P. 130 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 12901-13000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	4sqlem6 12901	Lemma for 4sq 12928. (Contributed by Mario Carneiro, 15-Jul-2014.)
|- ( ph -> A e. ZZ )   &    |- ( ph -> M e. NN )   &    |- B = ( ( ( A + ( M / 2 ) ) mod M ) - ( M / 2 ) )   =>    |- ( ph -> ( -u ( M / 2 ) <_ B /\ B < ( M / 2 ) ) )
Theorem	4sqlem7 12902	Lemma for 4sq 12928. (Contributed by Mario Carneiro, 15-Jul-2014.)
|- ( ph -> A e. ZZ )   &    |- ( ph -> M e. NN )   &    |- B = ( ( ( A + ( M / 2 ) ) mod M ) - ( M / 2 ) )   =>    |- ( ph -> ( B ^ 2 ) <_ ( ( ( M ^ 2 ) / 2 ) / 2 ) )
Theorem	4sqlem8 12903	Lemma for 4sq 12928. (Contributed by Mario Carneiro, 15-Jul-2014.)
|- ( ph -> A e. ZZ )   &    |- ( ph -> M e. NN )   &    |- B = ( ( ( A + ( M / 2 ) ) mod M ) - ( M / 2 ) )   =>    |- ( ph -> M || ( ( A ^ 2 ) - ( B ^ 2 ) ) )
Theorem	4sqlem9 12904	Lemma for 4sq 12928. (Contributed by Mario Carneiro, 15-Jul-2014.)
|- ( ph -> A e. ZZ )   &    |- ( ph -> M e. NN )   &    |- B = ( ( ( A + ( M / 2 ) ) mod M ) - ( M / 2 ) )   &    |- ( ( ph /\ ps ) -> ( B ^ 2 ) = 0 )   =>    |- ( ( ph /\ ps ) -> ( M ^ 2 ) || ( A ^ 2 ) )
Theorem	4sqlem10 12905	Lemma for 4sq 12928. (Contributed by Mario Carneiro, 16-Jul-2014.)
|- ( ph -> A e. ZZ )   &    |- ( ph -> M e. NN )   &    |- B = ( ( ( A + ( M / 2 ) ) mod M ) - ( M / 2 ) )   &    |- ( ( ph /\ ps ) -> ( ( ( ( M ^ 2 ) / 2 ) / 2 ) - ( B ^ 2 ) ) = 0 )   =>    |- ( ( ph /\ ps ) -> ( M ^ 2 ) || ( ( A ^ 2 ) - ( ( ( M ^ 2 ) / 2 ) / 2 ) ) )
Theorem	4sqlem1 12906*	Lemma for 4sq 12928. The set S is the set of all numbers that are expressible as a sum of four squares. Our goal is to show that S = NN0; here we show one subset direction. (Contributed by Mario Carneiro, 14-Jul-2014.)
|- S = { n | E. x e. ZZ E. y e. ZZ E. z e. ZZ E. w e. ZZ n = ( ( ( x ^ 2 ) + ( y ^ 2 ) ) + ( ( z ^ 2 ) + ( w ^ 2 ) ) ) }   =>    |- S C_ NN0
Theorem	4sqlem2 12907*	Lemma for 4sq 12928. Change bound variables in S. (Contributed by Mario Carneiro, 14-Jul-2014.)
|- S = { n | E. x e. ZZ E. y e. ZZ E. z e. ZZ E. w e. ZZ n = ( ( ( x ^ 2 ) + ( y ^ 2 ) ) + ( ( z ^ 2 ) + ( w ^ 2 ) ) ) }   =>    |- ( A e. S <-> E. a e. ZZ E. b e. ZZ E. c e. ZZ E. d e. ZZ A = ( ( ( a ^ 2 ) + ( b ^ 2 ) ) + ( ( c ^ 2 ) + ( d ^ 2 ) ) ) )
Theorem	4sqlem3 12908*	Lemma for 4sq 12928. Sufficient condition to be in S. (Contributed by Mario Carneiro, 14-Jul-2014.)
|- S = { n | E. x e. ZZ E. y e. ZZ E. z e. ZZ E. w e. ZZ n = ( ( ( x ^ 2 ) + ( y ^ 2 ) ) + ( ( z ^ 2 ) + ( w ^ 2 ) ) ) }   =>    |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( C e. ZZ /\ D e. ZZ ) ) -> ( ( ( A ^ 2 ) + ( B ^ 2 ) ) + ( ( C ^ 2 ) + ( D ^ 2 ) ) ) e. S )
Theorem	4sqlem4a 12909*	Lemma for 4sqlem4 12910. (Contributed by Mario Carneiro, 14-Jul-2014.)
|- S = { n | E. x e. ZZ E. y e. ZZ E. z e. ZZ E. w e. ZZ n = ( ( ( x ^ 2 ) + ( y ^ 2 ) ) + ( ( z ^ 2 ) + ( w ^ 2 ) ) ) }   =>    |- ( ( A e. ZZ[_i] /\ B e. ZZ[_i] ) -> ( ( ( abs ` A ) ^ 2 ) + ( ( abs ` B ) ^ 2 ) ) e. S )
Theorem	4sqlem4 12910*	Lemma for 4sq 12928. We can express the four-square property more compactly in terms of gaussian integers, because the norms of gaussian integers are exactly sums of two squares. (Contributed by Mario Carneiro, 14-Jul-2014.)
|- S = { n | E. x e. ZZ E. y e. ZZ E. z e. ZZ E. w e. ZZ n = ( ( ( x ^ 2 ) + ( y ^ 2 ) ) + ( ( z ^ 2 ) + ( w ^ 2 ) ) ) }   =>    |- ( A e. S <-> E. u e. ZZ[_i] E. v e. ZZ[_i] A = ( ( ( abs ` u ) ^ 2 ) + ( ( abs ` v ) ^ 2 ) ) )
Theorem	mul4sqlem 12911*	Lemma for mul4sq 12912: algebraic manipulations. The extra assumptions involving M would let us know not just that the product is a sum of squares, but also that it preserves divisibility by M. (Contributed by Mario Carneiro, 14-Jul-2014.)
|- S = { n | E. x e. ZZ E. y e. ZZ E. z e. ZZ E. w e. ZZ n = ( ( ( x ^ 2 ) + ( y ^ 2 ) ) + ( ( z ^ 2 ) + ( w ^ 2 ) ) ) }   &    |- ( ph -> A e. ZZ[_i] )   &    |- ( ph -> B e. ZZ[_i] )   &    |- ( ph -> C e. ZZ[_i] )   &    |- ( ph -> D e. ZZ[_i] )   &    |- X = ( ( ( abs ` A ) ^ 2 ) + ( ( abs ` B ) ^ 2 ) )   &    |- Y = ( ( ( abs ` C ) ^ 2 ) + ( ( abs ` D ) ^ 2 ) )   &    |- ( ph -> M e. NN )   &    |- ( ph -> ( ( A - C ) / M ) e. ZZ[_i] )   &    |- ( ph -> ( ( B - D ) / M ) e. ZZ[_i] )   &    |- ( ph -> ( X / M ) e. NN0 )   =>    |- ( ph -> ( ( X / M ) x. ( Y / M ) ) e. S )
Theorem	mul4sq 12912*	Euler's four-square identity: The product of two sums of four squares is also a sum of four squares. This is usually quoted as an explicit formula involving eight real variables; we save some time by working with complex numbers (gaussian integers) instead, so that we only have to work with four variables, and also hiding the actual formula for the product in the proof of mul4sqlem 12911. (For the curious, the explicit formula that is used is ( | a | ^ 2 + | b | ^ 2 ) ( | c | ^ 2 + | d | ^ 2 ) = | a * x. c + b x. d * | ^ 2 + | a * x. d - b x. c * | ^ 2.) (Contributed by Mario Carneiro, 14-Jul-2014.)
