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Theorem List for Metamath Proof Explorer - 14201-14300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremexpnlbnd 14201* The reciprocal of exponentiation with a base greater than 1 has no positive lower bound. (Contributed by NM, 18-Jul-2008.)
((๐ด โˆˆ โ„+ โˆง ๐ต โˆˆ โ„ โˆง 1 < ๐ต) โ†’ โˆƒ๐‘˜ โˆˆ โ„• (1 / (๐ตโ†‘๐‘˜)) < ๐ด)
 
Theoremexpnlbnd2 14202* The reciprocal of exponentiation with a base greater than 1 has no positive lower bound. (Contributed by NM, 18-Jul-2008.) (Proof shortened by Mario Carneiro, 5-Jun-2014.)
((๐ด โˆˆ โ„+ โˆง ๐ต โˆˆ โ„ โˆง 1 < ๐ต) โ†’ โˆƒ๐‘— โˆˆ โ„• โˆ€๐‘˜ โˆˆ (โ„คโ‰ฅโ€˜๐‘—)(1 / (๐ตโ†‘๐‘˜)) < ๐ด)
 
Theoremexpmulnbnd 14203* Exponentiation with a base greater than 1 is not bounded by any linear function. (Contributed by Mario Carneiro, 31-Mar-2015.)
((๐ด โˆˆ โ„ โˆง ๐ต โˆˆ โ„ โˆง 1 < ๐ต) โ†’ โˆƒ๐‘— โˆˆ โ„•0 โˆ€๐‘˜ โˆˆ (โ„คโ‰ฅโ€˜๐‘—)(๐ด ยท ๐‘˜) < (๐ตโ†‘๐‘˜))
 
Theoremdigit2 14204 Two ways to express the ๐พ th digit in the decimal (when base ๐ต = 10) expansion of a number ๐ด. ๐พ = 1 corresponds to the first digit after the decimal point. (Contributed by NM, 25-Dec-2008.)
((๐ด โˆˆ โ„ โˆง ๐ต โˆˆ โ„• โˆง ๐พ โˆˆ โ„•) โ†’ ((โŒŠโ€˜((๐ตโ†‘๐พ) ยท ๐ด)) mod ๐ต) = ((โŒŠโ€˜((๐ตโ†‘๐พ) ยท ๐ด)) โˆ’ (๐ต ยท (โŒŠโ€˜((๐ตโ†‘(๐พ โˆ’ 1)) ยท ๐ด)))))
 
Theoremdigit1 14205 Two ways to express the ๐พ th digit in the decimal expansion of a number ๐ด (when base ๐ต = 10). ๐พ = 1 corresponds to the first digit after the decimal point. (Contributed by NM, 3-Jan-2009.)
((๐ด โˆˆ โ„ โˆง ๐ต โˆˆ โ„• โˆง ๐พ โˆˆ โ„•) โ†’ ((โŒŠโ€˜((๐ตโ†‘๐พ) ยท ๐ด)) mod ๐ต) = (((โŒŠโ€˜((๐ตโ†‘๐พ) ยท ๐ด)) mod (๐ตโ†‘๐พ)) โˆ’ ((๐ต ยท (โŒŠโ€˜((๐ตโ†‘(๐พ โˆ’ 1)) ยท ๐ด))) mod (๐ตโ†‘๐พ))))
 
Theoremmodexp 14206 Exponentiation property of the modulo operation, see theorem 5.2(c) in [ApostolNT] p. 107. (Contributed by Mario Carneiro, 28-Feb-2014.)
(((๐ด โˆˆ โ„ค โˆง ๐ต โˆˆ โ„ค) โˆง (๐ถ โˆˆ โ„•0 โˆง ๐ท โˆˆ โ„+) โˆง (๐ด mod ๐ท) = (๐ต mod ๐ท)) โ†’ ((๐ดโ†‘๐ถ) mod ๐ท) = ((๐ตโ†‘๐ถ) mod ๐ท))
 
Theoremdiscr1 14207* A nonnegative quadratic form has nonnegative leading coefficient. (Contributed by Mario Carneiro, 4-Jun-2014.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„)    &   (๐œ‘ โ†’ ๐ต โˆˆ โ„)    &   (๐œ‘ โ†’ ๐ถ โˆˆ โ„)    &   ((๐œ‘ โˆง ๐‘ฅ โˆˆ โ„) โ†’ 0 โ‰ค (((๐ด ยท (๐‘ฅโ†‘2)) + (๐ต ยท ๐‘ฅ)) + ๐ถ))    &   ๐‘‹ = if(1 โ‰ค (((๐ต + if(0 โ‰ค ๐ถ, ๐ถ, 0)) + 1) / -๐ด), (((๐ต + if(0 โ‰ค ๐ถ, ๐ถ, 0)) + 1) / -๐ด), 1)    โ‡’   (๐œ‘ โ†’ 0 โ‰ค ๐ด)
 
Theoremdiscr 14208* If a quadratic polynomial with real coefficients is nonnegative for all values, then its discriminant is nonpositive. (Contributed by NM, 10-Aug-1999.) (Revised by Mario Carneiro, 4-Jun-2014.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„)    &   (๐œ‘ โ†’ ๐ต โˆˆ โ„)    &   (๐œ‘ โ†’ ๐ถ โˆˆ โ„)    &   ((๐œ‘ โˆง ๐‘ฅ โˆˆ โ„) โ†’ 0 โ‰ค (((๐ด ยท (๐‘ฅโ†‘2)) + (๐ต ยท ๐‘ฅ)) + ๐ถ))    โ‡’   (๐œ‘ โ†’ ((๐ตโ†‘2) โˆ’ (4 ยท (๐ด ยท ๐ถ))) โ‰ค 0)
 
