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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | bccm1k 43101 | Generalized binomial coefficient: πΆ choose (πΎ β 1), when πΆ is not (πΎ β 1). (Contributed by Steve Rodriguez, 22-Apr-2020.) |
β’ (π β πΆ β (β β {(πΎ β 1)})) & β’ (π β πΎ β β) β β’ (π β (πΆCπ(πΎ β 1)) = ((πΆCππΎ) / ((πΆ β (πΎ β 1)) / πΎ))) | ||
Theorem | bccn0 43102 | Generalized binomial coefficient: πΆ choose 0. (Contributed by Steve Rodriguez, 22-Apr-2020.) |
β’ (π β πΆ β β) β β’ (π β (πΆCπ0) = 1) | ||
Theorem | bccn1 43103 | Generalized binomial coefficient: πΆ choose 1. (Contributed by Steve Rodriguez, 22-Apr-2020.) |
β’ (π β πΆ β β) β β’ (π β (πΆCπ1) = πΆ) | ||
Theorem | bccbc 43104 | The binomial coefficient and generalized binomial coefficient are equal when their arguments are nonnegative integers. (Contributed by Steve Rodriguez, 22-Apr-2020.) |
β’ (π β π β β0) & β’ (π β πΎ β β0) β β’ (π β (πCππΎ) = (πCπΎ)) | ||
Theorem | uzmptshftfval 43105* | When πΉ is a maps-to function on some set of upper integers π that returns a set π΅, (πΉ shift π) is another maps-to function on the shifted set of upper integers π. (Contributed by Steve Rodriguez, 22-Apr-2020.) |
β’ πΉ = (π₯ β π β¦ π΅) & β’ π΅ β V & β’ (π₯ = (π¦ β π) β π΅ = πΆ) & β’ π = (β€β₯βπ) & β’ π = (β€β₯β(π + π)) & β’ (π β π β β€) & β’ (π β π β β€) β β’ (π β (πΉ shift π) = (π¦ β π β¦ πΆ)) | ||
Theorem | dvradcnv2 43106* | The radius of convergence of the (formal) derivative π» of the power series πΊ is (at least) as large as the radius of convergence of πΊ. This version of dvradcnv 25933 uses a shifted version of π» to match the sum form of (β D πΉ) in pserdv2 25942 (and shows how to use uzmptshftfval 43105 to shift a maps-to function on a set of upper integers). (Contributed by Steve Rodriguez, 22-Apr-2020.) |
β’ πΊ = (π₯ β β β¦ (π β β0 β¦ ((π΄βπ) Β· (π₯βπ)))) & β’ π = sup({π β β β£ seq0( + , (πΊβπ)) β dom β }, β*, < ) & β’ π» = (π β β β¦ ((π Β· (π΄βπ)) Β· (πβ(π β 1)))) & β’ (π β π΄:β0βΆβ) & β’ (π β π β β) & β’ (π β (absβπ) < π ) β β’ (π β seq1( + , π») β dom β ) | ||
Theorem | binomcxplemwb 43107 | Lemma for binomcxp 43116. The lemma in the Wikibooks proof. (Contributed by Steve Rodriguez, 22-Apr-2020.) |
β’ (π β πΆ β β) & β’ (π β πΎ β β) β β’ (π β (((πΆ β πΎ) Β· (πΆCππΎ)) + ((πΆ β (πΎ β 1)) Β· (πΆCπ(πΎ β 1)))) = (πΆ Β· (πΆCππΎ))) | ||
Theorem | binomcxplemnn0 43108* | Lemma for binomcxp 43116. When πΆ is a nonnegative integer, the binomial's finite sum value by the standard binomial theorem binom 15776 equals this generalized infinite sum: the generalized binomial coefficient and exponentiation operators give exactly the same values in the standard index set (0...πΆ), and when the index set is widened beyond πΆ the additional values are just zeroes. (Contributed by Steve Rodriguez, 22-Apr-2020.) |
β’ (π β π΄ β β+) & β’ (π β π΅ β β) & β’ (π β (absβπ΅) < (absβπ΄)) & β’ (π β πΆ β β) β β’ ((π β§ πΆ β β0) β ((π΄ + π΅)βππΆ) = Ξ£π β β0 ((πΆCππ) Β· ((π΄βπ(πΆ β π)) Β· (π΅βπ)))) | ||