|- S = { n | E. x e. ZZ E. y e. ZZ E. z e. ZZ E. w e. ZZ n = ( ( ( x ^ 2 ) + ( y ^ 2 ) ) + ( ( z ^ 2 ) + ( w ^ 2 ) ) ) }   =>    |- ( ( A e. S /\ B e. S ) -> ( A x. B ) e. S )
Theorem	4sqlemafi 12913*	Lemma for 4sq 12928. A is finite. (Contributed by Jim Kingdon, 24-May-2025.)
|- ( ph -> N e. NN )   &    |- ( ph -> P e. NN )   &    |- A = { u | E. m e. ( 0 ... N ) u = ( ( m ^ 2 ) mod P ) }   =>    |- ( ph -> A e. Fin )
Theorem	4sqlemffi 12914*	Lemma for 4sq 12928. ran F is finite. (Contributed by Jim Kingdon, 24-May-2025.)
|- ( ph -> N e. NN )   &    |- ( ph -> P e. NN )   &    |- A = { u | E. m e. ( 0 ... N ) u = ( ( m ^ 2 ) mod P ) }   &    |- F = ( v e. A |-> ( ( P - 1 ) - v ) )   =>    |- ( ph -> ran F e. Fin )
Theorem	4sqleminfi 12915*	Lemma for 4sq 12928. A i^i ran F is finite. (Contributed by Jim Kingdon, 24-May-2025.)
|- ( ph -> N e. NN )   &    |- ( ph -> P e. NN )   &    |- A = { u | E. m e. ( 0 ... N ) u = ( ( m ^ 2 ) mod P ) }   &    |- F = ( v e. A |-> ( ( P - 1 ) - v ) )   =>    |- ( ph -> ( A i^i ran F ) e. Fin )
Theorem	4sqexercise1 12916*	Exercise which may help in understanding the proof of 4sqlemsdc 12918. (Contributed by Jim Kingdon, 25-May-2025.)
|- S = { n | E. x e. ZZ n = ( x ^ 2 ) }   =>    |- ( A e. NN0 -> DECID A e. S )
Theorem	4sqexercise2 12917*	Exercise which may help in understanding the proof of 4sqlemsdc 12918. (Contributed by Jim Kingdon, 30-May-2025.)
|- S = { n | E. x e. ZZ E. y e. ZZ n = ( ( x ^ 2 ) + ( y ^ 2 ) ) }   =>    |- ( A e. NN0 -> DECID A e. S )
Theorem	4sqlemsdc 12918*	Lemma for 4sq 12928. The property of being the sum of four squares is decidable. The proof involves showing that (for a particular A) there are only a finite number of possible ways that it could be the sum of four squares, so checking each of those possibilities in turn decides whether the number is the sum of four squares. If this proof is hard to follow, especially because of its length, the simplified versions at 4sqexercise1 12916 and 4sqexercise2 12917 may help clarify, as they are using very much the same techniques on simplified versions of this lemma. (Contributed by Jim Kingdon, 25-May-2025.)
|- S = { n | E. x e. ZZ E. y e. ZZ E. z e. ZZ E. w e. ZZ n = ( ( ( x ^ 2 ) + ( y ^ 2 ) ) + ( ( z ^ 2 ) + ( w ^ 2 ) ) ) }   =>    |- ( A e. NN0 -> DECID A e. S )
Theorem	4sqlem11 12919*	Lemma for 4sq 12928. Use the pigeonhole principle to show that the sets { m ^ 2 | m e. ( 0 ... N ) } and { -u 1 - n ^ 2 | n e. ( 0 ... N ) } have a common element, mod P. Note that although the conclusion is stated in terms of A i^i ran F being nonempty, it is also inhabited by 4sqleminfi 12915 and fin0 7043. (Contributed by Mario Carneiro, 15-Jul-2014.)
|- S = { n | E. x e. ZZ E. y e. ZZ E. z e. ZZ E. w e. ZZ n = ( ( ( x ^ 2 ) + ( y ^ 2 ) ) + ( ( z ^ 2 ) + ( w ^ 2 ) ) ) }   &    |- ( ph -> N e. NN )   &    |- ( ph -> P = ( ( 2 x. N ) + 1 ) )   &    |- ( ph -> P e. Prime )   &    |- A = { u | E. m e. ( 0 ... N ) u = ( ( m ^ 2 ) mod P ) }   &    |- F = ( v e. A |-> ( ( P - 1 ) - v ) )   =>    |- ( ph -> ( A i^i ran F ) =/= (/) )
Theorem	4sqlem12 12920*	Lemma for 4sq 12928. For any odd prime P, there is a k < P such that k P - 1 is a sum of two squares. (Contributed by Mario Carneiro, 15-Jul-2014.)
|- S = { n | E. x e. ZZ E. y e. ZZ E. z e. ZZ E. w e. ZZ n = ( ( ( x ^ 2 ) + ( y ^ 2 ) ) + ( ( z ^ 2 ) + ( w ^ 2 ) ) ) }   &    |- ( ph -> N e. NN )   &    |- ( ph -> P = ( ( 2 x. N ) + 1 ) )   &    |- ( ph -> P e. Prime )   &    |- A = { u | E. m e. ( 0 ... N ) u = ( ( m ^ 2 ) mod P ) }   &    |- F = ( v e. A |-> ( ( P - 1 ) - v ) )   =>    |- ( ph -> E. k e. ( 1 ... ( P - 1 ) ) E. u e. ZZ[_i] ( ( ( abs ` u ) ^ 2 ) + 1 ) = ( k x. P ) )
Theorem	4sqlem13m 12921*	Lemma for 4sq 12928. (Contributed by Mario Carneiro, 16-Jul-2014.) (Revised by AV, 14-Sep-2020.)
|- S = { n | E. x e. ZZ E. y e. ZZ E. z e. ZZ E. w e. ZZ n = ( ( ( x ^ 2 ) + ( y ^ 2 ) ) + ( ( z ^ 2 ) + ( w ^ 2 ) ) ) }   &    |- ( ph -> N e. NN )   &    |- ( ph -> P = ( ( 2 x. N ) + 1 ) )   &    |- ( ph -> P e. Prime )   &    |- ( ph -> ( 0 ... ( 2 x. N ) ) C_ S )   &    |- T = { i e. NN | ( i x. P ) e. S }   &    |- M = inf ( T , RR , < )   =>    |- ( ph -> ( E. j j e. T /\ M < P ) )
Theorem	4sqlem14 12922*	Lemma for 4sq 12928. (Contributed by Mario Carneiro, 16-Jul-2014.) (Revised by AV, 14-Sep-2020.)
