Theorem meadjiun 40683

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
meadjiun.1	|- F/ k ph
meadjiun.m	|- ( ph -> M e. Meas )
meadjiun.s	|- S = dom M
meadjiun.b	|- ( ( ph /\ k e. A ) -> B e. S )
meadjiun.a	|- ( ph -> A ~<_ om )
meadjiun.dj	|- ( ph -> Disj k e. A B )

Assertion

Ref	Expression
meadjiun	|- ( ph -> ( M ` U_ k e. A B ) = (Σ^ ` ( k e. A |-> ( M ` B ) ) ) )

Distinct variable groups:   A, k   k, M   S, k

Allowed substitution hints:   ph( k)   B( k)

Proof of Theorem meadjiun

Dummy variables x i j y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		meadjiun.1	. . . . 5 |- F/ k ph
2		meadjiun.b	. . . . . 6 |- ( ( ph /\ k e. A ) -> B e. S )
3	2	ex 450	. . . . 5 |- ( ph -> ( k e. A -> B e. S ) )
4	1, 3	ralrimi 2957	. . . 4 |- ( ph -> A. k e. A B e. S )
5		dfiun3g 5378	. . . 4 |- ( A. k e. A B e. S -> U_ k e. A B = U. ran ( k e. A |-> B ) )
6	4, 5	syl 17	. . 3 |- ( ph -> U_ k e. A B = U. ran ( k e. A |-> B ) )
7	6	fveq2d 6195	. 2 |- ( ph -> ( M ` U_ k e. A B ) = ( M ` U. ran ( k e. A |-> B ) ) )
8		meadjiun.m	. . 3 |- ( ph -> M e. Meas )
9		meadjiun.s	. . 3 |- S = dom M
10		eqid 2622	. . . . 5 |- ( k e. A |-> B ) = ( k e. A |-> B )
11	10	rnmptss 6392	. . . 4 |- ( A. k e. A B e. S -> ran ( k e. A |-> B ) C_ S )
12	4, 11	syl 17	. . 3 |- ( ph -> ran ( k e. A |-> B ) C_ S )
13		meadjiun.a	. . . 4 |- ( ph -> A ~<_ om )
14		1stcrestlem 21255	. . . 4 |- ( A ~<_ om -> ran ( k e. A |-> B ) ~<_ om )
15	13, 14	syl 17	. . 3 |- ( ph -> ran ( k e. A |-> B ) ~<_ om )
16		meadjiun.dj	. . . 4 |- ( ph -> Disj k e. A B )
17	10	disjrnmpt2 39375	. . . 4 |- (Disj k e. A B -> Disj x e. ran ( k e. A |-> B ) x )
18	16, 17	syl 17	. . 3 |- ( ph -> Disj x e. ran ( k e. A |-> B ) x )
19	8, 9, 12, 15, 18	meadjuni 40674	. 2 |- ( ph -> ( M ` U. ran ( k e. A |-> B ) ) = (Σ^ ` ( M |` ran ( k e. A |-> B ) ) ) )
20		reldom 7961	. . . . . 6 |- Rel ~<_
21		brrelex 5156	. . . . . 6 |- ( ( Rel ~<_ /\ A ~<_ om ) -> A e. _V )
22	20, 21	mpan 706	. . . . 5 |- ( A ~<_ om -> A e. _V )
23	13, 22	syl 17	. . . 4 |- ( ph -> A e. _V )
24	1, 2, 10	fmptdf 6387	. . . 4 |- ( ph -> ( k e. A |-> B ) : A --> S )
25		fveq2 6191	. . . . . 6 |- ( j = i -> ( ( k e. A |-> B ) ` j ) = ( ( k e. A |-> B ) ` i ) )
26	25	neeq1d 2853	. . . . 5 |- ( j = i -> ( ( ( k e. A |-> B ) ` j ) =/= (/) <-> ( ( k e. A |-> B ) ` i ) =/= (/) ) )