Theorem | binomcxplemrat 43109* | Lemma for binomcxp 43116. As π increases, this ratio's absolute value converges to one. Part of equation "Since continuity of the absolute value..." in the Wikibooks proof (proven for the inverse ratio, which we later show is no problem). (Contributed by Steve Rodriguez, 22-Apr-2020.) |
β’ (π β π΄ β β+) & β’ (π β π΅ β β) & β’ (π β (absβπ΅) < (absβπ΄)) & β’ (π β πΆ β β) β β’ (π β (π β β0 β¦ (absβ((πΆ β π) / (π + 1)))) β 1) | ||
Theorem | binomcxplemfrat 43110* | Lemma for binomcxp 43116. binomcxplemrat 43109 implies that when πΆ is not a nonnegative integer, the absolute value of the ratio ((πΉβ(π + 1)) / (πΉβπ)) converges to one. The rest of equation "Since continuity of the absolute value..." in the Wikibooks proof. (Contributed by Steve Rodriguez, 22-Apr-2020.) |
β’ (π β π΄ β β+) & β’ (π β π΅ β β) & β’ (π β (absβπ΅) < (absβπ΄)) & β’ (π β πΆ β β) & β’ πΉ = (π β β0 β¦ (πΆCππ)) β β’ ((π β§ Β¬ πΆ β β0) β (π β β0 β¦ (absβ((πΉβ(π + 1)) / (πΉβπ)))) β 1) | ||
Theorem | binomcxplemradcnv 43111* | Lemma for binomcxp 43116. By binomcxplemfrat 43110 and radcnvrat 43073 the radius of convergence of power series Ξ£π β β0((πΉβπ) Β· (πβπ)) is one. (Contributed by Steve Rodriguez, 22-Apr-2020.) |
β’ (π β π΄ β β+) & β’ (π β π΅ β β) & β’ (π β (absβπ΅) < (absβπ΄)) & β’ (π β πΆ β β) & β’ πΉ = (π β β0 β¦ (πΆCππ)) & β’ π = (π β β β¦ (π β β0 β¦ ((πΉβπ) Β· (πβπ)))) & β’ π = sup({π β β β£ seq0( + , (πβπ)) β dom β }, β*, < ) β β’ ((π β§ Β¬ πΆ β β0) β π = 1) | ||
Theorem | binomcxplemdvbinom 43112* | Lemma for binomcxp 43116. By the power and chain rules, calculate the derivative of ((1 + π)βπ-πΆ), with respect to π in the disk of convergence π·. We later multiply the derivative in the later binomcxplemdvsum 43114 by this derivative to show that ((1 + π)βππΆ) (with a nonnegated πΆ) and the later sum, since both at π = 0 equal one, are the same. (Contributed by Steve Rodriguez, 22-Apr-2020.) |
β’ (π β π΄ β β+) & β’ (π β π΅ β β) & β’ (π β (absβπ΅) < (absβπ΄)) & β’ (π β πΆ β β) & β’ πΉ = (π β β0 β¦ (πΆCππ)) & β’ π = (π β β β¦ (π β β0 β¦ ((πΉβπ) Β· (πβπ)))) & β’ π = sup({π β β β£ seq0( + , (πβπ)) β dom β }, β*, < ) & β’ πΈ = (π β β β¦ (π β β β¦ ((π Β· (πΉβπ)) Β· (πβ(π β 1))))) & β’ π· = (β‘abs β (0[,)π )) β β’ ((π β§ Β¬ πΆ β β0) β (β D (π β π· β¦ ((1 + π)βπ-πΆ))) = (π β π· β¦ (-πΆ Β· ((1 + π)βπ(-πΆ β 1))))) | ||
Theorem | binomcxplemcvg 43113* | Lemma for binomcxp 43116. The sum in binomcxplemnn0 43108 and its derivative (see the next theorem, binomcxplemdvsum 43114) converge, as long as their base π½ is within the disk of convergence. Part of remark "This convergence allows us to apply term-by-term differentiation..." in the Wikibooks proof. (Contributed by Steve Rodriguez, 22-Apr-2020.) |
β’ (π β π΄ β β+) & β’ (π β π΅ β β) & β’ (π β (absβπ΅) < (absβπ΄)) & β’ (π β πΆ β β) & β’ πΉ = (π β β0 β¦ (πΆCππ)) & β’ π = (π β β β¦ (π β β0 β¦ ((πΉβπ) Β· (πβπ)))) & β’ π = sup({π β β β£ seq0( + , (πβπ)) β dom β }, β*, < ) & β’ πΈ = (π β β β¦ (π β β β¦ ((π Β· (πΉβπ)) Β· (πβ(π β 1))))) & β’ π· = (β‘abs β (0[,)π )) β β’ ((π β§ π½ β π·) β (seq0( + , (πβπ½)) β dom β β§ seq1( + , (πΈβπ½)) β dom β )) | ||
