Theorem ovoliun2 23274

Description: The Lebesgue outer measure function is countably sub-additive. (This version is a little easier to read, but does not allow infinite values like ovoliun 23273.) (Contributed by Mario Carneiro, 12-Jun-2014.)

Hypotheses

Ref	Expression
ovoliun.t	|- T = seq 1 ( + , G )
ovoliun.g	|- G = ( n e. NN |-> ( vol* ` A ) )
ovoliun.a	|- ( ( ph /\ n e. NN ) -> A C_ RR )
ovoliun.v	|- ( ( ph /\ n e. NN ) -> ( vol* ` A ) e. RR )
ovoliun2.t	|- ( ph -> T e. dom ~~> )

Assertion

Ref	Expression
ovoliun2	|- ( ph -> ( vol* ` U_ n e. NN A ) <_ sum_ n e. NN ( vol* ` A ) )

Distinct variable group:   ph, n

Allowed substitution hints:   A( n)   T( n)   G( n)

Proof of Theorem ovoliun2

Dummy variables k m x z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovoliun.t	. . 3 |- T = seq 1 ( + , G )
2		ovoliun.g	. . 3 |- G = ( n e. NN |-> ( vol* ` A ) )
3		ovoliun.a	. . 3 |- ( ( ph /\ n e. NN ) -> A C_ RR )
4		ovoliun.v	. . 3 |- ( ( ph /\ n e. NN ) -> ( vol* ` A ) e. RR )
5	1, 2, 3, 4	ovoliun 23273	. 2 |- ( ph -> ( vol* ` U_ n e. NN A ) <_ sup ( ran T , RR* , < ) )
6		nnuz 11723	. . . . . . . 8 |- NN = ( ZZ>= ` 1 )
7		1zzd 11408	. . . . . . . 8 |- ( ph -> 1 e. ZZ )
8		fvex 6201	. . . . . . . . . . 11 |- ( vol* ` [_ m / n ]_ A ) e. _V
9		nfcv 2764	. . . . . . . . . . . . . 14 |- F/_ m ( vol* ` A )
10		nfcv 2764	. . . . . . . . . . . . . . 15 |- F/_ n vol*
11		nfcsb1v 3549	. . . . . . . . . . . . . . 15 |- F/_ n [_ m / n ]_ A
12	10, 11	nffv 6198	. . . . . . . . . . . . . 14 |- F/_ n ( vol* ` [_ m / n ]_ A )
13		csbeq1a 3542	. . . . . . . . . . . . . . 15 |- ( n = m -> A = [_ m / n ]_ A )
14	13	fveq2d 6195	. . . . . . . . . . . . . 14 |- ( n = m -> ( vol* ` A ) = ( vol* ` [_ m / n ]_ A ) )
15	9, 12, 14	cbvmpt 4749	. . . . . . . . . . . . 13 |- ( n e. NN |-> ( vol* ` A ) ) = ( m e. NN |-> ( vol* ` [_ m / n ]_ A ) )
16	2, 15	eqtri 2644	. . . . . . . . . . . 12 |- G = ( m e. NN |-> ( vol* ` [_ m / n ]_ A ) )
17	16	fvmpt2 6291	. . . . . . . . . . 11 |- ( ( m e. NN /\ ( vol* ` [_ m / n ]_ A ) e. _V ) -> ( G ` m ) = ( vol* ` [_ m / n ]_ A ) )
18	8, 17	mpan2 707	. . . . . . . . . 10 |- ( m e. NN -> ( G ` m ) = ( vol* ` [_ m / n ]_ A ) )
19	18	adantl 482	. . . . . . . . 9 |- ( ( ph /\ m e. NN ) -> ( G ` m ) = ( vol* ` [_ m / n ]_ A ) )
20	4	ralrimiva 2966	. . . . . . . . . . 11 |- ( ph -> A. n e. NN ( vol* ` A ) e. RR )
21	9	nfel1 2779	. . . . . . . . . . . 12 |- F/ m ( vol* ` A ) e. RR
22	12	nfel1 2779	. . . . . . . . . . . 12 |- F/ n ( vol* ` [_ m / n ]_ A ) e. RR
