P. 233 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 23201-23300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	minveclem4a 23201*	Lemma for minvec 23207. F converges to a point P in Y. (Contributed by Mario Carneiro, 7-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.)
|- X = ( Base ` U )   &    |- .- = ( -g ` U )   &    |- N = ( norm ` U )   &    |- ( ph -> U e. CPreHil )   &    |- ( ph -> Y e. ( LSubSp ` U ) )   &    |- ( ph -> ( U ↾s Y ) e. CMetSp )   &    |- ( ph -> A e. X )   &    |- J = ( TopOpen ` U )   &    |- R = ran ( y e. Y |-> ( N ` ( A .- y ) ) )   &    |- S = inf ( R , RR , < )   &    |- D = ( ( dist ` U ) |` ( X X. X ) )   &    |- F = ran ( r e. RR+ |-> { y e. Y | ( ( A D y ) ^ 2 ) <_ ( ( S ^ 2 ) + r ) } )   &    |- P = U. ( J fLim ( X filGen F ) )   =>    |- ( ph -> P e. ( ( J fLim ( X filGen F ) ) i^i Y ) )
Theorem	minveclem4b 23202*	Lemma for minvec 23207. The convergent point of the Cauchy sequence F is a member of the base space. (Contributed by Mario Carneiro, 16-Jun-2014.) (Revised by Mario Carneiro, 15-Oct-2015.)
|- X = ( Base ` U )   &    |- .- = ( -g ` U )   &    |- N = ( norm ` U )   &    |- ( ph -> U e. CPreHil )   &    |- ( ph -> Y e. ( LSubSp ` U ) )   &    |- ( ph -> ( U ↾s Y ) e. CMetSp )   &    |- ( ph -> A e. X )   &    |- J = ( TopOpen ` U )   &    |- R = ran ( y e. Y |-> ( N ` ( A .- y ) ) )   &    |- S = inf ( R , RR , < )   &    |- D = ( ( dist ` U ) |` ( X X. X ) )   &    |- F = ran ( r e. RR+ |-> { y e. Y | ( ( A D y ) ^ 2 ) <_ ( ( S ^ 2 ) + r ) } )   &    |- P = U. ( J fLim ( X filGen F ) )   =>    |- ( ph -> P e. X )
Theorem	minveclem4 23203*	Lemma for minvec 23207. The convergent point of the Cauchy sequence F attains the minimum distance, and so is closer to A than any other point in Y. (Contributed by Mario Carneiro, 7-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.) (Revised by AV, 3-Oct-2020.)
|- X = ( Base ` U )   &    |- .- = ( -g ` U )   &    |- N = ( norm ` U )   &    |- ( ph -> U e. CPreHil )   &    |- ( ph -> Y e. ( LSubSp ` U ) )   &    |- ( ph -> ( U ↾s Y ) e. CMetSp )   &    |- ( ph -> A e. X )   &    |- J = ( TopOpen ` U )   &    |- R = ran ( y e. Y |-> ( N ` ( A .- y ) ) )   &    |- S = inf ( R , RR , < )   &    |- D = ( ( dist ` U ) |` ( X X. X ) )   &    |- F = ran ( r e. RR+ |-> { y e. Y | ( ( A D y ) ^ 2 ) <_ ( ( S ^ 2 ) + r ) } )   &    |- P = U. ( J fLim ( X filGen F ) )   &    |- T = ( ( ( ( ( A D P ) + S ) / 2 ) ^ 2 ) - ( S ^ 2 ) )   =>    |- ( ph -> E. x e. Y A. y e. Y ( N ` ( A .- x ) ) <_ ( N ` ( A .- y ) ) )
Theorem	minveclem5 23204*	Lemma for minvec 23207. Discharge the assumptions in minveclem4 23203. (Contributed by Mario Carneiro, 9-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.)
|- X = ( Base ` U )   &    |- .- = ( -g ` U )   &    |- N = ( norm ` U )   &    |- ( ph -> U e. CPreHil )   &    |- ( ph -> Y e. ( LSubSp ` U ) )   &    |- ( ph -> ( U ↾s Y ) e. CMetSp )   &    |- ( ph -> A e. X )   &    |- J = ( TopOpen ` U )   &    |- R = ran ( y e. Y |-> ( N ` ( A .- y ) ) )   &    |- S = inf ( R , RR , < )   &    |- D = ( ( dist ` U ) |` ( X X. X ) )   =>    |- ( ph -> E. x e. Y A. y e. Y ( N ` ( A .- x ) ) <_ ( N ` ( A .- y ) ) )
Theorem	minveclem6 23205*	Lemma for minvec 23207. Any minimal point is less than S away from A. (Contributed by Mario Carneiro, 9-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.) (Revised by AV, 3-Oct-2020.)
|- X = ( Base ` U )   &    |- .- = ( -g ` U )   &    |- N = ( norm ` U )   &    |- ( ph -> U e. CPreHil )   &    |- ( ph -> Y e. ( LSubSp ` U ) )   &    |- ( ph -> ( U ↾s Y ) e. CMetSp )   &    |- ( ph -> A e. X )   &    |- J = ( TopOpen ` U )   &    |- R = ran ( y e. Y |-> ( N ` ( A .- y ) ) )   &    |- S = inf ( R , RR , < )   &    |- D = ( ( dist ` U ) |` ( X X. X ) )   =>    |- ( ( ph /\ x e. Y ) -> ( ( ( A D x ) ^ 2 ) <_ ( ( S ^ 2 ) + 0 ) <-> A. y e. Y ( N ` ( A .- x ) ) <_ ( N ` ( A .- y ) ) ) )
Theorem	minveclem7 23206*	Lemma for minvec 23207. Since any two minimal points are distance zero away from each other, the minimal point is unique. (Contributed by Mario Carneiro, 9-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.)
|- X = ( Base ` U )   &    |- .- = ( -g ` U )   &    |- N = ( norm ` U )   &    |- ( ph -> U e. CPreHil )   &    |- ( ph -> Y e. ( LSubSp ` U ) )   &    |- ( ph -> ( U ↾s Y ) e. CMetSp )   &    |- ( ph -> A e. X )   &    |- J = ( TopOpen ` U )   &    |- R = ran ( y e. Y |-> ( N ` ( A .- y ) ) )   &    |- S = inf ( R , RR , < )   &    |- D = ( ( dist ` U ) |` ( X X. X ) )   =>    |- ( ph -> E! x e. Y A. y e. Y ( N ` ( A .- x ) ) <_ ( N ` ( A .- y ) ) )
Theorem	minvec 23207*	Minimizing vector theorem, or the Hilbert projection theorem. There is exactly one vector in a complete subspace W that minimizes the distance to an arbitrary vector A in a parent inner product space. Theorem 3.3-1 of [Kreyszig] p. 144, specialized to subspaces instead of convex subsets. (Contributed by NM, 11-Apr-2008.) (Proof shortened by Mario Carneiro, 9-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.) (Proof shortened by AV, 3-Oct-2020.)
|- X = ( Base ` U )   &    |- .- = ( -g ` U )   &    |- N = ( norm ` U )   &    |- ( ph -> U e. CPreHil )   &    |- ( ph -> Y e. ( LSubSp ` U ) )   &    |- ( ph -> ( U ↾s Y ) e. CMetSp )   &    |- ( ph -> A e. X )   =>    |- ( ph -> E! x e. Y A. y e. Y ( N ` ( A .- x ) ) <_ ( N ` ( A .- y ) ) )
12.5.10  Projection Theorem
Theorem	pjthlem1 23208*	Lemma for pjth 23210. (Contributed by NM, 10-Oct-1999.) (Revised by Mario Carneiro, 17-Oct-2015.)