Theoremexpnngt1 14209 If an integer power with a positive integer base is greater than 1, then the exponent is positive. (Contributed by AV, 28-Dec-2022.)
((๐ด โˆˆ โ„• โˆง ๐ต โˆˆ โ„ค โˆง 1 < (๐ดโ†‘๐ต)) โ†’ ๐ต โˆˆ โ„•)
 
Theoremexpnngt1b 14210 An integer power with an integer base greater than 1 is greater than 1 iff the exponent is positive. (Contributed by AV, 28-Dec-2022.)
((๐ด โˆˆ (โ„คโ‰ฅโ€˜2) โˆง ๐ต โˆˆ โ„ค) โ†’ (1 < (๐ดโ†‘๐ต) โ†” ๐ต โˆˆ โ„•))
 
Theoremsqoddm1div8 14211 A squared odd number minus 1 divided by 8 is the odd number multiplied with its successor divided by 2. (Contributed by AV, 19-Jul-2021.)
((๐‘ โˆˆ โ„ค โˆง ๐‘€ = ((2 ยท ๐‘) + 1)) โ†’ (((๐‘€โ†‘2) โˆ’ 1) / 8) = ((๐‘ ยท (๐‘ + 1)) / 2))
 
Theoremnnsqcld 14212 The naturals are closed under squaring. (Contributed by Mario Carneiro, 28-May-2016.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„•)    โ‡’   (๐œ‘ โ†’ (๐ดโ†‘2) โˆˆ โ„•)
 
Theoremnnexpcld 14213 Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 28-May-2016.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„•)    &   (๐œ‘ โ†’ ๐‘ โˆˆ โ„•0)    โ‡’   (๐œ‘ โ†’ (๐ดโ†‘๐‘) โˆˆ โ„•)
 
Theoremnn0expcld 14214 Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 28-May-2016.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„•0)    &   (๐œ‘ โ†’ ๐‘ โˆˆ โ„•0)    โ‡’   (๐œ‘ โ†’ (๐ดโ†‘๐‘) โˆˆ โ„•0)
 
Theoremrpexpcld 14215 Closure law for exponentiation of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„+)    &   (๐œ‘ โ†’ ๐‘ โˆˆ โ„ค)    โ‡’   (๐œ‘ โ†’ (๐ดโ†‘๐‘) โˆˆ โ„+)
 
Theoremltexp2rd 14216 The power of a positive number smaller than 1 decreases as its exponent increases. (Contributed by Mario Carneiro, 28-May-2016.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„+)    &   (๐œ‘ โ†’ ๐‘ โˆˆ โ„ค)    &   (๐œ‘ โ†’ ๐‘€ โˆˆ โ„ค)    &   (๐œ‘ โ†’ ๐ด < 1)    โ‡’   (๐œ‘ โ†’ (๐‘€ < ๐‘ โ†” (๐ดโ†‘๐‘) < (๐ดโ†‘๐‘€)))
 
Theoremreexpclzd 14217 Closure of exponentiation of reals. (Contributed by Mario Carneiro, 28-May-2016.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„)    &   (๐œ‘ โ†’ ๐ด โ‰  0)    &   (๐œ‘ โ†’ ๐‘ โˆˆ โ„ค)    โ‡’   (๐œ‘ โ†’ (๐ดโ†‘๐‘) โˆˆ โ„)
 
Theoremsqgt0d 14218 The square of a nonzero real is positive. (Contributed by Mario Carneiro, 28-May-2016.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„)    &   (๐œ‘ โ†’ ๐ด โ‰  0)    โ‡’   (๐œ‘ โ†’ 0 < (๐ดโ†‘2))
 
Theoremltexp2d 14219 Ordering relationship for exponentiation. (Contributed by Mario Carneiro, 28-May-2016.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„)    &   (๐œ‘ โ†’ ๐‘€ โˆˆ โ„ค)    &   (๐œ‘ โ†’ ๐‘ โˆˆ โ„ค)    &   (๐œ‘ โ†’ 1 < ๐ด)    โ‡’   (๐œ‘ โ†’ (๐‘€ < ๐‘ โ†” (๐ดโ†‘๐‘€) < (๐ดโ†‘๐‘)))
 
Theoremleexp2d 14220 Ordering law for exponentiation. (Contributed by Mario Carneiro, 28-May-2016.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„)    &   (๐œ‘ โ†’ ๐‘€ โˆˆ โ„ค)    &   (๐œ‘ โ†’ ๐‘ โˆˆ โ„ค)    &   (๐œ‘ โ†’ 1 < ๐ด)    โ‡’   (๐œ‘ โ†’ (๐‘€ โ‰ค ๐‘ โ†” (๐ดโ†‘๐‘€) โ‰ค (๐ดโ†‘๐‘)))
 
Theoremexpcand 14221 Ordering relationship for exponentiation. (Contributed by Mario Carneiro, 28-May-2016.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„)    &   (๐œ‘ โ†’ ๐‘€ โˆˆ โ„ค)    &   (๐œ‘ โ†’ ๐‘ โˆˆ โ„ค)    &   (๐œ‘ โ†’ 1 < ๐ด)    &   (๐œ‘ โ†’ (๐ดโ†‘๐‘€) = (๐ดโ†‘๐‘))    โ‡’   (๐œ‘ โ†’ ๐‘€ = ๐‘)
 