Theorem | binomcxplemdvsum 43114* | Lemma for binomcxp 43116. The derivative of the generalized sum in binomcxplemnn0 43108. Part of remark "This convergence allows to apply term-by-term differentiation..." in the Wikibooks proof. (Contributed by Steve Rodriguez, 22-Apr-2020.) |
β’ (π β π΄ β β+) & β’ (π β π΅ β β) & β’ (π β (absβπ΅) < (absβπ΄)) & β’ (π β πΆ β β) & β’ πΉ = (π β β0 β¦ (πΆCππ)) & β’ π = (π β β β¦ (π β β0 β¦ ((πΉβπ) Β· (πβπ)))) & β’ π = sup({π β β β£ seq0( + , (πβπ)) β dom β }, β*, < ) & β’ πΈ = (π β β β¦ (π β β β¦ ((π Β· (πΉβπ)) Β· (πβ(π β 1))))) & β’ π· = (β‘abs β (0[,)π )) & β’ π = (π β π· β¦ Ξ£π β β0 ((πβπ)βπ)) β β’ (π β (β D π) = (π β π· β¦ Ξ£π β β ((πΈβπ)βπ))) | ||
Theorem | binomcxplemnotnn0 43115* |
Lemma for binomcxp 43116. When πΆ is not a nonnegative integer, the
generalized sum in binomcxplemnn0 43108 βwhich we will call π
βis a convergent power series: its base π is always of
smaller absolute value than the radius of convergence.
pserdv2 25942 gives the derivative of π, which by dvradcnv 25933 also converges in that radius. When π΄ is fixed at one, (π΄ + π) times that derivative equals (πΆ Β· π) and fraction (π / ((π΄ + π)βππΆ)) is always defined with derivative zero, so the fraction is a constantβspecifically one, because ((1 + 0)βππΆ) = 1. Thus ((1 + π)βππΆ) = (πβπ). Finally, let π be (π΅ / π΄), and multiply both the binomial ((1 + (π΅ / π΄))βππΆ) and the sum (πβ(π΅ / π΄)) by (π΄βππΆ) to get the result. (Contributed by Steve Rodriguez, 22-Apr-2020.) |
β’ (π β π΄ β β+) & β’ (π β π΅ β β) & β’ (π β (absβπ΅) < (absβπ΄)) & β’ (π β πΆ β β) & β’ πΉ = (π β β0 β¦ (πΆCππ)) & β’ π = (π β β β¦ (π β β0 β¦ ((πΉβπ) Β· (πβπ)))) & β’ π = sup({π β β β£ seq0( + , (πβπ)) β dom β }, β*, < ) & β’ πΈ = (π β β β¦ (π β β β¦ ((π Β· (πΉβπ)) Β· (πβ(π β 1))))) & β’ π· = (β‘abs β (0[,)π )) & β’ π = (π β π· β¦ Ξ£π β β0 ((πβπ)βπ)) β β’ ((π β§ Β¬ πΆ β β0) β ((π΄ + π΅)βππΆ) = Ξ£π β β0 ((πΆCππ) Β· ((π΄βπ(πΆ β π)) Β· (π΅βπ)))) | ||
Theorem | binomcxp 43116* | Generalize the binomial theorem binom 15776 to positive real summand π΄, real summand π΅, and complex exponent πΆ. Proof in https://en.wikibooks.org/wiki/Advanced_Calculus 15776; see also https://en.wikipedia.org/wiki/Binomial_series 15776, https://en.wikipedia.org/wiki/Binomial_theorem 15776 (sections "Newton's generalized binomial theorem" and "Future generalizations"), and proof "General Binomial Theorem" in https://proofwiki.org/wiki/Binomial_Theorem 15776. (Contributed by Steve Rodriguez, 22-Apr-2020.) |
β’ (π β π΄ β β+) & β’ (π β π΅ β β) & β’ (π β (absβπ΅) < (absβπ΄)) & β’ (π β πΆ β β) β β’ (π β ((π΄ + π΅)βππΆ) = Ξ£π β β0 ((πΆCππ) Β· ((π΄βπ(πΆ β π)) Β· (π΅βπ)))) | ||
Theorem | pm10.12 43117* | Theorem *10.12 in [WhiteheadRussell] p. 146. In *10, this is treated as an axiom, and the proofs in *10 are based on this theorem. (Contributed by Andrew Salmon, 17-Jun-2011.) |
β’ (βπ₯(π β¨ π) β (π β¨ βπ₯π)) | ||
Theorem | pm10.14 43118 | Theorem *10.14 in [WhiteheadRussell] p. 146. (Contributed by Andrew Salmon, 17-Jun-2011.) |
β’ ((βπ₯π β§ βπ₯π) β ([π¦ / π₯]π β§ [π¦ / π₯]π)) | ||