|- S = { n | E. x e. ZZ E. y e. ZZ E. z e. ZZ E. w e. ZZ n = ( ( ( x ^ 2 ) + ( y ^ 2 ) ) + ( ( z ^ 2 ) + ( w ^ 2 ) ) ) }   &    |- ( ph -> N e. NN )   &    |- ( ph -> P = ( ( 2 x. N ) + 1 ) )   &    |- ( ph -> P e. Prime )   &    |- ( ph -> ( 0 ... ( 2 x. N ) ) C_ S )   &    |- T = { i e. NN | ( i x. P ) e. S }   &    |- M = inf ( T , RR , < )   &    |- ( ph -> M e. ( ZZ>= ` 2 ) )   &    |- ( ph -> A e. ZZ )   &    |- ( ph -> B e. ZZ )   &    |- ( ph -> C e. ZZ )   &    |- ( ph -> D e. ZZ )   &    |- E = ( ( ( A + ( M / 2 ) ) mod M ) - ( M / 2 ) )   &    |- F = ( ( ( B + ( M / 2 ) ) mod M ) - ( M / 2 ) )   &    |- G = ( ( ( C + ( M / 2 ) ) mod M ) - ( M / 2 ) )   &    |- H = ( ( ( D + ( M / 2 ) ) mod M ) - ( M / 2 ) )   &    |- R = ( ( ( ( E ^ 2 ) + ( F ^ 2 ) ) + ( ( G ^ 2 ) + ( H ^ 2 ) ) ) / M )   &    |- ( ph -> ( M x. P ) = ( ( ( A ^ 2 ) + ( B ^ 2 ) ) + ( ( C ^ 2 ) + ( D ^ 2 ) ) ) )   =>    |- ( ph -> R e. NN0 )
Theorem	4sqlem15 12923*	Lemma for 4sq 12928. (Contributed by Mario Carneiro, 16-Jul-2014.) (Revised by AV, 14-Sep-2020.)
|- S = { n | E. x e. ZZ E. y e. ZZ E. z e. ZZ E. w e. ZZ n = ( ( ( x ^ 2 ) + ( y ^ 2 ) ) + ( ( z ^ 2 ) + ( w ^ 2 ) ) ) }   &    |- ( ph -> N e. NN )   &    |- ( ph -> P = ( ( 2 x. N ) + 1 ) )   &    |- ( ph -> P e. Prime )   &    |- ( ph -> ( 0 ... ( 2 x. N ) ) C_ S )   &    |- T = { i e. NN | ( i x. P ) e. S }   &    |- M = inf ( T , RR , < )   &    |- ( ph -> M e. ( ZZ>= ` 2 ) )   &    |- ( ph -> A e. ZZ )   &    |- ( ph -> B e. ZZ )   &    |- ( ph -> C e. ZZ )   &    |- ( ph -> D e. ZZ )   &    |- E = ( ( ( A + ( M / 2 ) ) mod M ) - ( M / 2 ) )   &    |- F = ( ( ( B + ( M / 2 ) ) mod M ) - ( M / 2 ) )   &    |- G = ( ( ( C + ( M / 2 ) ) mod M ) - ( M / 2 ) )   &    |- H = ( ( ( D + ( M / 2 ) ) mod M ) - ( M / 2 ) )   &    |- R = ( ( ( ( E ^ 2 ) + ( F ^ 2 ) ) + ( ( G ^ 2 ) + ( H ^ 2 ) ) ) / M )   &    |- ( ph -> ( M x. P ) = ( ( ( A ^ 2 ) + ( B ^ 2 ) ) + ( ( C ^ 2 ) + ( D ^ 2 ) ) ) )   =>    |- ( ( ph /\ R = M ) -> ( ( ( ( ( ( M ^ 2 ) / 2 ) / 2 ) - ( E ^ 2 ) ) = 0 /\ ( ( ( ( M ^ 2 ) / 2 ) / 2 ) - ( F ^ 2 ) ) = 0 ) /\ ( ( ( ( ( M ^ 2 ) / 2 ) / 2 ) - ( G ^ 2 ) ) = 0 /\ ( ( ( ( M ^ 2 ) / 2 ) / 2 ) - ( H ^ 2 ) ) = 0 ) ) )
Theorem	4sqlem16 12924*	Lemma for 4sq 12928. (Contributed by Mario Carneiro, 16-Jul-2014.) (Revised by AV, 14-Sep-2020.)
|- S = { n | E. x e. ZZ E. y e. ZZ E. z e. ZZ E. w e. ZZ n = ( ( ( x ^ 2 ) + ( y ^ 2 ) ) + ( ( z ^ 2 ) + ( w ^ 2 ) ) ) }   &    |- ( ph -> N e. NN )   &    |- ( ph -> P = ( ( 2 x. N ) + 1 ) )   &    |- ( ph -> P e. Prime )   &    |- ( ph -> ( 0 ... ( 2 x. N ) ) C_ S )   &    |- T = { i e. NN | ( i x. P ) e. S }   &    |- M = inf ( T , RR , < )   &    |- ( ph -> M e. ( ZZ>= ` 2 ) )   &    |- ( ph -> A e. ZZ )   &    |- ( ph -> B e. ZZ )   &    |- ( ph -> C e. ZZ )   &    |- ( ph -> D e. ZZ )   &    |- E = ( ( ( A + ( M / 2 ) ) mod M ) - ( M / 2 ) )   &    |- F = ( ( ( B + ( M / 2 ) ) mod M ) - ( M / 2 ) )   &    |- G = ( ( ( C + ( M / 2 ) ) mod M ) - ( M / 2 ) )   &    |- H = ( ( ( D + ( M / 2 ) ) mod M ) - ( M / 2 ) )   &    |- R = ( ( ( ( E ^ 2 ) + ( F ^ 2 ) ) + ( ( G ^ 2 ) + ( H ^ 2 ) ) ) / M )   &    |- ( ph -> ( M x. P ) = ( ( ( A ^ 2 ) + ( B ^ 2 ) ) + ( ( C ^ 2 ) + ( D ^ 2 ) ) ) )   =>    |- ( ph -> ( R <_ M /\ ( ( R = 0 \/ R = M ) -> ( M ^ 2 ) || ( M x. P ) ) ) )
Theorem	4sqlem17 12925*	Lemma for 4sq 12928. (Contributed by Mario Carneiro, 16-Jul-2014.) (Revised by AV, 14-Sep-2020.)
|- S = { n | E. x e. ZZ E. y e. ZZ E. z e. ZZ E. w e. ZZ n = ( ( ( x ^ 2 ) + ( y ^ 2 ) ) + ( ( z ^ 2 ) + ( w ^ 2 ) ) ) }   &    |- ( ph -> N e. NN )   &    |- ( ph -> P = ( ( 2 x. N ) + 1 ) )   &    |- ( ph -> P e. Prime )   &    |- ( ph -> ( 0 ... ( 2 x. N ) ) C_ S )   &    |- T = { i e. NN | ( i x. P ) e. S }   &    |- M = inf ( T , RR , < )   &    |- ( ph -> M e. ( ZZ>= ` 2 ) )   &    |- ( ph -> A e. ZZ )   &    |- ( ph -> B e. ZZ )   &    |- ( ph -> C e. ZZ )   &    |- ( ph -> D e. ZZ )   &    |- E = ( ( ( A + ( M / 2 ) ) mod M ) - ( M / 2 ) )   &    |- F = ( ( ( B + ( M / 2 ) ) mod M ) - ( M / 2 ) )   &    |- G = ( ( ( C + ( M / 2 ) ) mod M ) - ( M / 2 ) )   &    |- H = ( ( ( D + ( M / 2 ) ) mod M ) - ( M / 2 ) )   &    |- R = ( ( ( ( E ^ 2 ) + ( F ^ 2 ) ) + ( ( G ^ 2 ) + ( H ^ 2 ) ) ) / M )   &    |- ( ph -> ( M x. P ) = ( ( ( A ^ 2 ) + ( B ^ 2 ) ) + ( ( C ^ 2 ) + ( D ^ 2 ) ) ) )   =>    |- -. ph
Theorem	4sqlem18 12926*	Lemma for 4sq 12928. Inductive step, odd prime case. (Contributed by Mario Carneiro, 16-Jul-2014.) (Revised by AV, 14-Sep-2020.)