|- V = ( Base ` W )   &    |- N = ( norm ` W )   &    |- .+ = ( +g ` W )   &    |- .- = ( -g ` W )   &    |- ., = ( .i ` W )   &    |- L = ( LSubSp ` W )   &    |- ( ph -> W e. CHil )   &    |- ( ph -> U e. L )   &    |- ( ph -> A e. V )   &    |- ( ph -> B e. U )   &    |- ( ph -> A. x e. U ( N ` A ) <_ ( N ` ( A .- x ) ) )   &    |- T = ( ( A ., B ) / ( ( B ., B ) + 1 ) )   =>    |- ( ph -> ( A ., B ) = 0 )
Theorem	pjthlem2 23209	Lemma for pjth 23210. (Contributed by NM, 10-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.)
|- V = ( Base ` W )   &    |- N = ( norm ` W )   &    |- .+ = ( +g ` W )   &    |- .- = ( -g ` W )   &    |- ., = ( .i ` W )   &    |- L = ( LSubSp ` W )   &    |- ( ph -> W e. CHil )   &    |- ( ph -> U e. L )   &    |- ( ph -> A e. V )   &    |- J = ( TopOpen ` W )   &    |- .(+) = ( LSSum ` W )   &    |- O = ( ocv ` W )   &    |- ( ph -> U e. ( Clsd ` J ) )   =>    |- ( ph -> A e. ( U .(+) ( O ` U ) ) )
Theorem	pjth 23210	Projection Theorem: Any Hilbert space vector A can be decomposed uniquely into a member x of a closed subspace H and a member y of the complement of the subspace. Theorem 3.7(i) of [Beran] p. 102 (existence part). (Contributed by NM, 23-Oct-1999.) (Revised by Mario Carneiro, 14-May-2014.)
|- V = ( Base ` W )   &    |- .(+) = ( LSSum ` W )   &    |- O = ( ocv ` W )   &    |- J = ( TopOpen ` W )   &    |- L = ( LSubSp ` W )   =>    |- ( ( W e. CHil /\ U e. L /\ U e. ( Clsd ` J ) ) -> ( U .(+) ( O ` U ) ) = V )
Theorem	pjth2 23211	Projection Theorem with abbreviations: A topologically closed subspace is a projection subspace. (Contributed by Mario Carneiro, 17-Oct-2015.)
|- J = ( TopOpen ` W )   &    |- L = ( LSubSp ` W )   &    |- K = ( proj ` W )   =>    |- ( ( W e. CHil /\ U e. L /\ U e. ( Clsd ` J ) ) -> U e. dom K )
Theorem	cldcss 23212	Corollary of the Projection Theorem: A topologically closed subspace is algebraically closed in Hilbert space. (Contributed by Mario Carneiro, 17-Oct-2015.)
|- V = ( Base ` W )   &    |- J = ( TopOpen ` W )   &    |- L = ( LSubSp ` W )   &    |- C = ( CSubSp ` W )   =>    |- ( W e. CHil -> ( U e. C <-> ( U e. L /\ U e. ( Clsd ` J ) ) ) )
Theorem	cldcss2 23213	Corollary of the Projection Theorem: A topologically closed subspace is algebraically closed in Hilbert space. (Contributed by Mario Carneiro, 17-Oct-2015.)
|- V = ( Base ` W )   &    |- J = ( TopOpen ` W )   &    |- L = ( LSubSp ` W )   &    |- C = ( CSubSp ` W )   =>    |- ( W e. CHil -> C = ( L i^i ( Clsd ` J ) ) )
Theorem	hlhil 23214	Corollary of the Projection Theorem: A subcomplex Hilbert space is a Hilbert space (in the algebraic sense, meaning that all algebraically closed subspaces have a projection decomposition). (Contributed by Mario Carneiro, 17-Oct-2015.)
|- ( W e. CHil -> W e. Hil )
PART 13  BASIC REAL AND COMPLEX ANALYSIS
13.1  Continuity
Theorem	mulcncf 23215*	The multiplication of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
|- ( ph -> ( x e. X |-> A ) e. ( X -cn-> CC ) )   &    |- ( ph -> ( x e. X |-> B ) e. ( X -cn-> CC ) )   =>    |- ( ph -> ( x e. X |-> ( A x. B ) ) e. ( X -cn-> CC ) )
Theorem	divcncf 23216*	The quotient of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ph -> ( x e. X |-> A ) e. ( X -cn-> CC ) )   &    |- ( ph -> ( x e. X |-> B ) e. ( X -cn-> ( CC \ { 0 } ) ) )   =>    |- ( ph -> ( x e. X |-> ( A / B ) ) e. ( X -cn-> CC ) )
13.1.1  Intermediate value theorem
Theorem	pmltpclem1 23217*	Lemma for pmltpc 23219. (Contributed by Mario Carneiro, 1-Jul-2014.)
|- ( ph -> A e. S )   &    |- ( ph -> B e. S )   &    |- ( ph -> C e. S )   &    |- ( ph -> A < B )   &    |- ( ph -> B < C )   &    |- ( ph -> ( ( ( F ` A ) < ( F ` B ) /\ ( F ` C ) < ( F ` B ) ) \/ ( ( F ` B ) < ( F ` A ) /\ ( F ` B ) < ( F ` C ) ) ) )   =>    |- ( ph -> E. a e. S E. b e. S E. c e. S ( a < b /\ b < c /\ ( ( ( F ` a ) < ( F ` b ) /\ ( F ` c ) < ( F ` b ) ) \/ ( ( F ` b ) < ( F ` a ) /\ ( F ` b ) < ( F ` c ) ) ) ) )
Theorem	pmltpclem2 23218*	Lemma for pmltpc 23219. (Contributed by Mario Carneiro, 1-Jul-2014.)
|- ( ph -> F e. ( RR ^pm RR ) )   &    |- ( ph -> A C_ dom F )   &    |- ( ph -> U e. A )   &    |- ( ph -> V e. A )   &    |- ( ph -> W e. A )   &    |- ( ph -> X e. A )   &    |- ( ph -> U <_ V )   &    |- ( ph -> W <_ X )   &    |- ( ph -> -. ( F ` U ) <_ ( F ` V ) )   &    |- ( ph -> -. ( F ` X ) <_ ( F ` W ) )   =>    |- ( ph -> E. a e. A E. b e. A E. c e. A ( a < b /\ b < c /\ ( ( ( F ` a ) < ( F ` b ) /\ ( F ` c ) < ( F ` b ) ) \/ ( ( F ` b ) < ( F ` a ) /\ ( F ` b ) < ( F ` c ) ) ) ) )
Theorem	pmltpc 23219*	Any function on the reals is either increasing, decreasing, or has a triple of points in a vee formation. (This theorem was created on demand by Mario Carneiro for the 6PCM conference in Bialystok, 1-Jul-2014.) (Contributed by Mario Carneiro, 1-Jul-2014.)
