P. 107 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 10601-10700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	nnenom 10601	The set of positive integers (as a subset of complex numbers) is equinumerous to omega (the set of natural numbers as ordinals). (Contributed by NM, 31-Jul-2004.) (Revised by Mario Carneiro, 15-Sep-2013.)
|- NN ~~ om
Theorem	nnct 10602	NN is dominated by om. (Contributed by Thierry Arnoux, 29-Dec-2016.)
|- NN ~<_ om
Theorem	uzennn 10603	An upper integer set is equinumerous to the set of natural numbers. (Contributed by Jim Kingdon, 30-Jul-2023.)
|- ( M e. ZZ -> ( ZZ>= ` M ) ~~ NN )
Theorem	xnn0nnen 10604	The set of extended nonnegative integers is equinumerous to the set of natural numbers. (Contributed by Jim Kingdon, 14-Jul-2025.)
|- NN0* ~~ NN
Theorem	fnn0nninf 10605*	A function from NN0 into ℕ∞. (Contributed by Jim Kingdon, 16-Jul-2022.)
|- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- F = ( n e. om |-> ( i e. om |-> if ( i e. n , 1o , (/) ) ) )   =>    |- ( F o. `' G ) : NN0 -->ℕ∞
Theorem	fxnn0nninf 10606*	A function from NN0* into ℕ∞. (Contributed by Jim Kingdon, 16-Jul-2022.) TODO: use infnninf 7241 instead of infnninfOLD 7242. More generally, this theorem and most theorems in this section could use an extended G defined by G = (frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 ) u. <. om , +oo >. ) and F = ( n e. suc om |-> ( i e. om |-> if ( i e. n , 1o , (/) ) ) ) as in nnnninf2 7244.
|- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- F = ( n e. om |-> ( i e. om |-> if ( i e. n , 1o , (/) ) ) )   &    |- I = ( ( F o. `' G ) u. { <. +oo , ( om X. { 1o } ) >. } )   =>    |- I :NN0* -->ℕ∞
Theorem	0tonninf 10607*	The mapping of zero into ℕ∞ is the sequence of all zeroes. (Contributed by Jim Kingdon, 17-Jul-2022.)
|- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- F = ( n e. om |-> ( i e. om |-> if ( i e. n , 1o , (/) ) ) )   &    |- I = ( ( F o. `' G ) u. { <. +oo , ( om X. { 1o } ) >. } )   =>    |- ( I ` 0 ) = ( x e. om |-> (/) )
Theorem	1tonninf 10608*	The mapping of one into ℕ∞ is a sequence which is a one followed by zeroes. (Contributed by Jim Kingdon, 17-Jul-2022.)
|- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- F = ( n e. om |-> ( i e. om |-> if ( i e. n , 1o , (/) ) ) )   &    |- I = ( ( F o. `' G ) u. { <. +oo , ( om X. { 1o } ) >. } )   =>    |- ( I ` 1 ) = ( x e. om |-> if ( x = (/) , 1o , (/) ) )
Theorem	inftonninf 10609*	The mapping of +oo into ℕ∞ is the sequence of all ones. (Contributed by Jim Kingdon, 17-Jul-2022.)
|- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- F = ( n e. om |-> ( i e. om |-> if ( i e. n , 1o , (/) ) ) )   &    |- I = ( ( F o. `' G ) u. { <. +oo , ( om X. { 1o } ) >. } )   =>    |- ( I ` +oo ) = ( x e. om |-> 1o )
Theorem	nninfinf 10610	ℕ∞ is infinte. (Contributed by Jim Kingdon, 8-Jul-2025.)
|- om ~<_ ℕ∞
4.6.4  Strong induction over upper sets of integers
Theorem	uzsinds 10611*	Strong (or "total") induction principle over an upper set of integers. (Contributed by Scott Fenton, 16-May-2014.)
|- ( x = y -> ( ph <-> ps ) )   &    |- ( x = N -> ( ph <-> ch ) )   &    |- ( x e. ( ZZ>= ` M ) -> ( A. y e. ( M ... ( x - 1 ) ) ps -> ph ) )   =>    |- ( N e. ( ZZ>= ` M ) -> ch )
Theorem	nnsinds 10612*	Strong (or "total") induction principle over the naturals. (Contributed by Scott Fenton, 16-May-2014.)
|- ( x = y -> ( ph <-> ps ) )   &    |- ( x = N -> ( ph <-> ch ) )   &    |- ( x e. NN -> ( A. y e. ( 1 ... ( x - 1 ) ) ps -> ph ) )   =>    |- ( N e. NN -> ch )
Theorem	nn0sinds 10613*	Strong (or "total") induction principle over the nonnegative integers. (Contributed by Scott Fenton, 16-May-2014.)
|- ( x = y -> ( ph <-> ps ) )   &    |- ( x = N -> ( ph <-> ch ) )   &    |- ( x e. NN0 -> ( A. y e. ( 0 ... ( x - 1 ) ) ps -> ph ) )   =>    |- ( N e. NN0 -> ch )
4.6.5  The infinite sequence builder "seq"
Syntax	cseq 10614	Extend class notation with recursive sequence builder.
class seq M ( .+ , F )
Definition	df-seqfrec 10615*	Define a general-purpose operation that builds a recursive sequence (i.e., a function on an upper integer set such as NN or NN0) whose value at an index is a function of its previous value and the value of an input sequence at that index. This definition is complicated, but fortunately it is not intended to be used directly. Instead, the only purpose of this definition is to provide us with an object that has the properties expressed by seqf 10631, seq3-1 10629 and seq3p1 10632. Typically, those are the main theorems that would be used in practice. The first operand in the parentheses is the operation that is applied to the previous value and the value of the input sequence (second operand). The operand to the left of the parenthesis is the integer to start from. For example, for the operation +, an input sequence F with values 1, 1/2, 1/4, 1/8,... would be transformed into the output sequence seq 1 ( + , F ) with values 1, 3/2, 7/4, 15/8,.., so that ( seq 1 ( + , F ) ` 1 ) = 1, ( seq 1 ( + , F ) ` 2 ) = 3/2, etc. In other words, seq M ( + , F ) transforms a sequence F into an infinite series. seq M ( + , F ) ~~> 2 means "the sum of F(n) from n = M to infinity is 2". Since limits are unique (climuni 11679), by climdm 11681 the "sum of F(n) from n = 1 to infinity" can be expressed as ( ~~> ` seq 1 ( + , F ) ) (provided the sequence converges) and evaluates to 2 in this example. Internally, the frec function generates as its values a set of ordered pairs starting at <. M , ( F ` M ) >., with the first member of each pair incremented by one in each successive value. So, the range of frec is exactly the sequence we want, and we just extract the range and throw away the domain. (Contributed by NM, 18-Apr-2005.) (Revised by Jim Kingdon, 4-Nov-2022.)