Theorem | pm10.251 43119 | Theorem *10.251 in [WhiteheadRussell] p. 149. (Contributed by Andrew Salmon, 17-Jun-2011.) |
β’ (βπ₯ Β¬ π β Β¬ βπ₯π) | ||
Theorem | pm10.252 43120 | Theorem *10.252 in [WhiteheadRussell] p. 149. (Contributed by Andrew Salmon, 17-Jun-2011.) (New usage is discouraged.) |
β’ (Β¬ βπ₯π β βπ₯ Β¬ π) | ||
Theorem | pm10.253 43121 | Theorem *10.253 in [WhiteheadRussell] p. 149. (Contributed by Andrew Salmon, 17-Jun-2011.) |
β’ (Β¬ βπ₯π β βπ₯ Β¬ π) | ||
Theorem | albitr 43122 | Theorem *10.301 in [WhiteheadRussell] p. 151. (Contributed by Andrew Salmon, 24-May-2011.) |
β’ ((βπ₯(π β π) β§ βπ₯(π β π)) β βπ₯(π β π)) | ||
Theorem | pm10.42 43123 | Theorem *10.42 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 17-Jun-2011.) |
β’ ((βπ₯π β¨ βπ₯π) β βπ₯(π β¨ π)) | ||
Theorem | pm10.52 43124* | Theorem *10.52 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 24-May-2011.) |
β’ (βπ₯π β (βπ₯(π β π) β π)) | ||
Theorem | pm10.53 43125 | Theorem *10.53 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 24-May-2011.) |
β’ (Β¬ βπ₯π β βπ₯(π β π)) | ||
Theorem | pm10.541 43126* | Theorem *10.541 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 24-May-2011.) |
β’ (βπ₯(π β (π β¨ π)) β (π β¨ βπ₯(π β π))) | ||
Theorem | pm10.542 43127* | Theorem *10.542 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.) |
β’ (βπ₯(π β (π β π)) β (π β βπ₯(π β π))) | ||
Theorem | pm10.55 43128 | Theorem *10.55 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.) |
β’ ((βπ₯(π β§ π) β§ βπ₯(π β π)) β (βπ₯π β§ βπ₯(π β π))) | ||
Theorem | pm10.56 43129 | Theorem *10.56 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.) |
β’ ((βπ₯(π β π) β§ βπ₯(π β§ π)) β βπ₯(π β§ π)) | ||
Theorem | pm10.57 43130 | Theorem *10.57 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.) |
β’ (βπ₯(π β (π β¨ π)) β (βπ₯(π β π) β¨ βπ₯(π β§ π))) | ||
Theorem | 2alanimi 43131 | Removes two universal quantifiers from a statement. (Contributed by Andrew Salmon, 24-May-2011.) |
β’ ((π β§ π) β π) β β’ ((βπ₯βπ¦π β§ βπ₯βπ¦π) β βπ₯βπ¦π) | ||
Theorem | 2al2imi 43132 | Removes two universal quantifiers from a statement. (Contributed by Andrew Salmon, 24-May-2011.) |
β’ (π β (π β π)) β β’ (βπ₯βπ¦π β (βπ₯βπ¦π β βπ₯βπ¦π)) | ||
Theorem | pm11.11 43133 | Theorem *11.11 in [WhiteheadRussell] p. 159. (Contributed by Andrew Salmon, 17-Jun-2011.) |
β’ π β β’ βπ§βπ€[π§ / π₯][π€ / π¦]π | ||
Theorem | pm11.12 43134* | Theorem *11.12 in [WhiteheadRussell] p. 159. (Contributed by Andrew Salmon, 17-Jun-2011.) |
β’ (βπ₯βπ¦(π β¨ π) β (π β¨ βπ₯βπ¦π)) | ||
Theorem | 19.21vv 43135* | Compare Theorem *11.3 in [WhiteheadRussell] p. 161. Special case of theorem 19.21 of [Margaris] p. 90 with two quantifiers. See 19.21v 1943. (Contributed by Andrew Salmon, 24-May-2011.) |
β’ (βπ₯βπ¦(π β π) β (π β βπ₯βπ¦π)) | ||
Theorem | 2alim 43136 | Theorem *11.32 in [WhiteheadRussell] p. 162. Theorem 19.20 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.) |
β’ (βπ₯βπ¦(π β π) β (βπ₯βπ¦π β βπ₯βπ¦π)) | ||