Theoremleexp2ad 14222 Ordering relationship for exponentiation. (Contributed by Mario Carneiro, 28-May-2016.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„)    &   (๐œ‘ โ†’ 1 โ‰ค ๐ด)    &   (๐œ‘ โ†’ ๐‘ โˆˆ (โ„คโ‰ฅโ€˜๐‘€))    โ‡’   (๐œ‘ โ†’ (๐ดโ†‘๐‘€) โ‰ค (๐ดโ†‘๐‘))
 
Theoremleexp2rd 14223 Ordering relationship for exponentiation. (Contributed by Mario Carneiro, 28-May-2016.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„)    &   (๐œ‘ โ†’ ๐‘€ โˆˆ โ„•0)    &   (๐œ‘ โ†’ ๐‘ โˆˆ (โ„คโ‰ฅโ€˜๐‘€))    &   (๐œ‘ โ†’ 0 โ‰ค ๐ด)    &   (๐œ‘ โ†’ ๐ด โ‰ค 1)    โ‡’   (๐œ‘ โ†’ (๐ดโ†‘๐‘) โ‰ค (๐ดโ†‘๐‘€))
 
Theoremlt2sqd 14224 The square function on nonnegative reals is strictly monotonic. (Contributed by Mario Carneiro, 28-May-2016.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„)    &   (๐œ‘ โ†’ ๐ต โˆˆ โ„)    &   (๐œ‘ โ†’ 0 โ‰ค ๐ด)    &   (๐œ‘ โ†’ 0 โ‰ค ๐ต)    โ‡’   (๐œ‘ โ†’ (๐ด < ๐ต โ†” (๐ดโ†‘2) < (๐ตโ†‘2)))
 
Theoremle2sqd 14225 The square function on nonnegative reals is monotonic. (Contributed by Mario Carneiro, 28-May-2016.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„)    &   (๐œ‘ โ†’ ๐ต โˆˆ โ„)    &   (๐œ‘ โ†’ 0 โ‰ค ๐ด)    &   (๐œ‘ โ†’ 0 โ‰ค ๐ต)    โ‡’   (๐œ‘ โ†’ (๐ด โ‰ค ๐ต โ†” (๐ดโ†‘2) โ‰ค (๐ตโ†‘2)))
 
Theoremsq11d 14226 The square function is one-to-one for nonnegative reals. (Contributed by Mario Carneiro, 28-May-2016.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„)    &   (๐œ‘ โ†’ ๐ต โˆˆ โ„)    &   (๐œ‘ โ†’ 0 โ‰ค ๐ด)    &   (๐œ‘ โ†’ 0 โ‰ค ๐ต)    &   (๐œ‘ โ†’ (๐ดโ†‘2) = (๐ตโ†‘2))    โ‡’   (๐œ‘ โ†’ ๐ด = ๐ต)
 
Theoremmulsubdivbinom2 14227 The square of a binomial with factor minus a number divided by a nonzero number. (Contributed by AV, 19-Jul-2021.)
(((๐ด โˆˆ โ„‚ โˆง ๐ต โˆˆ โ„‚ โˆง ๐ท โˆˆ โ„‚) โˆง (๐ถ โˆˆ โ„‚ โˆง ๐ถ โ‰  0)) โ†’ (((((๐ถ ยท ๐ด) + ๐ต)โ†‘2) โˆ’ ๐ท) / ๐ถ) = (((๐ถ ยท (๐ดโ†‘2)) + (2 ยท (๐ด ยท ๐ต))) + (((๐ตโ†‘2) โˆ’ ๐ท) / ๐ถ)))
 
Theoremmuldivbinom2 14228 The square of a binomial with factor divided by a nonzero number. (Contributed by AV, 19-Jul-2021.)
((๐ด โˆˆ โ„‚ โˆง ๐ต โˆˆ โ„‚ โˆง (๐ถ โˆˆ โ„‚ โˆง ๐ถ โ‰  0)) โ†’ ((((๐ถ ยท ๐ด) + ๐ต)โ†‘2) / ๐ถ) = (((๐ถ ยท (๐ดโ†‘2)) + (2 ยท (๐ด ยท ๐ต))) + ((๐ตโ†‘2) / ๐ถ)))
 
Theoremsq10 14229 The square of 10 is 100. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 1-Aug-2021.)
(10โ†‘2) = 100
 
Theoremsq10e99m1 14230 The square of 10 is 99 plus 1. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 1-Aug-2021.)
(10โ†‘2) = (99 + 1)
 
Theorem3dec 14231 A "decimal constructor" which is used to build up "decimal integers" or "numeric terms" in base 10 with 3 "digits". (Contributed by AV, 14-Jun-2021.) (Revised by AV, 1-Aug-2021.)
๐ด โˆˆ โ„•0    &   ๐ต โˆˆ โ„•0    โ‡’   ๐ด๐ต๐ถ = ((((10โ†‘2) ยท ๐ด) + (10 ยท ๐ต)) + ๐ถ)
 
5.6.8  Ordered pair theorem for nonnegative integers
 
Theoremnn0le2msqi 14232 The square function on nonnegative integers is monotonic. (Contributed by Raph Levien, 10-Dec-2002.)
๐ด โˆˆ โ„•0    &   ๐ต โˆˆ โ„•0    โ‡’   (๐ด โ‰ค ๐ต โ†” (๐ด ยท ๐ด) โ‰ค (๐ต ยท ๐ต))
 
Theoremnn0opthlem1 14233 A rather pretty lemma for nn0opthi 14235. (Contributed by Raph Levien, 10-Dec-2002.)
๐ด โˆˆ โ„•0    &   ๐ถ โˆˆ โ„•0    โ‡’   (๐ด < ๐ถ โ†” ((๐ด ยท ๐ด) + (2 ยท ๐ด)) < (๐ถ ยท ๐ถ))
 
