Theorem omeiunle 40731

Description: The outer measure of the indexed union of a countable set is the less than or equal to the extended sum of the outer measures. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
omeiunle.nph	|- F/ n ph
omeiunle.ne	|- F/_ n E
omeiunle.o	|- ( ph -> O e. OutMeas )
omeiunle.x	|- X = U. dom O
omeiunle.z	|- Z = ( ZZ>= ` N )
omeiunle.e	|- ( ph -> E : Z --> ~P X )

Assertion

Ref	Expression
omeiunle	|- ( ph -> ( O ` U_ n e. Z ( E ` n ) ) <_ (Σ^ ` ( n e. Z |-> ( O ` ( E ` n ) ) ) ) )

Distinct variable groups:   n, O   n, X   n, Z

Allowed substitution hints:   ph( n)   E( n)   N( n)

Proof of Theorem omeiunle

Dummy variable m is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iccssxr 12256	. . 3 |- ( 0 [,] +oo ) C_ RR*
2		omeiunle.o	. . . 4 |- ( ph -> O e. OutMeas )
3		omeiunle.x	. . . 4 |- X = U. dom O
4		omeiunle.nph	. . . . . 6 |- F/ n ph
5		omeiunle.e	. . . . . . . . 9 |- ( ph -> E : Z --> ~P X )
6	5	ffvelrnda 6359	. . . . . . . 8 |- ( ( ph /\ n e. Z ) -> ( E ` n ) e. ~P X )
7		elpwi 4168	. . . . . . . 8 |- ( ( E ` n ) e. ~P X -> ( E ` n ) C_ X )
8	6, 7	syl 17	. . . . . . 7 |- ( ( ph /\ n e. Z ) -> ( E ` n ) C_ X )
9	8	ex 450	. . . . . 6 |- ( ph -> ( n e. Z -> ( E ` n ) C_ X ) )
10	4, 9	ralrimi 2957	. . . . 5 |- ( ph -> A. n e. Z ( E ` n ) C_ X )
11		iunss 4561	. . . . 5 |- ( U_ n e. Z ( E ` n ) C_ X <-> A. n e. Z ( E ` n ) C_ X )
12	10, 11	sylibr 224	. . . 4 |- ( ph -> U_ n e. Z ( E ` n ) C_ X )
13	2, 3, 12	omecl 40717	. . 3 |- ( ph -> ( O ` U_ n e. Z ( E ` n ) ) e. ( 0 [,] +oo ) )
14	1, 13	sseldi 3601	. 2 |- ( ph -> ( O ` U_ n e. Z ( E ` n ) ) e. RR* )
15	5	ffnd 6046	. . . . 5 |- ( ph -> E Fn Z )
16		omeiunle.z	. . . . . . 7 |- Z = ( ZZ>= ` N )
17		fvex 6201	. . . . . . 7 |- ( ZZ>= ` N ) e. _V
18	16, 17	eqeltri 2697	. . . . . 6 |- Z e. _V
19	18	a1i 11	. . . . 5 |- ( ph -> Z e. _V )
20		fnex 6481	. . . . 5 |- ( ( E Fn Z /\ Z e. _V ) -> E e. _V )
21	15, 19, 20	syl2anc 693	. . . 4 |- ( ph -> E e. _V )
22		rnexg 7098	. . . 4 |- ( E e. _V -> ran E e. _V )
23	21, 22	syl 17	. . 3 |- ( ph -> ran E e. _V )
24	2, 3	omef 40710	. . . 4 |- ( ph -> O : ~P X --> ( 0 [,] +oo ) )
25		frn 6053	. . . . 5 |- ( E : Z --> ~P X -> ran E C_ ~P X )
26	5, 25	syl 17	. . . 4 |- ( ph -> ran E C_ ~P X )
27	24, 26	fssresd 6071	. . 3 |- ( ph -> ( O |` ran E ) : ran E --> ( 0 [,] +oo ) )
28	23, 27	sge0xrcl 40602	. 2 |- ( ph -> (Σ^ ` ( O |` ran E ) ) e. RR* )
29	2	adantr 481	. . . . 5 |- ( ( ph /\ n e. Z ) -> O e. OutMeas )
