Metamath Proof Explorer

Theorem measunl

Description: A measure is sub-additive with respect to union. (Contributed by Thierry Arnoux, 20-Oct-2017)

Ref Expression
Hypotheses measunl.1 
|- ( ph -> M e. ( measures ` S ) )
measunl.2 
|- ( ph -> A e. S )
measunl.3 
|- ( ph -> B e. S )
Assertion measunl 
|- ( ph -> ( M ` ( A u. B ) ) <_ ( ( M ` A ) +e ( M ` B ) ) )

Proof

Step Hyp Ref Expression
1 measunl.1 
 |-  ( ph -> M e. ( measures ` S ) )
2 measunl.2 
 |-  ( ph -> A e. S )
3 measunl.3 
 |-  ( ph -> B e. S )
4 undif1 
 |-  ( ( A \ B ) u. B ) = ( A u. B )
5 4 fveq2i 
 |-  ( M ` ( ( A \ B ) u. B ) ) = ( M ` ( A u. B ) )
6 measbase 
 |-  ( M e. ( measures ` S ) -> S e. U. ran sigAlgebra )
7 1 6 syl 
 |-  ( ph -> S e. U. ran sigAlgebra )
8 difelsiga 
 |-  ( ( S e. U. ran sigAlgebra /\ A e. S /\ B e. S ) -> ( A \ B ) e. S )
9 7 2 3 8 syl3anc 
 |-  ( ph -> ( A \ B ) e. S )
10 disjdifr 
 |-  ( ( A \ B ) i^i B ) = (/)
11 10 a1i 
 |-  ( ph -> ( ( A \ B ) i^i B ) = (/) )
12 measun 
 |-  ( ( M e. ( measures ` S ) /\ ( ( A \ B ) e. S /\ B e. S ) /\ ( ( A \ B ) i^i B ) = (/) ) -> ( M ` ( ( A \ B ) u. B ) ) = ( ( M ` ( A \ B ) ) +e ( M ` B ) ) )
13 1 9 3 11 12 syl121anc 
 |-  ( ph -> ( M ` ( ( A \ B ) u. B ) ) = ( ( M ` ( A \ B ) ) +e ( M ` B ) ) )
14 5 13 eqtr3id 
 |-  ( ph -> ( M ` ( A u. B ) ) = ( ( M ` ( A \ B ) ) +e ( M ` B ) ) )
15 iccssxr 
 |-  ( 0 [,] +oo ) C_ RR*
16 measvxrge0 
 |-  ( ( M e. ( measures ` S ) /\ ( A \ B ) e. S ) -> ( M ` ( A \ B ) ) e. ( 0 [,] +oo ) )
17 1 9 16 syl2anc 
 |-  ( ph -> ( M ` ( A \ B ) ) e. ( 0 [,] +oo ) )
18 15 17 sselid 
 |-  ( ph -> ( M ` ( A \ B ) ) e. RR* )
19 measvxrge0 
 |-  ( ( M e. ( measures ` S ) /\ A e. S ) -> ( M ` A ) e. ( 0 [,] +oo ) )
20 1 2 19 syl2anc 
 |-  ( ph -> ( M ` A ) e. ( 0 [,] +oo ) )
21 15 20 sselid 
 |-  ( ph -> ( M ` A ) e. RR* )
22 measvxrge0 
 |-  ( ( M e. ( measures ` S ) /\ B e. S ) -> ( M ` B ) e. ( 0 [,] +oo ) )
23 1 3 22 syl2anc 
 |-  ( ph -> ( M ` B ) e. ( 0 [,] +oo ) )
24 15 23 sselid 
 |-  ( ph -> ( M ` B ) e. RR* )
25 inelsiga 
 |-  ( ( S e. U. ran sigAlgebra /\ A e. S /\ B e. S ) -> ( A i^i B ) e. S )
26 7 2 3 25 syl3anc 
 |-  ( ph -> ( A i^i B ) e. S )
27 measvxrge0 
 |-  ( ( M e. ( measures ` S ) /\ ( A i^i B ) e. S ) -> ( M ` ( A i^i B ) ) e. ( 0 [,] +oo ) )
