Most Recent Proofs - Intuitionistic Logic Explorer

Most recent proofs    These are the 100 (Unicode, GIF) or 1000 (Unicode, GIF) most recent proofs in the iset.mm database for the Intuitionistic Logic Explorer. The iset.mm database is maintained on GitHub with master (stable) and develop (development) versions. This page was created from the commit given on the MPE Most Recent Proofs page. The database from that commit is also available here: iset.mm.

See the MPE Most Recent Proofs page for news and some useful links.

Last updated on 20-Sep-2025 at 6:42 AM ET.

Recent Additions to the Intuitionistic Logic Explorer

Date	Label	Description
Theorem
16-Sep-2025	opabfi 6992	Finiteness of an ordered pair abstraction which is a decidable subset of finite sets. (Contributed by Jim Kingdon, 16-Sep-2025.)
|- S = { <. x , y >. | ( ( x e. A /\ y e. B ) /\ ps ) }   &    |- ( ph -> A e. Fin )   &    |- ( ph -> B e. Fin )   &    |- ( ph -> A. x e. A A. y e. B DECID ps )   =>    |- ( ph -> S e. Fin )
13-Sep-2025	uchoice 6190	Principle of unique choice. This is also called non-choice. The name choice results in its similarity to something like acfun 7267 (with the key difference being the change of E. to E!) but unique choice in fact follows from the axiom of collection and our other axioms. This is somewhat similar to Corollary 3.9.2 of [HoTT], p. (varies) but is better described by the paragraph at the end of Section 3.9 which starts "A similar issue arises in set-theoretic mathematics". (Contributed by Jim Kingdon, 13-Sep-2025.)
|- ( ( A e. V /\ A. x e. A E! y ph ) -> E. f ( f Fn A /\ A. x e. A [. ( f ` x ) / y ]. ph ) )
11-Sep-2025	expghmap 14095	Exponentiation is a group homomorphism from addition to multiplication. (Contributed by Mario Carneiro, 18-Jun-2015.) (Revised by AV, 10-Jun-2019.) (Revised by Jim Kingdon, 11-Sep-2025.)
|- M = (mulGrp ` ℂfld )   &    |- U = ( M ↾s { z e. CC | z # 0 } )   =>    |- ( ( A e. CC /\ A # 0 ) -> ( x e. ZZ |-> ( A ^ x ) ) e. (ℤring GrpHom U ) )
11-Sep-2025	cnfldui 14077	The invertible complex numbers are exactly those apart from zero. This is recapb 8690 but expressed in terms of ℂfld. (Contributed by Jim Kingdon, 11-Sep-2025.)
|- { z e. CC | z # 0 } = (Unit ` ℂfld )
9-Sep-2025	gsumfzfsumlemm 14075	Lemma for gsumfzfsum 14076. The case where the sum is inhabited. (Contributed by Jim Kingdon, 9-Sep-2025.)
|- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( M ... N ) ) -> B e. CC )   =>    |- ( ph -> (ℂfld gsumg ( k e. ( M ... N ) |-> B ) ) = sum_ k e. ( M ... N ) B )
9-Sep-2025	gsumfzfsumlem0 14074	Lemma for gsumfzfsum 14076. The case where the sum is empty. (Contributed by Jim Kingdon, 9-Sep-2025.)
|- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ZZ )   &    |- ( ph -> N < M )   =>    |- ( ph -> (ℂfld gsumg ( k e. ( M ... N ) |-> B ) ) = sum_ k e. ( M ... N ) B )
9-Sep-2025	gsumfzmhm2 13414	Apply a group homomorphism to a group sum, mapping version with implicit substitution. (Contributed by Mario Carneiro, 5-May-2015.) (Revised by AV, 6-Jun-2019.) (Revised by Jim Kingdon, 9-Sep-2025.)
|- B = ( Base ` G )   &    |- .0. = ( 0g ` G )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> H e. Mnd )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ZZ )   &    |- ( ph -> ( x e. B |-> C ) e. ( G MndHom H ) )   &    |- ( ( ph /\ k e. ( M ... N ) ) -> X e. B )   &    |- ( x = X -> C = D )   &    |- ( x = ( G gsumg ( k e. ( M ... N ) |-> X ) ) -> C = E )   =>    |- ( ph -> ( H gsumg ( k e. ( M ... N ) |-> D ) ) = E )
8-Sep-2025	gsumfzmhm 13413	Apply a monoid homomorphism to a group sum. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by AV, 6-Jun-2019.) (Revised by Jim Kingdon, 8-Sep-2025.)
|- B = ( Base ` G )   &    |- .0. = ( 0g ` G )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> H e. Mnd )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ZZ )   &    |- ( ph -> K e. ( G MndHom H ) )   &    |- ( ph -> F : ( M ... N ) --> B )   =>    |- ( ph -> ( H gsumg ( K o. F ) ) = ( K ` ( G gsumg F ) ) )
6-Sep-2025	gsumfzconst 13411	Sum of a constant series. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by Jim Kingdon, 6-Sep-2025.)
