P. 124 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 12301-12400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	elgz 12301	Elementhood in the gaussian integers. (Contributed by Mario Carneiro, 14-Jul-2014.)
|- ( A e. ZZ[_i] <-> ( A e. CC /\ ( Re ` A ) e. ZZ /\ ( Im ` A ) e. ZZ ) )
Theorem	gzcn 12302	A gaussian integer is a complex number. (Contributed by Mario Carneiro, 14-Jul-2014.)
|- ( A e. ZZ[_i] -> A e. CC )
Theorem	zgz 12303	An integer is a gaussian integer. (Contributed by Mario Carneiro, 14-Jul-2014.)
|- ( A e. ZZ -> A e. ZZ[_i] )
Theorem	igz 12304	_i is a gaussian integer. (Contributed by Mario Carneiro, 14-Jul-2014.)
|- _i e. ZZ[_i]
Theorem	gznegcl 12305	The gaussian integers are closed under negation. (Contributed by Mario Carneiro, 14-Jul-2014.)
|- ( A e. ZZ[_i] -> -u A e. ZZ[_i] )
Theorem	gzcjcl 12306	The gaussian integers are closed under conjugation. (Contributed by Mario Carneiro, 14-Jul-2014.)
|- ( A e. ZZ[_i] -> ( * ` A ) e. ZZ[_i] )
Theorem	gzaddcl 12307	The gaussian integers are closed under addition. (Contributed by Mario Carneiro, 14-Jul-2014.)
|- ( ( A e. ZZ[_i] /\ B e. ZZ[_i] ) -> ( A + B ) e. ZZ[_i] )
Theorem	gzmulcl 12308	The gaussian integers are closed under multiplication. (Contributed by Mario Carneiro, 14-Jul-2014.)
|- ( ( A e. ZZ[_i] /\ B e. ZZ[_i] ) -> ( A x. B ) e. ZZ[_i] )
Theorem	gzreim 12309	Construct a gaussian integer from real and imaginary parts. (Contributed by Mario Carneiro, 16-Jul-2014.)
|- ( ( A e. ZZ /\ B e. ZZ ) -> ( A + ( _i x. B ) ) e. ZZ[_i] )
Theorem	gzsubcl 12310	The gaussian integers are closed under subtraction. (Contributed by Mario Carneiro, 14-Jul-2014.)
|- ( ( A e. ZZ[_i] /\ B e. ZZ[_i] ) -> ( A - B ) e. ZZ[_i] )
Theorem	gzabssqcl 12311	The squared norm of a gaussian integer is an integer. (Contributed by Mario Carneiro, 16-Jul-2014.)
|- ( A e. ZZ[_i] -> ( ( abs ` A ) ^ 2 ) e. NN0 )
Theorem	4sqlem5 12312	Lemma for 4sq (not yet proved here). (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 e. ZZ /\ ( ( A - B ) / M ) e. ZZ ) )
Theorem	4sqlem6 12313	Lemma for 4sq (not yet proved here) . (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 12314	Lemma for 4sq (not yet proved here) . (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 12315	Lemma for 4sq (not yet proved here) . (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 12316	Lemma for 4sq (not yet proved here) . (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 12317	Lemma for 4sq (not yet proved here) . (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 12318*	Lemma for 4sq (not yet proved here) . 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 12319*	Lemma for 4sq (not yet proved here) . 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 12320*	Lemma for 4sq (not yet proved here) . 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 12321*	Lemma for 4sqlem4 12322. (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 12322*	Lemma for 4sq (not yet proved here) . 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 12323*	Lemma for mul4sq 12324: 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 12324*	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 12323. (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 )
5.3  Cardinality of real and complex number subsets
5.3.1  Countability of integers and rationals
Theorem	oddennn 12325	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 12326	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 12327	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 12328	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 12329	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 12330	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 12331	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 12332*	Lemma for ennnfone 12358. A direct consequence of fidcenumlemrk 6919. (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 12333*	Lemma for ennnfone 12358. (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 12334*	Lemma for ennnfone 12358. 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 12335*	Lemma for ennnfone 12358. 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 12336*	Lemma for ennnfone 12358. 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 12337*	Lemma for ennnfone 12358. (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 12338*	Lemma for ennnfone 12358. 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 12339*	Lemma for ennnfone 12358. 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 12340*	Lemma for ennnfone 12358. 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 12341*	Lemma for ennnfone 12358. 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 12342*	Lemma for ennnfone 12358. 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 12343*	Lemma for ennnfone 12358. 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 12344*	Lemma for ennnfone 12358. 