Theoremnn0opthlem2 14234 Lemma for nn0opthi 14235. (Contributed by Raph Levien, 10-Dec-2002.) (Revised by Scott Fenton, 8-Sep-2010.)
๐ด โˆˆ โ„•0    &   ๐ต โˆˆ โ„•0    &   ๐ถ โˆˆ โ„•0    &   ๐ท โˆˆ โ„•0    โ‡’   ((๐ด + ๐ต) < ๐ถ โ†’ ((๐ถ ยท ๐ถ) + ๐ท) โ‰  (((๐ด + ๐ต) ยท (๐ด + ๐ต)) + ๐ต))
 
Theoremnn0opthi 14235 An ordered pair theorem for nonnegative integers. Theorem 17.3 of [Quine] p. 124. We can represent an ordered pair of nonnegative integers ๐ด and ๐ต by (((๐ด + ๐ต) ยท (๐ด + ๐ต)) + ๐ต). If two such ordered pairs are equal, their first elements are equal and their second elements are equal. Contrast this ordered pair representation with the standard one df-op 4635 that works for any set. (Contributed by Raph Levien, 10-Dec-2002.) (Proof shortened by Scott Fenton, 8-Sep-2010.)
๐ด โˆˆ โ„•0    &   ๐ต โˆˆ โ„•0    &   ๐ถ โˆˆ โ„•0    &   ๐ท โˆˆ โ„•0    โ‡’   ((((๐ด + ๐ต) ยท (๐ด + ๐ต)) + ๐ต) = (((๐ถ + ๐ท) ยท (๐ถ + ๐ท)) + ๐ท) โ†” (๐ด = ๐ถ โˆง ๐ต = ๐ท))
 
Theoremnn0opth2i 14236 An ordered pair theorem for nonnegative integers. Theorem 17.3 of [Quine] p. 124. See comments for nn0opthi 14235. (Contributed by NM, 22-Jul-2004.)
๐ด โˆˆ โ„•0    &   ๐ต โˆˆ โ„•0    &   ๐ถ โˆˆ โ„•0    &   ๐ท โˆˆ โ„•0    โ‡’   ((((๐ด + ๐ต)โ†‘2) + ๐ต) = (((๐ถ + ๐ท)โ†‘2) + ๐ท) โ†” (๐ด = ๐ถ โˆง ๐ต = ๐ท))
 
Theoremnn0opth2 14237 An ordered pair theorem for nonnegative integers. Theorem 17.3 of [Quine] p. 124. See nn0opthi 14235. (Contributed by NM, 22-Jul-2004.)
(((๐ด โˆˆ โ„•0 โˆง ๐ต โˆˆ โ„•0) โˆง (๐ถ โˆˆ โ„•0 โˆง ๐ท โˆˆ โ„•0)) โ†’ ((((๐ด + ๐ต)โ†‘2) + ๐ต) = (((๐ถ + ๐ท)โ†‘2) + ๐ท) โ†” (๐ด = ๐ถ โˆง ๐ต = ๐ท)))
 
5.6.9  Factorial function
 
Syntaxcfa 14238 Extend class notation to include the factorial of nonnegative integers.
class !
 
Definitiondf-fac 14239 Define the factorial function on nonnegative integers. For example, (!โ€˜5) = 120 because 1 ยท 2 ยท 3 ยท 4 ยท 5 = 120 (ex-fac 29972). In the literature, the factorial function is written as a postscript exclamation point. (Contributed by NM, 2-Dec-2004.)
! = ({โŸจ0, 1โŸฉ} โˆช seq1( ยท , I ))
 
Theoremfacnn 14240 Value of the factorial function for positive integers. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.)
(๐‘ โˆˆ โ„• โ†’ (!โ€˜๐‘) = (seq1( ยท , I )โ€˜๐‘))
 
Theoremfac0 14241 The factorial of 0. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.)
(!โ€˜0) = 1
 
Theoremfac1 14242 The factorial of 1. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.)
(!โ€˜1) = 1
 
Theoremfacp1 14243 The factorial of a successor. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.)
(๐‘ โˆˆ โ„•0 โ†’ (!โ€˜(๐‘ + 1)) = ((!โ€˜๐‘) ยท (๐‘ + 1)))
 
Theoremfac2 14244 The factorial of 2. (Contributed by NM, 17-Mar-2005.)
(!โ€˜2) = 2
 
Theoremfac3 14245 The factorial of 3. (Contributed by NM, 17-Mar-2005.)
(!โ€˜3) = 6
 
Theoremfac4 14246 The factorial of 4. (Contributed by Mario Carneiro, 18-Jun-2015.)
(!โ€˜4) = 24
 
Theoremfacnn2 14247 Value of the factorial function expressed recursively. (Contributed by NM, 2-Dec-2004.)
(๐‘ โˆˆ โ„• โ†’ (!โ€˜๐‘) = ((!โ€˜(๐‘ โˆ’ 1)) ยท ๐‘))
 
Theoremfaccl 14248 Closure of the factorial function. (Contributed by NM, 2-Dec-2004.)
(๐‘ โˆˆ โ„•0 โ†’ (!โ€˜๐‘) โˆˆ โ„•)
 
Theoremfaccld 14249 Closure of the factorial function, deduction version of faccl 14248. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(๐œ‘ โ†’ ๐‘ โˆˆ โ„•0)    โ‡’   (๐œ‘ โ†’ (!โ€˜๐‘) โˆˆ โ„•)
 
Theoremfacmapnn 14250 The factorial function restricted to positive integers is a mapping from the positive integers to the positive integers. (Contributed by AV, 8-Aug-2020.)
(๐‘› โˆˆ โ„• โ†ฆ (!โ€˜๐‘›)) โˆˆ (โ„• โ†‘m โ„•)
 