|- B = ( Base ` G )   &    |- .x. = (.g ` G )   =>    |- ( ( G e. Mnd /\ N e. ( ZZ>= ` M ) /\ X e. B ) -> ( G gsumg ( k e. ( M ... N ) |-> X ) ) = ( ( ( N - M ) + 1 ) .x. X ) )
31-Aug-2025	gsumfzmptfidmadd 13409	The sum of two group sums expressed as mappings with finite domain. (Contributed by AV, 23-Jul-2019.) (Revised by Jim Kingdon, 31-Aug-2025.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ZZ )   &    |- ( ( ph /\ x e. ( M ... N ) ) -> C e. B )   &    |- ( ( ph /\ x e. ( M ... N ) ) -> D e. B )   &    |- F = ( x e. ( M ... N ) |-> C )   &    |- H = ( x e. ( M ... N ) |-> D )   =>    |- ( ph -> ( G gsumg ( x e. ( M ... N ) |-> ( C .+ D ) ) ) = ( ( G gsumg F ) .+ ( G gsumg H ) ) )
30-Aug-2025	gsumfzsubmcl 13408	Closure of a group sum in a submonoid. (Contributed by Mario Carneiro, 10-Jan-2015.) (Revised by AV, 3-Jun-2019.) (Revised by Jim Kingdon, 30-Aug-2025.)
|- ( ph -> G e. Mnd )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ZZ )   &    |- ( ph -> S e. (SubMnd ` G ) )   &    |- ( ph -> F : ( M ... N ) --> S )   =>    |- ( ph -> ( G gsumg F ) e. S )
30-Aug-2025	seqm1g 10545	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 ) ) )
29-Aug-2025	seqf1og 10592	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 ) )
25-Aug-2025	irrmulap 9713	The product of an irrational with a nonzero rational is irrational. By irrational we mean apart from any rational number. For a similar theorem with not rational in place of irrational, see irrmul 9712. (Contributed by Jim Kingdon, 25-Aug-2025.)
|- ( ph -> A e. RR )   &    |- ( ph -> A. q e. QQ A # q )   &    |- ( ph -> B e. QQ )   &    |- ( ph -> B =/= 0 )   &    |- ( ph -> Q e. QQ )   =>    |- ( ph -> ( A x. B ) # Q )
19-Aug-2025	seqp1g 10537	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 ) ) ) )
19-Aug-2025	seq1g 10534	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 ) )
18-Aug-2025	iswrdiz 10921	A zero-based sequence is a word. In iswrdinn0 10919 we can specify a length as an nonnegative integer. However, it will occasionally be helpful to allow a negative length, as well as zero, to specify an empty sequence. (Contributed by Jim Kingdon, 18-Aug-2025.)
|- ( ( W : ( 0..^ L ) --> S /\ L e. ZZ ) -> W e. Word S )
16-Aug-2025	gsumfzcl 13071	Closure of a finite group sum. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by AV, 3-Jun-2019.) (Revised by Jim Kingdon, 16-Aug-2025.)
|- B = ( Base ` G )   &    |- .0. = ( 0g ` G )   &    |- ( ph -> G e. Mnd )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ZZ )   &    |- ( ph -> F : ( M ... N ) --> B )   =>    |- ( ph -> ( G gsumg F ) e. B )
16-Aug-2025	iswrdinn0 10919	A zero-based sequence is a word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.) (Revised by Jim Kingdon, 16-Aug-2025.)
|- ( ( W : ( 0..^ L ) --> S /\ L e. NN0 ) -> W e. Word S )
15-Aug-2025	gsumfzz 13067	Value of a group sum over the zero element. (Contributed by Mario Carneiro, 7-Dec-2014.) (Revised by Jim Kingdon, 15-Aug-2025.)
|- .0. = ( 0g ` G )   =>    |- ( ( G e. Mnd /\ M e. ZZ /\ N e. ZZ ) -> ( G gsumg ( k e. ( M ... N ) |-> .0. ) ) = .0. )
14-Aug-2025	gsumfzval 12974	An expression for gsumg when summing over a finite set of sequential integers. (Contributed by Jim Kingdon, 14-Aug-2025.)
|- B = ( Base ` G )   &    |- .0. = ( 0g ` G )   &    |- .+ = ( +g ` G )   &    |- ( ph -> G e. V )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ZZ )   &    |- ( ph -> F : ( M ... N ) --> B )   =>    |- ( ph -> ( G gsumg F ) = if ( N < M , .0. , ( seq M ( .+ , F ) ` N ) ) )
13-Aug-2025	znidom 14145	The ℤ/nℤ structure is an integral domain when n is prime. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by Jim Kingdon, 13-Aug-2025.)
|- Y = (ℤ/nℤ ` N )   =>    |- ( N e. Prime -> Y e. IDomn )
12-Aug-2025	rrgmex 13757	A structure whose set of left-regular elements is inhabited is a set. (Contributed by Jim Kingdon, 12-Aug-2025.)
