝜑 → ¬ 𝑋𝑈)

Theoremlshpnelb 33289 The subspace sum of a hyperplane and the span of an element equals the vector space iff the element is not in the hyperplane. (Contributed by NM, 2-Oct-2014.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &    = (LSSum‘𝑊)    &   𝐻 = (LSHyp‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑈𝐻)    &   (𝜑𝑋𝑉)       (𝜑 → (¬ 𝑋𝑈 ↔ (𝑈 (𝑁‘{𝑋})) = 𝑉))

Theoremlshpnel2N 33290 Condition that determines a hyperplane. (Contributed by NM, 3-Oct-2014.) (New usage is discouraged.)
𝑉 = (Base‘𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &    = (LSSum‘𝑊)    &   𝐻 = (LSHyp‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑈𝑆)    &   (𝜑𝑈𝑉)    &   (𝜑𝑋𝑉)    &   (𝜑 → ¬ 𝑋𝑈)       (𝜑 → (𝑈𝐻 ↔ (𝑈 (𝑁‘{𝑋})) = 𝑉))

Theoremlshpne0 33291 The member of the span in the hyperplane definition does not belong to the hyperplane. (Contributed by NM, 14-Jul-2014.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &    = (LSSum‘𝑊)    &   𝐻 = (LSHyp‘𝑊)    &    0 = (0g𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑈𝐻)    &   (𝜑𝑋𝑉)    &   (𝜑 → (𝑈 (𝑁‘{𝑋})) = 𝑉)       (𝜑𝑋0 )

Theoremlshpdisj 33292 A hyperplane and the span in the hyperplane definition are disjoint. (Contributed by NM, 3-Jul-2014.)
𝑉 = (Base‘𝑊)    &    0 = (0g𝑊)    &   𝑁 = (LSpan‘𝑊)    &    = (LSSum‘𝑊)    &   𝐻 = (LSHyp‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑈𝐻)    &   (𝜑𝑋𝑉)    &   (𝜑 → (𝑈 (𝑁‘{𝑋})) = 𝑉)       (𝜑 → (𝑈 ∩ (𝑁‘{𝑋})) = { 0 })

Theoremlshpcmp 33293 If two hyperplanes are comparable, they are equal. (Contributed by NM, 9-Oct-2014.)
𝐻 = (LSHyp‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑇𝐻)    &   (𝜑𝑈𝐻)       (𝜑 → (𝑇𝑈𝑇 = 𝑈))

TheoremlshpinN 33294 The intersection of two different hyperplanes is not a hyperplane. (Contributed by NM, 29-Oct-2014.) (New usage is discouraged.)
𝐻 = (LSHyp‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑇𝐻)    &   (𝜑𝑈𝐻)       (𝜑 → ((𝑇𝑈) ∈ 𝐻𝑇 = 𝑈))

Theoremlsatset 33295* The set of all 1-dim subspaces (atoms) of a left module or left vector space. (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 22-Sep-2015.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &    0 = (0g𝑊)    &   𝐴 = (LSAtoms‘𝑊)       (𝑊𝑋𝐴 = ran (𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣})))

Theoremislsat 33296* The predicate "is a 1-dim subspace (atom)" (of a left module or left vector space). (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &    0 = (0g𝑊)    &   𝐴 = (LSAtoms‘𝑊)       (𝑊𝑋 → (𝑈𝐴 ↔ ∃𝑥 ∈ (𝑉 ∖ { 0 })𝑈 = (𝑁‘{𝑥})))

Theoremlsatlspsn2 33297 The span of a nonzero singleton is an atom. TODO: make this obsolete and use lsatlspsn 33298 instead? (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &    0 = (0g𝑊)    &   𝐴 = (LSAtoms‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑋𝑉𝑋0 ) → (𝑁‘{𝑋}) ∈ 𝐴)

Theoremlsatlspsn 33298 The span of a nonzero singleton is an atom. (Contributed by NM, 16-Jan-2015.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &    0 = (0g𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))       (𝜑 → (𝑁‘{𝑋}) ∈ 𝐴)

Theoremislsati 33299* A 1-dim subspace (atom) (of a left module or left vector space) equals the span of some vector. (Contributed by NM, 1-Oct-2014.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   𝐴 = (LSAtoms‘𝑊)       ((𝑊𝑋𝑈𝐴) → ∃𝑣𝑉 𝑈 = (𝑁‘{𝑣}))

Theoremlsateln0 33300* A 1-dim subspace (atom) (of a left module or left vector space) contains a nonzero vector. (Contributed by NM, 2-Jan-2015.)
0 = (0g𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑈𝐴)       (𝜑 → ∃𝑣𝑈 𝑣0 )

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