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Theorem List for Metamath Proof Explorer - 33101-33200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsps-o 33101 Generalization of antecedent. (Contributed by NM, 5-Jan-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)       (∀𝑥𝜑𝜓)

Theoremhbequid 33102 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 33079.) (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 33103 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 1713, ax-7 1885, ax-c9 33083, and ax-gen 1700. This shows that this can be proved without ax6 2142, even though the theorem equid 1889 cannot be. A shorter proof using ax6 2142 is obtainable from equid 1889 and hbth 1706.) Remark added 2-Dec-2015 NM: This proof does implicitly use ax6v 1839, which is used for the derivation of axc9 2194, unless we consider ax-c9 33083 the starting axiom rather than ax-13 2137. (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 33104 Proof of a single axiom that can replace ax-c5 33076 and ax-c7 33078. See axc5c7toc5 33105 and axc5c7toc7 33106 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 33105 Rederivation of ax-c5 33076 from axc5c7 33104. 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 33106 Rederivation of ax-c7 33078 from axc5c7 33104. 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 33107 Proof of a single axiom that can replace both ax-c7 33078 and ax-11 1971. See axc711toc7 33109 and axc711to11 33110 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 33108 𝑥 is not free in 𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝑥𝜑

Theoremaxc711toc7 33109 Rederivation of ax-c7 33078 from axc711 33107. Note that ax-c7 33078 and ax-11 1971 are not used by the rederivation. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 ¬ ∀𝑥𝜑𝜑)

Theoremaxc711to11 33110 Rederivation of ax-11 1971 from axc711 33107. Note that ax-c7 33078 and ax-11 1971 are not used by the rederivation. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)

Theoremaxc5c711 33111 Proof of a single axiom that can replace ax-c5 33076, ax-c7 33078, and ax-11 1971 in a subsystem that includes these axioms plus ax-c4 33077 and ax-gen 1700 (and propositional calculus). See axc5c711toc5 33112, axc5c711toc7 33113, and axc5c711to11 33114 for the rederivation of those axioms. This theorem extends the idea in Scott Fenton's axc5c7 33104. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
((∀𝑥𝑦 ¬ ∀𝑥𝑦𝜑 → ∀𝑥𝜑) → 𝜑)

Theoremaxc5c711toc5 33112 Rederivation of ax-c5 33076 from axc5c711 33111. Only propositional calculus is used by the rederivation. (Contributed by NM, 19-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥𝜑𝜑)

Theoremaxc5c711toc7 33113 Rederivation of ax-c7 33078 from axc5c711 33111. Note that ax-c7 33078 and ax-11 1971 are not used by the rederivation. The use of alimi 1715 (which uses ax-c5 33076) is allowed since we have already proved axc5c711toc5 33112. (Contributed by NM, 19-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 ¬ ∀𝑥𝜑𝜑)

Theoremaxc5c711to11 33114 Rederivation of ax-11 1971 from axc5c711 33111. Note that ax-c7 33078 and ax-11 1971 are not used by the rederivation. The use of alimi 1715 (which uses ax-c5 33076) is allowed since we have already proved axc5c711toc5 33112. (Contributed by NM, 19-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)

Theoremequidqe 33115 equid 1889 with existential quantifier without using ax-c5 33076 or ax-5 1793. (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 33116 A special case of ax-c5 33076 without using ax-c5 33076 or ax-5 1793. (Contributed by NM, 13-Jan-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑦 ¬ 𝑥 = 𝑥 → ¬ 𝑥 = 𝑥)

Theoremequidq 33117 equid 1889 with universal quantifier without using ax-c5 33076 or ax-5 1793. (Contributed by NM, 13-Jan-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑦 𝑥 = 𝑥

Theoremequid1ALT 33118 Alternate proof of equid 1889 and equid1 33092 from older axioms ax-c7 33078, ax-c10 33079 and ax-c9 33083. (Contributed by NM, 10-Jan-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥 = 𝑥

Theoremaxc11nfromc11 33119 Rederivation of ax-c11n 33081 from original version ax-c11 33080. See theorem axc11 2206 for the derivation of ax-c11 33080 from ax-c11n 33081.