|- E = (RLReg ` R )   =>    |- ( A e. E -> R e. _V )
10-Aug-2025	gausslemma2dlem1cl 15175	Lemma for gausslemma2dlem1 15177. Closure of the body of the definition of R. (Contributed by Jim Kingdon, 10-Aug-2025.)
|- ( ph -> P e. ( Prime \ { 2 } ) )   &    |- H = ( ( P - 1 ) / 2 )   &    |- R = ( x e. ( 1 ... H ) |-> if ( ( x x. 2 ) < ( P / 2 ) , ( x x. 2 ) , ( P - ( x x. 2 ) ) ) )   &    |- ( ph -> A e. ( 1 ... H ) )   =>    |- ( ph -> if ( ( A x. 2 ) < ( P / 2 ) , ( A x. 2 ) , ( P - ( A x. 2 ) ) ) e. ZZ )
9-Aug-2025	gausslemma2dlem1f1o 15176	Lemma for gausslemma2dlem1 15177. (Contributed by Jim Kingdon, 9-Aug-2025.)
|- ( ph -> P e. ( Prime \ { 2 } ) )   &    |- H = ( ( P - 1 ) / 2 )   &    |- R = ( x e. ( 1 ... H ) |-> if ( ( x x. 2 ) < ( P / 2 ) , ( x x. 2 ) , ( P - ( x x. 2 ) ) ) )   =>    |- ( ph -> R : ( 1 ... H ) -1-1-onto-> ( 1 ... H ) )
7-Aug-2025	qdclt 10315	Rational < is decidable. (Contributed by Jim Kingdon, 7-Aug-2025.)
|- ( ( A e. QQ /\ B e. QQ ) -> DECID A < B )
22-Jul-2025	ivthdich 14807	The intermediate value theorem implies real number dichotomy. Because real number dichotomy (also known as analytic LLPO) is a constructive taboo, this means we will be unable to prove the intermediate value theorem as stated here (although versions with additional conditions, such as ivthinc 14797 for strictly monotonic functions, can be proved). The proof is via a function which we call the hover function and which is also described in Section 5.1 of [Bauer], p. 493. Consider any real number z. We want to show that z <_ 0 \/ 0 <_ z. Because of hovercncf 14800, hovera 14801, and hoverb 14802, we are able to apply the intermediate value theorem to get a value c such that the hover function at c equals z. By axltwlin 8087, c < 1 or 0 < c, and that leads to z <_ 0 by hoverlt1 14803 or 0 <_ z by hovergt0 14804. (Contributed by Jim Kingdon and Mario Carneiro, 22-Jul-2025.)
|- ( A. f ( f e. ( RR -cn-> RR ) -> A. a e. RR A. b e. RR ( ( a < b /\ ( f ` a ) < 0 /\ 0 < ( f ` b ) ) -> E. x e. RR ( a < x /\ x < b /\ ( f ` x ) = 0 ) ) ) -> A. r e. RR A. s e. RR ( r <_ s \/ s <_ r ) )
22-Jul-2025	dich0 14806	Real number dichotomy stated in terms of two real numbers or a real number and zero. (Contributed by Jim Kingdon, 22-Jul-2025.)
|- ( A. z e. RR ( z <_ 0 \/ 0 <_ z ) <-> A. x e. RR A. y e. RR ( x <_ y \/ y <_ x ) )
22-Jul-2025	ivthdichlem 14805	Lemma for ivthdich 14807. The result, with a few notational conveniences. (Contributed by Jim Kingdon, 22-Jul-2025.)
|- F = ( x e. RR |-> sup ( {inf ( { x , 0 } , RR , < ) , ( x - 1 ) } , RR , < ) )   &    |- ( ph -> Z e. RR )   &    |- ( ph -> A. f ( f e. ( RR -cn-> RR ) -> A. a e. RR A. b e. RR ( ( a < b /\ ( f ` a ) < 0 /\ 0 < ( f ` b ) ) -> E. x e. RR ( a < x /\ x < b /\ ( f ` x ) = 0 ) ) ) )   =>    |- ( ph -> ( Z <_ 0 \/ 0 <_ Z ) )
22-Jul-2025	hovergt0 14804	The hover function evaluated at a point greater than zero. (Contributed by Jim Kingdon, 22-Jul-2025.)
|- F = ( x e. RR |-> sup ( {inf ( { x , 0 } , RR , < ) , ( x - 1 ) } , RR , < ) )   =>    |- ( ( C e. RR /\ 0 < C ) -> 0 <_ ( F ` C ) )
22-Jul-2025	hoverlt1 14803	The hover function evaluated at a point less than one. (Contributed by Jim Kingdon, 22-Jul-2025.)
|- F = ( x e. RR |-> sup ( {inf ( { x , 0 } , RR , < ) , ( x - 1 ) } , RR , < ) )   =>    |- ( ( C e. RR /\ C < 1 ) -> ( F ` C ) <_ 0 )
21-Jul-2025	hoverb 14802	A point at which the hover function is greater than a given value. (Contributed by Jim Kingdon, 21-Jul-2025.)