Theoremfacne0 14251 The factorial function is nonzero. (Contributed by NM, 26-Apr-2005.)
(๐‘ โˆˆ โ„•0 โ†’ (!โ€˜๐‘) โ‰  0)
 
Theoremfacdiv 14252 A positive integer divides the factorial of an equal or larger number. (Contributed by NM, 2-May-2005.)
((๐‘€ โˆˆ โ„•0 โˆง ๐‘ โˆˆ โ„• โˆง ๐‘ โ‰ค ๐‘€) โ†’ ((!โ€˜๐‘€) / ๐‘) โˆˆ โ„•)
 
Theoremfacndiv 14253 No positive integer (greater than one) divides the factorial plus one of an equal or larger number. (Contributed by NM, 3-May-2005.)
(((๐‘€ โˆˆ โ„•0 โˆง ๐‘ โˆˆ โ„•) โˆง (1 < ๐‘ โˆง ๐‘ โ‰ค ๐‘€)) โ†’ ยฌ (((!โ€˜๐‘€) + 1) / ๐‘) โˆˆ โ„ค)
 
Theoremfacwordi 14254 Ordering property of factorial. (Contributed by NM, 9-Dec-2005.)
((๐‘€ โˆˆ โ„•0 โˆง ๐‘ โˆˆ โ„•0 โˆง ๐‘€ โ‰ค ๐‘) โ†’ (!โ€˜๐‘€) โ‰ค (!โ€˜๐‘))
 
Theoremfaclbnd 14255 A lower bound for the factorial function. (Contributed by NM, 17-Dec-2005.)
((๐‘€ โˆˆ โ„•0 โˆง ๐‘ โˆˆ โ„•0) โ†’ (๐‘€โ†‘(๐‘ + 1)) โ‰ค ((๐‘€โ†‘๐‘€) ยท (!โ€˜๐‘)))
 
Theoremfaclbnd2 14256 A lower bound for the factorial function. (Contributed by NM, 17-Dec-2005.)
(๐‘ โˆˆ โ„•0 โ†’ ((2โ†‘๐‘) / 2) โ‰ค (!โ€˜๐‘))
 
Theoremfaclbnd3 14257 A lower bound for the factorial function. (Contributed by NM, 19-Dec-2005.)
((๐‘€ โˆˆ โ„•0 โˆง ๐‘ โˆˆ โ„•0) โ†’ (๐‘€โ†‘๐‘) โ‰ค ((๐‘€โ†‘๐‘€) ยท (!โ€˜๐‘)))
 
Theoremfaclbnd4lem1 14258 Lemma for faclbnd4 14262. Prepare the induction step. (Contributed by NM, 20-Dec-2005.)
๐‘ โˆˆ โ„•    &   ๐พ โˆˆ โ„•0    &   ๐‘€ โˆˆ โ„•0    โ‡’   ((((๐‘ โˆ’ 1)โ†‘๐พ) ยท (๐‘€โ†‘(๐‘ โˆ’ 1))) โ‰ค (((2โ†‘(๐พโ†‘2)) ยท (๐‘€โ†‘(๐‘€ + ๐พ))) ยท (!โ€˜(๐‘ โˆ’ 1))) โ†’ ((๐‘โ†‘(๐พ + 1)) ยท (๐‘€โ†‘๐‘)) โ‰ค (((2โ†‘((๐พ + 1)โ†‘2)) ยท (๐‘€โ†‘(๐‘€ + (๐พ + 1)))) ยท (!โ€˜๐‘)))
 
Theoremfaclbnd4lem2 14259 Lemma for faclbnd4 14262. Use the weak deduction theorem to convert the hypotheses of faclbnd4lem1 14258 to antecedents. (Contributed by NM, 23-Dec-2005.)
((๐‘€ โˆˆ โ„•0 โˆง ๐พ โˆˆ โ„•0 โˆง ๐‘ โˆˆ โ„•) โ†’ ((((๐‘ โˆ’ 1)โ†‘๐พ) ยท (๐‘€โ†‘(๐‘ โˆ’ 1))) โ‰ค (((2โ†‘(๐พโ†‘2)) ยท (๐‘€โ†‘(๐‘€ + ๐พ))) ยท (!โ€˜(๐‘ โˆ’ 1))) โ†’ ((๐‘โ†‘(๐พ + 1)) ยท (๐‘€โ†‘๐‘)) โ‰ค (((2โ†‘((๐พ + 1)โ†‘2)) ยท (๐‘€โ†‘(๐‘€ + (๐พ + 1)))) ยท (!โ€˜๐‘))))
 
Theoremfaclbnd4lem3 14260 Lemma for faclbnd4 14262. The ๐‘ = 0 case. (Contributed by NM, 23-Dec-2005.)
(((๐‘€ โˆˆ โ„•0 โˆง ๐พ โˆˆ โ„•0) โˆง ๐‘ = 0) โ†’ ((๐‘โ†‘๐พ) ยท (๐‘€โ†‘๐‘)) โ‰ค (((2โ†‘(๐พโ†‘2)) ยท (๐‘€โ†‘(๐‘€ + ๐พ))) ยท (!โ€˜๐‘)))
 
Theoremfaclbnd4lem4 14261 Lemma for faclbnd4 14262. Prove the 0 < ๐‘ case by induction on ๐พ. (Contributed by NM, 19-Dec-2005.)
((๐‘ โˆˆ โ„• โˆง ๐พ โˆˆ โ„•0 โˆง ๐‘€ โˆˆ โ„•0) โ†’ ((๐‘โ†‘๐พ) ยท (๐‘€โ†‘๐‘)) โ‰ค (((2โ†‘(๐พโ†‘2)) ยท (๐‘€โ†‘(๐‘€ + ๐พ))) ยท (!โ€˜๐‘)))
 