|- seq M ( .+ , F ) = ran frec ( ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. )
Theorem	seqex 10616	Existence of the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
|- seq M ( .+ , F ) e. _V
Theorem	seqeq1 10617	Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
|- ( M = N -> seq M ( .+ , F ) = seq N ( .+ , F ) )
Theorem	seqeq2 10618	Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
|- ( .+ = Q -> seq M ( .+ , F ) = seq M ( Q , F ) )
Theorem	seqeq3 10619	Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
|- ( F = G -> seq M ( .+ , F ) = seq M ( .+ , G ) )
Theorem	seqeq1d 10620	Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.)
|- ( ph -> A = B )   =>    |- ( ph -> seq A ( .+ , F ) = seq B ( .+ , F ) )
Theorem	seqeq2d 10621	Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.)
|- ( ph -> A = B )   =>    |- ( ph -> seq M ( A , F ) = seq M ( B , F ) )
Theorem	seqeq3d 10622	Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.)
|- ( ph -> A = B )   =>    |- ( ph -> seq M ( .+ , A ) = seq M ( .+ , B ) )
Theorem	seqeq123d 10623	Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.)
|- ( ph -> M = N )   &    |- ( ph -> .+ = Q )   &    |- ( ph -> F = G )   =>    |- ( ph -> seq M ( .+ , F ) = seq N ( Q , G ) )
Theorem	nfseq 10624	Hypothesis builder for the sequence builder operation. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
|- F/_ x M   &    |- F/_ x .+   &    |- F/_ x F   =>    |- F/_ x seq M ( .+ , F )
Theorem	iseqovex 10625*	Closure of a function used in proving sequence builder theorems. This can be thought of as a lemma for the small number of sequence builder theorems which need it. (Contributed by Jim Kingdon, 31-May-2020.)
|- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   =>    |- ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. S ) ) -> ( x ( z e. ( ZZ>= ` M ) , w e. S |-> ( w .+ ( F ` ( z + 1 ) ) ) ) y ) e. S )
Theorem	iseqvalcbv 10626*	Changing the bound variables in an expression which appears in some seq related proofs. (Contributed by Jim Kingdon, 28-Apr-2022.)
|- frec ( ( x e. ( ZZ>= ` M ) , y e. T |-> <. ( x + 1 ) , ( x ( z e. ( ZZ>= ` M ) , w e. S |-> ( w .+ ( F ` ( z + 1 ) ) ) ) y ) >. ) , <. M , ( F ` M ) >. ) = frec ( ( a e. ( ZZ>= ` M ) , b e. T |-> <. ( a + 1 ) , ( a ( c e. ( ZZ>= ` M ) , d e. S |-> ( d .+ ( F ` ( c + 1 ) ) ) ) b ) >. ) , <. M , ( F ` M ) >. )
Theorem	seq3val 10627*	Value of the sequence builder function. This helps expand the definition although there should be little need for it once we have proved seqf 10631, seq3-1 10629 and seq3p1 10632, as further development can be done in terms of those. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Jim Kingdon, 4-Nov-2022.)
|- ( ph -> M e. ZZ )   &    |- R = frec ( ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( x ( z e. ( ZZ>= ` M ) , w e. S |-> ( w .+ ( F ` ( z + 1 ) ) ) ) y ) >. ) , <. M , ( F ` M ) >. )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   =>    |- ( ph -> seq M ( .+ , F ) = ran R )
Theorem	seqvalcd 10628*	Value of the sequence builder function. Similar to seq3val 10627 but the classes D (type of each term) and C (type of the value we are accumulating) do not need to be the same. (Contributed by Jim Kingdon, 9-Jul-2023.)
|- ( ph -> M e. ZZ )   &    |- R = frec ( ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( x ( z e. ( ZZ>= ` M ) , w e. C |-> ( w .+ ( F ` ( z + 1 ) ) ) ) y ) >. ) , <. M , ( F ` M ) >. )   &    |- ( ph -> ( F ` M ) e. C )   &    |- ( ( ph /\ ( x e. C /\ y e. D ) ) -> ( x .+ y ) e. C )   &    |- ( ( ph /\ x e. ( ZZ>= ` ( M + 1 ) ) ) -> ( F ` x ) e. D )   =>    |- ( ph -> seq M ( .+ , F ) = ran R )
Theorem	seq3-1 10629*	Value of the sequence builder function at its initial value. (Contributed by Jim Kingdon, 3-Oct-2022.)
|- ( ph -> M e. ZZ )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   =>    |- ( ph -> ( seq M ( .+ , F ) ` M ) = ( F ` M ) )
Theorem	seq1g 10630	Value of the sequence builder function at its initial value. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Jim Kingdon, 19-Aug-2025.)
|- ( ( M e. ZZ /\ F e. V /\ .+ e. W ) -> ( seq M ( .+ , F ) ` M ) = ( F ` M ) )
Theorem	seqf 10631*	Range of the recursive sequence builder. (Contributed by Mario Carneiro, 24-Jun-2013.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ( ph /\ x e. Z ) -> ( F ` x ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   =>    |- ( ph -> seq M ( .+ , F ) : Z --> S )
Theorem	seq3p1 10632*	Value of the sequence builder function at a successor. (Contributed by Jim Kingdon, 30-Apr-2022.)
|- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   =>    |- ( ph -> ( seq M ( .+ , F ) ` ( N + 1 ) ) = ( ( seq M ( .+ , F ) ` N ) .+ ( F ` ( N + 1 ) ) ) )
Theorem	seqp1g 10633	Value of the sequence builder function at a successor. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Jim Kingdon, 19-Aug-2025.)
|- ( ( N e. ( ZZ>= ` M ) /\ F e. V /\ .+ e. W ) -> ( seq M ( .+ , F ) ` ( N + 1 ) ) = ( ( seq M ( .+ , F ) ` N ) .+ ( F ` ( N + 1 ) ) ) )
Theorem	seqovcd 10634*	A closure law for the recursive sequence builder. This is a lemma for theorems such as seqf2 10635 and seq1cd 10636 and is unlikely to be needed once such theorems are proved. (Contributed by Jim Kingdon, 20-Jul-2023.)
|- ( ( ph /\ x e. ( ZZ>= ` ( M + 1 ) ) ) -> ( F ` x ) e. D )   &    |- ( ( ph /\ ( x e. C /\ y e. D ) ) -> ( x .+ y ) e. C )   =>    |- ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. C ) ) -> ( x ( z e. ( ZZ>= ` M ) , w e. C |-> ( w .+ ( F ` ( z + 1 ) ) ) ) y ) e. C )
Theorem	seqf2 10635*	Range of the recursive sequence builder. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Jim Kingdon, 7-Jul-2023.)
|- ( ph -> ( F ` M ) e. C )   &    |- ( ( ph /\ ( x e. C /\ y e. D ) ) -> ( x .+ y ) e. C )   &    |- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ( ph /\ x e. ( ZZ>= ` ( M + 1 ) ) ) -> ( F ` x ) e. D )   =>    |- ( ph -> seq M ( .+ , F ) : Z --> C )
Theorem	seq1cd 10636*	Initial value of the recursive sequence builder. A version of seq3-1 10629 which provides two classes D and C for the terms and the value being accumulated, respectively. (Contributed by Jim Kingdon, 19-Jul-2023.)
