Theorem meadjuni 40674

Description: The measure of the disjoint union of a countable set is the extended sum of the measures. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
meadjuni.m	|- ( ph -> M e. Meas )
meadjuni.s	|- S = dom M
meadjuni.x	|- ( ph -> X C_ S )
meadjuni.cnb	|- ( ph -> X ~<_ om )
meadjuni.dj	|- ( ph -> Disj x e. X x )

Assertion

Ref	Expression
meadjuni	|- ( ph -> ( M ` U. X ) = (Σ^ ` ( M |` X ) ) )

Distinct variable group:   x, X

Allowed substitution hints:   ph( x)   S( x)   M( x)

Proof of Theorem meadjuni

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		meadjuni.cnb	. . 3 |- ( ph -> X ~<_ om )
2		meadjuni.dj	. . 3 |- ( ph -> Disj x e. X x )
3	1, 2	jca 554	. 2 |- ( ph -> ( X ~<_ om /\ Disj x e. X x ) )
4		meadjuni.x	. . . . 5 |- ( ph -> X C_ S )
5		meadjuni.s	. . . . 5 |- S = dom M
6	4, 5	syl6sseq 3651	. . . 4 |- ( ph -> X C_ dom M )
7		meadjuni.m	. . . . . . 7 |- ( ph -> M e. Meas )
8	7, 5	dmmeasal 40669	. . . . . 6 |- ( ph -> S e. SAlg )
9	8, 4	ssexd 4805	. . . . 5 |- ( ph -> X e. _V )
10		elpwg 4166	. . . . 5 |- ( X e. _V -> ( X e. ~P dom M <-> X C_ dom M ) )
11	9, 10	syl 17	. . . 4 |- ( ph -> ( X e. ~P dom M <-> X C_ dom M ) )
12	6, 11	mpbird 247	. . 3 |- ( ph -> X e. ~P dom M )
13		ismea 40668	. . . . 5 |- ( M e. Meas <-> ( ( ( M : dom M --> ( 0 [,] +oo ) /\ dom M e. SAlg ) /\ ( M ` (/) ) = 0 ) /\ A. y e. ~P dom M ( ( y ~<_ om /\ Disj x e. y x ) -> ( M ` U. y ) = (Σ^ ` ( M |` y ) ) ) ) )
14	7, 13	sylib 208	. . . 4 |- ( ph -> ( ( ( M : dom M --> ( 0 [,] +oo ) /\ dom M e. SAlg ) /\ ( M ` (/) ) = 0 ) /\ A. y e. ~P dom M ( ( y ~<_ om /\ Disj x e. y x ) -> ( M ` U. y ) = (Σ^ ` ( M |` y ) ) ) ) )
15	14	simprd 479	. . 3 |- ( ph -> A. y e. ~P dom M ( ( y ~<_ om /\ Disj x e. y x ) -> ( M ` U. y ) = (Σ^ ` ( M |` y ) ) ) )
16		breq1 4656	. . . . . 6 |- ( y = X -> ( y ~<_ om <-> X ~<_ om ) )
17		disjeq1 4627	. . . . . 6 |- ( y = X -> (Disj x e. y x <-> Disj x e. X x ) )
18	16, 17	anbi12d 747	. . . . 5 |- ( y = X -> ( ( y ~<_ om /\ Disj x e. y x ) <-> ( X ~<_ om /\ Disj x e. X x ) ) )
19		unieq 4444	. . . . . . 7 |- ( y = X -> U. y = U. X )
20	19	fveq2d 6195	. . . . . 6 |- ( y = X -> ( M ` U. y ) = ( M ` U. X ) )
21		reseq2 5391	. . . . . . 7 |- ( y = X -> ( M |` y ) = ( M |` X ) )
22	21	fveq2d 6195	. . . . . 6 |- ( y = X -> (Σ^ ` ( M |` y ) ) = (Σ^ ` ( M |` X ) ) )
23	20, 22	eqeq12d 2637	. . . . 5 |- ( y = X -> ( ( M ` U. y ) = (Σ^ ` ( M |` y ) ) <-> ( M ` U. X ) = (Σ^ ` ( M |` X ) ) ) )
24	18, 23	imbi12d 334	. . . 4 |- ( y = X -> ( ( ( y ~<_ om /\ Disj x e. y x ) -> ( M ` U. y ) = (Σ^ ` ( M |` y ) ) ) <-> ( ( X ~<_ om /\ Disj x e. X x ) -> ( M ` U. X ) = (Σ^ ` ( M |` X ) ) ) ) )
25	24	rspcva 3307	. . 3 |- ( ( X e. ~P dom M /\ A. y e. ~P dom M ( ( y ~<_ om /\ Disj x e. y x ) -> ( M ` U. y ) = (Σ^ ` ( M |` y ) ) ) ) -> ( ( X ~<_ om /\ Disj x e. X x ) -> ( M ` U. X ) = (Σ^ ` ( M |` X ) ) ) )
26	12, 15, 25	syl2anc 693	. 2 |- ( ph -> ( ( X ~<_ om /\ Disj x e. X x ) -> ( M ` U. X ) = (Σ^ ` ( M |` X ) ) ) )
27	3, 26	mpd 15	1 |- ( ph -> ( M ` U. X ) = (Σ^ ` ( M |` X ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   U.cuni 4436  Disj wdisj 4620   class class class wbr 4653   dom cdm 5114   |` cres 5116   -->wf 5884   ` cfv 5888  (class class class)co 6650   omcom 7065   ~<_ cdom 7953   0cc0 9936   +oocpnf 10071   [,]cicc 12178  SAlgcsalg 40528  Σ^{^}csumge0 40579  Meascmea 40666

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-iun 4522  df-disj 4621  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-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-mea 40667

This theorem is referenced by:  meadjun  40679  meadjiun  40683