27	26	cbvrabv 3199	. . . 4 |- { j e. A | ( ( k e. A |-> B ) ` j ) =/= (/) } = { i e. A | ( ( k e. A |-> B ) ` i ) =/= (/) }
28		simpr 477	. . . . . . . 8 |- ( ( ph /\ i e. A ) -> i e. A )
29		nfv 1843	. . . . . . . . . . 11 |- F/ k i e. A
30	1, 29	nfan 1828	. . . . . . . . . 10 |- F/ k ( ph /\ i e. A )
31		nfcv 2764	. . . . . . . . . . . 12 |- F/_ k i
32	31	nfcsb1 3548	. . . . . . . . . . 11 |- F/_ k [_ i / k ]_ B
33		nfcv 2764	. . . . . . . . . . 11 |- F/_ k S
34	32, 33	nfel 2777	. . . . . . . . . 10 |- F/ k [_ i / k ]_ B e. S
35	30, 34	nfim 1825	. . . . . . . . 9 |- F/ k ( ( ph /\ i e. A ) -> [_ i / k ]_ B e. S )
36		eleq1 2689	. . . . . . . . . . 11 |- ( k = i -> ( k e. A <-> i e. A ) )
37	36	anbi2d 740	. . . . . . . . . 10 |- ( k = i -> ( ( ph /\ k e. A ) <-> ( ph /\ i e. A ) ) )
38		csbeq1a 3542	. . . . . . . . . . 11 |- ( k = i -> B = [_ i / k ]_ B )
39	38	eleq1d 2686	. . . . . . . . . 10 |- ( k = i -> ( B e. S <-> [_ i / k ]_ B e. S ) )
40	37, 39	imbi12d 334	. . . . . . . . 9 |- ( k = i -> ( ( ( ph /\ k e. A ) -> B e. S ) <-> ( ( ph /\ i e. A ) -> [_ i / k ]_ B e. S ) ) )
41	35, 40, 2	chvar 2262	. . . . . . . 8 |- ( ( ph /\ i e. A ) -> [_ i / k ]_ B e. S )
42	31, 32, 38, 10	fvmptf 6301	. . . . . . . 8 |- ( ( i e. A /\ [_ i / k ]_ B e. S ) -> ( ( k e. A |-> B ) ` i ) = [_ i / k ]_ B )
43	28, 41, 42	syl2anc 693	. . . . . . 7 |- ( ( ph /\ i e. A ) -> ( ( k e. A |-> B ) ` i ) = [_ i / k ]_ B )
44	43	disjeq2dv 4625	. . . . . 6 |- ( ph -> (Disj i e. A ( ( k e. A |-> B ) ` i ) <-> Disj i e. A [_ i / k ]_ B ) )
45		nfcv 2764	. . . . . . . . 9 |- F/_ i B
46	45, 32, 38	cbvdisj 4630	. . . . . . . 8 |- (Disj k e. A B <-> Disj i e. A [_ i / k ]_ B )
47	46	bicomi 214	. . . . . . 7 |- (Disj i e. A [_ i / k ]_ B <-> Disj k e. A B )
48	47	a1i 11	. . . . . 6 |- ( ph -> (Disj i e. A [_ i / k ]_ B <-> Disj k e. A B ) )
49	44, 48	bitrd 268	. . . . 5 |- ( ph -> (Disj i e. A ( ( k e. A |-> B ) ` i ) <-> Disj k e. A B ) )
50	16, 49	mpbird 247	. . . 4 |- ( ph -> Disj i e. A ( ( k e. A |-> B ) ` i ) )
51	8, 9, 23, 24, 27, 50	meadjiunlem 40682	. . 3 |- ( ph -> (Σ^ ` ( M |` ran ( k e. A |-> B ) ) ) = (Σ^ ` ( M o. ( k e. A |-> B ) ) ) )
52	45, 32, 38	cbvmpt 4749	. . . . . . 7 |- ( k e. A |-> B ) = ( i e. A |-> [_ i / k ]_ B )
53	52	coeq2i 5282	. . . . . 6 |- ( M o. ( k e. A |-> B ) ) = ( M o. ( i e. A |-> [_ i / k ]_ B ) )
54	53	a1i 11	. . . . 5 |- ( ph -> ( M o. ( k e. A |-> B ) ) = ( M o. ( i e. A |-> [_ i / k ]_ B ) ) )