|- ( ( F e. ( RR ^pm RR ) /\ A C_ dom F ) -> ( A. x e. A A. y e. A ( x <_ y -> ( F ` x ) <_ ( F ` y ) ) \/ A. x e. A A. y e. A ( x <_ y -> ( F ` y ) <_ ( F ` x ) ) \/ E. a e. A E. b e. A E. c e. A ( a < b /\ b < c /\ ( ( ( F ` a ) < ( F ` b ) /\ ( F ` c ) < ( F ` b ) ) \/ ( ( F ` b ) < ( F ` a ) /\ ( F ` b ) < ( F ` c ) ) ) ) ) )
Theorem	ivthlem1 23220*	Lemma for ivth 23223. The set S of all x values with ( F ` x ) less than U is lower bounded by A and upper bounded by B. (Contributed by Mario Carneiro, 17-Jun-2014.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> U e. RR )   &    |- ( ph -> A < B )   &    |- ( ph -> ( A [,] B ) C_ D )   &    |- ( ph -> F e. ( D -cn-> CC ) )   &    |- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. RR )   &    |- ( ph -> ( ( F ` A ) < U /\ U < ( F ` B ) ) )   &    |- S = { x e. ( A [,] B ) | ( F ` x ) <_ U }   =>    |- ( ph -> ( A e. S /\ A. z e. S z <_ B ) )
Theorem	ivthlem2 23221*	Lemma for ivth 23223. Show that the supremum of S cannot be less than U. If it was, continuity of F implies that there are points just above the supremum that are also less than U, a contradiction. (Contributed by Mario Carneiro, 17-Jun-2014.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> U e. RR )   &    |- ( ph -> A < B )   &    |- ( ph -> ( A [,] B ) C_ D )   &    |- ( ph -> F e. ( D -cn-> CC ) )   &    |- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. RR )   &    |- ( ph -> ( ( F ` A ) < U /\ U < ( F ` B ) ) )   &    |- S = { x e. ( A [,] B ) | ( F ` x ) <_ U }   &    |- C = sup ( S , RR , < )   =>    |- ( ph -> -. ( F ` C ) < U )
Theorem	ivthlem3 23222*	Lemma for ivth 23223, the intermediate value theorem. Show that ( F ` C ) cannot be greater than U, and so establish the existence of a root of the function. (Contributed by Mario Carneiro, 30-Apr-2014.) (Revised by Mario Carneiro, 17-Jun-2014.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> U e. RR )   &    |- ( ph -> A < B )   &    |- ( ph -> ( A [,] B ) C_ D )   &    |- ( ph -> F e. ( D -cn-> CC ) )   &    |- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. RR )   &    |- ( ph -> ( ( F ` A ) < U /\ U < ( F ` B ) ) )   &    |- S = { x e. ( A [,] B ) | ( F ` x ) <_ U }   &    |- C = sup ( S , RR , < )   =>    |- ( ph -> ( C e. ( A (,) B ) /\ ( F ` C ) = U ) )
Theorem	ivth 23223*	The intermediate value theorem, increasing case. This is Metamath 100 proof #79. (Contributed by Paul Chapman, 22-Jan-2008.) (Proof shortened by Mario Carneiro, 30-Apr-2014.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> U e. RR )   &    |- ( ph -> A < B )   &    |- ( ph -> ( A [,] B ) C_ D )   &    |- ( ph -> F e. ( D -cn-> CC ) )   &    |- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. RR )   &    |- ( ph -> ( ( F ` A ) < U /\ U < ( F ` B ) ) )   =>    |- ( ph -> E. c e. ( A (,) B ) ( F ` c ) = U )
Theorem	ivth2 23224*	The intermediate value theorem, decreasing case. (Contributed by Paul Chapman, 22-Jan-2008.) (Revised by Mario Carneiro, 30-Apr-2014.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> U e. RR )   &    |- ( ph -> A < B )   &    |- ( ph -> ( A [,] B ) C_ D )   &    |- ( ph -> F e. ( D -cn-> CC ) )   &    |- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. RR )   &    |- ( ph -> ( ( F ` B ) < U /\ U < ( F ` A ) ) )   =>    |- ( ph -> E. c e. ( A (,) B ) ( F ` c ) = U )
Theorem	ivthle 23225*	The intermediate value theorem with weak inequality, increasing case. (Contributed by Mario Carneiro, 12-Aug-2014.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> U e. RR )   &    |- ( ph -> A < B )   &    |- ( ph -> ( A [,] B ) C_ D )   &    |- ( ph -> F e. ( D -cn-> CC ) )   &    |- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. RR )   &    |- ( ph -> ( ( F ` A ) <_ U /\ U <_ ( F ` B ) ) )   =>    |- ( ph -> E. c e. ( A [,] B ) ( F ` c ) = U )
Theorem	ivthle2 23226*	The intermediate value theorem with weak inequality, decreasing case. (Contributed by Mario Carneiro, 12-May-2014.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> U e. RR )   &    |- ( ph -> A < B )   &    |- ( ph -> ( A [,] B ) C_ D )   &    |- ( ph -> F e. ( D -cn-> CC ) )   &    |- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. RR )   &    |- ( ph -> ( ( F ` B ) <_ U /\ U <_ ( F ` A ) ) )   =>    |- ( ph -> E. c e. ( A [,] B ) ( F ` c ) = U )
Theorem	ivthicc 23227*	The interval between any two points of a continuous real function is contained in the range of the function. Equivalently, the range of a continuous real function is convex. (Contributed by Mario Carneiro, 12-Aug-2014.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> M e. ( A [,] B ) )   &    |- ( ph -> N e. ( A [,] B ) )   &    |- ( ph -> ( A [,] B ) C_ D )   &    |- ( ph -> F e. ( D -cn-> CC ) )   &    |- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. RR )   =>    |- ( ph -> ( ( F ` M ) [,] ( F ` N ) ) C_ ran F )
Theorem	evthicc 23228*	Specialization of the Extreme Value Theorem to a closed interval of RR. (Contributed by Mario Carneiro, 12-Aug-2014.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> A <_ B )   &    |- ( ph -> F e. ( ( A [,] B ) -cn-> RR ) )   =>    |- ( ph -> ( E. x e. ( A [,] B ) A. y e. ( A [,] B ) ( F ` y ) <_ ( F ` x ) /\ E. z e. ( A [,] B ) A. w e. ( A [,] B ) ( F ` z ) <_ ( F ` w ) ) )
Theorem	evthicc2 23229*	Combine ivthicc 23227 with evthicc 23228 to exactly describe the image of a closed interval. (Contributed by Mario Carneiro, 19-Feb-2015.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> A <_ B )   &    |- ( ph -> F e. ( ( A [,] B ) -cn-> RR ) )   =>    |- ( ph -> E. x e. RR E. y e. RR ran F = ( x [,] y ) )
Theorem	cniccbdd 23230*	A continuous function on a closed interval is bounded. (Contributed by Mario Carneiro, 7-Sep-2014.)
|- ( ( A e. RR /\ B e. RR /\ F e. ( ( A [,] B ) -cn-> CC ) ) -> E. x e. RR A. y e. ( A [,] B ) ( abs ` ( F ` y ) ) <_ x )
13.2  Integrals
13.2.1  Lebesgue measure
Syntax	covol 23231	Extend class notation with the outer Lebesgue measure.
class vol*
Syntax	cvol 23232	Extend class notation with the Lebesgue measure.
