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Theorem List for Intuitionistic Logic Explorer - 14401-14500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlgsdir2lem3 14401 Lemma for lgsdir2 14404. (Contributed by Mario Carneiro, 4-Feb-2015.)
((๐ด โˆˆ โ„ค โˆง ยฌ 2 โˆฅ ๐ด) โ†’ (๐ด mod 8) โˆˆ ({1, 7} โˆช {3, 5}))
 
Theoremlgsdir2lem4 14402 Lemma for lgsdir2 14404. (Contributed by Mario Carneiro, 4-Feb-2015.)
(((๐ด โˆˆ โ„ค โˆง ๐ต โˆˆ โ„ค) โˆง (๐ด mod 8) โˆˆ {1, 7}) โ†’ (((๐ด ยท ๐ต) mod 8) โˆˆ {1, 7} โ†” (๐ต mod 8) โˆˆ {1, 7}))
 
Theoremlgsdir2lem5 14403 Lemma for lgsdir2 14404. (Contributed by Mario Carneiro, 4-Feb-2015.)
(((๐ด โˆˆ โ„ค โˆง ๐ต โˆˆ โ„ค) โˆง ((๐ด mod 8) โˆˆ {3, 5} โˆง (๐ต mod 8) โˆˆ {3, 5})) โ†’ ((๐ด ยท ๐ต) mod 8) โˆˆ {1, 7})
 
Theoremlgsdir2 14404 The Legendre symbol is completely multiplicative at 2. (Contributed by Mario Carneiro, 4-Feb-2015.)
((๐ด โˆˆ โ„ค โˆง ๐ต โˆˆ โ„ค) โ†’ ((๐ด ยท ๐ต) /L 2) = ((๐ด /L 2) ยท (๐ต /L 2)))
 
Theoremlgsdirprm 14405 The Legendre symbol is completely multiplicative at the primes. See theorem 9.3 in [ApostolNT] p. 180. (Contributed by Mario Carneiro, 4-Feb-2015.) (Proof shortened by AV, 18-Mar-2022.)
((๐ด โˆˆ โ„ค โˆง ๐ต โˆˆ โ„ค โˆง ๐‘ƒ โˆˆ โ„™) โ†’ ((๐ด ยท ๐ต) /L ๐‘ƒ) = ((๐ด /L ๐‘ƒ) ยท (๐ต /L ๐‘ƒ)))
 
Theoremlgsdir 14406 The Legendre symbol is completely multiplicative in its left argument. Generalization of theorem 9.9(a) in [ApostolNT] p. 188 (which assumes that ๐ด and ๐ต are odd positive integers). (Contributed by Mario Carneiro, 4-Feb-2015.)
(((๐ด โˆˆ โ„ค โˆง ๐ต โˆˆ โ„ค โˆง ๐‘ โˆˆ โ„ค) โˆง (๐ด โ‰  0 โˆง ๐ต โ‰  0)) โ†’ ((๐ด ยท ๐ต) /L ๐‘) = ((๐ด /L ๐‘) ยท (๐ต /L ๐‘)))
 
Theoremlgsdilem2 14407* Lemma for lgsdi 14408. (Contributed by Mario Carneiro, 4-Feb-2015.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„ค)    &   (๐œ‘ โ†’ ๐‘€ โˆˆ โ„ค)    &   (๐œ‘ โ†’ ๐‘ โˆˆ โ„ค)    &   (๐œ‘ โ†’ ๐‘€ โ‰  0)    &   (๐œ‘ โ†’ ๐‘ โ‰  0)    &   ๐น = (๐‘› โˆˆ โ„• โ†ฆ if(๐‘› โˆˆ โ„™, ((๐ด /L ๐‘›)โ†‘(๐‘› pCnt ๐‘€)), 1))    โ‡’   (๐œ‘ โ†’ (seq1( ยท , ๐น)โ€˜(absโ€˜๐‘€)) = (seq1( ยท , ๐น)โ€˜(absโ€˜(๐‘€ ยท ๐‘))))
 
Theoremlgsdi 14408 The Legendre symbol is completely multiplicative in its right argument. Generalization of theorem 9.9(b) in [ApostolNT] p. 188 (which assumes that ๐‘€ and ๐‘ are odd positive integers). (Contributed by Mario Carneiro, 5-Feb-2015.)
(((๐ด โˆˆ โ„ค โˆง ๐‘€ โˆˆ โ„ค โˆง ๐‘ โˆˆ โ„ค) โˆง (๐‘€ โ‰  0 โˆง ๐‘ โ‰  0)) โ†’ (๐ด /L (๐‘€ ยท ๐‘)) = ((๐ด /L ๐‘€) ยท (๐ด /L ๐‘)))
 
Theoremlgsne0 14409 The Legendre symbol is nonzero (and hence equal to 1 or -1) precisely when the arguments are coprime. (Contributed by Mario Carneiro, 5-Feb-2015.)
((๐ด โˆˆ โ„ค โˆง ๐‘ โˆˆ โ„ค) โ†’ ((๐ด /L ๐‘) โ‰  0 โ†” (๐ด gcd ๐‘) = 1))
 
Theoremlgsabs1 14410 The Legendre symbol is nonzero (and hence equal to 1 or -1) precisely when the arguments are coprime. (Contributed by Mario Carneiro, 5-Feb-2015.)
((๐ด โˆˆ โ„ค โˆง ๐‘ โˆˆ โ„ค) โ†’ ((absโ€˜(๐ด /L ๐‘)) = 1 โ†” (๐ด gcd ๐‘) = 1))
 
Theoremlgssq 14411 The Legendre symbol at a square is equal to 1. Together with lgsmod 14397 this implies that the Legendre symbol takes value 1 at every quadratic residue. (Contributed by Mario Carneiro, 5-Feb-2015.) (Revised by AV, 20-Jul-2021.)
(((๐ด โˆˆ โ„ค โˆง ๐ด โ‰  0) โˆง ๐‘ โˆˆ โ„ค โˆง (๐ด gcd ๐‘) = 1) โ†’ ((๐ดโ†‘2) /L ๐‘) = 1)
 
Theoremlgssq2 14412 The Legendre symbol at a square is equal to 1. (Contributed by Mario Carneiro, 5-Feb-2015.)
((๐ด โˆˆ โ„ค โˆง ๐‘ โˆˆ โ„• โˆง (๐ด gcd ๐‘) = 1) โ†’ (๐ด /L (๐‘โ†‘2)) = 1)
 
Theoremlgsprme0 14413 The Legendre symbol at any prime (even at 2) is 0 iff the prime does not divide the first argument. See definition in [ApostolNT] p. 179. (Contributed by AV, 20-Jul-2021.)
((๐ด โˆˆ โ„ค โˆง ๐‘ƒ โˆˆ โ„™) โ†’ ((๐ด /L ๐‘ƒ) = 0 โ†” (๐ด mod ๐‘ƒ) = 0))
 
Theorem1lgs 14414 The Legendre symbol at 1. See example 1 in [ApostolNT] p. 180. (Contributed by Mario Carneiro, 28-Apr-2016.)
(๐‘ โˆˆ โ„ค โ†’ (1 /L ๐‘) = 1)
 