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 12345*	Lemma for ennnfone 12358. 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 12346*	Lemma for ennnfone 12358. 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 12347*	Lemma for ennnfone 12358. 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 12348*	Lemma for ennnfone 12358. 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 12349*	Lemma for ennnfone 12358. A consequence of ennnfonelemss 12343. (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 12350*	Lemma for ennnfone 12358. 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 12351*	Lemma for ennnfone 12358. 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 12352*	Lemma for ennnfone 12358. 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 12353*	Lemma for ennnfone 12358. 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 12354*	Lemma for ennnfone 12358. 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 12355*	Lemma for ennnfone 12358. A version of ennnfonelemen 12354 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 12356*	Lemma for ennnfone 12358. 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 12357*	Lemma for ennnfone 12358. 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 12358*	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 7074), 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 12359*	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 12360*	Lemma for ctinfom 12361. 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 12361*	A condition for a set being countably infinite. Restates ennnfone 12358 in terms of om and function image. Like ennnfone 12358 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 12362*	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 12363*	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 12364	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 12365*	Lemma for enct 12366. 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 12366*	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 12367*	Lemma for ctiunct 12373. (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 )
Theorem	ctiunctlemu2nd 12368*	Lemma for ctiunct 12373. (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 -> ( 2nd ` ( J ` N ) ) e. [_ ( F ` ( 1st ` ( J ` N ) ) ) / x ]_ T )
Theorem	ctiunctlemuom 12369	Lemma for ctiunct 12373. (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 -> U C_ om )
Theorem	ctiunctlemudc 12370*	Lemma for ctiunct 12373. (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 -> A. n e. om DECID n e. U )
Theorem	ctiunctlemf 12371*	Lemma for ctiunct 12373. (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 ) }   &    |- H = ( n e. U |-> ( [_ ( F ` ( 1st ` ( J ` n ) ) ) / x ]_ G ` ( 2nd ` ( J ` n ) ) ) )   =>    |- ( ph -> H : U --> U_ x e. A B )
Theorem	ctiunctlemfo 12372*	Lemma for ctiunct 12373. (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 ) }   &    |- H = ( n e. U |-> ( [_ ( F ` ( 1st ` ( J ` n ) ) ) / x ]_ G ` ( 2nd ` ( J ` n ) ) ) )   &    |- F/_ x H   &    |- F/_ x U   =>    |- ( ph -> H : U -onto-> U_ x e. A B )
Theorem	ctiunct 12373*	A sequence of enumerations gives an enumeration of the union. We refer to "sequence of enumerations" rather than "countably many countable sets" because the hypothesis provides more than countability for each B ( x ): it refers to B ( x ) together with the G ( x ) which enumerates it. Theorem 8.1.19 of [AczelRathjen], p. 74. For "countably many countable sets" the key hypothesis would be ( ph /\ x e. A ) -> E. g g : om -onto-> ( B ⊔ 1o ). This is almost omiunct 12377 (which uses countable choice) although that is for a countably infinite collection not any countable collection. Compare with the case of two sets instead of countably many, as seen at unct 12375, which says that the union of two countable sets is countable . The proof proceeds by mapping a natural number to a pair of natural numbers (by xpomen 12328) and using the first number to map to an element x of A and the second number to map to an element of B(x) . In this way we are able to map to every element of U_ x e. A B. Although it would be possible to work directly with countability expressed as F : om -onto-> ( A ⊔ 1o ), we instead use functions from subsets of the natural numbers via ctssdccl 7076 and ctssdc 7078. (Contributed by Jim Kingdon, 31-Oct-2023.)
|- ( ph -> F : om -onto-> ( A ⊔ 1o ) )   &    |- ( ( ph /\ x e. A ) -> G : om -onto-> ( B ⊔ 1o ) )   =>    |- ( ph -> E. h h : om -onto-> ( U_ x e. A B ⊔ 1o ) )
Theorem	ctiunctal 12374*	Variation of ctiunct 12373 which allows x to be present in ph. (Contributed by Jim Kingdon, 5-May-2024.)
|- ( ph -> F : om -onto-> ( A ⊔ 1o ) )   &    |- ( ph -> A. x e. A G : om -onto-> ( B ⊔ 1o ) )   =>    |- ( ph -> E. h h : om -onto-> ( U_ x e. A B ⊔ 1o ) )
Theorem	unct 12375*	The union of two countable sets is countable. Corollary 8.1.20 of [AczelRathjen], p. 75. (Contributed by Jim Kingdon, 1-Nov-2023.)