|- S = { n | E. x e. ZZ E. y e. ZZ E. z e. ZZ E. w e. ZZ n = ( ( ( x ^ 2 ) + ( y ^ 2 ) ) + ( ( z ^ 2 ) + ( w ^ 2 ) ) ) }   &    |- ( ph -> N e. NN )   &    |- ( ph -> P = ( ( 2 x. N ) + 1 ) )   &    |- ( ph -> P e. Prime )   &    |- ( ph -> ( 0 ... ( 2 x. N ) ) C_ S )   &    |- T = { i e. NN | ( i x. P ) e. S }   &    |- M = inf ( T , RR , < )   =>    |- ( ph -> P e. S )
Theorem	4sqlem19 12927*	Lemma for 4sq 12928. The proof is by strong induction - we show that if all the integers less than k are in S, then k is as well. In this part of the proof we do the induction argument and dispense with all the cases except the odd prime case, which is sent to 4sqlem18 12926. If k is 0 , 1 , 2, we show k e. S directly; otherwise if k is composite, k is the product of two numbers less than it (and hence in S by assumption), so by mul4sq 12912 k e. S. (Contributed by Mario Carneiro, 14-Jul-2014.) (Revised by Mario Carneiro, 20-Jun-2015.)
|- S = { n | E. x e. ZZ E. y e. ZZ E. z e. ZZ E. w e. ZZ n = ( ( ( x ^ 2 ) + ( y ^ 2 ) ) + ( ( z ^ 2 ) + ( w ^ 2 ) ) ) }   =>    |- NN0 = S
Theorem	4sq 12928*	Lagrange's four-square theorem, or Bachet's conjecture: every nonnegative integer is expressible as a sum of four squares. This is Metamath 100 proof #19. (Contributed by Mario Carneiro, 16-Jul-2014.)
|- ( A e. NN0 <-> E. a e. ZZ E. b e. ZZ E. c e. ZZ E. d e. ZZ A = ( ( ( a ^ 2 ) + ( b ^ 2 ) ) + ( ( c ^ 2 ) + ( d ^ 2 ) ) ) )
5.2.13  Decimal arithmetic (cont.)
Theorem	dec2dvds 12929	Divisibility by two is obvious in base 10. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- A e. NN0   &    |- B e. NN0   &    |- ( B x. 2 ) = C   &    |- D = ( C + 1 )   =>    |- -. 2 || ; A D
Theorem	dec5dvds 12930	Divisibility by five is obvious in base 10. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- A e. NN0   &    |- B e. NN   &    |- B < 5   =>    |- -. 5 || ; A B
Theorem	dec5dvds2 12931	Divisibility by five is obvious in base 10. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- A e. NN0   &    |- B e. NN   &    |- B < 5   &    |- ( 5 + B ) = C   =>    |- -. 5 || ; A C
Theorem	dec5nprm 12932	A decimal number greater than 10 and ending with five is not a prime number. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- A e. NN   =>    |- -. ; A 5 e. Prime
Theorem	dec2nprm 12933	A decimal number greater than 10 and ending with an even digit is not a prime number. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- A e. NN   &    |- B e. NN0   &    |- ( B x. 2 ) = C   =>    |- -. ; A C e. Prime
Theorem	modxai 12934	Add exponents in a power mod calculation. (Contributed by Mario Carneiro, 21-Feb-2014.) (Revised by Mario Carneiro, 5-Feb-2015.)
|- N e. NN   &    |- A e. NN   &    |- B e. NN0   &    |- D e. ZZ   &    |- K e. NN0   &    |- M e. NN0   &    |- C e. NN0   &    |- L e. NN0   &    |- ( ( A ^ B ) mod N ) = ( K mod N )   &    |- ( ( A ^ C ) mod N ) = ( L mod N )   &    |- ( B + C ) = E   &    |- ( ( D x. N ) + M ) = ( K x. L )   =>    |- ( ( A ^ E ) mod N ) = ( M mod N )
Theorem	mod2xi 12935	Double exponents in a power mod calculation. (Contributed by Mario Carneiro, 21-Feb-2014.)
|- N e. NN   &    |- A e. NN   &    |- B e. NN0   &    |- D e. ZZ   &    |- K e. NN0   &    |- M e. NN0   &    |- ( ( A ^ B ) mod N ) = ( K mod N )   &    |- ( 2 x. B ) = E   &    |- ( ( D x. N ) + M ) = ( K x. K )   =>    |- ( ( A ^ E ) mod N ) = ( M mod N )
Theorem	modxp1i 12936	Add one to an exponent in a power mod calculation. (Contributed by Mario Carneiro, 21-Feb-2014.)
|- N e. NN   &    |- A e. NN   &    |- B e. NN0   &    |- D e. ZZ   &    |- K e. NN0   &    |- M e. NN0   &    |- ( ( A ^ B ) mod N ) = ( K mod N )   &    |- ( B + 1 ) = E   &    |- ( ( D x. N ) + M ) = ( K x. A )   =>    |- ( ( A ^ E ) mod N ) = ( M mod N )
Theorem	modsubi 12937	Subtract from within a mod calculation. (Contributed by Mario Carneiro, 18-Feb-2014.)
|- N e. NN   &    |- A e. NN   &    |- B e. NN0   &    |- M e. NN0   &    |- ( A mod N ) = ( K mod N )   &    |- ( M + B ) = K   =>    |- ( ( A - B ) mod N ) = ( M mod N )
Theorem	gcdi 12938	Calculate a GCD via Euclid's algorithm. (Contributed by Mario Carneiro, 19-Feb-2014.)
|- K e. NN0   &    |- R e. NN0   &    |- N e. NN0   &    |- ( N gcd R ) = G   &    |- ( ( K x. N ) + R ) = M   =>    |- ( M gcd N ) = G
Theorem	gcdmodi 12939	Calculate a GCD via Euclid's algorithm. Theorem 5.6 in [ApostolNT] p. 109. (Contributed by Mario Carneiro, 19-Feb-2014.)
|- K e. NN0   &    |- R e. NN0   &    |- N e. NN   &    |- ( K mod N ) = ( R mod N )   &    |- ( N gcd R ) = G   =>    |- ( K gcd N ) = G
Theorem	numexp0 12940	Calculate an integer power. (Contributed by Mario Carneiro, 17-Apr-2015.)
|- A e. NN0   =>    |- ( A ^ 0 ) = 1
Theorem	numexp1 12941	Calculate an integer power. (Contributed by Mario Carneiro, 17-Apr-2015.)
|- A e. NN0   =>    |- ( A ^ 1 ) = A
Theorem	numexpp1 12942	Calculate an integer power. (Contributed by Mario Carneiro, 17-Apr-2015.)
|- A e. NN0   &    |- M e. NN0   &    |- ( M + 1 ) = N   &    |- ( ( A ^ M ) x. A ) = C   =>    |- ( A ^ N ) = C
Theorem	numexp2x 12943	Double an integer power. (Contributed by Mario Carneiro, 17-Apr-2015.)
|- A e. NN0   &    |- M e. NN0   &    |- ( 2 x. M ) = N   &    |- ( A ^ M ) = D   &    |- ( D x. D ) = C   =>    |- ( A ^ N ) = C
Theorem	decsplit0b 12944	Split a decimal number into two parts. Base case: N = 0. (Contributed by Mario Carneiro, 16-Jul-2015.) (Revised by AV, 8-Sep-2021.)
|- A e. NN0   =>    |- ( ( A x. (; 1 0 ^ 0 ) ) + B ) = ( A + B )
Theorem	decsplit0 12945	Split a decimal number into two parts. Base case: N = 0. (Contributed by Mario Carneiro, 16-Jul-2015.) (Revised by AV, 8-Sep-2021.)