|- ( ph -> ( F ` M ) e. C )   &    |- ( ( ph /\ ( x e. C /\ y e. D ) ) -> ( x .+ y ) e. C )   &    |- ( ph -> M e. ZZ )   &    |- ( ( ph /\ x e. ( ZZ>= ` ( M + 1 ) ) ) -> ( F ` x ) e. D )   =>    |- ( ph -> ( seq M ( .+ , F ) ` M ) = ( F ` M ) )
Theorem	seqp1cd 10637*	Value of the sequence builder function at a successor. A version of seq3p1 10632 which provides two classes D and C for the terms and the value being accumulated, respectively. (Contributed by Jim Kingdon, 20-Jul-2023.)
|- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ph -> ( F ` M ) e. C )   &    |- ( ( ph /\ ( x e. C /\ y e. D ) ) -> ( x .+ y ) e. C )   &    |- ( ( ph /\ x e. ( ZZ>= ` ( M + 1 ) ) ) -> ( F ` x ) e. D )   =>    |- ( ph -> ( seq M ( .+ , F ) ` ( N + 1 ) ) = ( ( seq M ( .+ , F ) ` N ) .+ ( F ` ( N + 1 ) ) ) )
Theorem	seq3clss 10638*	Closure property of the recursive sequence builder. (Contributed by Jim Kingdon, 28-Sep-2022.)
|- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. T )   &    |- ( ( ph /\ x e. ( M ... N ) ) -> ( F ` x ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   &    |- ( ph -> S C_ T )   &    |- ( ( ph /\ ( x e. T /\ y e. T ) ) -> ( x .+ y ) e. T )   =>    |- ( ph -> ( seq M ( .+ , F ) ` N ) e. S )
Theorem	seqclg 10639*	Closure properties of the recursive sequence builder. (Contributed by Mario Carneiro, 2-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.)
|- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ x e. ( M ... N ) ) -> ( F ` x ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   &    |- ( ph -> F e. V )   &    |- ( ph -> .+ e. W )   =>    |- ( ph -> ( seq M ( .+ , F ) ` N ) e. S )
Theorem	seq3m1 10640*	Value of the sequence builder function at a successor. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Jim Kingdon, 3-Nov-2022.)
|- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ( ZZ>= ` ( M + 1 ) ) )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   =>    |- ( ph -> ( seq M ( .+ , F ) ` N ) = ( ( seq M ( .+ , F ) ` ( N - 1 ) ) .+ ( F ` N ) ) )
Theorem	seqm1g 10641	Value of the sequence builder function at a successor. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Jim Kingdon, 30-Aug-2025.)
|- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ( ZZ>= ` ( M + 1 ) ) )   &    |- ( ph -> .+ e. V )   &    |- ( ph -> F e. W )   =>    |- ( ph -> ( seq M ( .+ , F ) ` N ) = ( ( seq M ( .+ , F ) ` ( N - 1 ) ) .+ ( F ` N ) ) )
Theorem	seq3fveq2 10642*	Equality of sequences. (Contributed by Jim Kingdon, 3-Jun-2020.)
|- ( ph -> K e. ( ZZ>= ` M ) )   &    |- ( ph -> ( seq M ( .+ , F ) ` K ) = ( G ` K ) )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )   &    |- ( ( ph /\ x e. ( ZZ>= ` K ) ) -> ( G ` x ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   &    |- ( ph -> N e. ( ZZ>= ` K ) )   &    |- ( ( ph /\ k e. ( ( K + 1 ) ... N ) ) -> ( F ` k ) = ( G ` k ) )   =>    |- ( ph -> ( seq M ( .+ , F ) ` N ) = ( seq K ( .+ , G ) ` N ) )
Theorem	seq3feq2 10643*	Equality of sequences. (Contributed by Jim Kingdon, 3-Jun-2020.)
|- ( ph -> K e. ( ZZ>= ` M ) )   &    |- ( ph -> ( seq M ( .+ , F ) ` K ) = ( G ` K ) )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )   &    |- ( ( ph /\ x e. ( ZZ>= ` K ) ) -> ( G ` x ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   &    |- ( ( ph /\ k e. ( ZZ>= ` ( K + 1 ) ) ) -> ( F ` k ) = ( G ` k ) )   =>    |- ( ph -> ( seq M ( .+ , F ) |` ( ZZ>= ` K ) ) = seq K ( .+ , G ) )
Theorem	seqfveq2g 10644*	Equality of sequences. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 27-May-2014.)
|- ( ph -> K e. ( ZZ>= ` M ) )   &    |- ( ph -> ( seq M ( .+ , F ) ` K ) = ( G ` K ) )   &    |- ( ph -> .+ e. V )   &    |- ( ph -> F e. W )   &    |- ( ph -> G e. X )   &    |- ( ph -> N e. ( ZZ>= ` K ) )   &    |- ( ( ph /\ k e. ( ( K + 1 ) ... N ) ) -> ( F ` k ) = ( G ` k ) )   =>    |- ( ph -> ( seq M ( .+ , F ) ` N ) = ( seq K ( .+ , G ) ` N ) )
Theorem	seqfveqg 10645*	Equality of sequences. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 27-May-2014.)
|- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) = ( G ` k ) )   &    |- ( ph -> .+ e. V )   &    |- ( ph -> F e. W )   &    |- ( ph -> G e. X )   =>    |- ( ph -> ( seq M ( .+ , F ) ` N ) = ( seq M ( .+ , G ) ` N ) )
Theorem	seq3fveq 10646*	Equality of sequences. (Contributed by Jim Kingdon, 4-Jun-2020.)
|- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) = ( G ` k ) )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( G ` x ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   =>    |- ( ph -> ( seq M ( .+ , F ) ` N ) = ( seq M ( .+ , G ) ` N ) )
Theorem	seq3feq 10647*	Equality of sequences. (Contributed by Jim Kingdon, 15-Aug-2021.) (Revised by Jim Kingdon, 7-Apr-2023.)
|- ( ph -> M e. ZZ )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) = ( G ` k ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   =>    |- ( ph -> seq M ( .+ , F ) = seq M ( .+ , G ) )
Theorem	seq3shft2 10648*	Shifting the index set of a sequence. (Contributed by Jim Kingdon, 15-Aug-2021.) (Revised by Jim Kingdon, 7-Apr-2023.)
|- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ph -> K e. ZZ )   &    |- ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) = ( G ` ( k + K ) ) )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )   &    |- ( ( ph /\ x e. ( ZZ>= ` ( M + K ) ) ) -> ( G ` x ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   =>    |- ( ph -> ( seq M ( .+ , F ) ` N ) = ( seq ( M + K ) ( .+ , G ) ` ( N + K ) ) )
Theorem	seqshft2g 10649*	Shifting the index set of a sequence. (Contributed by Mario Carneiro, 27-Feb-2014.) (Revised by Mario Carneiro, 27-May-2014.)
|- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ph -> K e. ZZ )   &    |- ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) = ( G ` ( k + K ) ) )   &    |- ( ph -> .+ e. V )   &    |- ( ph -> F e. W )   &    |- ( ph -> G e. X )   =>    |- ( ph -> ( seq M ( .+ , F ) ` N ) = ( seq ( M + K ) ( .+ , G ) ` ( N + K ) ) )
Theorem	serf 10650*	An infinite series of complex terms is a function from NN to CC. (Contributed by NM, 18-Apr-2005.) (Revised by Mario Carneiro, 27-May-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )   =>    |- ( ph -> seq M ( + , F ) : Z --> CC )
Theorem	serfre 10651*	An infinite series of real numbers is a function from NN to RR. (Contributed by NM, 18-Apr-2005.) (Revised by Mario Carneiro, 27-May-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. RR )   =>    |- ( ph -> seq M ( + , F ) : Z --> RR )
Theorem	monoord 10652*	Ordering relation for a monotonic sequence, increasing case. (Contributed by NM, 13-Mar-2005.) (Revised by Mario Carneiro, 9-Feb-2014.)
|- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) e. RR )   &    |- ( ( ph /\ k e. ( M ... ( N - 1 ) ) ) -> ( F ` k ) <_ ( F ` ( k + 1 ) ) )   =>    |- ( ph -> ( F ` M ) <_ ( F ` N ) )
Theorem	monoord2 10653*	Ordering relation for a monotonic sequence, decreasing case. (Contributed by Mario Carneiro, 18-Jul-2014.)
|- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) e. RR )   &    |- ( ( ph /\ k e. ( M ... ( N - 1 ) ) ) -> ( F ` ( k + 1 ) ) <_ ( F ` k ) )   =>    |- ( ph -> ( F ` N ) <_ ( F ` M ) )
Theorem	ser3mono 10654*	The partial sums in an infinite series of positive terms form a monotonic sequence. (Contributed by NM, 17-Mar-2005.) (Revised by Jim Kingdon, 22-Apr-2023.)
|- ( ph -> K e. ( ZZ>= ` M ) )   &    |- ( ph -> N e. ( ZZ>= ` K ) )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. RR )   &    |- ( ( ph /\ x e. ( ( K + 1 ) ... N ) ) -> 0 <_ ( F ` x ) )   =>    |- ( ph -> ( seq M ( + , F ) ` K ) <_ ( seq M ( + , F ) ` N ) )
Theorem	seq3split 10655*	Split a sequence into two sequences. (Contributed by Jim Kingdon, 16-Aug-2021.) (Revised by Jim Kingdon, 21-Oct-2022.)
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) )   &    |- ( ph -> N e. ( ZZ>= ` ( M + 1 ) ) )   &    |- ( ph -> M e. ( ZZ>= ` K ) )   &    |- ( ( ph /\ x e. ( ZZ>= ` K ) ) -> ( F ` x ) e. S )   =>    |- ( ph -> ( seq K ( .+ , F ) ` N ) = ( ( seq K ( .+ , F ) ` M ) .+ ( seq ( M + 1 ) ( .+ , F ) ` N ) ) )
Theorem	seqsplitg 10656*	Split a sequence into two sequences. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 27-May-2014.)
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) )   &    |- ( ph -> N e. ( ZZ>= ` ( M + 1 ) ) )   &    |- ( ph -> .+ e. V )   &    |- ( ph -> F e. W )   &    |- ( ph -> M e. ( ZZ>= ` K ) )   &    |- ( ( ph /\ x e. ( K ... N ) ) -> ( F ` x ) e. S )   =>    |- ( ph -> ( seq K ( .+ , F ) ` N ) = ( ( seq K ( .+ , F ) ` M ) .+ ( seq ( M + 1 ) ( .+ , F ) ` N ) ) )
Theorem	seq3-1p 10657*	Removing the first term from a sequence. (Contributed by Jim Kingdon, 16-Aug-2021.)
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) )   &    |- ( ph -> N e. ( ZZ>= ` ( M + 1 ) ) )   &    |- ( ph -> M e. ZZ )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )   =>    |- ( ph -> ( seq M ( .+ , F ) ` N ) = ( ( F ` M ) .+ ( seq ( M + 1 ) ( .+ , F ) ` N ) ) )
Theorem	seq3caopr3 10658*	Lemma for seq3caopr2 10660. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by Jim Kingdon, 22-Apr-2023.)
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x Q y ) e. S )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. S )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( G ` k ) e. S )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( H ` k ) = ( ( F ` k ) Q ( G ` k ) ) )   &    |- ( ( ph /\ n e. ( M..^ N ) ) -> ( ( ( seq M ( .+ , F ) ` n ) Q ( seq M ( .+ , G ) ` n ) ) .+ ( ( F ` ( n + 1 ) ) Q ( G ` ( n + 1 ) ) ) ) = ( ( ( seq M ( .+ , F ) ` n ) .+ ( F ` ( n + 1 ) ) ) Q ( ( seq M ( .+ , G ) ` n ) .+ ( G ` ( n + 1 ) ) ) ) )   =>    |- ( ph -> ( seq M ( .+ , H ) ` N ) = ( ( seq M ( .+ , F ) ` N ) Q ( seq M ( .+ , G ) ` N ) ) )
Theorem	seqcaopr3g 10659*	Lemma for seqcaopr2g 10661. (Contributed by Mario Carneiro, 25-Apr-2016.)
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x Q y ) e. S )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) e. S )   &    |- ( ( ph /\ k e. ( M ... N ) ) -> ( G ` k ) e. S )   &    |- ( ( ph /\ k e. ( M ... N ) ) -> ( H ` k ) = ( ( F ` k ) Q ( G ` k ) ) )   &    |- ( ph -> .+ e. V )   &    |- ( ph -> F e. W )   &    |- ( ph -> G e. X )   &    |- ( ph -> H e. Y )   &    |- ( ( ph /\ n e. ( M..^ N ) ) -> ( ( ( seq M ( .+ , F ) ` n ) Q ( seq M ( .+ , G ) ` n ) ) .+ ( ( F ` ( n + 1 ) ) Q ( G ` ( n + 1 ) ) ) ) = ( ( ( seq M ( .+ , F ) ` n ) .+ ( F ` ( n + 1 ) ) ) Q ( ( seq M ( .+ , G ) ` n ) .+ ( G ` ( n + 1 ) ) ) ) )   =>    |- ( ph -> ( seq M ( .+ , H ) ` N ) = ( ( seq M ( .+ , F ) ` N ) Q ( seq M ( .+ , G ) ` N ) ) )
Theorem	seq3caopr2 10660*	The sum of two infinite series (generalized to an arbitrary commutative and associative operation). (Contributed by Mario Carneiro, 30-May-2014.) (Revised by Jim Kingdon, 23-Apr-2023.)
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x Q y ) e. S )   &    |- ( ( ph /\ ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) ) -> ( ( x Q z ) .+ ( y Q w ) ) = ( ( x .+ y ) Q ( z .+ w ) ) )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. S )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( G ` k ) e. S )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( H ` k ) = ( ( F ` k ) Q ( G ` k ) ) )   =>    |- ( ph -> ( seq M ( .+ , H ) ` N ) = ( ( seq M ( .+ , F ) ` N ) Q ( seq M ( .+ , G ) ` N ) ) )
Theorem	seqcaopr2g 10661*	The sum of two infinite series (generalized to an arbitrary commutative and associative operation). (Contributed by Mario Carneiro, 30-May-2014.)