|- ( ( E. f f : om -onto-> ( A ⊔ 1o ) /\ E. g g : om -onto-> ( B ⊔ 1o ) ) -> E. h h : om -onto-> ( ( A u. B ) ⊔ 1o ) )
Theorem	omctfn 12376*	Using countable choice to find a sequence of enumerations for a collection of countable sets. Lemma 8.1.27 of [AczelRathjen], p. 77. (Contributed by Jim Kingdon, 19-Apr-2024.)
|- ( ph -> CCHOICE )   &    |- ( ( ph /\ x e. om ) -> E. g g : om -onto-> ( B ⊔ 1o ) )   =>    |- ( ph -> E. f ( f Fn om /\ A. x e. om ( f ` x ) : om -onto-> ( B ⊔ 1o ) ) )
Theorem	omiunct 12377*	The union of a countably infinite collection of countable sets is countable. Theorem 8.1.28 of [AczelRathjen], p. 78. Compare with ctiunct 12373 which has a stronger hypothesis but does not require countable choice. (Contributed by Jim Kingdon, 5-May-2024.)
|- ( ph -> CCHOICE )   &    |- ( ( ph /\ x e. om ) -> E. g g : om -onto-> ( B ⊔ 1o ) )   =>    |- ( ph -> E. h h : om -onto-> ( U_ x e. om B ⊔ 1o ) )
Theorem	ssomct 12378*	A decidable subset of om is countable. (Contributed by Jim Kingdon, 19-Sep-2024.)
|- ( ( A C_ om /\ A. x e. om DECID x e. A ) -> E. f f : om -onto-> ( A ⊔ 1o ) )
Theorem	ssnnctlemct 12379*	Lemma for ssnnct 12380. The result. (Contributed by Jim Kingdon, 29-Sep-2024.)
|- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 1 )   =>    |- ( ( A C_ NN /\ A. x e. NN DECID x e. A ) -> E. f f : om -onto-> ( A ⊔ 1o ) )
Theorem	ssnnct 12380*	A decidable subset of NN is countable. (Contributed by Jim Kingdon, 29-Sep-2024.)
|- ( ( A C_ NN /\ A. x e. NN DECID x e. A ) -> E. f f : om -onto-> ( A ⊔ 1o ) )
Theorem	nninfdclemcl 12381*	Lemma for nninfdc 12386. (Contributed by Jim Kingdon, 25-Sep-2024.)
|- ( ph -> A C_ NN )   &    |- ( ph -> A. x e. NN DECID x e. A )   &    |- ( ph -> A. m e. NN E. n e. A m < n )   &    |- ( ph -> P e. A )   &    |- ( ph -> Q e. A )   =>    |- ( ph -> ( P ( y e. NN , z e. NN |-> inf ( ( A i^i ( ZZ>= ` ( y + 1 ) ) ) , RR , < ) ) Q ) e. A )
Theorem	nninfdclemf 12382*	Lemma for nninfdc 12386. A function from the natural numbers into A. (Contributed by Jim Kingdon, 23-Sep-2024.)
|- ( ph -> A C_ NN )   &    |- ( ph -> A. x e. NN DECID x e. A )   &    |- ( ph -> A. m e. NN E. n e. A m < n )   &    |- ( ph -> ( J e. A /\ 1 < J ) )   &    |- F = seq 1 ( ( y e. NN , z e. NN |-> inf ( ( A i^i ( ZZ>= ` ( y + 1 ) ) ) , RR , < ) ) , ( i e. NN |-> J ) )   =>    |- ( ph -> F : NN --> A )
Theorem	nninfdclemp1 12383*	Lemma for nninfdc 12386. Each element of the sequence F is greater than the previous element. (Contributed by Jim Kingdon, 26-Sep-2024.)
|- ( ph -> A C_ NN )   &    |- ( ph -> A. x e. NN DECID x e. A )   &    |- ( ph -> A. m e. NN E. n e. A m < n )   &    |- ( ph -> ( J e. A /\ 1 < J ) )   &    |- F = seq 1 ( ( y e. NN , z e. NN |-> inf ( ( A i^i ( ZZ>= ` ( y + 1 ) ) ) , RR , < ) ) , ( i e. NN |-> J ) )   &    |- ( ph -> U e. NN )   =>    |- ( ph -> ( F ` U ) < ( F ` ( U + 1 ) ) )
Theorem	nninfdclemlt 12384*	Lemma for nninfdc 12386. The function from nninfdclemf 12382 is strictly monotonic. (Contributed by Jim Kingdon, 24-Sep-2024.)