This theorem should not be referenced in any proof. Instead, use ax-c11n 33081 above so that uses of ax-c11n 33081 can be more easily identified, or use aecom-o 33094 when this form is needed for studies involving ax-c11 33080 and omitting ax-5 1793. (Contributed by NM, 16-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

(∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)

Theoremnaecoms-o 33120 A commutation rule for distinct variable specifiers. Version of naecoms 2205 using ax-c11 33080. (Contributed by NM, 2-Jan-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦𝜑)       (¬ ∀𝑦 𝑦 = 𝑥𝜑)

Theoremhbnae-o 33121 All variables are effectively bound in a distinct variable specifier. Lemma L19 in [Megill] p. 446 (p. 14 of the preprint). Version of hbnae 2209 using ax-c11 33080. (Contributed by NM, 13-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦)

Theoremdvelimf-o 33122 Proof of dvelimh 2228 that uses ax-c11 33080 but not ax-c15 33082, ax-c11n 33081, or ax-12 1983. Version of dvelimh 2228 using ax-c11 33080 instead of axc11 2206. (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 33123 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 2216 using ax-c11 33080. (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 33124* A "distinctor elimination" lemma with no restrictions on variables in the consequent, proved without using ax-c16 33085. Version of aev 1931 using ax-c11 33080. (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 33125* Theorem to add distinct quantifier to atomic formula. (This theorem demonstrates the induction basis for ax-5 1793 considered as a metatheorem. Do not use it for later proofs - use ax-5 1793 instead, to avoid reference to the redundant axiom ax-c16 33085.) (Contributed by NM, 10-Jan-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)

Theoremdveeq2-o 33126* Quantifier introduction when one pair of variables is distinct. Version of dveeq2 2190 using ax-c15 33082. (Contributed by NM, 2-Jan-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))

Theoremaxc16g-o 33127* A generalization of axiom ax-c16 33085. Version of axc16g 2071 using ax-c11 33080. (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 33128* Quantifier introduction when one pair of variables is distinct. Version of dveeq1 2192 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 33129* Version of dveeq1 2192 using ax-c16 33085 instead of ax-5 1793. (Contributed by NM, 29-Apr-2008.) TODO: Recover proof from older set.mm to remove use of ax-5 1793. (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))

Theoremax5el 33130* Theorem to add distinct quantifier to atomic formula. This theorem demonstrates the induction basis for ax-5 1793 considered as a metatheorem.) (Contributed by NM, 22-Jun-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥𝑦 → ∀𝑧 𝑥𝑦)

Theoremaxc11n-16 33131* This theorem shows that, given ax-c16 33085, we can derive a version of ax-c11n 33081. However, it is weaker than ax-c11n 33081 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 33132* Alternate proof of dveel2 2263 using ax-c16 33085 instead of ax-5 1793. (Contributed by NM, 10-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑧𝑦 → ∀𝑥 𝑧𝑦))

Theoremax12f 33133 Basis step for constructing a substitution instance of ax-c15 33082 without using ax-c15 33082. 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 33134 Basis step for constructing a substitution instance of ax-c15 33082 without using ax-c15 33082. 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 33135 Basis step for constructing a substitution instance of ax-c15 33082 without using ax-c15 33082. 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 33136 Induction step for constructing a substitution instance of ax-c15 33082 without using ax-c15 33082. 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 33137 Induction step for constructing a substitution instance of ax-c15 33082 without using ax-c15 33082. 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 33138 Lemma for ax12inda2 33140 and ax12inda 33141. (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 33139* Alternate proof of ax12inda2 33140, slightly more direct and not requiring ax-c16 33085. (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 33140* Induction step for constructing a substitution instance of ax-c15 33082 without using ax-c15 33082. Quantification case. When 𝑧 and 𝑦 are distinct, this theorem avoids the dummy variables needed by the more general ax12inda 33141. (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 33141* Induction step for constructing a substitution instance of ax-c15 33082 without using ax-c15 33082. Quantification case. (When 𝑧 and 𝑦 are distinct, ax12inda2 33140 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 33142* Rederivation of ax-c15 33082 from ax12v 1984 (without using ax-c15 33082 or the full ax-12 1983). Thus, the hypothesis (ax12v 1984) provides an alternate axiom that can be used in place of ax-c15 33082. See also axc15 2195. (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 33143* Derive ax-c15 33082 from a hypothesis in the form of ax-12 1983, without using ax-12 1983 or ax-c15 33082. The hypothesis is weaker than ax-12 1983, 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 1983, if we also have ax-c11 33080, which this proof uses. As theorem ax12 2196 shows, the distinct variable conditions are optional. An open problem is whether we can derive this with ax-c11n 33081 instead of ax-c11 33080. (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 33144 Show that ax-c11 33080 can be derived from ax-c11n 33081 and ax-12 1983. An open problem is whether this theorem can be derived from ax-c11n 33081 and the others when ax-12 1983 is replaced with ax-c15 33082 or ax12v 1984. See theorem axc11nfromc11 33119 for the rederivation of ax-c11n 33081 from axc11 2206.