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x Q y ) e. S )   &    |- ( ( ph /\ ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) ) -> ( ( x Q z ) .+ ( y Q w ) ) = ( ( x .+ y ) Q ( z .+ w ) ) )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) e. S )   &    |- ( ( ph /\ k e. ( M ... N ) ) -> ( G ` k ) e. S )   &    |- ( ( ph /\ k e. ( M ... N ) ) -> ( H ` k ) = ( ( F ` k ) Q ( G ` k ) ) )   &    |- ( ph -> .+ e. V )   &    |- ( ph -> F e. W )   &    |- ( ph -> G e. X )   &    |- ( ph -> H e. Y )   =>    |- ( ph -> ( seq M ( .+ , H ) ` N ) = ( ( seq M ( .+ , F ) ` N ) Q ( seq M ( .+ , G ) ` N ) ) )
Theorem	seq3caopr 10662*	The sum of two infinite series (generalized to an arbitrary commutative and associative operation). (Contributed by NM, 17-Mar-2005.) (Revised by Jim Kingdon, 23-Apr-2023.)
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) = ( y .+ x ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. S )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( G ` k ) e. S )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( H ` k ) = ( ( F ` k ) .+ ( G ` k ) ) )   =>    |- ( ph -> ( seq M ( .+ , H ) ` N ) = ( ( seq M ( .+ , F ) ` N ) .+ ( seq M ( .+ , G ) ` N ) ) )
Theorem	seqcaoprg 10663*	The sum of two infinite series (generalized to an arbitrary commutative and associative operation). (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 30-May-2014.)
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) = ( y .+ x ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) e. S )   &    |- ( ( ph /\ k e. ( M ... N ) ) -> ( G ` k ) e. S )   &    |- ( ( ph /\ k e. ( M ... N ) ) -> ( H ` k ) = ( ( F ` k ) .+ ( G ` k ) ) )   &    |- ( ph -> .+ e. V )   &    |- ( ph -> F e. W )   &    |- ( ph -> G e. X )   &    |- ( ph -> H e. Y )   =>    |- ( ph -> ( seq M ( .+ , H ) ` N ) = ( ( seq M ( .+ , F ) ` N ) .+ ( seq M ( .+ , G ) ` N ) ) )
Theorem	iseqf1olemkle 10664*	Lemma for seq3f1o 10684. (Contributed by Jim Kingdon, 21-Aug-2022.)
|- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ph -> K e. ( M ... N ) )   &    |- ( ph -> J : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |- ( ph -> A. x e. ( M..^ K ) ( J ` x ) = x )   =>    |- ( ph -> K <_ ( `' J ` K ) )
Theorem	iseqf1olemklt 10665*	Lemma for seq3f1o 10684. (Contributed by Jim Kingdon, 21-Aug-2022.)
|- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ph -> K e. ( M ... N ) )   &    |- ( ph -> J : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |- ( ph -> A. x e. ( M..^ K ) ( J ` x ) = x )   &    |- ( ph -> K =/= ( `' J ` K ) )   =>    |- ( ph -> K < ( `' J ` K ) )
Theorem	iseqf1olemqcl 10666	Lemma for seq3f1o 10684. (Contributed by Jim Kingdon, 27-Aug-2022.)
|- ( ph -> K e. ( M ... N ) )   &    |- ( ph -> J : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |- ( ph -> A e. ( M ... N ) )   =>    |- ( ph -> if ( A e. ( K ... ( `' J ` K ) ) , if ( A = K , K , ( J ` ( A - 1 ) ) ) , ( J ` A ) ) e. ( M ... N ) )
Theorem	iseqf1olemqval 10667*	Lemma for seq3f1o 10684. Value of the function Q. (Contributed by Jim Kingdon, 28-Aug-2022.)
|- ( ph -> K e. ( M ... N ) )   &    |- ( ph -> J : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |- ( ph -> A e. ( M ... N ) )   &    |- Q = ( u e. ( M ... N ) |-> if ( u e. ( K ... ( `' J ` K ) ) , if ( u = K , K , ( J ` ( u - 1 ) ) ) , ( J ` u ) ) )   =>    |- ( ph -> ( Q ` A ) = if ( A e. ( K ... ( `' J ` K ) ) , if ( A = K , K , ( J ` ( A - 1 ) ) ) , ( J ` A ) ) )
Theorem	iseqf1olemnab 10668*	Lemma for seq3f1o 10684. (Contributed by Jim Kingdon, 27-Aug-2022.)
|- ( ph -> K e. ( M ... N ) )   &    |- ( ph -> J : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |- ( ph -> A e. ( M ... N ) )   &    |- ( ph -> B e. ( M ... N ) )   &    |- ( ph -> ( Q ` A ) = ( Q ` B ) )   &    |- Q = ( u e. ( M ... N ) |-> if ( u e. ( K ... ( `' J ` K ) ) , if ( u = K , K , ( J ` ( u - 1 ) ) ) , ( J ` u ) ) )   =>    |- ( ph -> -. ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) )
Theorem	iseqf1olemab 10669*	Lemma for seq3f1o 10684. (Contributed by Jim Kingdon, 27-Aug-2022.)
|- ( ph -> K e. ( M ... N ) )   &    |- ( ph -> J : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |- ( ph -> A e. ( M ... N ) )   &    |- ( ph -> B e. ( M ... N ) )   &    |- ( ph -> ( Q ` A ) = ( Q ` B ) )   &    |- Q = ( u e. ( M ... N ) |-> if ( u e. ( K ... ( `' J ` K ) ) , if ( u = K , K , ( J ` ( u - 1 ) ) ) , ( J ` u ) ) )   &    |- ( ph -> A e. ( K ... ( `' J ` K ) ) )   &    |- ( ph -> B e. ( K ... ( `' J ` K ) ) )   =>    |- ( ph -> A = B )
Theorem	iseqf1olemnanb 10670*	Lemma for seq3f1o 10684. (Contributed by Jim Kingdon, 27-Aug-2022.)
|- ( ph -> K e. ( M ... N ) )   &    |- ( ph -> J : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |- ( ph -> A e. ( M ... N ) )   &    |- ( ph -> B e. ( M ... N ) )   &    |- ( ph -> ( Q ` A ) = ( Q ` B ) )   &    |- Q = ( u e. ( M ... N ) |-> if ( u e. ( K ... ( `' J ` K ) ) , if ( u = K , K , ( J ` ( u - 1 ) ) ) , ( J ` u ) ) )   &    |- ( ph -> -. A e. ( K ... ( `' J ` K ) ) )   &    |- ( ph -> -. B e. ( K ... ( `' J ` K ) ) )   =>    |- ( ph -> A = B )
Theorem	iseqf1olemqf 10671*	Lemma for seq3f1o 10684. Domain and codomain of Q. (Contributed by Jim Kingdon, 26-Aug-2022.)