class vol
Definition	df-ovol 23233*	Define the outer Lebesgue measure for subsets of the reals. Here f is a function from the positive integers to pairs <. a , b >. with a <_ b, and the outer volume of the set x is the infimum over all such functions such that the union of the open intervals ( a , b ) covers x of the sum of b - a. (Contributed by Mario Carneiro, 16-Mar-2014.) (Revised by AV, 17-Sep-2020.)
|- vol* = ( x e. ~P RR |-> inf ( { y e. RR* | E. f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) ( x C_ U. ran ( (,) o. f ) /\ y = sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) ) } , RR* , < ) )
Definition	df-vol 23234*	Define the Lebesgue measure, which is just the outer measure with a peculiar domain of definition. The property of being Lebesgue-measurable can be expressed as A e. dom vol. (Contributed by Mario Carneiro, 17-Mar-2014.)
|- vol = ( vol* |` { x | A. y e. ( `' vol* " RR ) ( vol* ` y ) = ( ( vol* ` ( y i^i x ) ) + ( vol* ` ( y \ x ) ) ) } )
Theorem	ovolfcl 23235	Closure for the interval endpoint function. (Contributed by Mario Carneiro, 16-Mar-2014.)
|- ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ N e. NN ) -> ( ( 1st ` ( F ` N ) ) e. RR /\ ( 2nd ` ( F ` N ) ) e. RR /\ ( 1st ` ( F ` N ) ) <_ ( 2nd ` ( F ` N ) ) ) )
Theorem	ovolfioo 23236*	Unpack the interval covering property of the outer measure definition. (Contributed by Mario Carneiro, 16-Mar-2014.)
|- ( ( A C_ RR /\ F : NN --> ( <_ i^i ( RR X. RR ) ) ) -> ( A C_ U. ran ( (,) o. F ) <-> A. z e. A E. n e. NN ( ( 1st ` ( F ` n ) ) < z /\ z < ( 2nd ` ( F ` n ) ) ) ) )
Theorem	ovolficc 23237*	Unpack the interval covering property using closed intervals. (Contributed by Mario Carneiro, 16-Mar-2014.)
|- ( ( A C_ RR /\ F : NN --> ( <_ i^i ( RR X. RR ) ) ) -> ( A C_ U. ran ( [,] o. F ) <-> A. z e. A E. n e. NN ( ( 1st ` ( F ` n ) ) <_ z /\ z <_ ( 2nd ` ( F ` n ) ) ) ) )
Theorem	ovolficcss 23238	Any (closed) interval covering is a subset of the reals. (Contributed by Mario Carneiro, 24-Mar-2015.)
|- ( F : NN --> ( <_ i^i ( RR X. RR ) ) -> U. ran ( [,] o. F ) C_ RR )
Theorem	ovolfsval 23239	The value of the interval length function. (Contributed by Mario Carneiro, 16-Mar-2014.)
|- G = ( ( abs o. - ) o. F )   =>    |- ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ N e. NN ) -> ( G ` N ) = ( ( 2nd ` ( F ` N ) ) - ( 1st ` ( F ` N ) ) ) )
Theorem	ovolfsf 23240	Closure for the interval length function. (Contributed by Mario Carneiro, 16-Mar-2014.)
|- G = ( ( abs o. - ) o. F )   =>    |- ( F : NN --> ( <_ i^i ( RR X. RR ) ) -> G : NN --> ( 0 [,) +oo ) )
Theorem	ovolsf 23241	Closure for the partial sums of the interval length function. (Contributed by Mario Carneiro, 16-Mar-2014.)
|- G = ( ( abs o. - ) o. F )   &    |- S = seq 1 ( + , G )   =>    |- ( F : NN --> ( <_ i^i ( RR X. RR ) ) -> S : NN --> ( 0 [,) +oo ) )
Theorem	ovolval 23242*	The value of the outer measure. (Contributed by Mario Carneiro, 16-Mar-2014.) (Revised by AV, 17-Sep-2020.)
|- M = { y e. RR* | E. f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) ( A C_ U. ran ( (,) o. f ) /\ y = sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) ) }   =>    |- ( A C_ RR -> ( vol* ` A ) = inf ( M , RR* , < ) )
Theorem	elovolm 23243*	Elementhood in the set M of approximations to the outer measure. (Contributed by Mario Carneiro, 16-Mar-2014.)
|- M = { y e. RR* | E. f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) ( A C_ U. ran ( (,) o. f ) /\ y = sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) ) }   =>    |- ( B e. M <-> E. f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) ( A C_ U. ran ( (,) o. f ) /\ B = sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) ) )
Theorem	elovolmr 23244*	Sufficient condition for elementhood in the set M. (Contributed by Mario Carneiro, 16-Mar-2014.)
|- M = { y e. RR* | E. f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) ( A C_ U. ran ( (,) o. f ) /\ y = sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) ) }   &    |- S = seq 1 ( + , ( ( abs o. - ) o. F ) )   =>    |- ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ A C_ U. ran ( (,) o. F ) ) -> sup ( ran S , RR* , < ) e. M )
Theorem	ovolmge0 23245*	The set M is composed of nonnegative extended real numbers. (Contributed by Mario Carneiro, 16-Mar-2014.)
|- M = { y e. RR* | E. f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) ( A C_ U. ran ( (,) o. f ) /\ y = sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) ) }   =>    |- ( B e. M -> 0 <_ B )
Theorem	ovolcl 23246	The volume of a set is an extended real number. (Contributed by Mario Carneiro, 16-Mar-2014.)
|- ( A C_ RR -> ( vol* ` A ) e. RR* )
Theorem	ovollb 23247	The outer volume is a lower bound on the sum of all interval coverings of A. (Contributed by Mario Carneiro, 15-Jun-2014.)
|- S = seq 1 ( + , ( ( abs o. - ) o. F ) )   =>    |- ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ A C_ U. ran ( (,) o. F ) ) -> ( vol* ` A ) <_ sup ( ran S , RR* , < ) )
Theorem	ovolgelb 23248*	The outer volume is the greatest lower bound on the sum of all interval coverings of A. (Contributed by Mario Carneiro, 15-Jun-2014.)
|- S = seq 1 ( + , ( ( abs o. - ) o. g ) )   =>    |- ( ( A C_ RR /\ ( vol* ` A ) e. RR /\ B e. RR+ ) -> E. g e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) ( A C_ U. ran ( (,) o. g ) /\ sup ( ran S , RR* , < ) <_ ( ( vol* ` A ) + B ) ) )
Theorem	ovolge0 23249	The volume of a set is always nonnegative. (Contributed by Mario Carneiro, 16-Mar-2014.)
|- ( A C_ RR -> 0 <_ ( vol* ` A ) )
Theorem	ovolf 23250	The domain and range of the outer volume function. (Contributed by Mario Carneiro, 16-Mar-2014.) (Proof shortened by AV, 17-Sep-2020.)
|- vol* : ~P RR --> ( 0 [,] +oo )
Theorem	ovollecl 23251	If an outer volume is bounded above, then it is real. (Contributed by Mario Carneiro, 18-Mar-2014.)
|- ( ( A C_ RR /\ B e. RR /\ ( vol* ` A ) <_ B ) -> ( vol* ` A ) e. RR )
Theorem	ovolsslem 23252*	Lemma for ovolss 23253. (Contributed by Mario Carneiro, 16-Mar-2014.) (Proof shortened by AV, 17-Sep-2020.)