Normally, axc11 2206 should be used rather than ax-c11 33080 or axc11-o 33144, 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 33145* Index shift of a finite sum with a weaker "implicit substitution" hypothesis than fsumshft 14223. The proof demonstrates how this can be derived starting from from fsumshft 14223. (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 33146* Obsolete version of fsumshftd 33145 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 33147 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 5653. (Contributed by NM, 15-Sep-2011.) (Revised by Mario Carneiro, 15-Oct-2016.) WARNING: THIS "AXIOM", WHICH IS THE OLD df-riota 6388, CONFLICTS WITH (THE NEW) df-riota 6388 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 33148* Closure of restricted iota. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 24-Dec-2016.)
(𝐴𝑉 → (∃!𝑥𝐴 𝜑 ↔ (𝑥𝐴 𝜑) ∈ 𝐴))

TheoremriotaclbBAD 33149* Closure of restricted iota. (Contributed by NM, 15-Sep-2011.)
𝐴 ∈ V       (∃!𝑥𝐴 𝜑 ↔ (𝑥𝐴 𝜑) ∈ 𝐴)

Theoremriotasvd 33150* Deduction version of riotasv 33153. (Contributed by NM, 4-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
(𝜑𝐷 = (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)))    &   (𝜑𝐷𝐴)       ((𝜑𝐴𝑉) → ((𝑦𝐵𝜓) → 𝐷 = 𝐶))

Theoremriotasv2d 33151* Value of description binder 𝐷 for a single-valued class expression 𝐶(𝑦) (as in e.g. reusv2 4699). Special case of riota2f 6409. (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 33152* The value of description binder 𝐷 for a single-valued class expression 𝐶(𝑦) (as in e.g. reusv2 4699) in the form of a substitution instance. Special case of riota2f 6409. (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 33153* Value of description binder 𝐷 for a single-valued class expression 𝐶(𝑦) (as in e.g. reusv2 4699). Special case of riota2f 6409. (Contributed by NM, 26-Jan-2013.) (Proof shortened by Mario Carneiro, 6-Dec-2016.)
𝐴 ∈ V    &   𝐷 = (𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶))       ((𝐷𝐴𝑦𝐵𝜑) → 𝐷 = 𝐶)

Theoremriotasv3d 33154* A property 𝜒 holding for a representative of a single-valued class expression 𝐶(𝑦) (see e.g. reusv2 4699) 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)

20.22.4  Experiments with weak deduction theorem

Theoremelimhyps 33155 A version of elimhyp 3999 using explicit substitution. (Contributed by NM, 15-Jun-2019.)
[𝐵 / 𝑥]𝜑       [if(𝜑, 𝑥, 𝐵) / 𝑥]𝜑

Theoremdedths 33156 A version of weak deduction theorem dedth 3992 using explicit substitution. (Contributed by NM, 15-Jun-2019.)
[if(𝜑, 𝑥, 𝐵) / 𝑥]𝜓       (𝜑𝜓)

TheoremrenegclALT 33157 Closure law for negative of reals. Demonstrates use of weak deduction theorem with explicit substitution. The proof is much longer than that of renegcl 10095. (Contributed by NM, 15-Jun-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴 ∈ ℝ → -𝐴 ∈ ℝ)

Theoremelimhyps2 33158 Generalization of elimhyps 33155 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 33159 Generalization of dedths 33156 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 33160 Version of 19.9t 2021 for universal quantifier. (Contributed by NM, 9-Nov-2020.)
(Ⅎ𝑥𝜑 → (∀𝑥𝜑𝜑))

Theoremnfcxfrdf 33161 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 33162 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 3246. The last is assigned to the inference form (e.g. 𝑥 {𝑦 ∣ ∀𝑥𝑦𝐴}) whose hypothesis is satisfied using nfaba1 2660. (Contributed by NM, 19-Nov-2020.)
(𝜑𝑥𝐴)    &   (𝑥𝐴𝐵 = 𝐶)    &   𝑥𝐵       (𝜑𝑥𝐶)

Theoremnfded2 33163 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 4255) that starts from abidnf 3246. The last is assigned to the inference form (e.g. 𝑥⟨{𝑦 ∣ ∀𝑥𝑦𝐴}, {𝑦 ∣ ∀𝑥𝑦𝐵}⟩ for nfop 4254) whose hypotheses are satisfied using nfaba1 2660. (Contributed by NM, 19-Nov-2020.)
(𝜑𝑥𝐴)    &   (𝜑𝑥𝐵)    &   ((𝑥𝐴𝑥𝐵) → 𝐶 = 𝐷)    &   𝑥𝐶       (𝜑𝑥𝐷)