Theoremfaclbnd4 14262 Variant of faclbnd5 14263 providing a non-strict lower bound. (Contributed by NM, 23-Dec-2005.)
((๐‘ โˆˆ โ„•0 โˆง ๐พ โˆˆ โ„•0 โˆง ๐‘€ โˆˆ โ„•0) โ†’ ((๐‘โ†‘๐พ) ยท (๐‘€โ†‘๐‘)) โ‰ค (((2โ†‘(๐พโ†‘2)) ยท (๐‘€โ†‘(๐‘€ + ๐พ))) ยท (!โ€˜๐‘)))
 
Theoremfaclbnd5 14263 The factorial function grows faster than powers and exponentiations. If we consider ๐พ and ๐‘€ to be constants, the right-hand side of the inequality is a constant times ๐‘-factorial. (Contributed by NM, 24-Dec-2005.)
((๐‘ โˆˆ โ„•0 โˆง ๐พ โˆˆ โ„•0 โˆง ๐‘€ โˆˆ โ„•) โ†’ ((๐‘โ†‘๐พ) ยท (๐‘€โ†‘๐‘)) < ((2 ยท ((2โ†‘(๐พโ†‘2)) ยท (๐‘€โ†‘(๐‘€ + ๐พ)))) ยท (!โ€˜๐‘)))
 
Theoremfaclbnd6 14264 Geometric lower bound for the factorial function, where N is usually held constant. (Contributed by Paul Chapman, 28-Dec-2007.)
((๐‘ โˆˆ โ„•0 โˆง ๐‘€ โˆˆ โ„•0) โ†’ ((!โ€˜๐‘) ยท ((๐‘ + 1)โ†‘๐‘€)) โ‰ค (!โ€˜(๐‘ + ๐‘€)))
 
Theoremfacubnd 14265 An upper bound for the factorial function. (Contributed by Mario Carneiro, 15-Apr-2016.)
(๐‘ โˆˆ โ„•0 โ†’ (!โ€˜๐‘) โ‰ค (๐‘โ†‘๐‘))
 
Theoremfacavg 14266 The product of two factorials is greater than or equal to the factorial of (the floor of) their average. (Contributed by NM, 9-Dec-2005.)
((๐‘€ โˆˆ โ„•0 โˆง ๐‘ โˆˆ โ„•0) โ†’ (!โ€˜(โŒŠโ€˜((๐‘€ + ๐‘) / 2))) โ‰ค ((!โ€˜๐‘€) ยท (!โ€˜๐‘)))
 
5.6.10  The binomial coefficient operation
 
Syntaxcbc 14267 Extend class notation to include the binomial coefficient operation (combinatorial choose operation).
class C
 
Definitiondf-bc 14268* Define the binomial coefficient operation. For example, (5C3) = 10 (ex-bc 29973).

In the literature, this function is often written as a column vector of the two arguments, or with the arguments as subscripts before and after the letter "C". The expression (๐‘C๐พ) is read "๐‘ choose ๐พ". Definition of binomial coefficient in [Gleason] p. 295. As suggested by Gleason, we define it to be 0 when 0 โ‰ค ๐‘˜ โ‰ค ๐‘› does not hold. (Contributed by NM, 10-Jul-2005.)

C = (๐‘› โˆˆ โ„•0, ๐‘˜ โˆˆ โ„ค โ†ฆ if(๐‘˜ โˆˆ (0...๐‘›), ((!โ€˜๐‘›) / ((!โ€˜(๐‘› โˆ’ ๐‘˜)) ยท (!โ€˜๐‘˜))), 0))
 
Theorembcval 14269 Value of the binomial coefficient, ๐‘ choose ๐พ. Definition of binomial coefficient in [Gleason] p. 295. As suggested by Gleason, we define it to be 0 when 0 โ‰ค ๐พ โ‰ค ๐‘ does not hold. See bcval2 14270 for the value in the standard domain. (Contributed by NM, 10-Jul-2005.) (Revised by Mario Carneiro, 7-Nov-2013.)
((๐‘ โˆˆ โ„•0 โˆง ๐พ โˆˆ โ„ค) โ†’ (๐‘C๐พ) = if(๐พ โˆˆ (0...๐‘), ((!โ€˜๐‘) / ((!โ€˜(๐‘ โˆ’ ๐พ)) ยท (!โ€˜๐พ))), 0))
 
Theorembcval2 14270 Value of the binomial coefficient, ๐‘ choose ๐พ, in its standard domain. (Contributed by NM, 9-Jun-2005.) (Revised by Mario Carneiro, 7-Nov-2013.)
(๐พ โˆˆ (0...๐‘) โ†’ (๐‘C๐พ) = ((!โ€˜๐‘) / ((!โ€˜(๐‘ โˆ’ ๐พ)) ยท (!โ€˜๐พ))))
 
Theorembcval3 14271 Value of the binomial coefficient, ๐‘ choose ๐พ, outside of its standard domain. Remark in [Gleason] p. 295. (Contributed by NM, 14-Jul-2005.) (Revised by Mario Carneiro, 8-Nov-2013.)
((๐‘ โˆˆ โ„•0 โˆง ๐พ โˆˆ โ„ค โˆง ยฌ ๐พ โˆˆ (0...๐‘)) โ†’ (๐‘C๐พ) = 0)
 