|- ( ph -> K e. ( M ... N ) )   &    |- ( ph -> J : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |- Q = ( u e. ( M ... N ) |-> if ( u e. ( K ... ( `' J ` K ) ) , if ( u = K , K , ( J ` ( u - 1 ) ) ) , ( J ` u ) ) )   =>    |- ( ph -> Q : ( M ... N ) --> ( M ... N ) )
Theorem	iseqf1olemmo 10672*	Lemma for seq3f1o 10684. Showing that Q is one-to-one. (Contributed by Jim Kingdon, 27-Aug-2022.)
|- ( ph -> K e. ( M ... N ) )   &    |- ( ph -> J : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |- Q = ( u e. ( M ... N ) |-> if ( u e. ( K ... ( `' J ` K ) ) , if ( u = K , K , ( J ` ( u - 1 ) ) ) , ( J ` u ) ) )   &    |- ( ph -> A e. ( M ... N ) )   &    |- ( ph -> B e. ( M ... N ) )   &    |- ( ph -> ( Q ` A ) = ( Q ` B ) )   =>    |- ( ph -> A = B )
Theorem	iseqf1olemqf1o 10673*	Lemma for seq3f1o 10684. Q is a permutation of ( M ... N ). Q is formed from the constant portion of J, followed by the single element K (at position K), followed by the rest of J (with the K deleted and the elements before K moved one position later to fill the gap). (Contributed by Jim Kingdon, 21-Aug-2022.)
|- ( ph -> K e. ( M ... N ) )   &    |- ( ph -> J : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |- Q = ( u e. ( M ... N ) |-> if ( u e. ( K ... ( `' J ` K ) ) , if ( u = K , K , ( J ` ( u - 1 ) ) ) , ( J ` u ) ) )   =>    |- ( ph -> Q : ( M ... N ) -1-1-onto-> ( M ... N ) )
Theorem	iseqf1olemqk 10674*	Lemma for seq3f1o 10684. Q is constant for one more position than J is. (Contributed by Jim Kingdon, 21-Aug-2022.)
|- ( ph -> K e. ( M ... N ) )   &    |- ( ph -> J : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |- Q = ( u e. ( M ... N ) |-> if ( u e. ( K ... ( `' J ` K ) ) , if ( u = K , K , ( J ` ( u - 1 ) ) ) , ( J ` u ) ) )   &    |- ( ph -> A. x e. ( M..^ K ) ( J ` x ) = x )   =>    |- ( ph -> A. x e. ( M ... K ) ( Q ` x ) = x )
Theorem	iseqf1olemjpcl 10675*	Lemma for seq3f1o 10684. A closure lemma involving J and P. (Contributed by Jim Kingdon, 29-Aug-2022.)
|- ( ph -> K e. ( M ... N ) )   &    |- ( ph -> J : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |- Q = ( u e. ( M ... N ) |-> if ( u e. ( K ... ( `' J ` K ) ) , if ( u = K , K , ( J ` ( u - 1 ) ) ) , ( J ` u ) ) )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( G ` x ) e. S )   &    |- P = ( x e. ( ZZ>= ` M ) |-> if ( x <_ N , ( G ` ( f ` x ) ) , ( G ` M ) ) )   =>    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( [_ J / f ]_ P ` x ) e. S )
Theorem	iseqf1olemqpcl 10676*	Lemma for seq3f1o 10684. A closure lemma involving Q and P. (Contributed by Jim Kingdon, 29-Aug-2022.)
|- ( ph -> K e. ( M ... N ) )   &    |- ( ph -> J : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |- Q = ( u e. ( M ... N ) |-> if ( u e. ( K ... ( `' J ` K ) ) , if ( u = K , K , ( J ` ( u - 1 ) ) ) , ( J ` u ) ) )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( G ` x ) e. S )   &    |- P = ( x e. ( ZZ>= ` M ) |-> if ( x <_ N , ( G ` ( f ` x ) ) , ( G ` M ) ) )   =>    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( [_ Q / f ]_ P ` x ) e. S )
Theorem	iseqf1olemfvp 10677*	Lemma for seq3f1o 10684. (Contributed by Jim Kingdon, 30-Aug-2022.)
|- ( ph -> K e. ( M ... N ) )   &    |- ( ph -> T : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |- ( ph -> A e. ( M ... N ) )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( G ` x ) e. S )   &    |- P = ( x e. ( ZZ>= ` M ) |-> if ( x <_ N , ( G ` ( f ` x ) ) , ( G ` M ) ) )   =>    |- ( ph -> ( [_ T / f ]_ P ` A ) = ( G ` ( T ` A ) ) )
Theorem	seq3f1olemqsumkj 10678*	Lemma for seq3f1o 10684. Q gives the same sum as J in the range ( K ... ( `' J ` K ) ). (Contributed by Jim Kingdon, 29-Aug-2022.)
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) = ( y .+ x ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ph -> F : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( G ` x ) e. S )   &    |- ( ph -> K e. ( M ... N ) )   &    |- ( ph -> J : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |- ( ph -> A. x e. ( M..^ K ) ( J ` x ) = x )   &    |- ( ph -> K =/= ( `' J ` K ) )   &    |- Q = ( u e. ( M ... N ) |-> if ( u e. ( K ... ( `' J ` K ) ) , if ( u = K , K , ( J ` ( u - 1 ) ) ) , ( J ` u ) ) )   &    |- P = ( x e. ( ZZ>= ` M ) |-> if ( x <_ N , ( G ` ( f ` x ) ) , ( G ` M ) ) )   =>    |- ( ph -> ( seq K ( .+ , [_ J / f ]_ P ) ` ( `' J ` K ) ) = ( seq K ( .+ , [_ Q / f ]_ P ) ` ( `' J ` K ) ) )
Theorem	seq3f1olemqsumk 10679*	Lemma for seq3f1o 10684. Q gives the same sum as J in the range ( K ... N ). (Contributed by Jim Kingdon, 22-Aug-2022.)
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) = ( y .+ x ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ph -> F : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( G ` x ) e. S )   &    |- ( ph -> K e. ( M ... N ) )   &    |- ( ph -> J : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |- ( ph -> A. x e. ( M..^ K ) ( J ` x ) = x )   &    |- ( ph -> K =/= ( `' J ` K ) )   &    |- Q = ( u e. ( M ... N ) |-> if ( u e. ( K ... ( `' J ` K ) ) , if ( u = K , K , ( J ` ( u - 1 ) ) ) , ( J ` u ) ) )   &    |- P = ( x e. ( ZZ>= ` M ) |-> if ( x <_ N , ( G ` ( f ` x ) ) , ( G ` M ) ) )   =>    |- ( ph -> ( seq K ( .+ , [_ J / f ]_ P ) ` N ) = ( seq K ( .+ , [_ Q / f ]_ P ) ` N ) )
Theorem	seq3f1olemqsum 10680*	Lemma for seq3f1o 10684. Q gives the same sum as J. (Contributed by Jim Kingdon, 21-Aug-2022.)