|- F = ( x e. RR |-> sup ( {inf ( { x , 0 } , RR , < ) , ( x - 1 ) } , RR , < ) )   =>    |- ( Z e. RR -> Z < ( F ` ( Z + 2 ) ) )
21-Jul-2025	hovera 14801	A point at which the hover function is less than a given value. (Contributed by Jim Kingdon, 21-Jul-2025.)
|- F = ( x e. RR |-> sup ( {inf ( { x , 0 } , RR , < ) , ( x - 1 ) } , RR , < ) )   =>    |- ( Z e. RR -> ( F ` ( Z - 1 ) ) < Z )
21-Jul-2025	rexeqtrrdv 2701	Substitution of equal classes into a restricted existential quantifier. (Contributed by Matthew House, 21-Jul-2025.)
|- ( ph -> E. x e. A ps )   &    |- ( ph -> B = A )   =>    |- ( ph -> E. x e. B ps )
21-Jul-2025	raleqtrrdv 2700	Substitution of equal classes into a restricted universal quantifier. (Contributed by Matthew House, 21-Jul-2025.)
|- ( ph -> A. x e. A ps )   &    |- ( ph -> B = A )   =>    |- ( ph -> A. x e. B ps )
21-Jul-2025	rexeqtrdv 2699	Substitution of equal classes into a restricted existential quantifier. (Contributed by Matthew House, 21-Jul-2025.)
|- ( ph -> E. x e. A ps )   &    |- ( ph -> A = B )   =>    |- ( ph -> E. x e. B ps )
21-Jul-2025	raleqtrdv 2698	Substitution of equal classes into a restricted universal quantifier. (Contributed by Matthew House, 21-Jul-2025.)
|- ( ph -> A. x e. A ps )   &    |- ( ph -> A = B )   =>    |- ( ph -> A. x e. B ps )
20-Jul-2025	hovercncf 14800	The hover function is continuous. By hover function, we mean a a function which starts out as a line of slope one, is constant at zero from zero to one, and then resumes as a slope of one. (Contributed by Jim Kingdon, 20-Jul-2025.)
|- F = ( x e. RR |-> sup ( {inf ( { x , 0 } , RR , < ) , ( x - 1 ) } , RR , < ) )   =>    |- F e. ( RR -cn-> RR )
19-Jul-2025	mincncf 14770	The minimum of two continuous real functions is continuous. (Contributed by Jim Kingdon, 19-Jul-2025.)
|- ( ph -> ( x e. X |-> A ) e. ( X -cn-> RR ) )   &    |- ( ph -> ( x e. X |-> B ) e. ( X -cn-> RR ) )   =>    |- ( ph -> ( x e. X |-> inf ( { A , B } , RR , < ) ) e. ( X -cn-> RR ) )
18-Jul-2025	maxcncf 14769	The maximum of two continuous real functions is continuous. (Contributed by Jim Kingdon, 18-Jul-2025.)
|- ( ph -> ( x e. X |-> A ) e. ( X -cn-> RR ) )   &    |- ( ph -> ( x e. X |-> B ) e. ( X -cn-> RR ) )   =>    |- ( ph -> ( x e. X |-> sup ( { A , B } , RR , < ) ) e. ( X -cn-> RR ) )
14-Jul-2025	xnn0nnen 10508	The set of extended nonnegative integers is equinumerous to the set of natural numbers. (Contributed by Jim Kingdon, 14-Jul-2025.)
|- NN0* ~~ NN
12-Jul-2025	nninfninc 7182	All values beyond a zero in an ℕ∞ sequence are zero. This is another way of stating that elements of ℕ∞ are nonincreasing. (Contributed by Jim Kingdon, 12-Jul-2025.)
|- ( ph -> A e. ℕ∞ )   &    |- ( ph -> X e. om )   &    |- ( ph -> Y e. om )   &    |- ( ph -> X C_ Y )   &    |- ( ph -> ( A ` X ) = (/) )   =>    |- ( ph -> ( A ` Y ) = (/) )
10-Jul-2025	nninfctlemfo 12177	Lemma for nninfct 12178. (Contributed by Jim Kingdon, 10-Jul-2025.)
|- 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 } ) >. } )   =>    |- ( om e. Omni -> I :NN0* -onto->ℕ∞ )
8-Jul-2025	nnnninfen 15511	Equinumerosity of the natural numbers and ℕ∞ is equivalent to the Limited Principle of Omniscience (LPO). Remark in Section 1.1 of [Pradic2025], p. 2. (Contributed by Jim Kingdon, 8-Jul-2025.)
|- ( om ~~ ℕ∞ <-> om e. Omni )
8-Jul-2025	nninfct 12178	The limited principle of omniscience (LPO) implies that ℕ∞ is countable. (Contributed by Jim Kingdon, 8-Jul-2025.)
|- ( om e. Omni -> E. f f : om -onto-> (ℕ∞ ⊔ 1o ) )
8-Jul-2025	nninfinf 10514	ℕ∞ is infinte. (Contributed by Jim Kingdon, 8-Jul-2025.)