Theorem | 2albi 43137 | Theorem *11.33 in [WhiteheadRussell] p. 162. Theorem 19.15 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.) |
β’ (βπ₯βπ¦(π β π) β (βπ₯βπ¦π β βπ₯βπ¦π)) | ||
Theorem | 2exim 43138 | Theorem *11.34 in [WhiteheadRussell] p. 162. Theorem 19.22 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.) |
β’ (βπ₯βπ¦(π β π) β (βπ₯βπ¦π β βπ₯βπ¦π)) | ||
Theorem | 2exbi 43139 | Theorem *11.341 in [WhiteheadRussell] p. 162. Theorem 19.18 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.) |
β’ (βπ₯βπ¦(π β π) β (βπ₯βπ¦π β βπ₯βπ¦π)) | ||
Theorem | spsbce-2 43140 | Theorem *11.36 in [WhiteheadRussell] p. 162. (Contributed by Andrew Salmon, 24-May-2011.) |
β’ ([π§ / π₯][π€ / π¦]π β βπ₯βπ¦π) | ||
Theorem | 19.33-2 43141 | Theorem *11.421 in [WhiteheadRussell] p. 163. Theorem 19.33 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.) |
β’ ((βπ₯βπ¦π β¨ βπ₯βπ¦π) β βπ₯βπ¦(π β¨ π)) | ||
Theorem | 19.36vv 43142* | Theorem *11.43 in [WhiteheadRussell] p. 163. Theorem 19.36 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 25-May-2011.) |
β’ (βπ₯βπ¦(π β π) β (βπ₯βπ¦π β π)) | ||
Theorem | 19.31vv 43143* | Theorem *11.44 in [WhiteheadRussell] p. 163. Theorem 19.31 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.) |
β’ (βπ₯βπ¦(π β¨ π) β (βπ₯βπ¦π β¨ π)) | ||
Theorem | 19.37vv 43144* | Theorem *11.46 in [WhiteheadRussell] p. 164. Theorem 19.37 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.) |
β’ (βπ₯βπ¦(π β π) β (π β βπ₯βπ¦π)) | ||
Theorem | 19.28vv 43145* | Theorem *11.47 in [WhiteheadRussell] p. 164. Theorem 19.28 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.) |
β’ (βπ₯βπ¦(π β§ π) β (π β§ βπ₯βπ¦π)) | ||
Theorem | pm11.52 43146 | Theorem *11.52 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.) |
β’ (βπ₯βπ¦(π β§ π) β Β¬ βπ₯βπ¦(π β Β¬ π)) | ||
Theorem | aaanv 43147* | Theorem *11.56 in [WhiteheadRussell] p. 165. Special case of aaan 2328. (Contributed by Andrew Salmon, 24-May-2011.) |
β’ ((βπ₯π β§ βπ¦π) β βπ₯βπ¦(π β§ π)) | ||
Theorem | pm11.57 43148* | Theorem *11.57 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 24-May-2011.) |
β’ (βπ₯π β βπ₯βπ¦(π β§ [π¦ / π₯]π)) | ||
Theorem | pm11.58 43149* | Theorem *11.58 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 24-May-2011.) |
β’ (βπ₯π β βπ₯βπ¦(π β§ [π¦ / π₯]π)) | ||
Theorem | pm11.59 43150* | Theorem *11.59 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 25-May-2011.) |
β’ (βπ₯(π β π) β βπ¦βπ₯((π β§ [π¦ / π₯]π) β (π β§ [π¦ / π₯]π))) | ||
Theorem | pm11.6 43151* | Theorem *11.6 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 25-May-2011.) |
β’ (βπ₯(βπ¦(π β§ π) β§ π) β βπ¦(βπ₯(π β§ π) β§ π)) | ||
Theorem | pm11.61 43152* | Theorem *11.61 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.) |
β’ (βπ¦βπ₯(π β π) β βπ₯(π β βπ¦π)) | ||
Theorem | pm11.62 43153* | Theorem *11.62 in [WhiteheadRussell] p. 166. Importation combined with the rearrangement with quantifiers. (Contributed by Andrew Salmon, 24-May-2011.) |
β’ (βπ₯βπ¦((π β§ π) β π) β βπ₯(π β βπ¦(π β π))) | ||
Theorem | pm11.63 43154 | Theorem *11.63 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.) |