|- A e. NN0   =>    |- ( ( A x. (; 1 0 ^ 0 ) ) + 0 ) = A
Theorem	decsplit1 12946	Split a decimal number into two parts. Base case: N = 1. (Contributed by Mario Carneiro, 16-Jul-2015.) (Revised by AV, 8-Sep-2021.)
|- A e. NN0   =>    |- ( ( A x. (; 1 0 ^ 1 ) ) + B ) = ; A B
Theorem	decsplit 12947	Split a decimal number into two parts. Inductive step. (Contributed by Mario Carneiro, 16-Jul-2015.) (Revised by AV, 8-Sep-2021.)
|- A e. NN0   &    |- B e. NN0   &    |- D e. NN0   &    |- M e. NN0   &    |- ( M + 1 ) = N   &    |- ( ( A x. (; 1 0 ^ M ) ) + B ) = C   =>    |- ( ( A x. (; 1 0 ^ N ) ) + ; B D ) = ; C D
Theorem	karatsuba 12948	The Karatsuba multiplication algorithm. If X and Y are decomposed into two groups of digits of length M (only the lower group is known to be this size but the algorithm is most efficient when the partition is chosen near the middle of the digit string), then X Y can be written in three groups of digits, where each group needs only one multiplication. Thus, we can halve both inputs with only three multiplications on the smaller operands, yielding an asymptotic improvement of n^(log2 3) instead of n^2 for the "naive" algorithm decmul1c 9638. (Contributed by Mario Carneiro, 16-Jul-2015.) (Revised by AV, 9-Sep-2021.)
|- A e. NN0   &    |- B e. NN0   &    |- C e. NN0   &    |- D e. NN0   &    |- S e. NN0   &    |- M e. NN0   &    |- ( A x. C ) = R   &    |- ( B x. D ) = T   &    |- ( ( A + B ) x. ( C + D ) ) = ( ( R + S ) + T )   &    |- ( ( A x. (; 1 0 ^ M ) ) + B ) = X   &    |- ( ( C x. (; 1 0 ^ M ) ) + D ) = Y   &    |- ( ( R x. (; 1 0 ^ M ) ) + S ) = W   &    |- ( ( W x. (; 1 0 ^ M ) ) + T ) = Z   =>    |- ( X x. Y ) = Z
Theorem	2exp4 12949	Two to the fourth power is 16. (Contributed by Mario Carneiro, 20-Apr-2015.)
|- ( 2 ^ 4 ) = ; 1 6
Theorem	2exp5 12950	Two to the fifth power is 32. (Contributed by AV, 16-Aug-2021.)
|- ( 2 ^ 5 ) = ; 3 2
Theorem	2exp6 12951	Two to the sixth power is 64. (Contributed by Mario Carneiro, 20-Apr-2015.) (Proof shortened by OpenAI, 25-Mar-2020.)
|- ( 2 ^ 6 ) = ; 6 4
Theorem	2exp7 12952	Two to the seventh power is 128. (Contributed by AV, 16-Aug-2021.)
|- ( 2 ^ 7 ) = ;; 1 2 8
Theorem	2exp8 12953	Two to the eighth power is 256. (Contributed by Mario Carneiro, 20-Apr-2015.)
|- ( 2 ^ 8 ) = ;; 2 5 6
Theorem	2exp11 12954	Two to the eleventh power is 2048. (Contributed by AV, 16-Aug-2021.)
|- ( 2 ^; 1 1 ) = ;;; 2 0 4 8
Theorem	2exp16 12955	Two to the sixteenth power is 65536. (Contributed by Mario Carneiro, 20-Apr-2015.)
|- ( 2 ^; 1 6 ) = ;;;; 6 5 5 3 6
Theorem	3exp3 12956	Three to the third power is 27. (Contributed by Mario Carneiro, 20-Apr-2015.)
|- ( 3 ^ 3 ) = ; 2 7
Theorem	2expltfac 12957	The factorial grows faster than two to the power N. (Contributed by Mario Carneiro, 15-Sep-2016.)
|- ( N e. ( ZZ>= ` 4 ) -> ( 2 ^ N ) < ( ! ` N ) )
5.3  Cardinality of real and complex number subsets
5.3.1  Countability of integers and rationals
Theorem	oddennn 12958	There are as many odd positive integers as there are positive integers. (Contributed by Jim Kingdon, 11-May-2022.)
|- { z e. NN | -. 2 || z } ~~ NN
Theorem	evenennn 12959	There are as many even positive integers as there are positive integers. (Contributed by Jim Kingdon, 12-May-2022.)
|- { z e. NN | 2 || z } ~~ NN
Theorem	xpnnen 12960	The Cartesian product of the set of positive integers with itself is equinumerous to the set of positive integers. (Contributed by NM, 1-Aug-2004.)
|- ( NN X. NN ) ~~ NN
Theorem	xpomen 12961	The Cartesian product of omega (the set of ordinal natural numbers) with itself is equinumerous to omega. Exercise 1 of [Enderton] p. 133. (Contributed by NM, 23-Jul-2004.)
|- ( om X. om ) ~~ om
Theorem	xpct 12962	The cartesian product of two sets dominated by om is dominated by om. (Contributed by Thierry Arnoux, 24-Sep-2017.)
|- ( ( A ~<_ om /\ B ~<_ om ) -> ( A X. B ) ~<_ om )
Theorem	unennn 12963	The union of two disjoint countably infinite sets is countably infinite. (Contributed by Jim Kingdon, 13-May-2022.)
|- ( ( A ~~ NN /\ B ~~ NN /\ ( A i^i B ) = (/) ) -> ( A u. B ) ~~ NN )
Theorem	znnen 12964	The set of integers and the set of positive integers are equinumerous. Corollary 8.1.23 of [AczelRathjen], p. 75. (Contributed by NM, 31-Jul-2004.)
|- ZZ ~~ NN
Theorem	ennnfonelemdc 12965*	Lemma for ennnfone 12991. A direct consequence of fidcenumlemrk 7117. (Contributed by Jim Kingdon, 15-Jul-2023.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : om -onto-> A )   &    |- ( ph -> P e. om )   =>    |- ( ph -> DECID ( F ` P ) e. ( F " P ) )
Theorem	ennnfonelemk 12966*	Lemma for ennnfone 12991. (Contributed by Jim Kingdon, 15-Jul-2023.)