|- om ~<_ ℕ∞
7-Jul-2025	ivthreinc 14799	Restating the intermediate value theorem. Given a hypothesis stating the intermediate value theorem (in a strong form which is not provable given our axioms alone), provide a conclusion similar to the theorem as stated in the Metamath Proof Explorer (which is also similar to how we state the theorem for a strictly monotonic function at ivthinc 14797). Being able to have a hypothesis stating the intermediate value theorem will be helpful when it comes time to show that it implies a constructive taboo. This version of the theorem requires that the function F is continuous on the entire real line, not just ( A [,] B ) which may be an unnecessary condition but which is sufficient for the way we want to use it. (Contributed by Jim Kingdon, 7-Jul-2025.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> U e. RR )   &    |- ( ph -> A < B )   &    |- ( ph -> F e. ( RR -cn-> RR ) )   &    |- ( ph -> ( ( F ` A ) < U /\ U < ( F ` B ) ) )   &    |- ( ph -> A. f ( f e. ( RR -cn-> RR ) -> A. a e. RR A. b e. RR ( ( a < b /\ ( f ` a ) < 0 /\ 0 < ( f ` b ) ) -> E. x e. RR ( a < x /\ x < b /\ ( f ` x ) = 0 ) ) ) )   =>    |- ( ph -> E. c e. ( A (,) B ) ( F ` c ) = U )
28-Jun-2025	fngsum 12971	Iterated sum has a universal domain. (Contributed by Jim Kingdon, 28-Jun-2025.)
|- gsumg Fn ( _V X. _V )
28-Jun-2025	iotaexel 5878	Set existence of an iota expression in which all values are contained within a set. (Contributed by Jim Kingdon, 28-Jun-2025.)
|- ( ( A e. V /\ A. x ( ph -> x e. A ) ) -> ( iota x ph ) e. _V )
27-Jun-2025	df-igsum 12870	Define a finite group sum (also called "iterated sum") of a structure. Given G gsumg F where F : A --> ( Base ` G ), the set of indices is A and the values are given by F at each index. A group sum over a multiplicative group may be viewed as a product. The definition is meaningful in different contexts, depending on the size of the index set A and each demanding different properties of G. 1. If A = (/) and G has an identity element, then the sum equals this identity. 2. If A = ( M ... N ) and G is any magma, then the sum is the sum of the elements, evaluated left-to-right, i.e., ( ( F ` 1 ) + ( F ` 2 ) ) + ( F ` 3 ), etc. 3. This definition does not handle other cases. (Contributed by FL, 5-Sep-2010.) (Revised by Mario Carneiro, 7-Dec-2014.) (Revised by Jim Kingdon, 27-Jun-2025.)
|- gsumg = ( w e. _V , f e. _V |-> ( iota x ( ( dom f = (/) /\ x = ( 0g ` w ) ) \/ E. m E. n e. ( ZZ>= ` m ) ( dom f = ( m ... n ) /\ x = ( seq m ( ( +g ` w ) , f ) ` n ) ) ) ) )
20-Jun-2025	opprnzrbg 13681	The opposite of a nonzero ring is nonzero, bidirectional form of opprnzr 13682. (Contributed by SN, 20-Jun-2025.)
|- O = (oppr ` R )   =>    |- ( R e. V -> ( R e. NzRing <-> O e. NzRing ) )
16-Jun-2025	fnpsr 14153	The multivariate power series constructor has a universal domain. (Contributed by Jim Kingdon, 16-Jun-2025.)
|- mPwSer Fn ( _V X. _V )
14-Jun-2025	basm 12679	A structure whose base is inhabited is inhabited. (Contributed by Jim Kingdon, 14-Jun-2025.)
|- B = ( Base ` G )   =>    |- ( A e. B -> E. j j e. G )
14-Jun-2025	elfvm 5587	If a function value has a member, the function is inhabited. (Contributed by Jim Kingdon, 14-Jun-2025.)
|- ( A e. ( F ` B ) -> E. j j e. F )
6-Jun-2025	pcxqcl 12450	The general prime count function is an integer or infinite. (Contributed by Jim Kingdon, 6-Jun-2025.)
|- ( ( P e. Prime /\ N e. QQ ) -> ( ( P pCnt N ) e. ZZ \/ ( P pCnt N ) = +oo ) )
5-Jun-2025	xqltnle 10336	"Less than" expressed in terms of "less than or equal to", for extended numbers which are rational or +oo. We have not yet had enough usage of such numbers to warrant fully developing the concept, as in NN0* or RR*, so for now we just have a handful of theorems for what we need. (Contributed by Jim Kingdon, 5-Jun-2025.)
|- ( ( ( A e. QQ \/ A = +oo ) /\ ( B e. QQ \/ B = +oo ) ) -> ( A < B <-> -. B <_ A ) )
30-May-2025	4sqexercise2 12537	Exercise which may help in understanding the proof of 4sqlemsdc 12538. (Contributed by Jim Kingdon, 30-May-2025.)