23	14	eleq1d 2686	. . . . . . . . . . . 12 |- ( n = m -> ( ( vol* ` A ) e. RR <-> ( vol* ` [_ m / n ]_ A ) e. RR ) )
24	21, 22, 23	cbvral 3167	. . . . . . . . . . 11 |- ( A. n e. NN ( vol* ` A ) e. RR <-> A. m e. NN ( vol* ` [_ m / n ]_ A ) e. RR )
25	20, 24	sylib 208	. . . . . . . . . 10 |- ( ph -> A. m e. NN ( vol* ` [_ m / n ]_ A ) e. RR )
26	25	r19.21bi 2932	. . . . . . . . 9 |- ( ( ph /\ m e. NN ) -> ( vol* ` [_ m / n ]_ A ) e. RR )
27	19, 26	eqeltrd 2701	. . . . . . . 8 |- ( ( ph /\ m e. NN ) -> ( G ` m ) e. RR )
28	6, 7, 27	serfre 12830	. . . . . . 7 |- ( ph -> seq 1 ( + , G ) : NN --> RR )
29	1	feq1i 6036	. . . . . . 7 |- ( T : NN --> RR <-> seq 1 ( + , G ) : NN --> RR )
30	28, 29	sylibr 224	. . . . . 6 |- ( ph -> T : NN --> RR )
31		frn 6053	. . . . . 6 |- ( T : NN --> RR -> ran T C_ RR )
32	30, 31	syl 17	. . . . 5 |- ( ph -> ran T C_ RR )
33		1nn 11031	. . . . . . . 8 |- 1 e. NN
34		fdm 6051	. . . . . . . . 9 |- ( T : NN --> RR -> dom T = NN )
35	30, 34	syl 17	. . . . . . . 8 |- ( ph -> dom T = NN )
36	33, 35	syl5eleqr 2708	. . . . . . 7 |- ( ph -> 1 e. dom T )
37		ne0i 3921	. . . . . . 7 |- ( 1 e. dom T -> dom T =/= (/) )
38	36, 37	syl 17	. . . . . 6 |- ( ph -> dom T =/= (/) )
39		dm0rn0 5342	. . . . . . 7 |- ( dom T = (/) <-> ran T = (/) )
40	39	necon3bii 2846	. . . . . 6 |- ( dom T =/= (/) <-> ran T =/= (/) )
41	38, 40	sylib 208	. . . . 5 |- ( ph -> ran T =/= (/) )
42		ovoliun2.t	. . . . . . . . 9 |- ( ph -> T e. dom ~~> )
43	1, 42	syl5eqelr 2706	. . . . . . . 8 |- ( ph -> seq 1 ( + , G ) e. dom ~~> )
44	6, 7, 19, 26, 43	isumrecl 14496	. . . . . . 7 |- ( ph -> sum_ m e. NN ( vol* ` [_ m / n ]_ A ) e. RR )
45		elfznn 12370	. . . . . . . . . . . . 13 |- ( m e. ( 1 ... k ) -> m e. NN )
46	45	adantl 482	. . . . . . . . . . . 12 |- ( ( ( ph /\ k e. NN ) /\ m e. ( 1 ... k ) ) -> m e. NN )
47	46, 18	syl 17	. . . . . . . . . . 11 |- ( ( ( ph /\ k e. NN ) /\ m e. ( 1 ... k ) ) -> ( G ` m ) = ( vol* ` [_ m / n ]_ A ) )
48		simpr 477	. . . . . . . . . . . 12 |- ( ( ph /\ k e. NN ) -> k e. NN )
49	48, 6	syl6eleq 2711	. . . . . . . . . . 11 |- ( ( ph /\ k e. NN ) -> k e. ( ZZ>= ` 1 ) )
50		simpl 473	. . . . . . . . . . . . 13 |- ( ( ph /\ k e. NN ) -> ph )
51	50, 45, 26	syl2an 494	. . . . . . . . . . . 12 |- ( ( ( ph /\ k e. NN ) /\ m e. ( 1 ... k ) ) -> ( vol* ` [_ m / n ]_ A ) e. RR )
52	51	recnd 10068	. . . . . . . . . . 11 |- ( ( ( ph /\ k e. NN ) /\ m e. ( 1 ... k ) ) -> ( vol* ` [_ m / n ]_ A ) e. CC )