30	29, 3, 8	omecl 40717	. . . 4 |- ( ( ph /\ n e. Z ) -> ( O ` ( E ` n ) ) e. ( 0 [,] +oo ) )
31		eqid 2622	. . . 4 |- ( n e. Z |-> ( O ` ( E ` n ) ) ) = ( n e. Z |-> ( O ` ( E ` n ) ) )
32	4, 30, 31	fmptdf 6387	. . 3 |- ( ph -> ( n e. Z |-> ( O ` ( E ` n ) ) ) : Z --> ( 0 [,] +oo ) )
33	19, 32	sge0xrcl 40602	. 2 |- ( ph -> (Σ^ ` ( n e. Z |-> ( O ` ( E ` n ) ) ) ) e. RR* )
34		fvex 6201	. . . . . . . 8 |- ( E ` n ) e. _V
35	34	rgenw 2924	. . . . . . 7 |- A. n e. Z ( E ` n ) e. _V
36		dfiun3g 5378	. . . . . . 7 |- ( A. n e. Z ( E ` n ) e. _V -> U_ n e. Z ( E ` n ) = U. ran ( n e. Z |-> ( E ` n ) ) )
37	35, 36	ax-mp 5	. . . . . 6 |- U_ n e. Z ( E ` n ) = U. ran ( n e. Z |-> ( E ` n ) )
38	37	a1i 11	. . . . 5 |- ( ph -> U_ n e. Z ( E ` n ) = U. ran ( n e. Z |-> ( E ` n ) ) )
39	5	feqmptd 6249	. . . . . . . 8 |- ( ph -> E = ( m e. Z |-> ( E ` m ) ) )
40		omeiunle.ne	. . . . . . . . . . 11 |- F/_ n E
41		nfcv 2764	. . . . . . . . . . 11 |- F/_ n m
42	40, 41	nffv 6198	. . . . . . . . . 10 |- F/_ n ( E ` m )
43		nfcv 2764	. . . . . . . . . 10 |- F/_ m ( E ` n )
44		fveq2 6191	. . . . . . . . . 10 |- ( m = n -> ( E ` m ) = ( E ` n ) )
45	42, 43, 44	cbvmpt 4749	. . . . . . . . 9 |- ( m e. Z |-> ( E ` m ) ) = ( n e. Z |-> ( E ` n ) )
46	45	a1i 11	. . . . . . . 8 |- ( ph -> ( m e. Z |-> ( E ` m ) ) = ( n e. Z |-> ( E ` n ) ) )
47	39, 46	eqtrd 2656	. . . . . . 7 |- ( ph -> E = ( n e. Z |-> ( E ` n ) ) )
48	47	rneqd 5353	. . . . . 6 |- ( ph -> ran E = ran ( n e. Z |-> ( E ` n ) ) )
49	48	unieqd 4446	. . . . 5 |- ( ph -> U. ran E = U. ran ( n e. Z |-> ( E ` n ) ) )
50	38, 49	eqtr4d 2659	. . . 4 |- ( ph -> U_ n e. Z ( E ` n ) = U. ran E )
51	50	fveq2d 6195	. . 3 |- ( ph -> ( O ` U_ n e. Z ( E ` n ) ) = ( O ` U. ran E ) )
52		fnrndomg 9358	. . . . . 6 |- ( Z e. _V -> ( E Fn Z -> ran E ~<_ Z ) )
53	19, 15, 52	sylc 65	. . . . 5 |- ( ph -> ran E ~<_ Z )
54	16	uzct 39232	. . . . . 6 |- Z ~<_ om
55	54	a1i 11	. . . . 5 |- ( ph -> Z ~<_ om )
56		domtr 8009	. . . . 5 |- ( ( ran E ~<_ Z /\ Z ~<_ om ) -> ran E ~<_ om )
57	53, 55, 56	syl2anc 693	. . . 4 |- ( ph -> ran E ~<_ om )
58	2, 3, 26, 57	omeunile 40719	. . 3 |- ( ph -> ( O ` U. ran E ) <_ (Σ^ ` ( O |` ran E ) ) )
59	51, 58	eqbrtrd 4675	. 2 |- ( ph -> ( O ` U_ n e. Z ( E ` n ) ) <_ (Σ^ ` ( O |` ran E ) ) )
60		ltweuz 12760	. . . . . 6 |- < We ( ZZ>= ` N )
61		weeq2 5103	. . . . . . 7 |- ( Z = ( ZZ>= ` N ) -> ( < We Z <-> < We ( ZZ>= ` N ) ) )