|- S = { n | E. x e. ZZ E. y e. ZZ n = ( ( x ^ 2 ) + ( y ^ 2 ) ) }   =>    |- ( A e. NN0 -> DECID A e. S )
27-May-2025	iotaexab 5233	Existence of the iota class when all the possible values are contained in a set. (Contributed by Jim Kingdon, 27-May-2025.)
|- ( { x | ph } e. V -> ( iota x ph ) e. _V )
25-May-2025	4sqlemsdc 12538	Lemma for 4sq 12548. The property of being the sum of four squares is decidable. The proof involves showing that (for a particular A) there are only a finite number of possible ways that it could be the sum of four squares, so checking each of those possibilities in turn decides whether the number is the sum of four squares. If this proof is hard to follow, especially because of its length, the simplified versions at 4sqexercise1 12536 and 4sqexercise2 12537 may help clarify, as they are using very much the same techniques on simplified versions of this lemma. (Contributed by Jim Kingdon, 25-May-2025.)
|- S = { n | E. x e. ZZ E. y e. ZZ E. z e. ZZ E. w e. ZZ n = ( ( ( x ^ 2 ) + ( y ^ 2 ) ) + ( ( z ^ 2 ) + ( w ^ 2 ) ) ) }   =>    |- ( A e. NN0 -> DECID A e. S )
25-May-2025	4sqexercise1 12536	Exercise which may help in understanding the proof of 4sqlemsdc 12538. (Contributed by Jim Kingdon, 25-May-2025.)
|- S = { n | E. x e. ZZ n = ( x ^ 2 ) }   =>    |- ( A e. NN0 -> DECID A e. S )
24-May-2025	4sqleminfi 12535	Lemma for 4sq 12548. A i^i ran F is finite. (Contributed by Jim Kingdon, 24-May-2025.)
|- ( ph -> N e. NN )   &    |- ( ph -> P e. NN )   &    |- A = { u | E. m e. ( 0 ... N ) u = ( ( m ^ 2 ) mod P ) }   &    |- F = ( v e. A |-> ( ( P - 1 ) - v ) )   =>    |- ( ph -> ( A i^i ran F ) e. Fin )
24-May-2025	4sqlemffi 12534	Lemma for 4sq 12548. ran F is finite. (Contributed by Jim Kingdon, 24-May-2025.)
|- ( ph -> N e. NN )   &    |- ( ph -> P e. NN )   &    |- A = { u | E. m e. ( 0 ... N ) u = ( ( m ^ 2 ) mod P ) }   &    |- F = ( v e. A |-> ( ( P - 1 ) - v ) )   =>    |- ( ph -> ran F e. Fin )
24-May-2025	4sqlemafi 12533	Lemma for 4sq 12548. A is finite. (Contributed by Jim Kingdon, 24-May-2025.)
|- ( ph -> N e. NN )   &    |- ( ph -> P e. NN )   &    |- A = { u | E. m e. ( 0 ... N ) u = ( ( m ^ 2 ) mod P ) }   =>    |- ( ph -> A e. Fin )
24-May-2025	infidc 6993	The intersection of two sets is finite if one of them is and the other is decidable. (Contributed by Jim Kingdon, 24-May-2025.)
|- ( ( A e. Fin /\ A. x e. A DECID x e. B ) -> ( A i^i B ) e. Fin )
19-May-2025	zrhex 14109	Set existence for ZRHom. (Contributed by Jim Kingdon, 19-May-2025.)
|- L = ( ZRHom ` R )   =>    |- ( R e. V -> L e. _V )
16-May-2025	rhmex 13653	Set existence for ring homomorphism. (Contributed by Jim Kingdon, 16-May-2025.)
|- ( ( R e. V /\ S e. W ) -> ( R RingHom S ) e. _V )
15-May-2025	ghmex 13325	The set of group homomorphisms exists. (Contributed by Jim Kingdon, 15-May-2025.)
|- ( ( S e. Grp /\ T e. Grp ) -> ( S GrpHom T ) e. _V )
15-May-2025	mhmex 13034	The set of monoid homomorphisms exists. (Contributed by Jim Kingdon, 15-May-2025.)
|- ( ( S e. Mnd /\ T e. Mnd ) -> ( S MndHom T ) e. _V )
14-May-2025	idomcringd 13774	An integral domain is a commutative ring with unity. (Contributed by Thierry Arnoux, 4-May-2025.) (Proof shortened by SN, 14-May-2025.)
|- ( ph -> R e. IDomn )   =>    |- ( ph -> R e. CRing )
6-May-2025	rrgnz 13764	In a nonzero ring, the zero is a left zero divisor (that is, not a left-regular element). (Contributed by Thierry Arnoux, 6-May-2025.)
|- E = (RLReg ` R )   &    |- .0. = ( 0g ` R )   =>    |- ( R e. NzRing -> -. .0. e. E )
5-May-2025	rngressid 13450	A non-unital ring restricted to its base set is a non-unital ring. It will usually be the original non-unital ring exactly, of course, but to show that needs additional conditions such as those in strressid 12689. (Contributed by Jim Kingdon, 5-May-2025.)