β’ (Β¬ βπ₯βπ¦π β βπ₯βπ¦(π β π)) | ||
Theorem | pm11.7 43155 | Theorem *11.7 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.) |
β’ (βπ₯βπ¦(π β¨ π) β βπ₯βπ¦π) | ||
Theorem | pm11.71 43156* | Theorem *11.71 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.) |
β’ ((βπ₯π β§ βπ¦π) β ((βπ₯(π β π) β§ βπ¦(π β π)) β βπ₯βπ¦((π β§ π) β (π β§ π)))) | ||
Theorem | sbeqal1 43157* | If π₯ = π¦ always implies π₯ = π§, then π¦ = π§. (Contributed by Andrew Salmon, 2-Jun-2011.) |
β’ (βπ₯(π₯ = π¦ β π₯ = π§) β π¦ = π§) | ||
Theorem | sbeqal1i 43158* | Suppose you know π₯ = π¦ implies π₯ = π§, assuming π₯ and π§ are distinct. Then, π¦ = π§. (Contributed by Andrew Salmon, 3-Jun-2011.) |
β’ (π₯ = π¦ β π₯ = π§) β β’ π¦ = π§ | ||
Theorem | sbeqal2i 43159* | If π₯ = π¦ implies π₯ = π§, then we can infer π§ = π¦. (Contributed by Andrew Salmon, 3-Jun-2011.) |
β’ (π₯ = π¦ β π₯ = π§) β β’ π§ = π¦ | ||
Theorem | axc5c4c711 43160 | Proof of a theorem that can act as a sole axiom for pure predicate calculus with ax-gen 1798 as the inference rule. This proof extends the idea of axc5c711 37788 and related theorems. (Contributed by Andrew Salmon, 14-Jul-2011.) |
β’ ((βπ₯βπ¦ Β¬ βπ₯βπ¦(βπ¦π β π) β (π β βπ¦(βπ¦π β π))) β (βπ¦π β βπ¦π)) | ||
Theorem | axc5c4c711toc5 43161 | Rederivation of sp 2177 from axc5c4c711 43160. Note that ax6 2384 is used for the rederivation. (Contributed by Andrew Salmon, 14-Jul-2011.) Revised to use ax6v 1973 instead of ax6 2384, so that this rederivation requires only ax6v 1973 and propositional calculus. (Revised by BJ, 14-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
β’ (βπ₯π β π) | ||
Theorem | axc5c4c711toc4 43162 | Rederivation of axc4 2315 from axc5c4c711 43160. 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.) |
β’ (βπ₯(βπ₯π β π) β (βπ₯π β βπ₯π)) | ||
Theorem | axc5c4c711toc7 43163 | Rederivation of axc7 2311 from axc5c4c711 43160. Note that neither axc7 2311 nor ax-11 2155 are required for the rederivation. (Contributed by Andrew Salmon, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
β’ (Β¬ βπ₯ Β¬ βπ₯π β π) | ||
Theorem | axc5c4c711to11 43164 | Rederivation of ax-11 2155 from axc5c4c711 43160. Note that ax-11 2155 is not required for the rederivation. (Contributed by Andrew Salmon, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
β’ (βπ₯βπ¦π β βπ¦βπ₯π) | ||
Theorem | axc11next 43165* | This theorem shows that, given axextb 2707, we can derive a version of axc11n 2426. However, it is weaker than axc11n 2426 because it has a distinct variable requirement. (Contributed by Andrew Salmon, 16-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
β’ (βπ₯ π₯ = π§ β βπ§ π§ = π₯) | ||
Theorem | pm13.13a 43166 | 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.) |
β’ ((π β§ π₯ = π΄) β [π΄ / π₯]π) | ||
Theorem | pm13.13b 43167 | Theorem *13.13 in [WhiteheadRussell] p. 178 with different variable substitution. (Contributed by Andrew Salmon, 3-Jun-2011.) |
β’ (([π΄ / π₯]π β§ π₯ = π΄) β π) | ||
Theorem | pm13.14 43168 | Theorem *13.14 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.) |
