Theorem caragensplit 40714

Description: If E is in the set generated by the Caratheodory's method, then it splits any set A in two parts such that the sum of the outer measures of the two parts is equal to the outer measure of the whole set A. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
caragensplit.o	|- ( ph -> O e. OutMeas )
caragensplit.s	|- S = (CaraGen ` O )
caragensplit.x	|- X = U. dom O
caragensplit.e	|- ( ph -> E e. S )
caragensplit.a	|- ( ph -> A C_ X )

Assertion

Ref	Expression
caragensplit	|- ( ph -> ( ( O ` ( A i^i E ) ) +e ( O ` ( A \ E ) ) ) = ( O ` A ) )

Proof of Theorem caragensplit

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		caragensplit.a	. . . 4 |- ( ph -> A C_ X )
2		caragensplit.o	. . . . . . 7 |- ( ph -> O e. OutMeas )
3		caragensplit.x	. . . . . . 7 |- X = U. dom O
4	2, 3	unidmex 39217	. . . . . 6 |- ( ph -> X e. _V )
5		ssexg 4804	. . . . . 6 |- ( ( A C_ X /\ X e. _V ) -> A e. _V )
6	1, 4, 5	syl2anc 693	. . . . 5 |- ( ph -> A e. _V )
7		elpwg 4166	. . . . 5 |- ( A e. _V -> ( A e. ~P X <-> A C_ X ) )
8	6, 7	syl 17	. . . 4 |- ( ph -> ( A e. ~P X <-> A C_ X ) )
9	1, 8	mpbird 247	. . 3 |- ( ph -> A e. ~P X )
10	3	pweqi 4162	. . 3 |- ~P X = ~P U. dom O
11	9, 10	syl6eleq 2711	. 2 |- ( ph -> A e. ~P U. dom O )
12		caragensplit.e	. . . 4 |- ( ph -> E e. S )
13		caragensplit.s	. . . . 5 |- S = (CaraGen ` O )
14	2, 13	caragenel 40709	. . . 4 |- ( ph -> ( E e. S <-> ( E e. ~P U. dom O /\ A. a e. ~P U. dom O ( ( O ` ( a i^i E ) ) +e ( O ` ( a \ E ) ) ) = ( O ` a ) ) ) )
15	12, 14	mpbid 222	. . 3 |- ( ph -> ( E e. ~P U. dom O /\ A. a e. ~P U. dom O ( ( O ` ( a i^i E ) ) +e ( O ` ( a \ E ) ) ) = ( O ` a ) ) )
16	15	simprd 479	. 2 |- ( ph -> A. a e. ~P U. dom O ( ( O ` ( a i^i E ) ) +e ( O ` ( a \ E ) ) ) = ( O ` a ) )
17		ineq1 3807	. . . . . 6 |- ( a = A -> ( a i^i E ) = ( A i^i E ) )
18	17	fveq2d 6195	. . . . 5 |- ( a = A -> ( O ` ( a i^i E ) ) = ( O ` ( A i^i E ) ) )
19		difeq1 3721	. . . . . 6 |- ( a = A -> ( a \ E ) = ( A \ E ) )
20	19	fveq2d 6195	. . . . 5 |- ( a = A -> ( O ` ( a \ E ) ) = ( O ` ( A \ E ) ) )
21	18, 20	oveq12d 6668	. . . 4 |- ( a = A -> ( ( O ` ( a i^i E ) ) +e ( O ` ( a \ E ) ) ) = ( ( O ` ( A i^i E ) ) +e ( O ` ( A \ E ) ) ) )
22		fveq2 6191	. . . 4 |- ( a = A -> ( O ` a ) = ( O ` A ) )
23	21, 22	eqeq12d 2637	. . 3 |- ( a = A -> ( ( ( O ` ( a i^i E ) ) +e ( O ` ( a \ E ) ) ) = ( O ` a ) <-> ( ( O ` ( A i^i E ) ) +e ( O ` ( A \ E ) ) ) = ( O ` A ) ) )
24	23	rspcva 3307	. 2 |- ( ( A e. ~P U. dom O /\ A. a e. ~P U. dom O ( ( O ` ( a i^i E ) ) +e ( O ` ( a \ E ) ) ) = ( O ` a ) ) -> ( ( O ` ( A i^i E ) ) +e ( O ` ( A \ E ) ) ) = ( O ` A ) )
25	11, 16, 24	syl2anc 693	1 |- ( ph -> ( ( O ` ( A i^i E ) ) +e ( O ` ( A \ E ) ) ) = ( O ` A ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   \ cdif 3571   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   U.cuni 4436   dom cdm 5114   ` cfv 5888  (class class class)co 6650   +ecxad 11944  OutMeascome 40703  CaraGenccaragen 40705

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-caragen 40706

This theorem is referenced by:  caragenuncllem  40726  carageniuncllem1  40735  carageniuncllem2  40736  caratheodorylem1  40740