|- ( ph -> A C_ NN )   &    |- ( ph -> A. x e. NN DECID x e. A )   &    |- ( ph -> A. m e. NN E. n e. A m < n )   &    |- ( ph -> ( J e. A /\ 1 < J ) )   &    |- F = seq 1 ( ( y e. NN , z e. NN |-> inf ( ( A i^i ( ZZ>= ` ( y + 1 ) ) ) , RR , < ) ) , ( i e. NN |-> J ) )   &    |- ( ph -> U e. NN )   &    |- ( ph -> V e. NN )   &    |- ( ph -> U < V )   =>    |- ( ph -> ( F ` U ) < ( F ` V ) )
Theorem	nninfdclemf1 12385*	Lemma for nninfdc 12386. The function from nninfdclemf 12382 is one-to-one. (Contributed by Jim Kingdon, 23-Sep-2024.)
|- ( ph -> A C_ NN )   &    |- ( ph -> A. x e. NN DECID x e. A )   &    |- ( ph -> A. m e. NN E. n e. A m < n )   &    |- ( ph -> ( J e. A /\ 1 < J ) )   &    |- F = seq 1 ( ( y e. NN , z e. NN |-> inf ( ( A i^i ( ZZ>= ` ( y + 1 ) ) ) , RR , < ) ) , ( i e. NN |-> J ) )   =>    |- ( ph -> F : NN -1-1-> A )
Theorem	nninfdc 12386*	An unbounded decidable set of positive integers is infinite. (Contributed by Jim Kingdon, 23-Sep-2024.)
|- ( ( A C_ NN /\ A. x e. NN DECID x e. A /\ A. m e. NN E. n e. A m < n ) -> om ~<_ A )
Theorem	unbendc 12387*	An unbounded decidable set of positive integers is infinite. (Contributed by NM, 5-May-2005.) (Revised by Jim Kingdon, 30-Sep-2024.)
|- ( ( A C_ NN /\ A. x e. NN DECID x e. A /\ A. m e. NN E. n e. A m < n ) -> A ~~ NN )
Theorem	prminf 12388	There are an infinite number of primes. Theorem 1.7 in [ApostolNT] p. 16. (Contributed by Paul Chapman, 28-Nov-2012.)
|- Prime ~~ NN
Theorem	infpn2 12389*	There exist infinitely many prime numbers: the set of all primes S is unbounded by infpn 12291, so by unbendc 12387 it is infinite. This is Metamath 100 proof #11. (Contributed by NM, 5-May-2005.)
|- S = { n e. NN | ( 1 < n /\ A. m e. NN ( ( n / m ) e. NN -> ( m = 1 \/ m = n ) ) ) }   =>    |- S ~~ NN
PART 6  BASIC STRUCTURES
6.1  Extensible structures
6.1.1  Basic definitions An "extensible structure" (or "structure" in short, at least in this section) is used to define a specific group, ring, poset, and so on. An extensible structure can contain many components. For example, a group will have at least two components (base set and operation), although it can be further specialized by adding other components such as a multiplicative operation for rings (and still remain a group per our definition). Thus, every ring is also a group. This extensible structure approach allows theorems from more general structures (such as groups) to be reused for more specialized structures (such as rings) without having to reprove anything. Structures are common in mathematics, but in informal (natural language) proofs the details are assumed in ways that we must make explicit. An extensible structure is implemented as a function (a set of ordered pairs) on a finite (and not necessarily sequential) subset of NN. The function's argument is the index of a structure component (such as 1 for the base set of a group), and its value is the component (such as the base set). By convention, we normally avoid direct reference to the hard-coded numeric index and instead use structure component extractors such as ndxid 12418 and strslfv 12438. Using extractors makes it easier to change numeric indices and also makes the components' purpose clearer. See the comment of basendx 12448 for more details on numeric indices versus the structure component extractors. There are many other possible ways to handle structures. We chose this extensible structure approach because this approach (1) results in simpler notation than other approaches we are aware of, and (2) is easier to do proofs with. We cannot use an approach that uses "hidden" arguments; Metamath does not support hidden arguments, and in any case we want nothing hidden. It would be possible to use a categorical approach (e.g., something vaguely similar to Lean's mathlib). However, instances (the chain of proofs that an X is a Y via a bunch of forgetful functors) can cause serious performance problems for automated tooling, and the resulting proofs would be painful to look at directly (in the case of Lean, they are long past the level where people would find it acceptable to look at them directly). Metamath is working under much stricter conditions than this, and it has still managed to achieve about the same level of flexibility through this "extensible structure" approach. To create a substructure of a given extensible structure, you can simply use the multifunction restriction operator for extensible structures ↾s as defined in df-ress 12402. This can be used to turn statements about rings into statements about subrings, modules into submodules, etc. This definition knows nothing about individual structures and merely truncates the Base set while leaving operators alone. Individual kinds of structures will need to handle this behavior by ignoring operators' values outside the range, defining a function using the base set and applying that, or explicitly truncating the slot before use. Extensible structures only work well when they represent concrete categories, where there is a "base set", morphisms are functions, and subobjects are subsets with induced operations. In short, they primarily work well for "sets with (some) extra structure". Extensible structures may not suffice for more complicated situations. For example, in manifolds, ↾s would not work. That said, extensible structures are sufficient for many of the structures that set.mm currently considers, and offer a good compromise for a goal-oriented formalization.