|- ( ph -> F : om -onto-> A )   &    |- ( ph -> K e. om )   &    |- ( ph -> N e. om )   &    |- ( ph -> A. j e. suc N ( F ` K ) =/= ( F ` j ) )   =>    |- ( ph -> N e. K )
Theorem	ennnfonelemj0 12967*	Lemma for ennnfone 12991. Initial state for J. (Contributed by Jim Kingdon, 20-Jul-2023.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : om -onto-> A )   &    |- ( ph -> A. n e. om E. k e. om A. j e. suc n ( F ` k ) =/= ( F ` j ) )   &    |- G = ( x e. ( A ^pm om ) , y e. om |-> if ( ( F ` y ) e. ( F " y ) , x , ( x u. { <. dom x , ( F ` y ) >. } ) ) )   &    |- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- J = ( x e. NN0 |-> if ( x = 0 , (/) , ( `' N ` ( x - 1 ) ) ) )   &    |- H = seq 0 ( G , J )   =>    |- ( ph -> ( J ` 0 ) e. { g e. ( A ^pm om ) | dom g e. om } )
Theorem	ennnfonelemjn 12968*	Lemma for ennnfone 12991. Non-initial state for J. (Contributed by Jim Kingdon, 20-Jul-2023.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : om -onto-> A )   &    |- ( ph -> A. n e. om E. k e. om A. j e. suc n ( F ` k ) =/= ( F ` j ) )   &    |- G = ( x e. ( A ^pm om ) , y e. om |-> if ( ( F ` y ) e. ( F " y ) , x , ( x u. { <. dom x , ( F ` y ) >. } ) ) )   &    |- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- J = ( x e. NN0 |-> if ( x = 0 , (/) , ( `' N ` ( x - 1 ) ) ) )   &    |- H = seq 0 ( G , J )   =>    |- ( ( ph /\ f e. ( ZZ>= ` ( 0 + 1 ) ) ) -> ( J ` f ) e. om )
Theorem	ennnfonelemg 12969*	Lemma for ennnfone 12991. Closure for G. (Contributed by Jim Kingdon, 20-Jul-2023.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : om -onto-> A )   &    |- ( ph -> A. n e. om E. k e. om A. j e. suc n ( F ` k ) =/= ( F ` j ) )   &    |- G = ( x e. ( A ^pm om ) , y e. om |-> if ( ( F ` y ) e. ( F " y ) , x , ( x u. { <. dom x , ( F ` y ) >. } ) ) )   &    |- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- J = ( x e. NN0 |-> if ( x = 0 , (/) , ( `' N ` ( x - 1 ) ) ) )   &    |- H = seq 0 ( G , J )   =>    |- ( ( ph /\ ( f e. { g e. ( A ^pm om ) | dom g e. om } /\ j e. om ) ) -> ( f G j ) e. { g e. ( A ^pm om ) | dom g e. om } )
Theorem	ennnfonelemh 12970*	Lemma for ennnfone 12991. (Contributed by Jim Kingdon, 8-Jul-2023.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : om -onto-> A )   &    |- ( ph -> A. n e. om E. k e. om A. j e. suc n ( F ` k ) =/= ( F ` j ) )   &    |- G = ( x e. ( A ^pm om ) , y e. om |-> if ( ( F ` y ) e. ( F " y ) , x , ( x u. { <. dom x , ( F ` y ) >. } ) ) )   &    |- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- J = ( x e. NN0 |-> if ( x = 0 , (/) , ( `' N ` ( x - 1 ) ) ) )   &    |- H = seq 0 ( G , J )   =>    |- ( ph -> H : NN0 --> ( A ^pm om ) )
Theorem	ennnfonelem0 12971*	Lemma for ennnfone 12991. Initial value. (Contributed by Jim Kingdon, 15-Jul-2023.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : om -onto-> A )   &    |- ( ph -> A. n e. om E. k e. om A. j e. suc n ( F ` k ) =/= ( F ` j ) )   &    |- G = ( x e. ( A ^pm om ) , y e. om |-> if ( ( F ` y ) e. ( F " y ) , x , ( x u. { <. dom x , ( F ` y ) >. } ) ) )   &    |- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- J = ( x e. NN0 |-> if ( x = 0 , (/) , ( `' N ` ( x - 1 ) ) ) )   &    |- H = seq 0 ( G , J )   =>    |- ( ph -> ( H ` 0 ) = (/) )
Theorem	ennnfonelemp1 12972*	Lemma for ennnfone 12991. Value of H at a successor. (Contributed by Jim Kingdon, 23-Jul-2023.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : om -onto-> A )   &    |- ( ph -> A. n e. om E. k e. om A. j e. suc n ( F ` k ) =/= ( F ` j ) )   &    |- G = ( x e. ( A ^pm om ) , y e. om |-> if ( ( F ` y ) e. ( F " y ) , x , ( x u. { <. dom x , ( F ` y ) >. } ) ) )   &    |- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- J = ( x e. NN0 |-> if ( x = 0 , (/) , ( `' N ` ( x - 1 ) ) ) )   &    |- H = seq 0 ( G , J )   &    |- ( ph -> P e. NN0 )   =>    |- ( ph -> ( H ` ( P + 1 ) ) = if ( ( F ` ( `' N ` P ) ) e. ( F " ( `' N ` P ) ) , ( H ` P ) , ( ( H ` P ) u. { <. dom ( H ` P ) , ( F ` ( `' N ` P ) ) >. } ) ) )
Theorem	ennnfonelem1 12973*	Lemma for ennnfone 12991. Second value. (Contributed by Jim Kingdon, 19-Jul-2023.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : om -onto-> A )   &    |- ( ph -> A. n e. om E. k e. om A. j e. suc n ( F ` k ) =/= ( F ` j ) )   &    |- G = ( x e. ( A ^pm om ) , y e. om |-> if ( ( F ` y ) e. ( F " y ) , x , ( x u. { <. dom x , ( F ` y ) >. } ) ) )   &    |- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- J = ( x e. NN0 |-> if ( x = 0 , (/) , ( `' N ` ( x - 1 ) ) ) )   &    |- H = seq 0 ( G , J )   =>    |- ( ph -> ( H ` 1 ) = { <. (/) , ( F ` (/) ) >. } )
Theorem	ennnfonelemom 12974*	Lemma for ennnfone 12991. H yields finite sequences. (Contributed by Jim Kingdon, 19-Jul-2023.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : om -onto-> A )   &    |- ( ph -> A. n e. om E. k e. om A. j e. suc n ( F ` k ) =/= ( F ` j ) )   &    |- G = ( x e. ( A ^pm om ) , y e. om |-> if ( ( F ` y ) e. ( F " y ) , x , ( x u. { <. dom x , ( F ` y ) >. } ) ) )   &    |- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- J = ( x e. NN0 |-> if ( x = 0 , (/) , ( `' N ` ( x - 1 ) ) ) )   &    |- H = seq 0 ( G , J )   &    |- ( ph -> P e. NN0 )   =>    |- ( ph -> dom ( H ` P ) e. om )
Theorem	ennnfonelemhdmp1 12975*	Lemma for ennnfone 12991. Domain at a successor where we need to add an element to the sequence. (Contributed by Jim Kingdon, 23-Jul-2023.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : om -onto-> A )   &    |- ( ph -> A. n e. om E. k e. om A. j e. suc n ( F ` k ) =/= ( F ` j ) )   &    |- G = ( x e. ( A ^pm om ) , y e. om |-> if ( ( F ` y ) e. ( F " y ) , x , ( x u. { <. dom x , ( F ` y ) >. } ) ) )   &    |- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- J = ( x e. NN0 |-> if ( x = 0 , (/) , ( `' N ` ( x - 1 ) ) ) )   &    |- H = seq 0 ( G , J )   &    |- ( ph -> P e. NN0 )   &    |- ( ph -> -. ( F ` ( `' N ` P ) ) e. ( F " ( `' N ` P ) ) )   =>    |- ( ph -> dom ( H ` ( P + 1 ) ) = suc dom ( H ` P ) )