Theoremlgs1 14415 The Legendre symbol at 1. See definition in [ApostolNT] p. 188. (Contributed by Mario Carneiro, 28-Apr-2016.)
(๐ด โˆˆ โ„ค โ†’ (๐ด /L 1) = 1)
 
Theoremlgsmodeq 14416 The Legendre (Jacobi) symbol is preserved under reduction mod ๐‘› when ๐‘› is odd. Theorem 9.9(c) in [ApostolNT] p. 188. (Contributed by AV, 20-Jul-2021.)
((๐ด โˆˆ โ„ค โˆง ๐ต โˆˆ โ„ค โˆง (๐‘ โˆˆ โ„• โˆง ยฌ 2 โˆฅ ๐‘)) โ†’ ((๐ด mod ๐‘) = (๐ต mod ๐‘) โ†’ (๐ด /L ๐‘) = (๐ต /L ๐‘)))
 
Theoremlgsmulsqcoprm 14417 The Legendre (Jacobi) symbol is preserved under multiplication with a square of an integer coprime to the second argument. Theorem 9.9(d) in [ApostolNT] p. 188. (Contributed by AV, 20-Jul-2021.)
(((๐ด โˆˆ โ„ค โˆง ๐ด โ‰  0) โˆง (๐ต โˆˆ โ„ค โˆง ๐ต โ‰  0) โˆง (๐‘ โˆˆ โ„ค โˆง (๐ด gcd ๐‘) = 1)) โ†’ (((๐ดโ†‘2) ยท ๐ต) /L ๐‘) = (๐ต /L ๐‘))
 
Theoremlgsdirnn0 14418 Variation on lgsdir 14406 valid for all ๐ด, ๐ต but only for positive ๐‘. (The exact location of the failure of this law is for ๐ด = 0, ๐ต < 0, ๐‘ = -1 in which case (0 /L -1) = 1 but (๐ต /L -1) = -1.) (Contributed by Mario Carneiro, 28-Apr-2016.)
((๐ด โˆˆ โ„ค โˆง ๐ต โˆˆ โ„ค โˆง ๐‘ โˆˆ โ„•0) โ†’ ((๐ด ยท ๐ต) /L ๐‘) = ((๐ด /L ๐‘) ยท (๐ต /L ๐‘)))
 
Theoremlgsdinn0 14419 Variation on lgsdi 14408 valid for all ๐‘€, ๐‘ but only for positive ๐ด. (The exact location of the failure of this law is for ๐ด = -1, ๐‘€ = 0, and some ๐‘ in which case (-1 /L 0) = 1 but (-1 /L ๐‘) = -1 when -1 is not a quadratic residue mod ๐‘.) (Contributed by Mario Carneiro, 28-Apr-2016.)
((๐ด โˆˆ โ„•0 โˆง ๐‘€ โˆˆ โ„ค โˆง ๐‘ โˆˆ โ„ค) โ†’ (๐ด /L (๐‘€ ยท ๐‘)) = ((๐ด /L ๐‘€) ยท (๐ด /L ๐‘)))
 
10.2.2  Quadratic reciprocity
 
Theoremlgseisenlem1 14420* Lemma for Eisenstein's lemma. If ๐‘…(๐‘ข) = (๐‘„ ยท ๐‘ข) mod ๐‘ƒ and ๐‘€(๐‘ข) = (-1โ†‘๐‘…(๐‘ข)) ยท ๐‘…(๐‘ข), then for any even 1 โ‰ค ๐‘ข โ‰ค ๐‘ƒ โˆ’ 1, ๐‘€(๐‘ข) is also an even integer 1 โ‰ค ๐‘€(๐‘ข) โ‰ค ๐‘ƒ โˆ’ 1. To simplify these statements, we divide all the even numbers by 2, so that it becomes the statement that ๐‘€(๐‘ฅ / 2) = (-1โ†‘๐‘…(๐‘ฅ / 2)) ยท ๐‘…(๐‘ฅ / 2) / 2 is an integer between 1 and (๐‘ƒ โˆ’ 1) / 2. (Contributed by Mario Carneiro, 17-Jun-2015.)
(๐œ‘ โ†’ ๐‘ƒ โˆˆ (โ„™ โˆ– {2}))    &   (๐œ‘ โ†’ ๐‘„ โˆˆ (โ„™ โˆ– {2}))    &   (๐œ‘ โ†’ ๐‘ƒ โ‰  ๐‘„)    &   ๐‘… = ((๐‘„ ยท (2 ยท ๐‘ฅ)) mod ๐‘ƒ)    &   ๐‘€ = (๐‘ฅ โˆˆ (1...((๐‘ƒ โˆ’ 1) / 2)) โ†ฆ ((((-1โ†‘๐‘…) ยท ๐‘…) mod ๐‘ƒ) / 2))    โ‡’   (๐œ‘ โ†’ ๐‘€:(1...((๐‘ƒ โˆ’ 1) / 2))โŸถ(1...((๐‘ƒ โˆ’ 1) / 2)))
 
Theoremlgseisenlem2 14421* Lemma for Eisenstein's lemma. The function ๐‘€ is an injection (and hence a bijection by the pigeonhole principle). (Contributed by Mario Carneiro, 17-Jun-2015.)
(๐œ‘ โ†’ ๐‘ƒ โˆˆ (โ„™ โˆ– {2}))    &   (๐œ‘ โ†’ ๐‘„ โˆˆ (โ„™ โˆ– {2}))    &   (๐œ‘ โ†’ ๐‘ƒ โ‰  ๐‘„)    &   ๐‘… = ((๐‘„ ยท (2 ยท ๐‘ฅ)) mod ๐‘ƒ)    &   ๐‘€ = (๐‘ฅ โˆˆ (1...((๐‘ƒ โˆ’ 1) / 2)) โ†ฆ ((((-1โ†‘๐‘…) ยท ๐‘…) mod ๐‘ƒ) / 2))    &   ๐‘† = ((๐‘„ ยท (2 ยท ๐‘ฆ)) mod ๐‘ƒ)    โ‡’   (๐œ‘ โ†’ ๐‘€:(1...((๐‘ƒ โˆ’ 1) / 2))โ€“1-1-ontoโ†’(1...((๐‘ƒ โˆ’ 1) / 2)))
 
Theoremm1lgs 14422 The first supplement to the law of quadratic reciprocity. Negative one is a square mod an odd prime ๐‘ƒ iff ๐‘ƒโ‰ก1 (mod 4). See first case of theorem 9.4 in [ApostolNT] p. 181. (Contributed by Mario Carneiro, 19-Jun-2015.)
(๐‘ƒ โˆˆ (โ„™ โˆ– {2}) โ†’ ((-1 /L ๐‘ƒ) = 1 โ†” (๐‘ƒ mod 4) = 1))
 
Theorem2lgsoddprmlem1 14423 Lemma 1 for 2lgsoddprm . (Contributed by AV, 19-Jul-2021.)
((๐ด โˆˆ โ„ค โˆง ๐ต โˆˆ โ„ค โˆง ๐‘ = ((8 ยท ๐ด) + ๐ต)) โ†’ (((๐‘โ†‘2) โˆ’ 1) / 8) = (((8 ยท (๐ดโ†‘2)) + (2 ยท (๐ด ยท ๐ต))) + (((๐ตโ†‘2) โˆ’ 1) / 8)))
 