TheoremnfunidALT2 33164 Deduction version of nfuni 4276. (Contributed by NM, 19-Nov-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝑥𝐴)       (𝜑𝑥 𝐴)

TheoremnfunidALT 33165 Deduction version of nfuni 4276. (Contributed by NM, 19-Nov-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝑥𝐴)       (𝜑𝑥 𝐴)

TheoremnfopdALT 33166 Deduction version of bound-variable hypothesis builder nfop 4254. This shows how the deduction version of a not-free theorem such as nfop 4254 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⟩)

20.22.5  Miscellanea

Theoremcnaddcom 33167 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 33168* 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))

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

Syntaxclsa 33169 Extend class notation with all 1-dim subspaces (atoms) of a left module or left vector space.
class LSAtoms

Syntaxclsh 33170 Extend class notation with all subspaces of a left module or left vector space that are hyperplanes.
class LSHyp

Definitiondf-lsatoms 33171* 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 33172* 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 33173* 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 33174* 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 33175* 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 33176 A hyperplane is a subspace. (Contributed by NM, 3-Jul-2014.)
𝑆 = (LSubSp‘𝑊)    &   𝐻 = (LSHyp‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑈𝐻)       (𝜑𝑈𝑆)

Theoremlshpne 33177 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 33178 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 33179 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 33180 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 33181 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 33182 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 33183 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 33184 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 33185* 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 33186* 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 33187 The span of a nonzero singleton is an atom. TODO: make this obsolete and use lsatlspsn 33188 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 33188 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 33189* 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 33190* 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 )

Theoremlsatlss 33191 The set of 1-dim subspaces is a set of subspaces. (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
𝑆 = (LSubSp‘𝑊)    &   𝐴 = (LSAtoms‘𝑊)       (𝑊 ∈ LMod → 𝐴𝑆)

Theoremlsatlssel 33192 An atom is a subspace. (Contributed by NM, 25-Aug-2014.)
𝑆 = (LSubSp‘𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑈𝐴)       (𝜑𝑈𝑆)

Theoremlsatssv 33193 An atom is a set of vectors. (Contributed by NM, 27-Feb-2015.)
𝑉 = (Base‘𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑄𝐴)       (𝜑𝑄𝑉)

Theoremlsatn0 33194 A 1-dim subspace (atom) of a left module or left vector space is nonzero. (atne0 28376 analog.) (Contributed by NM, 25-Aug-2014.)
0 = (0g𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑈𝐴)       (𝜑𝑈 ≠ { 0 })

Theoremlsatspn0 33195 The span of a vector is an atom iff the vector is nonzero. (Contributed by NM, 4-Feb-2015.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &    0 = (0g𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑋𝑉)       (𝜑 → ((𝑁‘{𝑋}) ∈ 𝐴𝑋0 ))

Theoremlsator0sp 33196 The span of a vector is either an atom or the zero subspace. (Contributed by NM, 15-Mar-2015.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &    0 = (0g𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑋𝑉)       (𝜑 → ((𝑁‘{𝑋}) ∈ 𝐴 ∨ (𝑁‘{𝑋}) = { 0 }))

Theoremlsatssn0 33197 A subspace (or any class) including an atom is nonzero. (Contributed by NM, 3-Feb-2015.)
0 = (0g𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑄𝐴)    &   (𝜑𝑄𝑈)       (𝜑𝑈 ≠ { 0 })

Theoremlsatcmp 33198 If two atoms are comparable, they are equal. (atsseq 28378 analog.) TODO: can lspsncmp 18841 shorten this? (Contributed by NM, 25-Aug-2014.)
𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑇𝐴)    &   (𝜑𝑈𝐴)       (𝜑 → (𝑇𝑈𝑇 = 𝑈))

Theoremlsatcmp2 33199 If an atom is included in at-most an atom, they are equal. More general version of lsatcmp 33198. TODO: can lspsncmp 18841 shorten this? (Contributed by NM, 3-Feb-2015.)
0 = (0g𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑇𝐴)    &   (𝜑 → (𝑈𝐴𝑈 = { 0 }))       (𝜑 → (𝑇𝑈𝑇 = 𝑈))

Theoremlsatel 33200 A nonzero vector in an atom determines the atom. (Contributed by NM, 25-Aug-2014.)
0 = (0g𝑊)    &   𝑁 = (LSpan‘𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑈𝐴)    &   (𝜑𝑋𝑈)    &   (𝜑𝑋0 )       (𝜑𝑈 = (𝑁‘{𝑋}))

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