Theorem	ennnfonelemss 12976*	Lemma for ennnfone 12991. We only add elements to H as the index increases. (Contributed by Jim Kingdon, 15-Jul-2023.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : om -onto-> A )   &    |- ( ph -> A. n e. om E. k e. om A. j e. suc n ( F ` k ) =/= ( F ` j ) )   &    |- G = ( x e. ( A ^pm om ) , y e. om |-> if ( ( F ` y ) e. ( F " y ) , x , ( x u. { <. dom x , ( F ` y ) >. } ) ) )   &    |- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- J = ( x e. NN0 |-> if ( x = 0 , (/) , ( `' N ` ( x - 1 ) ) ) )   &    |- H = seq 0 ( G , J )   &    |- ( ph -> P e. NN0 )   =>    |- ( ph -> ( H ` P ) C_ ( H ` ( P + 1 ) ) )
Theorem	ennnfoneleminc 12977*	Lemma for ennnfone 12991. We only add elements to H as the index increases. (Contributed by Jim Kingdon, 21-Jul-2023.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : om -onto-> A )   &    |- ( ph -> A. n e. om E. k e. om A. j e. suc n ( F ` k ) =/= ( F ` j ) )   &    |- G = ( x e. ( A ^pm om ) , y e. om |-> if ( ( F ` y ) e. ( F " y ) , x , ( x u. { <. dom x , ( F ` y ) >. } ) ) )   &    |- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- J = ( x e. NN0 |-> if ( x = 0 , (/) , ( `' N ` ( x - 1 ) ) ) )   &    |- H = seq 0 ( G , J )   &    |- ( ph -> P e. NN0 )   &    |- ( ph -> Q e. NN0 )   &    |- ( ph -> P <_ Q )   =>    |- ( ph -> ( H ` P ) C_ ( H ` Q ) )
Theorem	ennnfonelemkh 12978*	Lemma for ennnfone 12991. Because we add zero or one entries for each new index, the length of each sequence is no greater than its index. (Contributed by Jim Kingdon, 19-Jul-2023.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : om -onto-> A )   &    |- ( ph -> A. n e. om E. k e. om A. j e. suc n ( F ` k ) =/= ( F ` j ) )   &    |- G = ( x e. ( A ^pm om ) , y e. om |-> if ( ( F ` y ) e. ( F " y ) , x , ( x u. { <. dom x , ( F ` y ) >. } ) ) )   &    |- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- J = ( x e. NN0 |-> if ( x = 0 , (/) , ( `' N ` ( x - 1 ) ) ) )   &    |- H = seq 0 ( G , J )   &    |- ( ph -> P e. NN0 )   =>    |- ( ph -> dom ( H ` P ) C_ ( `' N ` P ) )
Theorem	ennnfonelemhf1o 12979*	Lemma for ennnfone 12991. Each of the functions in H is one to one and onto an image of F. (Contributed by Jim Kingdon, 17-Jul-2023.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : om -onto-> A )   &    |- ( ph -> A. n e. om E. k e. om A. j e. suc n ( F ` k ) =/= ( F ` j ) )   &    |- G = ( x e. ( A ^pm om ) , y e. om |-> if ( ( F ` y ) e. ( F " y ) , x , ( x u. { <. dom x , ( F ` y ) >. } ) ) )   &    |- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- J = ( x e. NN0 |-> if ( x = 0 , (/) , ( `' N ` ( x - 1 ) ) ) )   &    |- H = seq 0 ( G , J )   &    |- ( ph -> P e. NN0 )   =>    |- ( ph -> ( H ` P ) : dom ( H ` P ) -1-1-onto-> ( F " ( `' N ` P ) ) )
Theorem	ennnfonelemex 12980*	Lemma for ennnfone 12991. Extending the sequence ( H ` P ) to include an additional element. (Contributed by Jim Kingdon, 19-Jul-2023.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : om -onto-> A )   &    |- ( ph -> A. n e. om E. k e. om A. j e. suc n ( F ` k ) =/= ( F ` j ) )   &    |- G = ( x e. ( A ^pm om ) , y e. om |-> if ( ( F ` y ) e. ( F " y ) , x , ( x u. { <. dom x , ( F ` y ) >. } ) ) )   &    |- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- J = ( x e. NN0 |-> if ( x = 0 , (/) , ( `' N ` ( x - 1 ) ) ) )   &    |- H = seq 0 ( G , J )   &    |- ( ph -> P e. NN0 )   =>    |- ( ph -> E. i e. NN0 dom ( H ` P ) e. dom ( H ` i ) )
Theorem	ennnfonelemhom 12981*	Lemma for ennnfone 12991. The sequences in H increase in length without bound if you go out far enough. (Contributed by Jim Kingdon, 19-Jul-2023.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : om -onto-> A )   &    |- ( ph -> A. n e. om E. k e. om A. j e. suc n ( F ` k ) =/= ( F ` j ) )   &    |- G = ( x e. ( A ^pm om ) , y e. om |-> if ( ( F ` y ) e. ( F " y ) , x , ( x u. { <. dom x , ( F ` y ) >. } ) ) )   &    |- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- J = ( x e. NN0 |-> if ( x = 0 , (/) , ( `' N ` ( x - 1 ) ) ) )   &    |- H = seq 0 ( G , J )   &    |- ( ph -> M e. om )   =>    |- ( ph -> E. i e. NN0 M e. dom ( H ` i ) )
Theorem	ennnfonelemrnh 12982*	Lemma for ennnfone 12991. A consequence of ennnfonelemss 12976. (Contributed by Jim Kingdon, 16-Jul-2023.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : om -onto-> A )   &    |- ( ph -> A. n e. om E. k e. om A. j e. suc n ( F ` k ) =/= ( F ` j ) )   &    |- G = ( x e. ( A ^pm om ) , y e. om |-> if ( ( F ` y ) e. ( F " y ) , x , ( x u. { <. dom x , ( F ` y ) >. } ) ) )   &    |- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- J = ( x e. NN0 |-> if ( x = 0 , (/) , ( `' N ` ( x - 1 ) ) ) )   &    |- H = seq 0 ( G , J )   &    |- ( ph -> X e. ran H )   &    |- ( ph -> Y e. ran H )   =>    |- ( ph -> ( X C_ Y \/ Y C_ X ) )
Theorem	ennnfonelemfun 12983*	Lemma for ennnfone 12991. L is a function. (Contributed by Jim Kingdon, 16-Jul-2023.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : om -onto-> A )   &    |- ( ph -> A. n e. om E. k e. om A. j e. suc n ( F ` k ) =/= ( F ` j ) )   &    |- G = ( x e. ( A ^pm om ) , y e. om |-> if ( ( F ` y ) e. ( F " y ) , x , ( x u. { <. dom x , ( F ` y ) >. } ) ) )   &    |- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- J = ( x e. NN0 |-> if ( x = 0 , (/) , ( `' N ` ( x - 1 ) ) ) )   &    |- H = seq 0 ( G , J )   &    |- L = U_ i e. NN0 ( H ` i )   =>    |- ( ph -> Fun L )
Theorem	ennnfonelemf1 12984*	Lemma for ennnfone 12991. L is one-to-one. (Contributed by Jim Kingdon, 16-Jul-2023.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : om -onto-> A )   &    |- ( ph -> A. n e. om E. k e. om A. j e. suc n ( F ` k ) =/= ( F ` j ) )   &    |- G = ( x e. ( A ^pm om ) , y e. om |-> if ( ( F ` y ) e. ( F " y ) , x , ( x u. { <. dom x , ( F ` y ) >. } ) ) )   &    |- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- J = ( x e. NN0 |-> if ( x = 0 , (/) , ( `' N ` ( x - 1 ) ) ) )   &    |- H = seq 0 ( G , J )   &    |- L = U_ i e. NN0 ( H ` i )   =>    |- ( ph -> L : dom L -1-1-> A )