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) = ( y .+ x ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ph -> F : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( G ` x ) e. S )   &    |- ( ph -> K e. ( M ... N ) )   &    |- ( ph -> J : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |- ( ph -> A. x e. ( M..^ K ) ( J ` x ) = x )   &    |- ( ph -> K =/= ( `' J ` K ) )   &    |- Q = ( u e. ( M ... N ) |-> if ( u e. ( K ... ( `' J ` K ) ) , if ( u = K , K , ( J ` ( u - 1 ) ) ) , ( J ` u ) ) )   &    |- P = ( x e. ( ZZ>= ` M ) |-> if ( x <_ N , ( G ` ( f ` x ) ) , ( G ` M ) ) )   =>    |- ( ph -> ( seq M ( .+ , [_ J / f ]_ P ) ` N ) = ( seq M ( .+ , [_ Q / f ]_ P ) ` N ) )
Theorem	seq3f1olemstep 10681*	Lemma for seq3f1o 10684. Given a permutation which is constant up to a point, supply a new one which is constant for one more position. (Contributed by Jim Kingdon, 19-Aug-2022.)
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) = ( y .+ x ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ph -> F : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( G ` x ) e. S )   &    |- ( ph -> K e. ( M ... N ) )   &    |- ( ph -> J : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |- ( ph -> A. x e. ( M..^ K ) ( J ` x ) = x )   &    |- ( ph -> ( seq M ( .+ , [_ J / f ]_ P ) ` N ) = ( seq M ( .+ , L ) ` N ) )   &    |- P = ( x e. ( ZZ>= ` M ) |-> if ( x <_ N , ( G ` ( f ` x ) ) , ( G ` M ) ) )   =>    |- ( ph -> E. f ( f : ( M ... N ) -1-1-onto-> ( M ... N ) /\ A. x e. ( M ... K ) ( f ` x ) = x /\ ( seq M ( .+ , P ) ` N ) = ( seq M ( .+ , L ) ` N ) ) )
Theorem	seq3f1olemp 10682*	Lemma for seq3f1o 10684. Existence of a constant permutation of ( M ... N ) which leads to the same sum as the permutation F itself. (Contributed by Jim Kingdon, 18-Aug-2022.)
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) = ( y .+ x ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ph -> F : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( G ` x ) e. S )   &    |- L = ( x e. ( ZZ>= ` M ) |-> if ( x <_ N , ( G ` ( F ` x ) ) , ( G ` M ) ) )   &    |- P = ( x e. ( ZZ>= ` M ) |-> if ( x <_ N , ( G ` ( f ` x ) ) , ( G ` M ) ) )   =>    |- ( ph -> E. f ( f : ( M ... N ) -1-1-onto-> ( M ... N ) /\ A. x e. ( M ... N ) ( f ` x ) = x /\ ( seq M ( .+ , P ) ` N ) = ( seq M ( .+ , L ) ` N ) ) )
Theorem	seq3f1oleml 10683*	Lemma for seq3f1o 10684. This is more or less the result, but stated in terms of F and G without H. L and H may differ in terms of what happens to terms after N. The terms after N don't matter for the value at N but we need some definition given the way our theorems concerning seq work. (Contributed by Jim Kingdon, 17-Aug-2022.)
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) = ( y .+ x ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ph -> F : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( G ` x ) e. S )   &    |- L = ( x e. ( ZZ>= ` M ) |-> if ( x <_ N , ( G ` ( F ` x ) ) , ( G ` M ) ) )   =>    |- ( ph -> ( seq M ( .+ , L ) ` N ) = ( seq M ( .+ , G ) ` N ) )
Theorem	seq3f1o 10684*	Rearrange a sum via an arbitrary bijection on ( M ... N ). (Contributed by Mario Carneiro, 27-Feb-2014.) (Revised by Jim Kingdon, 3-Nov-2022.)
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) = ( y .+ x ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ph -> F : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( G ` x ) e. S )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( H ` x ) e. S )   &    |- ( ( ph /\ k e. ( M ... N ) ) -> ( H ` k ) = ( G ` ( F ` k ) ) )   =>    |- ( ph -> ( seq M ( .+ , H ) ` N ) = ( seq M ( .+ , G ) ` N ) )
Theorem	seqf1oglem2a 10685*	Lemma for seqf1og 10688. (Contributed by Mario Carneiro, 24-Apr-2016.)
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   &    |- ( ( ph /\ ( x e. C /\ y e. C ) ) -> ( x .+ y ) = ( y .+ x ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ph -> C C_ S )   &    |- ( ph -> .+ e. V )   &    |- ( ph -> G : A --> C )   &    |- ( ph -> K e. A )   &    |- ( ph -> ( M ... N ) C_ A )   &    |- ( ph -> A e. W )   =>    |- ( ph -> ( ( G ` K ) .+ ( seq M ( .+ , G ) ` N ) ) = ( ( seq M ( .+ , G ) ` N ) .+ ( G ` K ) ) )
Theorem	seqf1oglem1 10686*	Lemma for seqf1og 10688. (Contributed by Mario Carneiro, 26-Feb-2014.) (Revised by Mario Carneiro, 27-May-2014.)
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   &    |- ( ( ph /\ ( x e. C /\ y e. C ) ) -> ( x .+ y ) = ( y .+ x ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ph -> C C_ S )   &    |- ( ph -> .+ e. V )   &    |- ( ph -> F : ( M ... ( N + 1 ) ) -1-1-onto-> ( M ... ( N + 1 ) ) )   &    |- ( ph -> G : ( M ... ( N + 1 ) ) --> C )   &    |- J = ( k e. ( M ... N ) |-> ( F ` if ( k < K , k , ( k + 1 ) ) ) )   &    |- K = ( `' F ` ( N + 1 ) )   =>    |- ( ph -> J : ( M ... N ) -1-1-onto-> ( M ... N ) )
Theorem	seqf1oglem2 10687*	Lemma for seqf1og 10688. (Contributed by Mario Carneiro, 27-Feb-2014.) (Revised by Mario Carneiro, 24-Apr-2016.)
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   &    |- ( ( ph /\ ( x e. C /\ y e. C ) ) -> ( x .+ y ) = ( y .+ x ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ph -> C C_ S )   &    |- ( ph -> .+ e. V )   &    |- ( ph -> F : ( M ... ( N + 1 ) ) -1-1-onto-> ( M ... ( N + 1 ) ) )   &    |- ( ph -> G : ( M ... ( N + 1 ) ) --> C )   &    |- J = ( k e. ( M ... N ) |-> ( F ` if ( k < K , k , ( k + 1 ) ) ) )   &    |- K = ( `' F ` ( N + 1 ) )   &    |- ( ph -> A. g A. f ( ( f : ( M ... N ) -1-1-onto-> ( M ... N ) /\ g : ( M ... N ) --> C ) -> ( seq M ( .+ , ( g o. f ) ) ` N ) = ( seq M ( .+ , g ) ` N ) ) )   =>    |- ( ph -> ( seq M ( .+ , ( G o. F ) ) ` ( N + 1 ) ) = ( seq M ( .+ , G ) ` ( N + 1 ) ) )
Theorem	seqf1og 10688*	Rearrange a sum via an arbitrary bijection on ( M ... N ). (Contributed by Mario Carneiro, 27-Feb-2014.) (Revised by Jim Kingdon, 29-Aug-2025.)