Theorem2lgsoddprmlem2 14424 Lemma 2 for 2lgsoddprm . (Contributed by AV, 19-Jul-2021.)
((๐‘ โˆˆ โ„ค โˆง ยฌ 2 โˆฅ ๐‘ โˆง ๐‘… = (๐‘ mod 8)) โ†’ (2 โˆฅ (((๐‘โ†‘2) โˆ’ 1) / 8) โ†” 2 โˆฅ (((๐‘…โ†‘2) โˆ’ 1) / 8)))
 
Theorem2lgsoddprmlem3a 14425 Lemma 1 for 2lgsoddprmlem3 14429. (Contributed by AV, 20-Jul-2021.)
(((1โ†‘2) โˆ’ 1) / 8) = 0
 
Theorem2lgsoddprmlem3b 14426 Lemma 2 for 2lgsoddprmlem3 14429. (Contributed by AV, 20-Jul-2021.)
(((3โ†‘2) โˆ’ 1) / 8) = 1
 
Theorem2lgsoddprmlem3c 14427 Lemma 3 for 2lgsoddprmlem3 14429. (Contributed by AV, 20-Jul-2021.)
(((5โ†‘2) โˆ’ 1) / 8) = 3
 
Theorem2lgsoddprmlem3d 14428 Lemma 4 for 2lgsoddprmlem3 14429. (Contributed by AV, 20-Jul-2021.)
(((7โ†‘2) โˆ’ 1) / 8) = (2 ยท 3)
 
Theorem2lgsoddprmlem3 14429 Lemma 3 for 2lgsoddprm . (Contributed by AV, 20-Jul-2021.)
((๐‘ โˆˆ โ„ค โˆง ยฌ 2 โˆฅ ๐‘ โˆง ๐‘… = (๐‘ mod 8)) โ†’ (2 โˆฅ (((๐‘…โ†‘2) โˆ’ 1) / 8) โ†” ๐‘… โˆˆ {1, 7}))
 
Theorem2lgsoddprmlem4 14430 Lemma 4 for 2lgsoddprm . (Contributed by AV, 20-Jul-2021.)
((๐‘ โˆˆ โ„ค โˆง ยฌ 2 โˆฅ ๐‘) โ†’ (2 โˆฅ (((๐‘โ†‘2) โˆ’ 1) / 8) โ†” (๐‘ mod 8) โˆˆ {1, 7}))
 
10.2.3  All primes 4n+1 are the sum of two squares
 
Theorem2sqlem1 14431* Lemma for 2sq . (Contributed by Mario Carneiro, 19-Jun-2015.)
๐‘† = ran (๐‘ค โˆˆ โ„ค[i] โ†ฆ ((absโ€˜๐‘ค)โ†‘2))    โ‡’   (๐ด โˆˆ ๐‘† โ†” โˆƒ๐‘ฅ โˆˆ โ„ค[i] ๐ด = ((absโ€˜๐‘ฅ)โ†‘2))
 
Theorem2sqlem2 14432* Lemma for 2sq . (Contributed by Mario Carneiro, 19-Jun-2015.)
๐‘† = ran (๐‘ค โˆˆ โ„ค[i] โ†ฆ ((absโ€˜๐‘ค)โ†‘2))    โ‡’   (๐ด โˆˆ ๐‘† โ†” โˆƒ๐‘ฅ โˆˆ โ„ค โˆƒ๐‘ฆ โˆˆ โ„ค ๐ด = ((๐‘ฅโ†‘2) + (๐‘ฆโ†‘2)))
 
Theoremmul2sq 14433 Fibonacci's identity (actually due to Diophantus). The product of two sums of two squares is also a sum of two squares. We can take advantage of Gaussian integers here to trivialize the proof. (Contributed by Mario Carneiro, 19-Jun-2015.)
๐‘† = ran (๐‘ค โˆˆ โ„ค[i] โ†ฆ ((absโ€˜๐‘ค)โ†‘2))    โ‡’   ((๐ด โˆˆ ๐‘† โˆง ๐ต โˆˆ ๐‘†) โ†’ (๐ด ยท ๐ต) โˆˆ ๐‘†)
 
Theorem2sqlem3 14434 Lemma for 2sqlem5 14436. (Contributed by Mario Carneiro, 20-Jun-2015.)
๐‘† = ran (๐‘ค โˆˆ โ„ค[i] โ†ฆ ((absโ€˜๐‘ค)โ†‘2))    &   (๐œ‘ โ†’ ๐‘ โˆˆ โ„•)    &   (๐œ‘ โ†’ ๐‘ƒ โˆˆ โ„™)    &   (๐œ‘ โ†’ ๐ด โˆˆ โ„ค)    &   (๐œ‘ โ†’ ๐ต โˆˆ โ„ค)    &   (๐œ‘ โ†’ ๐ถ โˆˆ โ„ค)    &   (๐œ‘ โ†’ ๐ท โˆˆ โ„ค)    &   (๐œ‘ โ†’ (๐‘ ยท ๐‘ƒ) = ((๐ดโ†‘2) + (๐ตโ†‘2)))    &   (๐œ‘ โ†’ ๐‘ƒ = ((๐ถโ†‘2) + (๐ทโ†‘2)))    &   (๐œ‘ โ†’ ๐‘ƒ โˆฅ ((๐ถ ยท ๐ต) + (๐ด ยท ๐ท)))    โ‡’   (๐œ‘ โ†’ ๐‘ โˆˆ ๐‘†)
 
Theorem2sqlem4 14435 Lemma for 2sqlem5 14436. (Contributed by Mario Carneiro, 20-Jun-2015.)
๐‘† = ran (๐‘ค โˆˆ โ„ค[i] โ†ฆ ((absโ€˜๐‘ค)โ†‘2))    &   (๐œ‘ โ†’ ๐‘ โˆˆ โ„•)    &   (๐œ‘ โ†’ ๐‘ƒ โˆˆ โ„™)    &   (๐œ‘ โ†’ ๐ด โˆˆ โ„ค)    &   (๐œ‘ โ†’ ๐ต โˆˆ โ„ค)    &   (๐œ‘ โ†’ ๐ถ โˆˆ โ„ค)    &   (๐œ‘ โ†’ ๐ท โˆˆ โ„ค)    &   (๐œ‘ โ†’ (๐‘ ยท ๐‘ƒ) = ((๐ดโ†‘2) + (๐ตโ†‘2)))    &   (๐œ‘ โ†’ ๐‘ƒ = ((๐ถโ†‘2) + (๐ทโ†‘2)))    โ‡’   (๐œ‘ โ†’ ๐‘ โˆˆ ๐‘†)
 
Theorem2sqlem5 14436 Lemma for 2sq . If a number that is a sum of two squares is divisible by a prime that is a sum of two squares, then the quotient is a sum of two squares. (Contributed by Mario Carneiro, 20-Jun-2015.)
๐‘† = ran (๐‘ค โˆˆ โ„ค[i] โ†ฆ ((absโ€˜๐‘ค)โ†‘2))    &   (๐œ‘ โ†’ ๐‘ โˆˆ โ„•)    &   (๐œ‘ โ†’ ๐‘ƒ โˆˆ โ„™)    &   (๐œ‘ โ†’ (๐‘ ยท ๐‘ƒ) โˆˆ ๐‘†)    &   (๐œ‘ โ†’ ๐‘ƒ โˆˆ ๐‘†)    โ‡’   (๐œ‘ โ†’ ๐‘ โˆˆ ๐‘†)
 