Theorembcval4 14272 Value of the binomial coefficient, ๐‘ choose ๐พ, outside of its standard domain. Remark in [Gleason] p. 295. (Contributed by NM, 14-Jul-2005.) (Revised by Mario Carneiro, 7-Nov-2013.)
((๐‘ โˆˆ โ„•0 โˆง ๐พ โˆˆ โ„ค โˆง (๐พ < 0 โˆจ ๐‘ < ๐พ)) โ†’ (๐‘C๐พ) = 0)
 
Theorembcrpcl 14273 Closure of the binomial coefficient in the positive reals. (This is mostly a lemma before we have bccl2 14288.) (Contributed by Mario Carneiro, 10-Mar-2014.)
(๐พ โˆˆ (0...๐‘) โ†’ (๐‘C๐พ) โˆˆ โ„+)
 
Theorembccmpl 14274 "Complementing" its second argument doesn't change a binary coefficient. (Contributed by NM, 21-Jun-2005.) (Revised by Mario Carneiro, 5-Mar-2014.)
((๐‘ โˆˆ โ„•0 โˆง ๐พ โˆˆ โ„ค) โ†’ (๐‘C๐พ) = (๐‘C(๐‘ โˆ’ ๐พ)))
 
Theorembcn0 14275 ๐‘ choose 0 is 1. Remark in [Gleason] p. 296. (Contributed by NM, 17-Jun-2005.) (Revised by Mario Carneiro, 8-Nov-2013.)
(๐‘ โˆˆ โ„•0 โ†’ (๐‘C0) = 1)
 
Theorembc0k 14276 The binomial coefficient " 0 choose ๐พ " is 0 for a positive integer K. Note that (0C0) = 1 (see bcn0 14275). (Contributed by Alexander van der Vekens, 1-Jan-2018.)
(๐พ โˆˆ โ„• โ†’ (0C๐พ) = 0)
 
Theorembcnn 14277 ๐‘ choose ๐‘ is 1. Remark in [Gleason] p. 296. (Contributed by NM, 17-Jun-2005.) (Revised by Mario Carneiro, 8-Nov-2013.)
(๐‘ โˆˆ โ„•0 โ†’ (๐‘C๐‘) = 1)
 
Theorembcn1 14278 Binomial coefficient: ๐‘ choose 1. (Contributed by NM, 21-Jun-2005.) (Revised by Mario Carneiro, 8-Nov-2013.)
(๐‘ โˆˆ โ„•0 โ†’ (๐‘C1) = ๐‘)
 
Theorembcnp1n 14279 Binomial coefficient: ๐‘ + 1 choose ๐‘. (Contributed by NM, 20-Jun-2005.) (Revised by Mario Carneiro, 8-Nov-2013.)
(๐‘ โˆˆ โ„•0 โ†’ ((๐‘ + 1)C๐‘) = (๐‘ + 1))
 
Theorembcm1k 14280 The proportion of one binomial coefficient to another with ๐พ decreased by 1. (Contributed by Mario Carneiro, 10-Mar-2014.)
(๐พ โˆˆ (1...๐‘) โ†’ (๐‘C๐พ) = ((๐‘C(๐พ โˆ’ 1)) ยท ((๐‘ โˆ’ (๐พ โˆ’ 1)) / ๐พ)))
 
Theorembcp1n 14281 The proportion of one binomial coefficient to another with ๐‘ increased by 1. (Contributed by Mario Carneiro, 10-Mar-2014.)
(๐พ โˆˆ (0...๐‘) โ†’ ((๐‘ + 1)C๐พ) = ((๐‘C๐พ) ยท ((๐‘ + 1) / ((๐‘ + 1) โˆ’ ๐พ))))
 
Theorembcp1nk 14282 The proportion of one binomial coefficient to another with ๐‘ and ๐พ increased by 1. (Contributed by Mario Carneiro, 16-Jan-2015.)
(๐พ โˆˆ (0...๐‘) โ†’ ((๐‘ + 1)C(๐พ + 1)) = ((๐‘C๐พ) ยท ((๐‘ + 1) / (๐พ + 1))))
 
Theorembcval5 14283 Write out the top and bottom parts of the binomial coefficient (๐‘C๐พ) = (๐‘ ยท (๐‘ โˆ’ 1) ยท ... ยท ((๐‘ โˆ’ ๐พ) + 1)) / ๐พ! explicitly. In this form, it is valid even for ๐‘ < ๐พ, although it is no longer valid for nonpositive ๐พ. (Contributed by Mario Carneiro, 22-May-2014.)
((๐‘ โˆˆ โ„•0 โˆง ๐พ โˆˆ โ„•) โ†’ (๐‘C๐พ) = ((seq((๐‘ โˆ’ ๐พ) + 1)( ยท , I )โ€˜๐‘) / (!โ€˜๐พ)))
 
Theorembcn2 14284 Binomial coefficient: ๐‘ choose 2. (Contributed by Mario Carneiro, 22-May-2014.)
(๐‘ โˆˆ โ„•0 โ†’ (๐‘C2) = ((๐‘ ยท (๐‘ โˆ’ 1)) / 2))
 
Theorembcp1m1 14285 Compute the binomial coefficient of (๐‘ + 1) over (๐‘ โˆ’ 1) (Contributed by Scott Fenton, 11-May-2014.) (Revised by Mario Carneiro, 22-May-2014.)
(๐‘ โˆˆ โ„•0 โ†’ ((๐‘ + 1)C(๐‘ โˆ’ 1)) = (((๐‘ + 1) ยท ๐‘) / 2))
 