|- B = ( Base ` G )   =>    |- ( G e. Rng -> ( G ↾s B ) e. Rng )
5-May-2025	ablressid 13405	A commutative group restricted to its base set is a commutative group. It will usually be the original group exactly, of course, but to show that needs additional conditions such as those in strressid 12689. (Contributed by Jim Kingdon, 5-May-2025.)
|- B = ( Base ` G )   =>    |- ( G e. Abel -> ( G ↾s B ) e. Abel )
29-Apr-2025	rlmscabas 13956	Scalars in the ring module have the same base set. (Contributed by Jim Kingdon, 29-Apr-2025.)
|- ( R e. X -> ( Base ` R ) = ( Base ` (Scalar ` (ringLMod ` R ) ) ) )
29-Apr-2025	ressbasid 12688	The trivial structure restriction leaves the base set unchanged. (Contributed by Jim Kingdon, 29-Apr-2025.)
|- B = ( Base ` W )   =>    |- ( W e. V -> ( Base ` ( W ↾s B ) ) = B )
28-Apr-2025	lssmex 13851	If a linear subspace is inhabited, the class it is built from is a set. (Contributed by Jim Kingdon, 28-Apr-2025.)
|- S = ( LSubSp ` W )   =>    |- ( U e. S -> W e. _V )
27-Apr-2025	lidlex 13969	Existence of the set of left ideals. (Contributed by Jim Kingdon, 27-Apr-2025.)
|- ( W e. V -> (LIdeal ` W ) e. _V )
27-Apr-2025	lssex 13850	Existence of a linear subspace. (Contributed by Jim Kingdon, 27-Apr-2025.)
|- ( W e. V -> ( LSubSp ` W ) e. _V )
25-Apr-2025	rspex 13970	Existence of the ring span. (Contributed by Jim Kingdon, 25-Apr-2025.)
|- ( W e. V -> (RSpan ` W ) e. _V )
25-Apr-2025	lspex 13891	Existence of the span of a set of vectors. (Contributed by Jim Kingdon, 25-Apr-2025.)
|- ( W e. X -> ( LSpan ` W ) e. _V )
25-Apr-2025	eqgex 13291	The left coset equivalence relation exists. (Contributed by Jim Kingdon, 25-Apr-2025.)
|- ( ( G e. V /\ S e. W ) -> ( G ~QG S ) e. _V )
25-Apr-2025	qusex 12908	Existence of a quotient structure. (Contributed by Jim Kingdon, 25-Apr-2025.)
|- ( ( R e. V /\ .~ e. W ) -> ( R /.s .~ ) e. _V )
23-Apr-2025	1dom1el 15483	If a set is dominated by one, then any two of its elements are equal. (Contributed by Jim Kingdon, 23-Apr-2025.)
|- ( ( A ~<_ 1o /\ B e. A /\ C e. A ) -> B = C )
22-Apr-2025	mulgex 13193	Existence of the group multiple operation. (Contributed by Jim Kingdon, 22-Apr-2025.)
|- ( G e. V -> (.g ` G ) e. _V )
20-Apr-2025	elovmpod 6116	Utility lemma for two-parameter classes. (Contributed by Stefan O'Rear, 21-Jan-2015.) Variant of elovmpo 6117 in deduction form. (Revised by AV, 20-Apr-2025.)
|- O = ( a e. A , b e. B |-> C )   &    |- ( ph -> X e. A )   &    |- ( ph -> Y e. B )   &    |- ( ph -> D e. V )   &    |- ( ( a = X /\ b = Y ) -> C = D )   =>    |- ( ph -> ( E e. ( X O Y ) <-> E e. D ) )
18-Apr-2025	df2idl2 14005	Alternate (the usual textbook) definition of a two-sided ideal of a ring to be a subgroup of the additive group of the ring which is closed under left- and right-multiplication by elements of the full ring. (Contributed by AV, 13-Feb-2025.) (Proof shortened by AV, 18-Apr-2025.)
|- U = (2Ideal ` R )   &    |- B = ( Base ` R )   &    |- .x. = ( .r ` R )   =>    |- ( R e. Ring -> ( I e. U <-> ( I e. (SubGrp ` R ) /\ A. x e. B A. y e. I ( ( x .x. y ) e. I /\ ( y .x. x ) e. I ) ) ) )
18-Apr-2025	2idlmex 13997	Existence of the set a two-sided ideal is built from (when the ideal is inhabited). (Contributed by Jim Kingdon, 18-Apr-2025.)
|- T = (2Ideal ` W )   =>    |- ( U e. T -> W e. _V )
18-Apr-2025	dflidl2 13984	Alternate (the usual textbook) definition of a (left) ideal of a ring to be a subgroup of the additive group of the ring which is closed under left-multiplication by elements of the full ring. (Contributed by AV, 13-Feb-2025.) (Proof shortened by AV, 18-Apr-2025.)