Theorem2sqlem6 14437* Lemma for 2sq . If a number that is a sum of two squares is divisible by a number whose prime divisors are all sums of two squares, then the quotient is a sum of two squares. (Contributed by Mario Carneiro, 20-Jun-2015.)
๐‘† = ran (๐‘ค โˆˆ โ„ค[i] โ†ฆ ((absโ€˜๐‘ค)โ†‘2))    &   (๐œ‘ โ†’ ๐ด โˆˆ โ„•)    &   (๐œ‘ โ†’ ๐ต โˆˆ โ„•)    &   (๐œ‘ โ†’ โˆ€๐‘ โˆˆ โ„™ (๐‘ โˆฅ ๐ต โ†’ ๐‘ โˆˆ ๐‘†))    &   (๐œ‘ โ†’ (๐ด ยท ๐ต) โˆˆ ๐‘†)    โ‡’   (๐œ‘ โ†’ ๐ด โˆˆ ๐‘†)
 
Theorem2sqlem7 14438* Lemma for 2sq . (Contributed by Mario Carneiro, 19-Jun-2015.)
๐‘† = ran (๐‘ค โˆˆ โ„ค[i] โ†ฆ ((absโ€˜๐‘ค)โ†‘2))    &   ๐‘Œ = {๐‘ง โˆฃ โˆƒ๐‘ฅ โˆˆ โ„ค โˆƒ๐‘ฆ โˆˆ โ„ค ((๐‘ฅ gcd ๐‘ฆ) = 1 โˆง ๐‘ง = ((๐‘ฅโ†‘2) + (๐‘ฆโ†‘2)))}    โ‡’   ๐‘Œ โŠ† (๐‘† โˆฉ โ„•)
 
Theorem2sqlem8a 14439* Lemma for 2sqlem8 14440. (Contributed by Mario Carneiro, 4-Jun-2016.)
๐‘† = ran (๐‘ค โˆˆ โ„ค[i] โ†ฆ ((absโ€˜๐‘ค)โ†‘2))    &   ๐‘Œ = {๐‘ง โˆฃ โˆƒ๐‘ฅ โˆˆ โ„ค โˆƒ๐‘ฆ โˆˆ โ„ค ((๐‘ฅ gcd ๐‘ฆ) = 1 โˆง ๐‘ง = ((๐‘ฅโ†‘2) + (๐‘ฆโ†‘2)))}    &   (๐œ‘ โ†’ โˆ€๐‘ โˆˆ (1...(๐‘€ โˆ’ 1))โˆ€๐‘Ž โˆˆ ๐‘Œ (๐‘ โˆฅ ๐‘Ž โ†’ ๐‘ โˆˆ ๐‘†))    &   (๐œ‘ โ†’ ๐‘€ โˆฅ ๐‘)    &   (๐œ‘ โ†’ ๐‘ โˆˆ โ„•)    &   (๐œ‘ โ†’ ๐‘€ โˆˆ (โ„คโ‰ฅโ€˜2))    &   (๐œ‘ โ†’ ๐ด โˆˆ โ„ค)    &   (๐œ‘ โ†’ ๐ต โˆˆ โ„ค)    &   (๐œ‘ โ†’ (๐ด gcd ๐ต) = 1)    &   (๐œ‘ โ†’ ๐‘ = ((๐ดโ†‘2) + (๐ตโ†‘2)))    &   ๐ถ = (((๐ด + (๐‘€ / 2)) mod ๐‘€) โˆ’ (๐‘€ / 2))    &   ๐ท = (((๐ต + (๐‘€ / 2)) mod ๐‘€) โˆ’ (๐‘€ / 2))    โ‡’   (๐œ‘ โ†’ (๐ถ gcd ๐ท) โˆˆ โ„•)
 
Theorem2sqlem8 14440* Lemma for 2sq . (Contributed by Mario Carneiro, 20-Jun-2015.)
๐‘† = ran (๐‘ค โˆˆ โ„ค[i] โ†ฆ ((absโ€˜๐‘ค)โ†‘2))    &   ๐‘Œ = {๐‘ง โˆฃ โˆƒ๐‘ฅ โˆˆ โ„ค โˆƒ๐‘ฆ โˆˆ โ„ค ((๐‘ฅ gcd ๐‘ฆ) = 1 โˆง ๐‘ง = ((๐‘ฅโ†‘2) + (๐‘ฆโ†‘2)))}    &   (๐œ‘ โ†’ โˆ€๐‘ โˆˆ (1...(๐‘€ โˆ’ 1))โˆ€๐‘Ž โˆˆ ๐‘Œ (๐‘ โˆฅ ๐‘Ž โ†’ ๐‘ โˆˆ ๐‘†))    &   (๐œ‘ โ†’ ๐‘€ โˆฅ ๐‘)    &   (๐œ‘ โ†’ ๐‘ โˆˆ โ„•)    &   (๐œ‘ โ†’ ๐‘€ โˆˆ (โ„คโ‰ฅโ€˜2))    &   (๐œ‘ โ†’ ๐ด โˆˆ โ„ค)    &   (๐œ‘ โ†’ ๐ต โˆˆ โ„ค)    &   (๐œ‘ โ†’ (๐ด gcd ๐ต) = 1)    &   (๐œ‘ โ†’ ๐‘ = ((๐ดโ†‘2) + (๐ตโ†‘2)))    &   ๐ถ = (((๐ด + (๐‘€ / 2)) mod ๐‘€) โˆ’ (๐‘€ / 2))    &   ๐ท = (((๐ต + (๐‘€ / 2)) mod ๐‘€) โˆ’ (๐‘€ / 2))    &   ๐ธ = (๐ถ / (๐ถ gcd ๐ท))    &   ๐น = (๐ท / (๐ถ gcd ๐ท))    โ‡’   (๐œ‘ โ†’ ๐‘€ โˆˆ ๐‘†)
 
Theorem2sqlem9 14441* Lemma for 2sq . (Contributed by Mario Carneiro, 19-Jun-2015.)
๐‘† = ran (๐‘ค โˆˆ โ„ค[i] โ†ฆ ((absโ€˜๐‘ค)โ†‘2))    &   ๐‘Œ = {๐‘ง โˆฃ โˆƒ๐‘ฅ โˆˆ โ„ค โˆƒ๐‘ฆ โˆˆ โ„ค ((๐‘ฅ gcd ๐‘ฆ) = 1 โˆง ๐‘ง = ((๐‘ฅโ†‘2) + (๐‘ฆโ†‘2)))}    &   (๐œ‘ โ†’ โˆ€๐‘ โˆˆ (1...(๐‘€ โˆ’ 1))โˆ€๐‘Ž โˆˆ ๐‘Œ (๐‘ โˆฅ ๐‘Ž โ†’ ๐‘ โˆˆ ๐‘†))    &   (๐œ‘ โ†’ ๐‘€ โˆฅ ๐‘)    &   (๐œ‘ โ†’ ๐‘€ โˆˆ โ„•)    &   (๐œ‘ โ†’ ๐‘ โˆˆ ๐‘Œ)    โ‡’   (๐œ‘ โ†’ ๐‘€ โˆˆ ๐‘†)
 