53	47, 49, 52	fsumser 14461	. . . . . . . . . 10 |- ( ( ph /\ k e. NN ) -> sum_ m e. ( 1 ... k ) ( vol* ` [_ m / n ]_ A ) = ( seq 1 ( + , G ) ` k ) )
54	1	fveq1i 6192	. . . . . . . . . 10 |- ( T ` k ) = ( seq 1 ( + , G ) ` k )
55	53, 54	syl6eqr 2674	. . . . . . . . 9 |- ( ( ph /\ k e. NN ) -> sum_ m e. ( 1 ... k ) ( vol* ` [_ m / n ]_ A ) = ( T ` k ) )
56		fzfid 12772	. . . . . . . . . . 11 |- ( ph -> ( 1 ... k ) e. Fin )
57		elfznn 12370	. . . . . . . . . . . . 13 |- ( n e. ( 1 ... k ) -> n e. NN )
58	57	ssriv 3607	. . . . . . . . . . . 12 |- ( 1 ... k ) C_ NN
59	58	a1i 11	. . . . . . . . . . 11 |- ( ph -> ( 1 ... k ) C_ NN )
60	3	ralrimiva 2966	. . . . . . . . . . . . . 14 |- ( ph -> A. n e. NN A C_ RR )
61		nfv 1843	. . . . . . . . . . . . . . 15 |- F/ m A C_ RR
62		nfcv 2764	. . . . . . . . . . . . . . . 16 |- F/_ n RR
63	11, 62	nfss 3596	. . . . . . . . . . . . . . 15 |- F/ n [_ m / n ]_ A C_ RR
64	13	sseq1d 3632	. . . . . . . . . . . . . . 15 |- ( n = m -> ( A C_ RR <-> [_ m / n ]_ A C_ RR ) )
65	61, 63, 64	cbvral 3167	. . . . . . . . . . . . . 14 |- ( A. n e. NN A C_ RR <-> A. m e. NN [_ m / n ]_ A C_ RR )
66	60, 65	sylib 208	. . . . . . . . . . . . 13 |- ( ph -> A. m e. NN [_ m / n ]_ A C_ RR )
67	66	r19.21bi 2932	. . . . . . . . . . . 12 |- ( ( ph /\ m e. NN ) -> [_ m / n ]_ A C_ RR )
68		ovolge0 23249	. . . . . . . . . . . 12 |- ( [_ m / n ]_ A C_ RR -> 0 <_ ( vol* ` [_ m / n ]_ A ) )
69	67, 68	syl 17	. . . . . . . . . . 11 |- ( ( ph /\ m e. NN ) -> 0 <_ ( vol* ` [_ m / n ]_ A ) )
70	6, 7, 56, 59, 19, 26, 69, 43	isumless 14577	. . . . . . . . . 10 |- ( ph -> sum_ m e. ( 1 ... k ) ( vol* ` [_ m / n ]_ A ) <_ sum_ m e. NN ( vol* ` [_ m / n ]_ A ) )
71	70	adantr 481	. . . . . . . . 9 |- ( ( ph /\ k e. NN ) -> sum_ m e. ( 1 ... k ) ( vol* ` [_ m / n ]_ A ) <_ sum_ m e. NN ( vol* ` [_ m / n ]_ A ) )
72	55, 71	eqbrtrrd 4677	. . . . . . . 8 |- ( ( ph /\ k e. NN ) -> ( T ` k ) <_ sum_ m e. NN ( vol* ` [_ m / n ]_ A ) )
73	72	ralrimiva 2966	. . . . . . 7 |- ( ph -> A. k e. NN ( T ` k ) <_ sum_ m e. NN ( vol* ` [_ m / n ]_ A ) )
74		breq2 4657	. . . . . . . . 9 |- ( x = sum_ m e. NN ( vol* ` [_ m / n ]_ A ) -> ( ( T ` k ) <_ x <-> ( T ` k ) <_ sum_ m e. NN ( vol* ` [_ m / n ]_ A ) ) )
75	74	ralbidv 2986	. . . . . . . 8 |- ( x = sum_ m e. NN ( vol* ` [_ m / n ]_ A ) -> ( A. k e. NN ( T ` k ) <_ x <-> A. k e. NN ( T ` k ) <_ sum_ m e. NN ( vol* ` [_ m / n ]_ A ) ) )
76	75	rspcev 3309	. . . . . . 7 |- ( ( sum_ m e. NN ( vol* ` [_ m / n ]_ A ) e. RR /\ A. k e. NN ( T ` k ) <_ sum_ m e. NN ( vol* ` [_ m / n ]_ A ) ) -> E. x e. RR A. k e. NN ( T ` k ) <_ x )