|- M = { y e. RR* | E. f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) ( A C_ U. ran ( (,) o. f ) /\ y = sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) ) }   &    |- N = { y e. RR* | E. f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) ( B C_ U. ran ( (,) o. f ) /\ y = sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) ) }   =>    |- ( ( A C_ B /\ B C_ RR ) -> ( vol* ` A ) <_ ( vol* ` B ) )
Theorem	ovolss 23253	The volume of a set is monotone with respect to set inclusion. (Contributed by Mario Carneiro, 16-Mar-2014.)
|- ( ( A C_ B /\ B C_ RR ) -> ( vol* ` A ) <_ ( vol* ` B ) )
Theorem	ovolsscl 23254	If a set is contained in another of bounded measure, it too is bounded. (Contributed by Mario Carneiro, 18-Mar-2014.)
|- ( ( A C_ B /\ B C_ RR /\ ( vol* ` B ) e. RR ) -> ( vol* ` A ) e. RR )
Theorem	ovolssnul 23255	A subset of a nullset is null. (Contributed by Mario Carneiro, 19-Mar-2014.)
|- ( ( A C_ B /\ B C_ RR /\ ( vol* ` B ) = 0 ) -> ( vol* ` A ) = 0 )
Theorem	ovollb2lem 23256*	Lemma for ovollb2 23257. (Contributed by Mario Carneiro, 24-Mar-2015.)
|- S = seq 1 ( + , ( ( abs o. - ) o. F ) )   &    |- G = ( n e. NN |-> <. ( ( 1st ` ( F ` n ) ) - ( ( B / 2 ) / ( 2 ^ n ) ) ) , ( ( 2nd ` ( F ` n ) ) + ( ( B / 2 ) / ( 2 ^ n ) ) ) >. )   &    |- T = seq 1 ( + , ( ( abs o. - ) o. G ) )   &    |- ( ph -> F : NN --> ( <_ i^i ( RR X. RR ) ) )   &    |- ( ph -> A C_ U. ran ( [,] o. F ) )   &    |- ( ph -> B e. RR+ )   &    |- ( ph -> sup ( ran S , RR* , < ) e. RR )   =>    |- ( ph -> ( vol* ` A ) <_ ( sup ( ran S , RR* , < ) + B ) )
Theorem	ovollb2 23257	It is often more convenient to do calculations with *closed* coverings rather than open ones; here we show that it makes no difference (compare ovollb 23247). (Contributed by Mario Carneiro, 24-Mar-2015.)
|- S = seq 1 ( + , ( ( abs o. - ) o. F ) )   =>    |- ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ A C_ U. ran ( [,] o. F ) ) -> ( vol* ` A ) <_ sup ( ran S , RR* , < ) )
Theorem	ovolctb 23258	The volume of a denumerable set is 0. (Contributed by Mario Carneiro, 17-Mar-2014.) (Proof shortened by Mario Carneiro, 25-Mar-2015.)
|- ( ( A C_ RR /\ A ~~ NN ) -> ( vol* ` A ) = 0 )
Theorem	ovolq 23259	The rational numbers have 0 outer Lebesgue measure. (Contributed by Mario Carneiro, 17-Mar-2014.)
|- ( vol* ` QQ ) = 0
Theorem	ovolctb2 23260	The volume of a countable set is 0. (Contributed by Mario Carneiro, 17-Mar-2014.)
|- ( ( A C_ RR /\ A ~<_ NN ) -> ( vol* ` A ) = 0 )
Theorem	ovol0 23261	The empty set has 0 outer Lebesgue measure. (Contributed by Mario Carneiro, 17-Mar-2014.)
|- ( vol* ` (/) ) = 0
Theorem	ovolfi 23262	A finite set has 0 outer Lebesgue measure. (Contributed by Mario Carneiro, 13-Aug-2014.)
|- ( ( A e. Fin /\ A C_ RR ) -> ( vol* ` A ) = 0 )
Theorem	ovolsn 23263	A singleton has 0 outer Lebesgue measure. (Contributed by Mario Carneiro, 15-Aug-2014.)
|- ( A e. RR -> ( vol* ` { A } ) = 0 )
Theorem	ovolunlem1a 23264*	Lemma for ovolun 23267. (Contributed by Mario Carneiro, 7-May-2015.)
|- ( ph -> ( A C_ RR /\ ( vol* ` A ) e. RR ) )   &    |- ( ph -> ( B C_ RR /\ ( vol* ` B ) e. RR ) )   &    |- ( ph -> C e. RR+ )   &    |- S = seq 1 ( + , ( ( abs o. - ) o. F ) )   &    |- T = seq 1 ( + , ( ( abs o. - ) o. G ) )   &    |- U = seq 1 ( + , ( ( abs o. - ) o. H ) )   &    |- ( ph -> F e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) )   &    |- ( ph -> A C_ U. ran ( (,) o. F ) )   &    |- ( ph -> sup ( ran S , RR* , < ) <_ ( ( vol* ` A ) + ( C / 2 ) ) )   &    |- ( ph -> G e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) )   &    |- ( ph -> B C_ U. ran ( (,) o. G ) )   &    |- ( ph -> sup ( ran T , RR* , < ) <_ ( ( vol* ` B ) + ( C / 2 ) ) )   &    |- H = ( n e. NN |-> if ( ( n / 2 ) e. NN , ( G ` ( n / 2 ) ) , ( F ` ( ( n + 1 ) / 2 ) ) ) )   =>    |- ( ( ph /\ k e. NN ) -> ( U ` k ) <_ ( ( ( vol* ` A ) + ( vol* ` B ) ) + C ) )
Theorem	ovolunlem1 23265*	Lemma for ovolun 23267. (Contributed by Mario Carneiro, 12-Jun-2014.)
|- ( ph -> ( A C_ RR /\ ( vol* ` A ) e. RR ) )   &    |- ( ph -> ( B C_ RR /\ ( vol* ` B ) e. RR ) )   &    |- ( ph -> C e. RR+ )   &    |- S = seq 1 ( + , ( ( abs o. - ) o. F ) )   &    |- T = seq 1 ( + , ( ( abs o. - ) o. G ) )   &    |- U = seq 1 ( + , ( ( abs o. - ) o. H ) )   &    |- ( ph -> F e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) )   &    |- ( ph -> A C_ U. ran ( (,) o. F ) )   &    |- ( ph -> sup ( ran S , RR* , < ) <_ ( ( vol* ` A ) + ( C / 2 ) ) )   &    |- ( ph -> G e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) )   &    |- ( ph -> B C_ U. ran ( (,) o. G ) )   &    |- ( ph -> sup ( ran T , RR* , < ) <_ ( ( vol* ` B ) + ( C / 2 ) ) )   &    |- H = ( n e. NN |-> if ( ( n / 2 ) e. NN , ( G ` ( n / 2 ) ) , ( F ` ( ( n + 1 ) / 2 ) ) ) )   =>    |- ( ph -> ( vol* ` ( A u. B ) ) <_ ( ( ( vol* ` A ) + ( vol* ` B ) ) + C ) )
Theorem	ovolunlem2 23266	Lemma for ovolun 23267. (Contributed by Mario Carneiro, 12-Jun-2014.)
|- ( ph -> ( A C_ RR /\ ( vol* ` A ) e. RR ) )   &    |- ( ph -> ( B C_ RR /\ ( vol* ` B ) e. RR ) )   &    |- ( ph -> C e. RR+ )   =>    |- ( ph -> ( vol* ` ( A u. B ) ) <_ ( ( ( vol* ` A ) + ( vol* ` B ) ) + C ) )
Theorem	ovolun 23267	The Lebesgue outer measure function is finitely sub-additive. (Unlike the stronger ovoliun 23273, this does not require any choice principles.) (Contributed by Mario Carneiro, 12-Jun-2014.)