Theorem2sqlem10 14442* Lemma for 2sq . Every factor of a "proper" sum of two squares (where the summands are coprime) is a sum of two squares. (Contributed by Mario Carneiro, 19-Jun-2015.)
๐‘† = ran (๐‘ค โˆˆ โ„ค[i] โ†ฆ ((absโ€˜๐‘ค)โ†‘2))    &   ๐‘Œ = {๐‘ง โˆฃ โˆƒ๐‘ฅ โˆˆ โ„ค โˆƒ๐‘ฆ โˆˆ โ„ค ((๐‘ฅ gcd ๐‘ฆ) = 1 โˆง ๐‘ง = ((๐‘ฅโ†‘2) + (๐‘ฆโ†‘2)))}    โ‡’   ((๐ด โˆˆ ๐‘Œ โˆง ๐ต โˆˆ โ„• โˆง ๐ต โˆฅ ๐ด) โ†’ ๐ต โˆˆ ๐‘†)
 
PART 11  GUIDES AND MISCELLANEA
 
11.1  Guides (conventions, explanations, and examples)
 
11.1.1  Conventions

This section describes the conventions we use. These conventions often refer to existing mathematical practices, which are discussed in more detail in other references. The following sources lay out how mathematics is developed without the law of the excluded middle. Of course, there are a greater number of sources which assume excluded middle and most of what is in them applies here too (especially in a treatment such as ours which is built on first-order logic and set theory, rather than, say, type theory). Studying how a topic is treated in the Metamath Proof Explorer and the references therein is often a good place to start (and is easy to compare with the Intuitionistic Logic Explorer). The textbooks provide a motivation for what we are doing, whereas Metamath lets you see in detail all hidden and implicit steps. Most standard theorems are accompanied by citations. Some closely followed texts include the following:

  • Axioms of propositional calculus - Stanford Encyclopedia of Philosophy or [Heyting].
  • Axioms of predicate calculus - our axioms are adapted from the ones in the Metamath Proof Explorer.
  • Theorems of propositional calculus - [Heyting].
  • Theorems of pure predicate calculus - Metamath Proof Explorer.
  • Theorems of equality and substitution - Metamath Proof Explorer.
  • Axioms of set theory - [Crosilla].
  • Development of set theory - Chapter 10 of [HoTT].
  • Construction of real and complex numbers - Chapter 11 of [HoTT]; [BauerTaylor].
  • Theorems about real numbers - [Geuvers].
 
Theoremconventions 14443 Unless there is a reason to diverge, we follow the conventions of the Metamath Proof Explorer (MPE, set.mm). This list of conventions is intended to be read in conjunction with the corresponding conventions in the Metamath Proof Explorer, and only the differences are described below.
  • Minimizing axioms and the axiom of choice. We prefer proofs that depend on fewer and/or weaker axioms, even if the proofs are longer. In particular, our choice of IZF (Intuitionistic Zermelo-Fraenkel) over CZF (Constructive Zermelo-Fraenkel, a weaker system) was just an expedient choice because IZF is easier to formalize in Metamath. You can find some development using CZF in BJ's mathbox starting at wbd 14534 (and the section header just above it). As for the axiom of choice, the full axiom of choice implies excluded middle as seen at acexmid 5873, although some authors will use countable choice or dependent choice. For example, countable choice or excluded middle is needed to show that the Cauchy reals coincide with the Dedekind reals - Corollary 11.4.3 of [HoTT], p. (varies).
  • Junk/undefined results. Much of the discussion of this topic in the Metamath Proof Explorer applies except that certain techniques are not available to us. For example, the Metamath Proof Explorer will often say "if a function is evaluated within its domain, a certain result follows; if the function is evaluated outside its domain, the same result follows. Since the function must be evaluated within its domain or outside it, the result follows unconditionally" (the use of excluded middle in this argument is perhaps obvious when stated this way). Often, the easiest fix will be to prove we are evaluating functions within their domains, other times it will be possible to use a theorem like relelfvdm 5547 which says that if a function value produces an inhabited set, then the function is being evaluated within its domain.
  • Bibliography references. The bibliography for the Intuitionistic Logic Explorer is separate from the one for the Metamath Proof Explorer but feel free to copy-paste a citation in either direction in order to cite it.

Label naming conventions

Here are a few of the label naming conventions:

  • Suffixes. We follow the conventions of the Metamath Proof Explorer with a few additions. A biconditional in set.mm which is an implication in iset.mm should have a "r" (for the reverse direction), or "i"/"im" (for the forward direction) appended. A theorem in set.mm which has a decidability condition added should add "dc" to the theorem name. A theorem in set.mm where "nonempty class" is changed to "inhabited class" should add "m" (for member) to the theorem name.
  • iset.mm versus set.mm names

    Theorems which are the same as in set.mm should be named the same (that is, where the statement of the theorem is the same; the proof can differ without a new name being called for). Theorems which are different should be named differently (we do have a small number of intentional exceptions to this rule but on the whole it serves us well).

    As for how to choose names so they are different between iset.mm and set.mm, when possible choose a name which reflect the difference in the theorems. For example, if a theorem in set.mm is an equality and the iset.mm analogue is a subset, add "ss" to the iset.mm name. If need be, add "i" to the iset.mm name (usually as a prefix to some portion of the name).

    As with set.mm, we welcome suggestions for better names (such as names which are more consistent with naming conventions).

    We do try to keep set.mm and iset.mm similar where we can. For example, if a theorem exists in both places but the name in set.mm isn't great, we tend to keep that name for iset.mm, or change it in both files together. This is mainly to make it easier to copy theorems, but also to generally help people browse proofs, find theorems, write proofs, etc.

The following table shows some commonly-used abbreviations in labels which are not found in the Metamath Proof Explorer, in alphabetical order. For each abbreviation we provide a mnenomic to help you remember it, the source theorem/assumption defining it, an expression showing what it looks like, whether or not it is a "syntax fragment" (an abbreviation that indicates a particular kind of syntax), and hyperlinks to label examples that use the abbreviation. The abbreviation is bolded if there is a df-NAME definition but the label fragment is not NAME.

For the "g" abbreviation, this is related to the set.mm usage, in which "is a set" conditions are converted from hypotheses to antecedents, but is also used where "is a set" conditions are added relative to similar set.mm theorems.

AbbreviationMnenomic/MeaningSource ExpressionSyntax?Example(s)
apapart df-pap 7246, df-ap 8538 Yes apadd1 8564, apne 8579
gwith "is a set" condition No 1stvalg 6142, brtposg 6254, setsmsbasg 13949
seq3, sum3recursive sequence df-seqfrec 10445 Yes seq3-1 10459, fsum3 11394
taptight apartness df-tap 7248 Yes df-tap 7248

(Contributed by Jim Kingdon, 24-Feb-2020.) (New usage is discouraged.)