β’ (([π΄ / π₯]π β§ Β¬ π) β π₯ β π΄) | ||
Theorem | pm13.192 43169* | Theorem *13.192 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.) (Revised by NM, 4-Jan-2017.) |
β’ (βπ¦(βπ₯(π₯ = π΄ β π₯ = π¦) β§ π) β [π΄ / π¦]π) | ||
Theorem | pm13.193 43170 | Theorem *13.193 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.) |
β’ ((π β§ π₯ = π¦) β ([π¦ / π₯]π β§ π₯ = π¦)) | ||
Theorem | pm13.194 43171 | Theorem *13.194 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.) |
β’ ((π β§ π₯ = π¦) β ([π¦ / π₯]π β§ π β§ π₯ = π¦)) | ||
Theorem | pm13.195 43172* | Theorem *13.195 in [WhiteheadRussell] p. 179. This theorem is very similar to sbc5 3806. (Contributed by Andrew Salmon, 3-Jun-2011.) (Revised by NM, 4-Jan-2017.) |
β’ (βπ¦(π¦ = π΄ β§ π) β [π΄ / π¦]π) | ||
Theorem | pm13.196a 43173* | Theorem *13.196 in [WhiteheadRussell] p. 179. The only difference is the position of the substituted variable. (Contributed by Andrew Salmon, 3-Jun-2011.) |
β’ (Β¬ π β βπ¦([π¦ / π₯]π β π¦ β π₯)) | ||
Theorem | 2sbc6g 43174* | Theorem *13.21 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.) |
β’ ((π΄ β πΆ β§ π΅ β π·) β (βπ§βπ€((π§ = π΄ β§ π€ = π΅) β π) β [π΄ / π§][π΅ / π€]π)) | ||
Theorem | 2sbc5g 43175* | Theorem *13.22 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.) |
β’ ((π΄ β πΆ β§ π΅ β π·) β (βπ§βπ€((π§ = π΄ β§ π€ = π΅) β§ π) β [π΄ / π§][π΅ / π€]π)) | ||
Theorem | iotain 43176 | Equivalence between two different forms of β©. (Contributed by Andrew Salmon, 15-Jul-2011.) |
β’ (β!π₯π β β© {π₯ β£ π} = (β©π₯π)) | ||
Theorem | iotaexeu 43177 | The iota class exists. This theorem does not require ax-nul 5307 for its proof. (Contributed by Andrew Salmon, 11-Jul-2011.) |
β’ (β!π₯π β (β©π₯π) β V) | ||
Theorem | iotasbc 43178* | Definition *14.01 in [WhiteheadRussell] p. 184. In Principia Mathematica, Russell and Whitehead define β© in terms of a function of (β©π₯π). Their definition differs in that a function of (β©π₯π) evaluates to "false" when there isn't a single π₯ that satisfies π. (Contributed by Andrew Salmon, 11-Jul-2011.) |
β’ (β!π₯π β ([(β©π₯π) / π¦]π β βπ¦(βπ₯(π β π₯ = π¦) β§ π))) | ||
Theorem | iotasbc2 43179* | Theorem *14.111 in [WhiteheadRussell] p. 184. (Contributed by Andrew Salmon, 11-Jul-2011.) |
β’ ((β!π₯π β§ β!π₯π) β ([(β©π₯π) / π¦][(β©π₯π) / π§]π β βπ¦βπ§(βπ₯(π β π₯ = π¦) β§ βπ₯(π β π₯ = π§) β§ π))) | ||
Theorem | pm14.12 43180* | Theorem *14.12 in [WhiteheadRussell] p. 184. (Contributed by Andrew Salmon, 11-Jul-2011.) |
β’ (β!π₯π β βπ₯βπ¦((π β§ [π¦ / π₯]π) β π₯ = π¦)) | ||
Theorem | pm14.122a 43181* | Theorem *14.122 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.) |
β’ (π΄ β π β (βπ₯(π β π₯ = π΄) β (βπ₯(π β π₯ = π΄) β§ [π΄ / π₯]π))) | ||
Theorem | pm14.122b 43182* | Theorem *14.122 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.) |
β’ (π΄ β π β ((βπ₯(π β π₯ = π΄) β§ [π΄ / π₯]π) β (βπ₯(π β π₯ = π΄) β§ βπ₯π))) | ||
Theorem | pm14.122c 43183* | Theorem *14.122 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.) |
β’ (π΄ β π β (βπ₯(π β π₯ = π΄) β (βπ₯(π β π₯ = π΄) β§ βπ₯π))) | ||