|- U = (LIdeal ` R )   &    |- B = ( Base ` R )   &    |- .x. = ( .r ` R )   =>    |- ( R e. Ring -> ( I e. U <-> ( I e. (SubGrp ` R ) /\ A. x e. B A. y e. I ( x .x. y ) e. I ) ) )
18-Apr-2025	lidlmex 13971	Existence of the set a left ideal is built from (when the ideal is inhabited). (Contributed by Jim Kingdon, 18-Apr-2025.)
|- I = (LIdeal ` W )   =>    |- ( U e. I -> W e. _V )
18-Apr-2025	lsslsp 13925	Spans in submodules correspond to spans in the containing module. (Contributed by Stefan O'Rear, 12-Dec-2014.) Terms in the equation were swapped as proposed by NM on 15-Mar-2015. (Revised by AV, 18-Apr-2025.)
|- X = ( W ↾s U )   &    |- M = ( LSpan ` W )   &    |- N = ( LSpan ` X )   &    |- L = ( LSubSp ` W )   =>    |- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> ( N ` G ) = ( M ` G ) )
16-Apr-2025	sraex 13942	Existence of a subring algebra. (Contributed by Jim Kingdon, 16-Apr-2025.)
|- ( ph -> A = ( (subringAlg ` W ) ` S ) )   &    |- ( ph -> S C_ ( Base ` W ) )   &    |- ( ph -> W e. X )   =>    |- ( ph -> A e. _V )
14-Apr-2025	grpmgmd 13098	A group is a magma, deduction form. (Contributed by SN, 14-Apr-2025.)
|- ( ph -> G e. Grp )   =>    |- ( ph -> G e. Mgm )
12-Apr-2025	psraddcl 14164	Closure of the power series addition operation. (Contributed by Mario Carneiro, 28-Dec-2014.) Generalize to magmas. (Revised by SN, 12-Apr-2025.)
|- S = ( I mPwSer R )   &    |- B = ( Base ` S )   &    |- .+ = ( +g ` S )   &    |- ( ph -> R e. Mgm )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   =>    |- ( ph -> ( X .+ Y ) e. B )
10-Apr-2025	cndcap 15549	Real number trichotomy is equivalent to decidability of complex number apartness. (Contributed by Jim Kingdon, 10-Apr-2025.)
|- ( A. x e. RR A. y e. RR ( x < y \/ x = y \/ y < x ) <-> A. z e. CC A. w e. CC DECID z # w )
4-Apr-2025	ghmf1 13343	Two ways of saying a group homomorphism is 1-1 into its codomain. (Contributed by Paul Chapman, 3-Mar-2008.) (Revised by Mario Carneiro, 13-Jan-2015.) (Proof shortened by AV, 4-Apr-2025.)
|- A = ( Base ` R )   &    |- B = ( Base ` S )   &    |- N = ( 0g ` R )   &    |- .0. = ( 0g ` S )   =>    |- ( F e. ( R GrpHom S ) -> ( F : A -1-1-> B <-> A. x e. A ( ( F ` x ) = .0. -> x = N ) ) )
3-Apr-2025	quscrng 14029	The quotient of a commutative ring by an ideal is a commutative ring. (Contributed by Mario Carneiro, 15-Jun-2015.) (Proof shortened by AV, 3-Apr-2025.)
|- U = ( R /.s ( R ~QG S ) )   &    |- I = (LIdeal ` R )   =>    |- ( ( R e. CRing /\ S e. I ) -> U e. CRing )
31-Mar-2025	mpocnfldmul 14055	The multiplication operation of the field of complex numbers. Version of cnfldmul 14054 using maps-to notation. (Contributed by GG, 31-Mar-2025.)
|- ( x e. CC , y e. CC |-> ( x x. y ) ) = ( .r ` ℂfld )
31-Mar-2025	mpocnfldadd 14053	The addition operation of the field of complex numbers. Version of cnfldadd 14052 using maps-to notation. (Contributed by GG, 31-Mar-2025.)
|- ( x e. CC , y e. CC |-> ( x + y ) ) = ( +g ` ℂfld )
31-Mar-2025	2idlcpbl 14020	The coset equivalence relation for a two-sided ideal is compatible with ring multiplication. (Contributed by Mario Carneiro, 14-Jun-2015.) (Proof shortened by AV, 31-Mar-2025.)
|- X = ( Base ` R )   &    |- E = ( R ~QG S )   &    |- I = (2Ideal ` R )   &    |- .x. = ( .r ` R )   =>    |- ( ( R e. Ring /\ S e. I ) -> ( ( A E C /\ B E D ) -> ( A .x. B ) E ( C .x. D ) ) )
22-Mar-2025	idomringd 13775	An integral domain is a ring. (Contributed by Thierry Arnoux, 22-Mar-2025.)
|- ( ph -> R e. IDomn )   =>    |- ( ph -> R e. Ring )
22-Mar-2025	idomdomd 13773	An integral domain is a domain. (Contributed by Thierry Arnoux, 22-Mar-2025.)
|- ( ph -> R e. IDomn )   =>    |- ( ph -> R e. Domn )