|- ( ( ( A C_ RR /\ ( vol* ` A ) e. RR ) /\ ( B C_ RR /\ ( vol* ` B ) e. RR ) ) -> ( vol* ` ( A u. B ) ) <_ ( ( vol* ` A ) + ( vol* ` B ) ) )
Theorem	ovolunnul 23268	Adding a nullset does not change the measure of a set. (Contributed by Mario Carneiro, 25-Mar-2015.)
|- ( ( A C_ RR /\ B C_ RR /\ ( vol* ` B ) = 0 ) -> ( vol* ` ( A u. B ) ) = ( vol* ` A ) )
Theorem	ovolfiniun 23269*	The Lebesgue outer measure function is finitely sub-additive. Finite sum version. (Contributed by Mario Carneiro, 19-Jun-2014.)
|- ( ( A e. Fin /\ A. k e. A ( B C_ RR /\ ( vol* ` B ) e. RR ) ) -> ( vol* ` U_ k e. A B ) <_ sum_ k e. A ( vol* ` B ) )
Theorem	ovoliunlem1 23270*	Lemma for ovoliun 23273. (Contributed by Mario Carneiro, 12-Jun-2014.)
|- T = seq 1 ( + , G )   &    |- G = ( n e. NN |-> ( vol* ` A ) )   &    |- ( ( ph /\ n e. NN ) -> A C_ RR )   &    |- ( ( ph /\ n e. NN ) -> ( vol* ` A ) e. RR )   &    |- ( ph -> sup ( ran T , RR* , < ) e. RR )   &    |- ( ph -> B e. RR+ )   &    |- S = seq 1 ( + , ( ( abs o. - ) o. ( F ` n ) ) )   &    |- U = seq 1 ( + , ( ( abs o. - ) o. H ) )   &    |- H = ( k e. NN |-> ( ( F ` ( 1st ` ( J ` k ) ) ) ` ( 2nd ` ( J ` k ) ) ) )   &    |- ( ph -> J : NN -1-1-onto-> ( NN X. NN ) )   &    |- ( ph -> F : NN --> ( ( <_ i^i ( RR X. RR ) ) ^m NN ) )   &    |- ( ( ph /\ n e. NN ) -> A C_ U. ran ( (,) o. ( F ` n ) ) )   &    |- ( ( ph /\ n e. NN ) -> sup ( ran S , RR* , < ) <_ ( ( vol* ` A ) + ( B / ( 2 ^ n ) ) ) )   &    |- ( ph -> K e. NN )   &    |- ( ph -> L e. ZZ )   &    |- ( ph -> A. w e. ( 1 ... K ) ( 1st ` ( J ` w ) ) <_ L )   =>    |- ( ph -> ( U ` K ) <_ ( sup ( ran T , RR* , < ) + B ) )
Theorem	ovoliunlem2 23271*	Lemma for ovoliun 23273. (Contributed by Mario Carneiro, 12-Jun-2014.)
|- T = seq 1 ( + , G )   &    |- G = ( n e. NN |-> ( vol* ` A ) )   &    |- ( ( ph /\ n e. NN ) -> A C_ RR )   &    |- ( ( ph /\ n e. NN ) -> ( vol* ` A ) e. RR )   &    |- ( ph -> sup ( ran T , RR* , < ) e. RR )   &    |- ( ph -> B e. RR+ )   &    |- S = seq 1 ( + , ( ( abs o. - ) o. ( F ` n ) ) )   &    |- U = seq 1 ( + , ( ( abs o. - ) o. H ) )   &    |- H = ( k e. NN |-> ( ( F ` ( 1st ` ( J ` k ) ) ) ` ( 2nd ` ( J ` k ) ) ) )   &    |- ( ph -> J : NN -1-1-onto-> ( NN X. NN ) )   &    |- ( ph -> F : NN --> ( ( <_ i^i ( RR X. RR ) ) ^m NN ) )   &    |- ( ( ph /\ n e. NN ) -> A C_ U. ran ( (,) o. ( F ` n ) ) )   &    |- ( ( ph /\ n e. NN ) -> sup ( ran S , RR* , < ) <_ ( ( vol* ` A ) + ( B / ( 2 ^ n ) ) ) )   =>    |- ( ph -> ( vol* ` U_ n e. NN A ) <_ ( sup ( ran T , RR* , < ) + B ) )
Theorem	ovoliunlem3 23272*	Lemma for ovoliun 23273. (Contributed by Mario Carneiro, 12-Jun-2014.)
|- T = seq 1 ( + , G )   &    |- G = ( n e. NN |-> ( vol* ` A ) )   &    |- ( ( ph /\ n e. NN ) -> A C_ RR )   &    |- ( ( ph /\ n e. NN ) -> ( vol* ` A ) e. RR )   &    |- ( ph -> sup ( ran T , RR* , < ) e. RR )   &    |- ( ph -> B e. RR+ )   =>    |- ( ph -> ( vol* ` U_ n e. NN A ) <_ ( sup ( ran T , RR* , < ) + B ) )
Theorem	ovoliun 23273*	The Lebesgue outer measure function is countably sub-additive. (Many books allow +oo as a value for one of the sets in the sum, but in our setup we can't do arithmetic on infinity, and in any case the volume of a union containing an infinitely large set is already infinitely large by monotonicity ovolss 23253, so we need not consider this case here, although we do allow the sum itself to be infinite.) (Contributed by Mario Carneiro, 12-Jun-2014.)
|- T = seq 1 ( + , G )   &    |- G = ( n e. NN |-> ( vol* ` A ) )   &    |- ( ( ph /\ n e. NN ) -> A C_ RR )   &    |- ( ( ph /\ n e. NN ) -> ( vol* ` A ) e. RR )   =>    |- ( ph -> ( vol* ` U_ n e. NN A ) <_ sup ( ran T , RR* , < ) )
Theorem	ovoliun2 23274*	The Lebesgue outer measure function is countably sub-additive. (This version is a little easier to read, but does not allow infinite values like ovoliun 23273.) (Contributed by Mario Carneiro, 12-Jun-2014.)
|- T = seq 1 ( + , G )   &    |- G = ( n e. NN |-> ( vol* ` A ) )   &    |- ( ( ph /\ n e. NN ) -> A C_ RR )   &    |- ( ( ph /\ n e. NN ) -> ( vol* ` A ) e. RR )   &    |- ( ph -> T e. dom ~~> )   =>    |- ( ph -> ( vol* ` U_ n e. NN A ) <_ sum_ n e. NN ( vol* ` A ) )
Theorem	ovoliunnul 23275*	A countable union of nullsets is null. (Contributed by Mario Carneiro, 8-Apr-2015.)
|- ( ( A ~<_ NN /\ A. n e. A ( B C_ RR /\ ( vol* ` B ) = 0 ) ) -> ( vol* ` U_ n e. A B ) = 0 )
Theorem	shft2rab 23276*	If B is a shift of A by C, then A is a shift of B by -u C. (Contributed by Mario Carneiro, 22-Mar-2014.) (Revised by Mario Carneiro, 6-Apr-2015.)