55		eqidd 2623	. . . . . 6 |- ( ph -> ( i e. A |-> [_ i / k ]_ B ) = ( i e. A |-> [_ i / k ]_ B ) )
56	8, 9	meaf 40670	. . . . . . 7 |- ( ph -> M : S --> ( 0 [,] +oo ) )
57	56	feqmptd 6249	. . . . . 6 |- ( ph -> M = ( y e. S |-> ( M ` y ) ) )
58		fveq2 6191	. . . . . 6 |- ( y = [_ i / k ]_ B -> ( M ` y ) = ( M ` [_ i / k ]_ B ) )
59	41, 55, 57, 58	fmptco 6396	. . . . 5 |- ( ph -> ( M o. ( i e. A |-> [_ i / k ]_ B ) ) = ( i e. A |-> ( M ` [_ i / k ]_ B ) ) )
60		nfcv 2764	. . . . . . . 8 |- F/_ i ( M ` B )
61		nfcv 2764	. . . . . . . . 9 |- F/_ k M
62	61, 32	nffv 6198	. . . . . . . 8 |- F/_ k ( M ` [_ i / k ]_ B )
63	38	fveq2d 6195	. . . . . . . 8 |- ( k = i -> ( M ` B ) = ( M ` [_ i / k ]_ B ) )
64	60, 62, 63	cbvmpt 4749	. . . . . . 7 |- ( k e. A |-> ( M ` B ) ) = ( i e. A |-> ( M ` [_ i / k ]_ B ) )
65	64	eqcomi 2631	. . . . . 6 |- ( i e. A |-> ( M ` [_ i / k ]_ B ) ) = ( k e. A |-> ( M ` B ) )
66	65	a1i 11	. . . . 5 |- ( ph -> ( i e. A |-> ( M ` [_ i / k ]_ B ) ) = ( k e. A |-> ( M ` B ) ) )
67	54, 59, 66	3eqtrd 2660	. . . 4 |- ( ph -> ( M o. ( k e. A |-> B ) ) = ( k e. A |-> ( M ` B ) ) )
68	67	fveq2d 6195	. . 3 |- ( ph -> (Σ^ ` ( M o. ( k e. A |-> B ) ) ) = (Σ^ ` ( k e. A |-> ( M ` B ) ) ) )
69	51, 68	eqtrd 2656	. 2 |- ( ph -> (Σ^ ` ( M |` ran ( k e. A |-> B ) ) ) = (Σ^ ` ( k e. A |-> ( M ` B ) ) ) )
70	7, 19, 69	3eqtrd 2660	1 |- ( ph -> ( M ` U_ k e. A B ) = (Σ^ ` ( k e. A |-> ( M ` B ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   F/wnf 1708   e. wcel 1990   =/= wne 2794   A.wral 2912   {crab 2916   _Vcvv 3200   [_csb 3533   C_ wss 3574   (/)c0 3915   U.cuni 4436   U_ciun 4520  Disj wdisj 4620   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   ran crn 5115   |` cres 5116   o. ccom 5118   Rel wrel 5119   ` cfv 5888  (class class class)co 6650   omcom 7065   ~<_ cdom 7953   0cc0 9936   +oocpnf 10071   [,]cicc 12178  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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-nel 2898  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-disj 4621  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-oi 8415  df-card 8765  df-acn 8768  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-xadd 11947  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-sumge0 40580  df-mea 40667

This theorem is referenced by:  meaiunlelem  40685  meaiuninclem  40697  vonct  40907