๐œ‘    โ‡’   ๐œ‘
 
11.1.2  Definitional examples
 
Theoremex-or 14444 Example for ax-io 709. Example by David A. Wheeler. (Contributed by Mario Carneiro, 9-May-2015.)
(2 = 3 โˆจ 4 = 4)
 
Theoremex-an 14445 Example for ax-ia1 106. Example by David A. Wheeler. (Contributed by Mario Carneiro, 9-May-2015.)
(2 = 2 โˆง 3 = 3)
 
Theorem1kp2ke3k 14446 Example for df-dec 9384, 1000 + 2000 = 3000.

This proof disproves (by counterexample) the assertion of Hao Wang, who stated, "There is a theorem in the primitive notation of set theory that corresponds to the arithmetic theorem 1000 + 2000 = 3000. The formula would be forbiddingly long... even if (one) knows the definitions and is asked to simplify the long formula according to them, chances are he will make errors and arrive at some incorrect result." (Hao Wang, "Theory and practice in mathematics" , In Thomas Tymoczko, editor, New Directions in the Philosophy of Mathematics, pp 129-152, Birkauser Boston, Inc., Boston, 1986. (QA8.6.N48). The quote itself is on page 140.)

This is noted in Metamath: A Computer Language for Pure Mathematics by Norman Megill (2007) section 1.1.3. Megill then states, "A number of writers have conveyed the impression that the kind of absolute rigor provided by Metamath is an impossible dream, suggesting that a complete, formal verification of a typical theorem would take millions of steps in untold volumes of books... These writers assume, however, that in order to achieve the kind of complete formal verification they desire one must break down a proof into individual primitive steps that make direct reference to the axioms. This is not necessary. There is no reason not to make use of previously proved theorems rather than proving them over and over... A hierarchy of theorems and definitions permits an exponential growth in the formula sizes and primitive proof steps to be described with only a linear growth in the number of symbols used. Of course, this is how ordinary informal mathematics is normally done anyway, but with Metamath it can be done with absolute rigor and precision."

The proof here starts with (2 + 1) = 3, commutes it, and repeatedly multiplies both sides by ten. This is certainly longer than traditional mathematical proofs, e.g., there are a number of steps explicitly shown here to show that we're allowed to do operations such as multiplication. However, while longer, the proof is clearly a manageable size - even though every step is rigorously derived all the way back to the primitive notions of set theory and logic. And while there's a risk of making errors, the many independent verifiers make it much less likely that an incorrect result will be accepted.

This proof heavily relies on the decimal constructor df-dec 9384 developed by Mario Carneiro in 2015. The underlying Metamath language has an intentionally very small set of primitives; it doesn't even have a built-in construct for numbers. Instead, the digits are defined using these primitives, and the decimal constructor is used to make it easy to express larger numbers as combinations of digits.

(Contributed by David A. Wheeler, 29-Jun-2016.) (Shortened by Mario Carneiro using the arithmetic algorithm in mmj2, 30-Jun-2016.)

(1000 + 2000) = 3000
 
Theoremex-fl 14447 Example for df-fl 10269. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)
((โŒŠโ€˜(3 / 2)) = 1 โˆง (โŒŠโ€˜-(3 / 2)) = -2)
 
Theoremex-ceil 14448 Example for df-ceil 10270. (Contributed by AV, 4-Sep-2021.)
((โŒˆโ€˜(3 / 2)) = 2 โˆง (โŒˆโ€˜-(3 / 2)) = -1)
 
Theoremex-exp 14449 Example for df-exp 10519. (Contributed by AV, 4-Sep-2021.)
((5โ†‘2) = 25 โˆง (-3โ†‘-2) = (1 / 9))
 
Theoremex-fac 14450 Example for df-fac 10705. (Contributed by AV, 4-Sep-2021.)
(!โ€˜5) = 120
 
Theoremex-bc 14451 Example for df-bc 10727. (Contributed by AV, 4-Sep-2021.)
(5C3) = 10
 
Theoremex-dvds 14452 Example for df-dvds 11794: 3 divides into 6. (Contributed by David A. Wheeler, 19-May-2015.)
3 โˆฅ 6
 
Theoremex-gcd 14453 Example for df-gcd 11943. (Contributed by AV, 5-Sep-2021.)
(-6 gcd 9) = 3
 
PART 12  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
 
12.1  Mathboxes for user contributions
 
12.1.1  Mathbox guidelines
 
Theoremmathbox 14454 (This theorem is a dummy placeholder for these guidelines. The label of this theorem, "mathbox", is hard-coded into the Metamath program to identify the start of the mathbox section for web page generation.)

A "mathbox" is a user-contributed section that is maintained by its contributor independently from the main part of iset.mm.

For contributors:

By making a contribution, you agree to release it into the public domain, according to the statement at the beginning of iset.mm.

Guidelines:

Mathboxes in iset.mm follow the same practices as in set.mm, so refer to the mathbox guidelines there for more details.

(Contributed by NM, 20-Feb-2007.) (Revised by the Metamath team, 9-Sep-2023.) (New usage is discouraged.)

๐œ‘    โ‡’   ๐œ‘
 
12.2  Mathbox for BJ
 
12.2.1  Propositional calculus
 
Theorembj-nnsn 14455 As far as implying a negated formula is concerned, a formula is equivalent to its double negation. (Contributed by BJ, 24-Nov-2023.)
((๐œ‘ โ†’ ยฌ ๐œ“) โ†” (ยฌ ยฌ ๐œ‘ โ†’ ยฌ ๐œ“))
 
Theorembj-nnor 14456 Double negation of a disjunction in terms of implication. (Contributed by BJ, 9-Oct-2019.)
(ยฌ ยฌ (๐œ‘ โˆจ ๐œ“) โ†” (ยฌ ๐œ‘ โ†’ ยฌ ยฌ ๐œ“))
 
Theorembj-nnim 14457 The double negation of an implication implies the implication with the consequent doubly negated. (Contributed by BJ, 24-Nov-2023.)
(ยฌ ยฌ (๐œ‘ โ†’ ๐œ“) โ†’ (๐œ‘ โ†’ ยฌ ยฌ ๐œ“))
 
Theorembj-nnan 14458 The double negation of a conjunction implies the conjunction of the double negations. (Contributed by BJ, 24-Nov-2023.)
(ยฌ ยฌ (๐œ‘ โˆง ๐œ“) โ†’ (ยฌ ยฌ ๐œ‘ โˆง ยฌ ยฌ ๐œ“))
 
Theorembj-nnclavius 14459 Clavius law with doubly negated consequent. (Contributed by BJ, 4-Dec-2023.)
((ยฌ ๐œ‘ โ†’ ๐œ‘) โ†’ ยฌ ยฌ ๐œ‘)
 
Theorembj-imnimnn 14460 If a formula is implied by both a formula and its negation, then it is not refutable. There is another proof using the inference associated with bj-nnclavius 14459 as its last step. (Contributed by BJ, 27-Oct-2024.)
(๐œ‘ โ†’ ๐œ“)    &   (ยฌ ๐œ‘ โ†’ ๐œ“)    โ‡’    ยฌ ยฌ ๐œ“
 
12.2.1.1  Stable formulas

Some of the following theorems, like bj-sttru 14462 or bj-stfal 14464 could be deduced from their analogues for decidability, but stability is not provable from decidability in minimal calculus, so direct proofs have their interest.