Syntax	cstr 12390	Extend class notation with the class of structures with components numbered below A.
class Struct
Syntax	cnx 12391	Extend class notation with the structure component index extractor.
class ndx
Syntax	csts 12392	Set components of a structure.
class sSet
Syntax	cslot 12393	Extend class notation with the slot function.
class Slot A
Syntax	cbs 12394	Extend class notation with the class of all base set extractors.
class Base
Syntax	cress 12395	Extend class notation with the extensible structure builder restriction operator.
class ↾s
Definition	df-struct 12396*	Define a structure with components in M ... N. This is not a requirement for groups, posets, etc., but it is a useful assumption for component extraction theorems. As mentioned in the section header, an "extensible structure should be implemented as a function (a set of ordered pairs)". The current definition, however, is less restrictive: it allows for classes which contain the empty set (/) to be extensible structures. Because of 0nelfun 5206, such classes cannot be functions. Without the empty set, however, a structure must be a function, see structn0fun 12407: F Struct X -> Fun ( F \ { (/) } ). Allowing an extensible structure to contain the empty set ensures that expressions like { <. A , B >. , <. C , D >. } are structures without asserting or implying that A, B, C and D are sets (if A or B is a proper class, then <. A , B >. = (/), see opprc 3779). (Contributed by Mario Carneiro, 29-Aug-2015.)
|- Struct = { <. f , x >. | ( x e. ( <_ i^i ( NN X. NN ) ) /\ Fun ( f \ { (/) } ) /\ dom f C_ ( ... ` x ) ) }
Definition	df-ndx 12397	Define the structure component index extractor. See Theorem ndxarg 12417 to understand its purpose. The restriction to NN ensures that ndx is a set. The restriction to some set is necessary since _I is a proper class. In principle, we could have chosen CC or (if we revise all structure component definitions such as df-base 12400) another set such as the set of finite ordinals om (df-iom 4568). (Contributed by NM, 4-Sep-2011.)
|- ndx = ( _I |` NN )
Definition	df-slot 12398*	Define the slot extractor for extensible structures. The class Slot A is a function whose argument can be any set, although it is meaningful only if that set is a member of an extensible structure (such as a partially ordered set or a group). Note that Slot A is implemented as "evaluation at A". That is, (Slot A ` S ) is defined to be ( S ` A ), where A will typically be a small nonzero natural number. Each extensible structure S is a function defined on specific natural number "slots", and this function extracts the value at a particular slot. The special "structure" ndx, defined as the identity function restricted to NN, can be used to extract the number A from a slot, since (Slot A ` ndx ) = A (see ndxarg 12417). This is typically used to refer to the number of a slot when defining structures without having to expose the detail of what that number is (for instance, we use the expression ( Base ` ndx ) in theorems and proofs instead of its value 1). The class Slot cannot be defined as ( x e. _V |-> ( f e. _V |-> ( f ` x ) ) ) because each Slot A is a function on the proper class _V so is itself a proper class, and the values of functions are sets (fvex 5506). It is necessary to allow proper classes as values of Slot A since for instance the class of all (base sets of) groups is proper. (Contributed by Mario Carneiro, 22-Sep-2015.)
|- Slot A = ( x e. _V |-> ( x ` A ) )
Theorem	sloteq 12399	Equality theorem for the Slot construction. The converse holds if A (or B) is a set. (Contributed by BJ, 27-Dec-2021.)
|- ( A = B -> Slot A = Slot B )
Definition	df-base 12400	Define the base set (also called underlying set, ground set, carrier set, or carrier) extractor for extensible structures. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
|- Base = Slot 1