62	16, 61	ax-mp 5	. . . . . 6 |- ( < We Z <-> < We ( ZZ>= ` N ) )
63	60, 62	mpbir 221	. . . . 5 |- < We Z
64	63	a1i 11	. . . 4 |- ( ph -> < We Z )
65	19, 24, 5, 64	sge0resrn 40621	. . 3 |- ( ph -> (Σ^ ` ( O |` ran E ) ) <_ (Σ^ ` ( O o. E ) ) )
66		fcompt 6400	. . . . . 6 |- ( ( O : ~P X --> ( 0 [,] +oo ) /\ E : Z --> ~P X ) -> ( O o. E ) = ( m e. Z |-> ( O ` ( E ` m ) ) ) )
67		nfcv 2764	. . . . . . . . 9 |- F/_ n O
68	67, 42	nffv 6198	. . . . . . . 8 |- F/_ n ( O ` ( E ` m ) )
69		nfcv 2764	. . . . . . . 8 |- F/_ m ( O ` ( E ` n ) )
70	44	fveq2d 6195	. . . . . . . 8 |- ( m = n -> ( O ` ( E ` m ) ) = ( O ` ( E ` n ) ) )
71	68, 69, 70	cbvmpt 4749	. . . . . . 7 |- ( m e. Z |-> ( O ` ( E ` m ) ) ) = ( n e. Z |-> ( O ` ( E ` n ) ) )
72	71	a1i 11	. . . . . 6 |- ( ( O : ~P X --> ( 0 [,] +oo ) /\ E : Z --> ~P X ) -> ( m e. Z |-> ( O ` ( E ` m ) ) ) = ( n e. Z |-> ( O ` ( E ` n ) ) ) )
73	66, 72	eqtrd 2656	. . . . 5 |- ( ( O : ~P X --> ( 0 [,] +oo ) /\ E : Z --> ~P X ) -> ( O o. E ) = ( n e. Z |-> ( O ` ( E ` n ) ) ) )
74	24, 5, 73	syl2anc 693	. . . 4 |- ( ph -> ( O o. E ) = ( n e. Z |-> ( O ` ( E ` n ) ) ) )
75	74	fveq2d 6195	. . 3 |- ( ph -> (Σ^ ` ( O o. E ) ) = (Σ^ ` ( n e. Z |-> ( O ` ( E ` n ) ) ) ) )
76	65, 75	breqtrd 4679	. 2 |- ( ph -> (Σ^ ` ( O |` ran E ) ) <_ (Σ^ ` ( n e. Z |-> ( O ` ( E ` n ) ) ) ) )
77	14, 28, 33, 59, 76	xrletrd 11993	1 |- ( ph -> ( O ` U_ n e. Z ( E ` n ) ) <_ (Σ^ ` ( n e. Z |-> ( O ` ( E ` n ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   F/wnf 1708   e. wcel 1990   F/_wnfc 2751   A.wral 2912   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   U.cuni 4436   U_ciun 4520   class class class wbr 4653   |-> cmpt 4729   We wwe 5072   dom cdm 5114   ran crn 5115   |` cres 5116   o. ccom 5118   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   omcom 7065   ~<_ cdom 7953   0cc0 9936   +oocpnf 10071   RR*cxr 10073   < clt 10074   <_ cle 10075   ZZ>=cuz 11687   [,]cicc 12178  Σ^{^}csumge0 40579  OutMeascome 40703

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-ac2 9285  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-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-omul 7565  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-ac 8939  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-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-ome 40704

This theorem is referenced by:  omeiunltfirp  40733  omeiunlempt  40734  caratheodorylem2  40741