 
Theorembj-trst 14461 A provable formula is stable. (Contributed by BJ, 24-Nov-2023.)
(๐œ‘ โ†’ STAB ๐œ‘)
 
Theorembj-sttru 14462 The true truth value is stable. (Contributed by BJ, 5-Aug-2024.)
STAB โŠค
 
Theorembj-fast 14463 A refutable formula is stable. (Contributed by BJ, 24-Nov-2023.)
(ยฌ ๐œ‘ โ†’ STAB ๐œ‘)
 
Theorembj-stfal 14464 The false truth value is stable. (Contributed by BJ, 5-Aug-2024.)
STAB โŠฅ
 
Theorembj-nnst 14465 Double negation of stability of a formula. Intuitionistic logic refutes unstability (but does not prove stability) of any formula. This theorem can also be proved in classical refutability calculus (see https://us.metamath.org/mpeuni/bj-peircestab.html) but not in minimal calculus (see https://us.metamath.org/mpeuni/bj-stabpeirce.html). See nnnotnotr 14712 for the version not using the definition of stability. (Contributed by BJ, 9-Oct-2019.) Prove it in ( โ†’ , ยฌ ) -intuitionistic calculus with definitions (uses of ax-ia1 106, ax-ia2 107, ax-ia3 108 are via sylibr 134, necessary for definition unpackaging), and in ( โ†’ , โ†” , ยฌ )-intuitionistic calculus, following a discussion with Jim Kingdon. (Revised by BJ, 27-Oct-2024.)
ยฌ ยฌ STAB ๐œ‘
 
Theorembj-nnbist 14466 If a formula is not refutable, then it is stable if and only if it is provable. By double-negation translation, if ๐œ‘ is a classical tautology, then ยฌ ยฌ ๐œ‘ is an intuitionistic tautology. Therefore, if ๐œ‘ is a classical tautology, then ๐œ‘ is intuitionistically equivalent to its stability (and to its decidability, see bj-nnbidc 14479). (Contributed by BJ, 24-Nov-2023.)
(ยฌ ยฌ ๐œ‘ โ†’ (STAB ๐œ‘ โ†” ๐œ‘))
 
Theorembj-stst 14467 Stability of a proposition is stable if and only if that proposition is stable. STAB is idempotent. (Contributed by BJ, 9-Oct-2019.)
(STAB STAB ๐œ‘ โ†” STAB ๐œ‘)
 
Theorembj-stim 14468 A conjunction with a stable consequent is stable. See stabnot 833 for negation , bj-stan 14469 for conjunction , and bj-stal 14471 for universal quantification. (Contributed by BJ, 24-Nov-2023.)
(STAB ๐œ“ โ†’ STAB (๐œ‘ โ†’ ๐œ“))
 
Theorembj-stan 14469 The conjunction of two stable formulas is stable. See bj-stim 14468 for implication, stabnot 833 for negation, and bj-stal 14471 for universal quantification. (Contributed by BJ, 24-Nov-2023.)
((STAB ๐œ‘ โˆง STAB ๐œ“) โ†’ STAB (๐œ‘ โˆง ๐œ“))
 
Theorembj-stand 14470 The conjunction of two stable formulas is stable. Deduction form of bj-stan 14469. Its proof is shorter (when counting all steps, including syntactic steps), so one could prove it first and then bj-stan 14469 from it, the usual way. (Contributed by BJ, 24-Nov-2023.) (Proof modification is discouraged.)
(๐œ‘ โ†’ STAB ๐œ“)    &   (๐œ‘ โ†’ STAB ๐œ’)    โ‡’   (๐œ‘ โ†’ STAB (๐œ“ โˆง ๐œ’))
 
Theorembj-stal 14471 The universal quantification of a stable formula is stable. See bj-stim 14468 for implication, stabnot 833 for negation, and bj-stan 14469 for conjunction. (Contributed by BJ, 24-Nov-2023.)
(โˆ€๐‘ฅSTAB ๐œ‘ โ†’ STAB โˆ€๐‘ฅ๐œ‘)
 
Theorembj-pm2.18st 14472 Clavius law for stable formulas. See pm2.18dc 855. (Contributed by BJ, 4-Dec-2023.)
(STAB ๐œ‘ โ†’ ((ยฌ ๐œ‘ โ†’ ๐œ‘) โ†’ ๐œ‘))
 
Theorembj-con1st 14473 Contraposition when the antecedent is a negated stable proposition. See con1dc 856. (Contributed by BJ, 11-Nov-2024.)
(STAB ๐œ‘ โ†’ ((ยฌ ๐œ‘ โ†’ ๐œ“) โ†’ (ยฌ ๐œ“ โ†’ ๐œ‘)))
 
12.2.1.2  Decidable formulas
 
Theorembj-trdc 14474 A provable formula is decidable. (Contributed by BJ, 24-Nov-2023.)
(๐œ‘ โ†’ DECID ๐œ‘)
 
Theorembj-dctru 14475 The true truth value is decidable. (Contributed by BJ, 5-Aug-2024.)
DECID โŠค
 
Theorembj-fadc 14476 A refutable formula is decidable. (Contributed by BJ, 24-Nov-2023.)
(ยฌ ๐œ‘ โ†’ DECID ๐œ‘)
 
Theorembj-dcfal 14477 The false truth value is decidable. (Contributed by BJ, 5-Aug-2024.)
DECID โŠฅ
 
Theorembj-dcstab 14478 A decidable formula is stable. (Contributed by BJ, 24-Nov-2023.) (Proof modification is discouraged.)
(DECID ๐œ‘ โ†’ STAB ๐œ‘)
 
Theorembj-nnbidc 14479 If a formula is not refutable, then it is decidable if and only if it is provable. See also comment of bj-nnbist 14466. (Contributed by BJ, 24-Nov-2023.)
(ยฌ ยฌ ๐œ‘ โ†’ (DECID ๐œ‘ โ†” ๐œ‘))
 
Theorembj-nndcALT 14480 Alternate proof of nndc 851. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by BJ, 9-Oct-2019.)
ยฌ ยฌ DECID ๐œ‘
 
Theorembj-dcdc 14481 Decidability of a proposition is decidable if and only if that proposition is decidable. DECID is idempotent. (Contributed by BJ, 9-Oct-2019.)
(DECID DECID ๐œ‘ โ†” DECID ๐œ‘)
 
Theorembj-stdc 14482 Decidability of a proposition is stable if and only if that proposition is decidable. In particular, the assumption that every formula is stable implies that every formula is decidable, hence classical logic. (Contributed by BJ, 9-Oct-2019.)
(STAB DECID ๐œ‘ โ†” DECID ๐œ‘)
 
Theorembj-dcst 14483 Stability of a proposition is decidable if and only if that proposition is stable. (Contributed by BJ, 24-Nov-2023.)
(DECID STAB ๐œ‘ โ†” STAB ๐œ‘)
 
12.2.2  Predicate calculus
 
Theorembj-ex 14484* Existential generalization. (Contributed by BJ, 8-Dec-2019.) Proof modification is discouraged because there are shorter proofs, but using less basic results (like exlimiv 1598 and 19.9ht 1641 or 19.23ht 1497). (Proof modification is discouraged.)
(โˆƒ๐‘ฅ๐œ‘ โ†’ ๐œ‘)
 