Theorem	ennnfonelemrn 12985*	Lemma for ennnfone 12991. L is onto A. (Contributed by Jim Kingdon, 16-Jul-2023.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : om -onto-> A )   &    |- ( ph -> A. n e. om E. k e. om A. j e. suc n ( F ` k ) =/= ( F ` j ) )   &    |- G = ( x e. ( A ^pm om ) , y e. om |-> if ( ( F ` y ) e. ( F " y ) , x , ( x u. { <. dom x , ( F ` y ) >. } ) ) )   &    |- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- J = ( x e. NN0 |-> if ( x = 0 , (/) , ( `' N ` ( x - 1 ) ) ) )   &    |- H = seq 0 ( G , J )   &    |- L = U_ i e. NN0 ( H ` i )   =>    |- ( ph -> ran L = A )
Theorem	ennnfonelemdm 12986*	Lemma for ennnfone 12991. The function L is defined everywhere. (Contributed by Jim Kingdon, 16-Jul-2023.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : om -onto-> A )   &    |- ( ph -> A. n e. om E. k e. om A. j e. suc n ( F ` k ) =/= ( F ` j ) )   &    |- G = ( x e. ( A ^pm om ) , y e. om |-> if ( ( F ` y ) e. ( F " y ) , x , ( x u. { <. dom x , ( F ` y ) >. } ) ) )   &    |- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- J = ( x e. NN0 |-> if ( x = 0 , (/) , ( `' N ` ( x - 1 ) ) ) )   &    |- H = seq 0 ( G , J )   &    |- L = U_ i e. NN0 ( H ` i )   =>    |- ( ph -> dom L = om )
Theorem	ennnfonelemen 12987*	Lemma for ennnfone 12991. The result. (Contributed by Jim Kingdon, 16-Jul-2023.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : om -onto-> A )   &    |- ( ph -> A. n e. om E. k e. om A. j e. suc n ( F ` k ) =/= ( F ` j ) )   &    |- G = ( x e. ( A ^pm om ) , y e. om |-> if ( ( F ` y ) e. ( F " y ) , x , ( x u. { <. dom x , ( F ` y ) >. } ) ) )   &    |- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- J = ( x e. NN0 |-> if ( x = 0 , (/) , ( `' N ` ( x - 1 ) ) ) )   &    |- H = seq 0 ( G , J )   &    |- L = U_ i e. NN0 ( H ` i )   =>    |- ( ph -> A ~~ NN )
Theorem	ennnfonelemnn0 12988*	Lemma for ennnfone 12991. A version of ennnfonelemen 12987 expressed in terms of NN0 instead of om. (Contributed by Jim Kingdon, 27-Oct-2022.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : NN0 -onto-> A )   &    |- ( ph -> A. n e. NN0 E. k e. NN0 A. j e. ( 0 ... n ) ( F ` k ) =/= ( F ` j ) )   &    |- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   =>    |- ( ph -> A ~~ NN )
Theorem	ennnfonelemr 12989*	Lemma for ennnfone 12991. The interesting direction, expressed in deduction form. (Contributed by Jim Kingdon, 27-Oct-2022.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : NN0 -onto-> A )   &    |- ( ph -> A. n e. NN0 E. k e. NN0 A. j e. ( 0 ... n ) ( F ` k ) =/= ( F ` j ) )   =>    |- ( ph -> A ~~ NN )
Theorem	ennnfonelemim 12990*	Lemma for ennnfone 12991. The trivial direction. (Contributed by Jim Kingdon, 27-Oct-2022.)
|- ( A ~~ NN -> ( A. x e. A A. y e. A DECID x = y /\ E. f ( f : NN0 -onto-> A /\ A. n e. NN0 E. k e. NN0 A. j e. ( 0 ... n ) ( f ` k ) =/= ( f ` j ) ) ) )
Theorem	ennnfone 12991*	A condition for a set being countably infinite. Corollary 8.1.13 of [AczelRathjen], p. 73. Roughly speaking, the condition says that A is countable (that's the f : NN0 -onto-> A part, as seen in theorems like ctm 7272), infinite (that's the part about being able to find an element of A distinct from any mapping of a natural number via f), and has decidable equality. (Contributed by Jim Kingdon, 27-Oct-2022.)
|- ( A ~~ NN <-> ( A. x e. A A. y e. A DECID x = y /\ E. f ( f : NN0 -onto-> A /\ A. n e. NN0 E. k e. NN0 A. j e. ( 0 ... n ) ( f ` k ) =/= ( f ` j ) ) ) )
Theorem	exmidunben 12992*	If any unbounded set of positive integers is equinumerous to NN, then the Limited Principle of Omniscience (LPO) implies excluded middle. (Contributed by Jim Kingdon, 29-Jul-2023.)
|- ( ( A. x ( ( x C_ NN /\ A. m e. NN E. n e. x m < n ) -> x ~~ NN ) /\ om e. Omni ) -> EXMID )
Theorem	ctinfomlemom 12993*	Lemma for ctinfom 12994. Converting between om and NN0. (Contributed by Jim Kingdon, 10-Aug-2023.)
|- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- G = ( F o. `' N )   &    |- ( ph -> F : om -onto-> A )   &    |- ( ph -> A. n e. om E. k e. om -. ( F ` k ) e. ( F " n ) )   =>    |- ( ph -> ( G : NN0 -onto-> A /\ A. m e. NN0 E. j e. NN0 A. i e. ( 0 ... m ) ( G ` j ) =/= ( G ` i ) ) )
Theorem	ctinfom 12994*	A condition for a set being countably infinite. Restates ennnfone 12991 in terms of om and function image. Like ennnfone 12991 the condition can be summarized as A being countable, infinite, and having decidable equality. (Contributed by Jim Kingdon, 7-Aug-2023.)
|- ( A ~~ NN <-> ( A. x e. A A. y e. A DECID x = y /\ E. f ( f : om -onto-> A /\ A. n e. om E. k e. om -. ( f ` k ) e. ( f " n ) ) ) )
Theorem	inffinp1 12995*	An infinite set contains an element not contained in a given finite subset. (Contributed by Jim Kingdon, 7-Aug-2023.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> om ~<_ A )   &    |- ( ph -> B C_ A )   &    |- ( ph -> B e. Fin )   =>    |- ( ph -> E. x e. A -. x e. B )
Theorem	ctinf 12996*	A set is countably infinite if and only if it has decidable equality, is countable, and is infinite. (Contributed by Jim Kingdon, 7-Aug-2023.)
|- ( A ~~ NN <-> ( A. x e. A A. y e. A DECID x = y /\ E. f f : om -onto-> A /\ om ~<_ A ) )
Theorem	qnnen 12997	The rational numbers are countably infinite. Corollary 8.1.23 of [AczelRathjen], p. 75. This is Metamath 100 proof #3. (Contributed by Jim Kingdon, 11-Aug-2023.)
|- QQ ~~ NN
Theorem	enctlem 12998*	Lemma for enct 12999. One direction of the biconditional. (Contributed by Jim Kingdon, 23-Dec-2023.)
|- ( A ~~ B -> ( E. f f : om -onto-> ( A ⊔ 1o ) -> E. g g : om -onto-> ( B ⊔ 1o ) ) )
Theorem	enct 12999*	Countability is invariant relative to equinumerosity. (Contributed by Jim Kingdon, 23-Dec-2023.)
|- ( A ~~ B -> ( E. f f : om -onto-> ( A ⊔ 1o ) <-> E. g g : om -onto-> ( B ⊔ 1o ) ) )
Theorem	ctiunctlemu1st 13000*	Lemma for ctiunct 13006. (Contributed by Jim Kingdon, 28-Oct-2023.)
|- ( ph -> S C_ om )   &    |- ( ph -> A. n e. om DECID n e. S )   &    |- ( ph -> F : S -onto-> A )   &    |- ( ( ph /\ x e. A ) -> T C_ om )   &    |- ( ( ph /\ x e. A ) -> A. n e. om DECID n e. T )   &    |- ( ( ph /\ x e. A ) -> G : T -onto-> B )   &    |- ( ph -> J : om -1-1-onto-> ( om X. om ) )   &    |- U = { z e. om | ( ( 1st ` ( J ` z ) ) e. S /\ ( 2nd ` ( J ` z ) ) e. [_ ( F ` ( 1st ` ( J ` z ) ) ) / x ]_ T ) }   &    |- ( ph -> N e. U )   =>    |- ( ph -> ( 1st ` ( J ` N ) ) e. S )