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   &    |- ( ( ph /\ ( x e. C /\ y e. C ) ) -> ( x .+ y ) = ( y .+ x ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ph -> C C_ S )   &    |- ( ph -> .+ e. V )   &    |- ( ph -> F : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |- ( ( ph /\ x e. ( M ... N ) ) -> ( G ` x ) e. C )   &    |- ( ( ph /\ k e. ( M ... N ) ) -> ( H ` k ) = ( G ` ( F ` k ) ) )   &    |- ( ph -> G e. W )   &    |- ( ph -> H e. X )   =>    |- ( ph -> ( seq M ( .+ , H ) ` N ) = ( seq M ( .+ , G ) ` N ) )
Theorem	ser3add 10689*	The sum of two infinite series. (Contributed by NM, 17-Mar-2005.) (Revised by Jim Kingdon, 4-Oct-2022.)
|- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( G ` k ) e. CC )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( H ` k ) = ( ( F ` k ) + ( G ` k ) ) )   =>    |- ( ph -> ( seq M ( + , H ) ` N ) = ( ( seq M ( + , F ) ` N ) + ( seq M ( + , G ) ` N ) ) )
Theorem	ser3sub 10690*	The difference of two infinite series. (Contributed by NM, 17-Mar-2005.) (Revised by Jim Kingdon, 22-Apr-2023.)
|- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( G ` k ) e. CC )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( H ` k ) = ( ( F ` k ) - ( G ` k ) ) )   =>    |- ( ph -> ( seq M ( + , H ) ` N ) = ( ( seq M ( + , F ) ` N ) - ( seq M ( + , G ) ` N ) ) )
Theorem	seq3id3 10691*	A sequence that consists entirely of "zeroes" sums to "zero". More precisely, a constant sequence with value an element which is a .+ -idempotent sums (or " .+'s") to that element. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by Jim Kingdon, 8-Apr-2023.)
|- ( ph -> ( Z .+ Z ) = Z )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ x e. ( M ... N ) ) -> ( F ` x ) = Z )   &    |- ( ph -> Z e. S )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   =>    |- ( ph -> ( seq M ( .+ , F ) ` N ) = Z )
Theorem	seq3id 10692*	Discarding the first few terms of a sequence that starts with all zeroes (or any element which is a left-identity for .+) has no effect on its sum. (Contributed by Mario Carneiro, 13-Jul-2013.) (Revised by Jim Kingdon, 8-Apr-2023.)
|- ( ( ph /\ x e. S ) -> ( Z .+ x ) = x )   &    |- ( ph -> Z e. S )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ph -> ( F ` N ) e. S )   &    |- ( ( ph /\ x e. ( M ... ( N - 1 ) ) ) -> ( F ` x ) = Z )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   =>    |- ( ph -> ( seq M ( .+ , F ) |` ( ZZ>= ` N ) ) = seq N ( .+ , F ) )
Theorem	seq3id2 10693*	The last few partial sums of a sequence that ends with all zeroes (or any element which is a right-identity for .+) are all the same. (Contributed by Mario Carneiro, 13-Jul-2013.) (Revised by Jim Kingdon, 12-Nov-2022.)
|- ( ( ph /\ x e. S ) -> ( x .+ Z ) = x )   &    |- ( ph -> K e. ( ZZ>= ` M ) )   &    |- ( ph -> N e. ( ZZ>= ` K ) )   &    |- ( ph -> ( seq M ( .+ , F ) ` K ) e. S )   &    |- ( ( ph /\ x e. ( ( K + 1 ) ... N ) ) -> ( F ` x ) = Z )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   =>    |- ( ph -> ( seq M ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` N ) )
Theorem	seq3homo 10694*	Apply a homomorphism to a sequence. (Contributed by Jim Kingdon, 10-Oct-2022.)
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( H ` ( x .+ y ) ) = ( ( H ` x ) Q ( H ` y ) ) )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( H ` ( F ` x ) ) = ( G ` x ) )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( G ` x ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x Q y ) e. S )   =>    |- ( ph -> ( H ` ( seq M ( .+ , F ) ` N ) ) = ( seq M ( Q , G ) ` N ) )
Theorem	seq3z 10695*	If the operation .+ has an absorbing element Z (a.k.a. zero element), then any sequence containing a Z evaluates to Z. (Contributed by Mario Carneiro, 27-May-2014.) (Revised by Jim Kingdon, 23-Apr-2023.)
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )   &    |- ( ( ph /\ x e. S ) -> ( Z .+ x ) = Z )   &    |- ( ( ph /\ x e. S ) -> ( x .+ Z ) = Z )   &    |- ( ph -> K e. ( M ... N ) )   &    |- ( ph -> ( F ` K ) = Z )   =>    |- ( ph -> ( seq M ( .+ , F ) ` N ) = Z )
Theorem	seqfeq3 10696*	Equality of series under different addition operations which agree on an additively closed subset. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 25-Apr-2016.)
|- ( ph -> M e. ZZ )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) = ( x Q y ) )   =>    |- ( ph -> seq M ( .+ , F ) = seq M ( Q , F ) )
Theorem	seqhomog 10697*	Apply a homomorphism to a sequence. (Contributed by Mario Carneiro, 28-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.)
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   &    |- ( ( ph /\ x e. ( M ... N ) ) -> ( F ` x ) e. S )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( H ` ( x .+ y ) ) = ( ( H ` x ) Q ( H ` y ) ) )   &    |- ( ( ph /\ x e. ( M ... N ) ) -> ( H ` ( F ` x ) ) = ( G ` x ) )   &    |- ( ph -> F e. V )   &    |- ( ph -> G e. W )   &    |- ( ph -> .+ e. X )   &    |- ( ph -> Q e. Y )   =>    |- ( ph -> ( H ` ( seq M ( .+ , F ) ` N ) ) = ( seq M ( Q , G ) ` N ) )
Theorem	seqfeq4g 10698*	Equality of series under different addition operations which agree on an additively closed subset. (Contributed by Mario Carneiro, 25-Apr-2016.)
|- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ x e. ( M ... N ) ) -> ( F ` x ) e. S )   &    |- ( ph -> F e. V )   &    |- ( ph -> .+ e. W )   &    |- ( ph -> Q e. X )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) = ( x Q y ) )   =>    |- ( ph -> ( seq M ( .+ , F ) ` N ) = ( seq M ( Q , F ) ` N ) )
Theorem	seq3distr 10699*	The distributive property for series. (Contributed by Jim Kingdon, 10-Oct-2022.)
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( C T ( x .+ y ) ) = ( ( C T x ) .+ ( C T y ) ) )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( G ` x ) e. S )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) = ( C T ( G ` x ) ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x T y ) e. S )   &    |- ( ph -> C e. S )   =>    |- ( ph -> ( seq M ( .+ , F ) ` N ) = ( C T ( seq M ( .+ , G ) ` N ) ) )
Theorem	ser0 10700	The value of the partial sums in a zero-valued infinite series. (Contributed by Mario Carneiro, 31-Aug-2013.) (Revised by Mario Carneiro, 15-Dec-2014.)
|- Z = ( ZZ>= ` M )   =>    |- ( N e. Z -> ( seq M ( + , ( Z X. { 0 } ) ) ` N ) = 0 )