Theorembj-hbalt 14485 Closed form of hbal 1477 (copied from set.mm). (Contributed by BJ, 2-May-2019.)
(โˆ€๐‘ฆ(๐œ‘ โ†’ โˆ€๐‘ฅ๐œ‘) โ†’ (โˆ€๐‘ฆ๐œ‘ โ†’ โˆ€๐‘ฅโˆ€๐‘ฆ๐œ‘))
 
Theorembj-nfalt 14486 Closed form of nfal 1576 (copied from set.mm). (Contributed by BJ, 2-May-2019.) (Proof modification is discouraged.)
(โˆ€๐‘ฅโ„ฒ๐‘ฆ๐œ‘ โ†’ โ„ฒ๐‘ฆโˆ€๐‘ฅ๐œ‘)
 
Theoremspimd 14487 Deduction form of spim 1738. (Contributed by BJ, 17-Oct-2019.)
(๐œ‘ โ†’ โ„ฒ๐‘ฅ๐œ’)    &   (๐œ‘ โ†’ โˆ€๐‘ฅ(๐‘ฅ = ๐‘ฆ โ†’ (๐œ“ โ†’ ๐œ’)))    โ‡’   (๐œ‘ โ†’ (โˆ€๐‘ฅ๐œ“ โ†’ ๐œ’))
 
Theorem2spim 14488* Double substitution, as in spim 1738. (Contributed by BJ, 17-Oct-2019.)
โ„ฒ๐‘ฅ๐œ’    &   โ„ฒ๐‘ง๐œ’    &   ((๐‘ฅ = ๐‘ฆ โˆง ๐‘ง = ๐‘ก) โ†’ (๐œ“ โ†’ ๐œ’))    โ‡’   (โˆ€๐‘งโˆ€๐‘ฅ๐œ“ โ†’ ๐œ’)
 
Theoremch2var 14489* Implicit substitution of ๐‘ฆ for ๐‘ฅ and ๐‘ก for ๐‘ง into a theorem. (Contributed by BJ, 17-Oct-2019.)
โ„ฒ๐‘ฅ๐œ“    &   โ„ฒ๐‘ง๐œ“    &   ((๐‘ฅ = ๐‘ฆ โˆง ๐‘ง = ๐‘ก) โ†’ (๐œ‘ โ†” ๐œ“))    &   ๐œ‘    โ‡’   ๐œ“
 
Theoremch2varv 14490* Version of ch2var 14489 with nonfreeness hypotheses replaced with disjoint variable conditions. (Contributed by BJ, 17-Oct-2019.)
((๐‘ฅ = ๐‘ฆ โˆง ๐‘ง = ๐‘ก) โ†’ (๐œ‘ โ†” ๐œ“))    &   ๐œ‘    โ‡’   ๐œ“
 
Theorembj-exlimmp 14491 Lemma for bj-vtoclgf 14498. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
โ„ฒ๐‘ฅ๐œ“    &   (๐œ’ โ†’ ๐œ‘)    โ‡’   (โˆ€๐‘ฅ(๐œ’ โ†’ (๐œ‘ โ†’ ๐œ“)) โ†’ (โˆƒ๐‘ฅ๐œ’ โ†’ ๐œ“))
 
Theorembj-exlimmpi 14492 Lemma for bj-vtoclgf 14498. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
โ„ฒ๐‘ฅ๐œ“    &   (๐œ’ โ†’ ๐œ‘)    &   (๐œ’ โ†’ (๐œ‘ โ†’ ๐œ“))    โ‡’   (โˆƒ๐‘ฅ๐œ’ โ†’ ๐œ“)
 
Theorembj-sbimedh 14493 A strengthening of sbiedh 1787 (same proof). (Contributed by BJ, 16-Dec-2019.)
(๐œ‘ โ†’ โˆ€๐‘ฅ๐œ‘)    &   (๐œ‘ โ†’ (๐œ’ โ†’ โˆ€๐‘ฅ๐œ’))    &   (๐œ‘ โ†’ (๐‘ฅ = ๐‘ฆ โ†’ (๐œ“ โ†’ ๐œ’)))    โ‡’   (๐œ‘ โ†’ ([๐‘ฆ / ๐‘ฅ]๐œ“ โ†’ ๐œ’))
 
Theorembj-sbimeh 14494 A strengthening of sbieh 1790 (same proof). (Contributed by BJ, 16-Dec-2019.)
(๐œ“ โ†’ โˆ€๐‘ฅ๐œ“)    &   (๐‘ฅ = ๐‘ฆ โ†’ (๐œ‘ โ†’ ๐œ“))    โ‡’   ([๐‘ฆ / ๐‘ฅ]๐œ‘ โ†’ ๐œ“)
 
Theorembj-sbime 14495 A strengthening of sbie 1791 (same proof). (Contributed by BJ, 16-Dec-2019.)
โ„ฒ๐‘ฅ๐œ“    &   (๐‘ฅ = ๐‘ฆ โ†’ (๐œ‘ โ†’ ๐œ“))    โ‡’   ([๐‘ฆ / ๐‘ฅ]๐œ‘ โ†’ ๐œ“)
 
12.2.3  Set theorey miscellaneous
 
Theorembj-el2oss1o 14496 Shorter proof of el2oss1o 6443 using more axioms. (Contributed by BJ, 21-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
(๐ด โˆˆ 2o โ†’ ๐ด โŠ† 1o)
 
12.2.4  Extensionality

Various utility theorems using FOL and extensionality.

 
Theorembj-vtoclgft 14497 Weakening two hypotheses of vtoclgf 2795. (Contributed by BJ, 21-Nov-2019.)
โ„ฒ๐‘ฅ๐ด    &   โ„ฒ๐‘ฅ๐œ“    &   (๐‘ฅ = ๐ด โ†’ ๐œ‘)    โ‡’   (โˆ€๐‘ฅ(๐‘ฅ = ๐ด โ†’ (๐œ‘ โ†’ ๐œ“)) โ†’ (๐ด โˆˆ ๐‘‰ โ†’ ๐œ“))
 
Theorembj-vtoclgf 14498 Weakening two hypotheses of vtoclgf 2795. (Contributed by BJ, 21-Nov-2019.)
โ„ฒ๐‘ฅ๐ด    &   โ„ฒ๐‘ฅ๐œ“    &   (๐‘ฅ = ๐ด โ†’ ๐œ‘)    &   (๐‘ฅ = ๐ด โ†’ (๐œ‘ โ†’ ๐œ“))    โ‡’   (๐ด โˆˆ ๐‘‰ โ†’ ๐œ“)
 
Theoremelabgf0 14499 Lemma for elabgf 2879. (Contributed by BJ, 21-Nov-2019.)
(๐‘ฅ = ๐ด โ†’ (๐ด โˆˆ {๐‘ฅ โˆฃ ๐œ‘} โ†” ๐œ‘))
 
Theoremelabgft1 14500 One implication of elabgf 2879, in closed form. (Contributed by BJ, 21-Nov-2019.)
โ„ฒ๐‘ฅ๐ด    &   โ„ฒ๐‘ฅ๐œ“    โ‡’   (โˆ€๐‘ฅ(๐‘ฅ = ๐ด โ†’ (๐œ‘ โ†’ ๐œ“)) โ†’ (๐ด โˆˆ {๐‘ฅ โˆฃ ๐œ‘} โ†’ ๐œ“))
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