77	44, 73, 76	syl2anc 693	. . . . . 6 |- ( ph -> E. x e. RR A. k e. NN ( T ` k ) <_ x )
78		ffn 6045	. . . . . . . . 9 |- ( T : NN --> RR -> T Fn NN )
79	30, 78	syl 17	. . . . . . . 8 |- ( ph -> T Fn NN )
80		breq1 4656	. . . . . . . . 9 |- ( z = ( T ` k ) -> ( z <_ x <-> ( T ` k ) <_ x ) )
81	80	ralrn 6362	. . . . . . . 8 |- ( T Fn NN -> ( A. z e. ran T z <_ x <-> A. k e. NN ( T ` k ) <_ x ) )
82	79, 81	syl 17	. . . . . . 7 |- ( ph -> ( A. z e. ran T z <_ x <-> A. k e. NN ( T ` k ) <_ x ) )
83	82	rexbidv 3052	. . . . . 6 |- ( ph -> ( E. x e. RR A. z e. ran T z <_ x <-> E. x e. RR A. k e. NN ( T ` k ) <_ x ) )
84	77, 83	mpbird 247	. . . . 5 |- ( ph -> E. x e. RR A. z e. ran T z <_ x )
85		supxrre 12157	. . . . 5 |- ( ( ran T C_ RR /\ ran T =/= (/) /\ E. x e. RR A. z e. ran T z <_ x ) -> sup ( ran T , RR* , < ) = sup ( ran T , RR , < ) )
86	32, 41, 84, 85	syl3anc 1326	. . . 4 |- ( ph -> sup ( ran T , RR* , < ) = sup ( ran T , RR , < ) )
87	6, 1, 7, 19, 26, 69, 77	isumsup 14579	. . . 4 |- ( ph -> sum_ m e. NN ( vol* ` [_ m / n ]_ A ) = sup ( ran T , RR , < ) )
88	86, 87	eqtr4d 2659	. . 3 |- ( ph -> sup ( ran T , RR* , < ) = sum_ m e. NN ( vol* ` [_ m / n ]_ A ) )
89	9, 12, 14	cbvsumi 14427	. . 3 |- sum_ n e. NN ( vol* ` A ) = sum_ m e. NN ( vol* ` [_ m / n ]_ A )
90	88, 89	syl6eqr 2674	. 2 |- ( ph -> sup ( ran T , RR* , < ) = sum_ n e. NN ( vol* ` A ) )
91	5, 90	breqtrd 4679	1 |- ( ph -> ( vol* ` U_ n e. NN A ) <_ sum_ n e. NN ( vol* ` A ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   _Vcvv 3200   [_csb 3533   C_ wss 3574   (/)c0 3915   U_ciun 4520   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   ran crn 5115   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   supcsup 8346   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   RR*cxr 10073   < clt 10074   <_ cle 10075   NNcn 11020   ZZ>=cuz 11687   ...cfz 12326   seqcseq 12801   ~~> cli 14215   sum_csu 14416   vol*covol 23231

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-cc 9257  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-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  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-q 11789  df-rp 11833  df-ioo 12179  df-ico 12181  df-fz 12327  df-fzo 12466  df-fl 12593  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-rlim 14220  df-sum 14417  df-ovol 23233

This theorem is referenced by:  ovoliunnul  23275  vitalilem5  23381