|- ( ph -> A C_ RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> B = { x e. RR | ( x - C ) e. A } )   =>    |- ( ph -> A = { y e. RR | ( y - -u C ) e. B } )
Theorem	ovolshftlem1 23277*	Lemma for ovolshft 23279. (Contributed by Mario Carneiro, 22-Mar-2014.)
|- ( ph -> A C_ RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> B = { x e. RR | ( x - C ) e. A } )   &    |- M = { y e. RR* | E. f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) ( B C_ U. ran ( (,) o. f ) /\ y = sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) ) }   &    |- S = seq 1 ( + , ( ( abs o. - ) o. F ) )   &    |- G = ( n e. NN |-> <. ( ( 1st ` ( F ` n ) ) + C ) , ( ( 2nd ` ( F ` n ) ) + C ) >. )   &    |- ( ph -> F : NN --> ( <_ i^i ( RR X. RR ) ) )   &    |- ( ph -> A C_ U. ran ( (,) o. F ) )   =>    |- ( ph -> sup ( ran S , RR* , < ) e. M )
Theorem	ovolshftlem2 23278*	Lemma for ovolshft 23279. (Contributed by Mario Carneiro, 22-Mar-2014.)
|- ( ph -> A C_ RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> B = { x e. RR | ( x - C ) e. A } )   &    |- M = { y e. RR* | E. f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) ( B C_ U. ran ( (,) o. f ) /\ y = sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) ) }   =>    |- ( ph -> { z e. RR* | E. g e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) ( A C_ U. ran ( (,) o. g ) /\ z = sup ( ran seq 1 ( + , ( ( abs o. - ) o. g ) ) , RR* , < ) ) } C_ M )
Theorem	ovolshft 23279*	The Lebesgue outer measure function is shift-invariant. (Contributed by Mario Carneiro, 22-Mar-2014.) (Proof shortened by AV, 17-Sep-2020.)
|- ( ph -> A C_ RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> B = { x e. RR | ( x - C ) e. A } )   =>    |- ( ph -> ( vol* ` A ) = ( vol* ` B ) )
Theorem	sca2rab 23280*	If B is a scale of A by C, then A is a scale of B by 1 / C. (Contributed by Mario Carneiro, 22-Mar-2014.)
|- ( ph -> A C_ RR )   &    |- ( ph -> C e. RR+ )   &    |- ( ph -> B = { x e. RR | ( C x. x ) e. A } )   =>    |- ( ph -> A = { y e. RR | ( ( 1 / C ) x. y ) e. B } )
Theorem	ovolscalem1 23281*	Lemma for ovolsca 23283. (Contributed by Mario Carneiro, 6-Apr-2015.)
|- ( ph -> A C_ RR )   &    |- ( ph -> C e. RR+ )   &    |- ( ph -> B = { x e. RR | ( C x. x ) e. A } )   &    |- ( ph -> ( vol* ` A ) e. RR )   &    |- S = seq 1 ( + , ( ( abs o. - ) o. F ) )   &    |- G = ( n e. NN |-> <. ( ( 1st ` ( F ` n ) ) / C ) , ( ( 2nd ` ( F ` n ) ) / C ) >. )   &    |- ( ph -> F : NN --> ( <_ i^i ( RR X. RR ) ) )   &    |- ( ph -> A C_ U. ran ( (,) o. F ) )   &    |- ( ph -> R e. RR+ )   &    |- ( ph -> sup ( ran S , RR* , < ) <_ ( ( vol* ` A ) + ( C x. R ) ) )   =>    |- ( ph -> ( vol* ` B ) <_ ( ( ( vol* ` A ) / C ) + R ) )
Theorem	ovolscalem2 23282*	Lemma for ovolshft 23279. (Contributed by Mario Carneiro, 22-Mar-2014.)
|- ( ph -> A C_ RR )   &    |- ( ph -> C e. RR+ )   &    |- ( ph -> B = { x e. RR | ( C x. x ) e. A } )   &    |- ( ph -> ( vol* ` A ) e. RR )   =>    |- ( ph -> ( vol* ` B ) <_ ( ( vol* ` A ) / C ) )
Theorem	ovolsca 23283*	The Lebesgue outer measure function respects scaling of sets by positive reals. (Contributed by Mario Carneiro, 6-Apr-2015.)
|- ( ph -> A C_ RR )   &    |- ( ph -> C e. RR+ )   &    |- ( ph -> B = { x e. RR | ( C x. x ) e. A } )   &    |- ( ph -> ( vol* ` A ) e. RR )   =>    |- ( ph -> ( vol* ` B ) = ( ( vol* ` A ) / C ) )
Theorem	ovolicc1 23284*	The measure of a closed interval is lower bounded by its length. (Contributed by Mario Carneiro, 13-Jun-2014.) (Proof shortened by Mario Carneiro, 25-Mar-2015.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> A <_ B )   &    |- G = ( n e. NN |-> if ( n = 1 , <. A , B >. , <. 0 , 0 >. ) )   =>    |- ( ph -> ( vol* ` ( A [,] B ) ) <_ ( B - A ) )
Theorem	ovolicc2lem1 23285*	Lemma for ovolicc2 23290. (Contributed by Mario Carneiro, 14-Jun-2014.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> A <_ B )   &    |- S = seq 1 ( + , ( ( abs o. - ) o. F ) )   &    |- ( ph -> F : NN --> ( <_ i^i ( RR X. RR ) ) )   &    |- ( ph -> U e. ( ~P ran ( (,) o. F ) i^i Fin ) )   &    |- ( ph -> ( A [,] B ) C_ U. U )   &    |- ( ph -> G : U --> NN )   &    |- ( ( ph /\ t e. U ) -> ( ( (,) o. F ) ` ( G ` t ) ) = t )   =>    |- ( ( ph /\ X e. U ) -> ( P e. X <-> ( P e. RR /\ ( 1st ` ( F ` ( G ` X ) ) ) < P /\ P < ( 2nd ` ( F ` ( G ` X ) ) ) ) ) )
Theorem	ovolicc2lem2 23286*	Lemma for ovolicc2 23290. (Contributed by Mario Carneiro, 14-Jun-2014.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> A <_ B )   &    |- S = seq 1 ( + , ( ( abs o. - ) o. F ) )   &    |- ( ph -> F : NN --> ( <_ i^i ( RR X. RR ) ) )   &    |- ( ph -> U e. ( ~P ran ( (,) o. F ) i^i Fin ) )   &    |- ( ph -> ( A [,] B ) C_ U. U )   &    |- ( ph -> G : U --> NN )   &    |- ( ( ph /\ t e. U ) -> ( ( (,) o. F ) ` ( G ` t ) ) = t )   &    |- T = { u e. U | ( u i^i ( A [,] B ) ) =/= (/) }   &    |- ( ph -> H : T --> T )   &    |- ( ( ph /\ t e. T ) -> if ( ( 2nd ` ( F ` ( G ` t ) ) ) <_ B , ( 2nd ` ( F ` ( G ` t ) ) ) , B ) e. ( H ` t ) )   &    |- ( ph -> A e. C )   &    |- ( ph -> C e. T )   &    |- K = seq 1 ( ( H o. 1st ) , ( NN X. { C } ) )   &    |- W = { n e. NN | B e. ( K ` n ) }   =>    |- ( ( ph /\ ( N e. NN /\ -. N e. W ) ) -> ( 2nd ` ( F ` ( G ` ( K ` N ) ) ) ) <_ B )