Theorem | pm14.123a 43184* | Theorem *14.123 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.) |
β’ ((π΄ β π β§ π΅ β π) β (βπ§βπ€(π β (π§ = π΄ β§ π€ = π΅)) β (βπ§βπ€(π β (π§ = π΄ β§ π€ = π΅)) β§ [π΄ / π§][π΅ / π€]π))) | ||
Theorem | pm14.123b 43185* | Theorem *14.123 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.) |
β’ ((π΄ β π β§ π΅ β π) β ((βπ§βπ€(π β (π§ = π΄ β§ π€ = π΅)) β§ [π΄ / π§][π΅ / π€]π) β (βπ§βπ€(π β (π§ = π΄ β§ π€ = π΅)) β§ βπ§βπ€π))) | ||
Theorem | pm14.123c 43186* | Theorem *14.123 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.) |
β’ ((π΄ β π β§ π΅ β π) β (βπ§βπ€(π β (π§ = π΄ β§ π€ = π΅)) β (βπ§βπ€(π β (π§ = π΄ β§ π€ = π΅)) β§ βπ§βπ€π))) | ||
Theorem | pm14.18 43187 | Theorem *14.18 in [WhiteheadRussell] p. 189. (Contributed by Andrew Salmon, 11-Jul-2011.) |
β’ (β!π₯π β (βπ₯π β [(β©π₯π) / π₯]π)) | ||
Theorem | iotaequ 43188* | Theorem *14.2 in [WhiteheadRussell] p. 189. (Contributed by Andrew Salmon, 11-Jul-2011.) |
β’ (β©π₯π₯ = π¦) = π¦ | ||
Theorem | iotavalb 43189* | Theorem *14.202 in [WhiteheadRussell] p. 189. A biconditional version of iotaval 6515. (Contributed by Andrew Salmon, 11-Jul-2011.) |
β’ (β!π₯π β (βπ₯(π β π₯ = π¦) β (β©π₯π) = π¦)) | ||
Theorem | iotasbc5 43190* | Theorem *14.205 in [WhiteheadRussell] p. 190. (Contributed by Andrew Salmon, 11-Jul-2011.) |
β’ (β!π₯π β ([(β©π₯π) / π¦]π β βπ¦(π¦ = (β©π₯π) β§ π))) | ||
Theorem | pm14.24 43191* | Theorem *14.24 in [WhiteheadRussell] p. 191. (Contributed by Andrew Salmon, 12-Jul-2011.) |
β’ (β!π₯π β βπ¦([π¦ / π₯]π β π¦ = (β©π₯π))) | ||
Theorem | iotavalsb 43192* | Theorem *14.242 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.) |
β’ (βπ₯(π β π₯ = π¦) β ([π¦ / π§]π β [(β©π₯π) / π§]π)) | ||
Theorem | sbiota1 43193 | Theorem *14.25 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 12-Jul-2011.) |
β’ (β!π₯π β (βπ₯(π β π) β [(β©π₯π) / π₯]π)) | ||
Theorem | sbaniota 43194 | Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 12-Jul-2011.) |
β’ (β!π₯π β (βπ₯(π β§ π) β [(β©π₯π) / π₯]π)) | ||
Theorem | eubiOLD 43195 | Obsolete proof of eubi 2579 as of 7-Oct-2022. (Contributed by Andrew Salmon, 11-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
β’ (βπ₯(π β π) β (β!π₯π β β!π₯π)) | ||
Theorem | iotasbcq 43196 | Theorem *14.272 in [WhiteheadRussell] p. 193. (Contributed by Andrew Salmon, 11-Jul-2011.) |
β’ (βπ₯(π β π) β ([(β©π₯π) / π¦]π β [(β©π₯π) / π¦]π)) | ||
Theorem | elnev 43197* | Any set that contains one element less than the universe is not equal to it. (Contributed by Andrew Salmon, 16-Jun-2011.) |
β’ (π΄ β V β {π₯ β£ Β¬ π₯ = π΄} β V) | ||
Theorem | rusbcALT 43198 | 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.) |
β’ {π₯ β£ π₯ β π₯} β V | ||
Theorem | compeq 43199* | Equality between two ways of saying "the complement of π΄". (Contributed by Andrew Salmon, 15-Jul-2011.) |
β’ (V β π΄) = {π₯ β£ Β¬ π₯ β π΄} | ||
Theorem | compne 43200 | The complement of π΄ is not equal to π΄. (Contributed by Andrew Salmon, 15-Jul-2011.) (Proof shortened by BJ, 11-Nov-2021.) |
β’ (V β π΄) β π΄ |
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