Theorembcpasc 14286 Pascal's rule for the binomial coefficient, generalized to all integers ๐พ. Equation 2 of [Gleason] p. 295. (Contributed by NM, 13-Jul-2005.) (Revised by Mario Carneiro, 10-Mar-2014.)
((๐‘ โˆˆ โ„•0 โˆง ๐พ โˆˆ โ„ค) โ†’ ((๐‘C๐พ) + (๐‘C(๐พ โˆ’ 1))) = ((๐‘ + 1)C๐พ))
 
Theorembccl 14287 A binomial coefficient, in its extended domain, is a nonnegative integer. (Contributed by NM, 10-Jul-2005.) (Revised by Mario Carneiro, 9-Nov-2013.)
((๐‘ โˆˆ โ„•0 โˆง ๐พ โˆˆ โ„ค) โ†’ (๐‘C๐พ) โˆˆ โ„•0)
 
Theorembccl2 14288 A binomial coefficient, in its standard domain, is a positive integer. (Contributed by NM, 3-Jan-2006.) (Revised by Mario Carneiro, 10-Mar-2014.)
(๐พ โˆˆ (0...๐‘) โ†’ (๐‘C๐พ) โˆˆ โ„•)
 
Theorembcn2m1 14289 Compute the binomial coefficient "๐‘ choose 2 " from "(๐‘ โˆ’ 1) choose 2 ": (N-1) + ( (N-1) 2 ) = ( N 2 ). (Contributed by Alexander van der Vekens, 7-Jan-2018.)
(๐‘ โˆˆ โ„• โ†’ ((๐‘ โˆ’ 1) + ((๐‘ โˆ’ 1)C2)) = (๐‘C2))
 
Theorembcn2p1 14290 Compute the binomial coefficient "(๐‘ + 1) choose 2 " from "๐‘ choose 2 ": N + ( N 2 ) = ( (N+1) 2 ). (Contributed by Alexander van der Vekens, 8-Jan-2018.)
(๐‘ โˆˆ โ„•0 โ†’ (๐‘ + (๐‘C2)) = ((๐‘ + 1)C2))
 
Theorempermnn 14291 The number of permutations of ๐‘ โˆ’ ๐‘… objects from a collection of ๐‘ objects is a positive integer. (Contributed by Jason Orendorff, 24-Jan-2007.)
(๐‘… โˆˆ (0...๐‘) โ†’ ((!โ€˜๐‘) / (!โ€˜๐‘…)) โˆˆ โ„•)
 
Theorembcnm1 14292 The binomial coefficent of (๐‘ โˆ’ 1) is ๐‘. (Contributed by Scott Fenton, 16-May-2014.)
(๐‘ โˆˆ โ„•0 โ†’ (๐‘C(๐‘ โˆ’ 1)) = ๐‘)
 
Theorem4bc3eq4 14293 The value of four choose three. (Contributed by Scott Fenton, 11-Jun-2016.)
(4C3) = 4
 
Theorem4bc2eq6 14294 The value of four choose two. (Contributed by Scott Fenton, 9-Jan-2017.)
(4C2) = 6
 
5.6.11  The ` # ` (set size) function
 
Syntaxchash 14295 Extend the definition of a class to include the set size function.
class โ™ฏ
 
Definitiondf-hash 14296 Define the set size function โ™ฏ, which gives the cardinality of a finite set as a member of โ„•0, and assigns all infinite sets the value +โˆž. For example, (โ™ฏโ€˜{0, 1, 2}) = 3 (ex-hash 29974). (Contributed by Paul Chapman, 22-Jun-2011.)
โ™ฏ = (((rec((๐‘ฅ โˆˆ V โ†ฆ (๐‘ฅ + 1)), 0) โ†พ ฯ‰) โˆ˜ card) โˆช ((V โˆ– Fin) ร— {+โˆž}))
 
Theoremhashkf 14297 The finite part of the size function maps all finite sets to their cardinality, as members of โ„•0. (Contributed by Mario Carneiro, 13-Sep-2013.) (Revised by Mario Carneiro, 26-Dec-2014.)
๐บ = (rec((๐‘ฅ โˆˆ V โ†ฆ (๐‘ฅ + 1)), 0) โ†พ ฯ‰)    &   ๐พ = (๐บ โˆ˜ card)    โ‡’   ๐พ:FinโŸถโ„•0
 
Theoremhashgval 14298* The value of the โ™ฏ function in terms of the mapping ๐บ from ฯ‰ to โ„•0. The proof avoids the use of ax-ac 10458. (Contributed by Paul Chapman, 22-Jun-2011.) (Revised by Mario Carneiro, 26-Dec-2014.)
๐บ = (rec((๐‘ฅ โˆˆ V โ†ฆ (๐‘ฅ + 1)), 0) โ†พ ฯ‰)    โ‡’   (๐ด โˆˆ Fin โ†’ (๐บโ€˜(cardโ€˜๐ด)) = (โ™ฏโ€˜๐ด))
 
Theoremhashginv 14299* The converse of ๐บ maps the size function's value to card. (Contributed by Paul Chapman, 22-Jun-2011.) (Revised by Mario Carneiro, 15-Sep-2013.)
๐บ = (rec((๐‘ฅ โˆˆ V โ†ฆ (๐‘ฅ + 1)), 0) โ†พ ฯ‰)    โ‡’   (๐ด โˆˆ Fin โ†’ (โ—ก๐บโ€˜(โ™ฏโ€˜๐ด)) = (cardโ€˜๐ด))
 
Theoremhashinf 14300 The value of the โ™ฏ function on an infinite set. (Contributed by Mario Carneiro, 13-Jul-2014.)
((๐ด โˆˆ ๐‘‰ โˆง ยฌ ๐ด โˆˆ Fin) โ†’ (โ™ฏโ€˜๐ด) = +โˆž)
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