Theorem ovolsca

Description: The Lebesgue outer measure function respects scaling of sets by positive reals. (Contributed by Mario Carneiro, 6-Apr-2015)

	Ref	Expression
Hypotheses	ovolsca.1	|- ( ph -> A C_ RR )
	ovolsca.2	|- ( ph -> C e. RR+ )
	ovolsca.3	|- ( ph -> B = { x e. RR | ( C x. x ) e. A } )
	ovolsca.4	|- ( ph -> ( vol* ` A ) e. RR )
Assertion	ovolsca	|- ( ph -> ( vol* ` B ) = ( ( vol* ` A ) / C ) )

Proof

Step	Hyp	Ref	Expression
1		ovolsca.1	|- ( ph -> A C_ RR )
2		ovolsca.2	|- ( ph -> C e. RR+ )
3		ovolsca.3	|- ( ph -> B = { x e. RR | ( C x. x ) e. A } )
4		ovolsca.4	|- ( ph -> ( vol* ` A ) e. RR )
5	1 2 3 4	ovolscalem2	|- ( ph -> ( vol* ` B ) <_ ( ( vol* ` A ) / C ) )
6	4	recnd	|- ( ph -> ( vol* ` A ) e. CC )
7	2	rpcnd	|- ( ph -> C e. CC )
8	2	rpne0d	|- ( ph -> C =/= 0 )
9	6 7 8	divrecd	|- ( ph -> ( ( vol* ` A ) / C ) = ( ( vol* ` A ) x. ( 1 / C ) ) )
10		ssrab2	|- { x e. RR | ( C x. x ) e. A } C_ RR
11	3 10	eqsstrdi	|- ( ph -> B C_ RR )
12	2	rpreccld	|- ( ph -> ( 1 / C ) e. RR+ )
13	1 2 3	sca2rab	|- ( ph -> A = { y e. RR | ( ( 1 / C ) x. y ) e. B } )
14	4 2	rerpdivcld	|- ( ph -> ( ( vol* ` A ) / C ) e. RR )
15		ovollecl	|- ( ( B C_ RR /\ ( ( vol* ` A ) / C ) e. RR /\ ( vol* ` B ) <_ ( ( vol* ` A ) / C ) ) -> ( vol* ` B ) e. RR )
16	11 14 5 15	syl3anc	|- ( ph -> ( vol* ` B ) e. RR )
17	11 12 13 16	ovolscalem2	|- ( ph -> ( vol* ` A ) <_ ( ( vol* ` B ) / ( 1 / C ) ) )
18	4 16 12	lemuldivd	|- ( ph -> ( ( ( vol* ` A ) x. ( 1 / C ) ) <_ ( vol* ` B ) <-> ( vol* ` A ) <_ ( ( vol* ` B ) / ( 1 / C ) ) ) )
19	17 18	mpbird	|- ( ph -> ( ( vol* ` A ) x. ( 1 / C ) ) <_ ( vol* ` B ) )
20	9 19	eqbrtrd	|- ( ph -> ( ( vol* ` A ) / C ) <_ ( vol* ` B ) )
21	16 14	letri3d	|- ( ph -> ( ( vol* ` B ) = ( ( vol* ` A ) / C ) <-> ( ( vol* ` B ) <_ ( ( vol* ` A ) / C ) /\ ( ( vol* ` A ) / C ) <_ ( vol* ` B ) ) ) )
22	5 20 21	mpbir2and	|- ( ph -> ( vol* ` B ) = ( ( vol* ` A ) / C ) )