Theorem	ovolicc2lem3 23287*	Lemma for ovolicc2 23290. (Contributed by Mario Carneiro, 14-Jun-2014.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> A <_ B )   &    |- S = seq 1 ( + , ( ( abs o. - ) o. F ) )   &    |- ( ph -> F : NN --> ( <_ i^i ( RR X. RR ) ) )   &    |- ( ph -> U e. ( ~P ran ( (,) o. F ) i^i Fin ) )   &    |- ( ph -> ( A [,] B ) C_ U. U )   &    |- ( ph -> G : U --> NN )   &    |- ( ( ph /\ t e. U ) -> ( ( (,) o. F ) ` ( G ` t ) ) = t )   &    |- T = { u e. U | ( u i^i ( A [,] B ) ) =/= (/) }   &    |- ( ph -> H : T --> T )   &    |- ( ( ph /\ t e. T ) -> if ( ( 2nd ` ( F ` ( G ` t ) ) ) <_ B , ( 2nd ` ( F ` ( G ` t ) ) ) , B ) e. ( H ` t ) )   &    |- ( ph -> A e. C )   &    |- ( ph -> C e. T )   &    |- K = seq 1 ( ( H o. 1st ) , ( NN X. { C } ) )   &    |- W = { n e. NN | B e. ( K ` n ) }   =>    |- ( ( ph /\ ( N e. { n e. NN | A. m e. W n <_ m } /\ P e. { n e. NN | A. m e. W n <_ m } ) ) -> ( N = P <-> ( 2nd ` ( F ` ( G ` ( K ` N ) ) ) ) = ( 2nd ` ( F ` ( G ` ( K ` P ) ) ) ) ) )
Theorem	ovolicc2lem4 23288*	Lemma for ovolicc2 23290. (Contributed by Mario Carneiro, 14-Jun-2014.) (Revised by AV, 17-Sep-2020.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> A <_ B )   &    |- S = seq 1 ( + , ( ( abs o. - ) o. F ) )   &    |- ( ph -> F : NN --> ( <_ i^i ( RR X. RR ) ) )   &    |- ( ph -> U e. ( ~P ran ( (,) o. F ) i^i Fin ) )   &    |- ( ph -> ( A [,] B ) C_ U. U )   &    |- ( ph -> G : U --> NN )   &    |- ( ( ph /\ t e. U ) -> ( ( (,) o. F ) ` ( G ` t ) ) = t )   &    |- T = { u e. U | ( u i^i ( A [,] B ) ) =/= (/) }   &    |- ( ph -> H : T --> T )   &    |- ( ( ph /\ t e. T ) -> if ( ( 2nd ` ( F ` ( G ` t ) ) ) <_ B , ( 2nd ` ( F ` ( G ` t ) ) ) , B ) e. ( H ` t ) )   &    |- ( ph -> A e. C )   &    |- ( ph -> C e. T )   &    |- K = seq 1 ( ( H o. 1st ) , ( NN X. { C } ) )   &    |- W = { n e. NN | B e. ( K ` n ) }   &    |- M = inf ( W , RR , < )   =>    |- ( ph -> ( B - A ) <_ sup ( ran S , RR* , < ) )
Theorem	ovolicc2lem5 23289*	Lemma for ovolicc2 23290. (Contributed by Mario Carneiro, 14-Jun-2014.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> A <_ B )   &    |- S = seq 1 ( + , ( ( abs o. - ) o. F ) )   &    |- ( ph -> F : NN --> ( <_ i^i ( RR X. RR ) ) )   &    |- ( ph -> U e. ( ~P ran ( (,) o. F ) i^i Fin ) )   &    |- ( ph -> ( A [,] B ) C_ U. U )   &    |- ( ph -> G : U --> NN )   &    |- ( ( ph /\ t e. U ) -> ( ( (,) o. F ) ` ( G ` t ) ) = t )   &    |- T = { u e. U | ( u i^i ( A [,] B ) ) =/= (/) }   =>    |- ( ph -> ( B - A ) <_ sup ( ran S , RR* , < ) )
Theorem	ovolicc2 23290*	The measure of a closed interval is upper bounded by its length. (Contributed by Mario Carneiro, 14-Jun-2014.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> A <_ B )   &    |- M = { y e. RR* | E. f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) ( ( A [,] B ) C_ U. ran ( (,) o. f ) /\ y = sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) ) }   =>    |- ( ph -> ( B - A ) <_ ( vol* ` ( A [,] B ) ) )
Theorem	ovolicc 23291	The measure of a closed interval. (Contributed by Mario Carneiro, 14-Jun-2014.)
|- ( ( A e. RR /\ B e. RR /\ A <_ B ) -> ( vol* ` ( A [,] B ) ) = ( B - A ) )
Theorem	ovolicopnf 23292	The measure of a right-unbounded interval. (Contributed by Mario Carneiro, 14-Jun-2014.)
|- ( A e. RR -> ( vol* ` ( A [,) +oo ) ) = +oo )
Theorem	ovolre 23293	The measure of the real numbers. (Contributed by Mario Carneiro, 14-Jun-2014.)
|- ( vol* ` RR ) = +oo
Theorem	ismbl 23294*	The predicate " A is Lebesgue-measurable". A set is measurable if it splits every other set x in a "nice" way, that is, if the measure of the pieces x i^i A and x \ A sum up to the measure of x (assuming that the measure of x is a real number, so that this addition makes sense). (Contributed by Mario Carneiro, 17-Mar-2014.)
|- ( A e. dom vol <-> ( A C_ RR /\ A. x e. ~P RR ( ( vol* ` x ) e. RR -> ( vol* ` x ) = ( ( vol* ` ( x i^i A ) ) + ( vol* ` ( x \ A ) ) ) ) ) )
Theorem	ismbl2 23295*	From ovolun 23267, it suffices to show that the measure of x is at least the sum of the measures of x i^i A and x \ A. (Contributed by Mario Carneiro, 15-Jun-2014.)
|- ( A e. dom vol <-> ( A C_ RR /\ A. x e. ~P RR ( ( vol* ` x ) e. RR -> ( ( vol* ` ( x i^i A ) ) + ( vol* ` ( x \ A ) ) ) <_ ( vol* ` x ) ) ) )
Theorem	volres 23296	A self-referencing abbreviated definition of the Lebesgue measure. (Contributed by Mario Carneiro, 19-Mar-2014.)
|- vol = ( vol* |` dom vol )
Theorem	volf 23297	The domain and range of the Lebesgue measure function. (Contributed by Mario Carneiro, 19-Mar-2014.)
|- vol : dom vol --> ( 0 [,] +oo )
Theorem	mblvol 23298	The volume of a measurable set is the same as its outer volume. (Contributed by Mario Carneiro, 17-Mar-2014.)
|- ( A e. dom vol -> ( vol ` A ) = ( vol* ` A ) )
Theorem	mblss 23299	A measurable set is a subset of the reals. (Contributed by Mario Carneiro, 17-Mar-2014.)
|- ( A e. dom vol -> A C_ RR )
Theorem	mblsplit 23300	The defining property of measurability. (Contributed by Mario Carneiro, 17-Mar-2014.)
|- ( ( A e. dom vol /\ B C_ RR /\ ( vol* ` B ) e. RR ) -> ( vol* ` B ) = ( ( vol* ` ( B i^i A ) ) + ( vol* ` ( B \ A ) ) ) )