28 1 26 27 syl2anc 
 |-  ( ph -> ( M ` ( A i^i B ) ) e. ( 0 [,] +oo ) )
29 elxrge0 
 |-  ( ( M ` ( A i^i B ) ) e. ( 0 [,] +oo ) <-> ( ( M ` ( A i^i B ) ) e. RR* /\ 0 <_ ( M ` ( A i^i B ) ) ) )
30 28 29 sylib 
 |-  ( ph -> ( ( M ` ( A i^i B ) ) e. RR* /\ 0 <_ ( M ` ( A i^i B ) ) ) )
31 30 simprd 
 |-  ( ph -> 0 <_ ( M ` ( A i^i B ) ) )
32 15 28 sselid 
 |-  ( ph -> ( M ` ( A i^i B ) ) e. RR* )
33 xraddge02 
 |-  ( ( ( M ` ( A \ B ) ) e. RR* /\ ( M ` ( A i^i B ) ) e. RR* ) -> ( 0 <_ ( M ` ( A i^i B ) ) -> ( M ` ( A \ B ) ) <_ ( ( M ` ( A \ B ) ) +e ( M ` ( A i^i B ) ) ) ) )
34 18 32 33 syl2anc 
 |-  ( ph -> ( 0 <_ ( M ` ( A i^i B ) ) -> ( M ` ( A \ B ) ) <_ ( ( M ` ( A \ B ) ) +e ( M ` ( A i^i B ) ) ) ) )
35 31 34 mpd 
 |-  ( ph -> ( M ` ( A \ B ) ) <_ ( ( M ` ( A \ B ) ) +e ( M ` ( A i^i B ) ) ) )
36 uncom 
 |-  ( ( A i^i B ) u. ( A \ B ) ) = ( ( A \ B ) u. ( A i^i B ) )
37 inundif 
 |-  ( ( A i^i B ) u. ( A \ B ) ) = A
38 36 37 eqtr3i 
 |-  ( ( A \ B ) u. ( A i^i B ) ) = A
39 38 fveq2i 
 |-  ( M ` ( ( A \ B ) u. ( A i^i B ) ) ) = ( M ` A )
40 incom 
 |-  ( ( A i^i B ) i^i ( A \ B ) ) = ( ( A \ B ) i^i ( A i^i B ) )
41 inindif 
 |-  ( ( A i^i B ) i^i ( A \ B ) ) = (/)
42 40 41 eqtr3i 
 |-  ( ( A \ B ) i^i ( A i^i B ) ) = (/)
43 42 a1i 
 |-  ( ph -> ( ( A \ B ) i^i ( A i^i B ) ) = (/) )
44 measun 
 |-  ( ( M e. ( measures ` S ) /\ ( ( A \ B ) e. S /\ ( A i^i B ) e. S ) /\ ( ( A \ B ) i^i ( A i^i B ) ) = (/) ) -> ( M ` ( ( A \ B ) u. ( A i^i B ) ) ) = ( ( M ` ( A \ B ) ) +e ( M ` ( A i^i B ) ) ) )
45 1 9 26 43 44 syl121anc 
 |-  ( ph -> ( M ` ( ( A \ B ) u. ( A i^i B ) ) ) = ( ( M ` ( A \ B ) ) +e ( M ` ( A i^i B ) ) ) )
46 39 45 eqtr3id 
 |-  ( ph -> ( M ` A ) = ( ( M ` ( A \ B ) ) +e ( M ` ( A i^i B ) ) ) )
47 35 46 breqtrrd 
 |-  ( ph -> ( M ` ( A \ B ) ) <_ ( M ` A ) )
48 xleadd1a 
 |-  ( ( ( ( M ` ( A \ B ) ) e. RR* /\ ( M ` A ) e. RR* /\ ( M ` B ) e. RR* ) /\ ( M ` ( A \ B ) ) <_ ( M ` A ) ) -> ( ( M ` ( A \ B ) ) +e ( M ` B ) ) <_ ( ( M ` A ) +e ( M ` B ) ) )
49 18 21 24 47 48 syl31anc 
 |-  ( ph -> ( ( M ` ( A \ B ) ) +e ( M ` B ) ) <_ ( ( M ` A ) +e ( M ` B ) ) )
50 14 49 eqbrtrd 
 |-  ( ph -> ( M ` ( A u. B ) ) <_ ( ( M ` A ) +e ( M ` B ) ) )