Theorem ovoliun 21646

Description: The Lebesgue outer measure function is countably sub-additive. (Many books allow +oo as a value for one of the sets in the sum, but in our setup we can't do arithmetic on infinity, and in any case the volume of a union containing an infinitely large set is already infinitely large by monotonicity ovolss 21626, so we need not consider this case here, although we do allow the sum itself to be infinite.) (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 )

Assertion

Ref	Expression
ovoliun	|- ( ph -> ( vol* ` U_ n e. NN A ) <_ sup ( ran T , RR* , < ) )

Distinct variable group:   ph, n

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

Proof of Theorem ovoliun

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

Step	Hyp	Ref	Expression
1		nnuz 11108	. . . . . . . . . 10 |- NN = ( ZZ>= ` 1 )
2		1zzd 10886	. . . . . . . . . 10 |- ( ph -> 1 e. ZZ )
3		ovoliun.v	. . . . . . . . . . . 12 |- ( ( ph /\ n e. NN ) -> ( vol* ` A ) e. RR )
4		ovoliun.g	. . . . . . . . . . . 12 |- G = ( n e. NN |-> ( vol* ` A ) )
5	3, 4	fmptd 6038	. . . . . . . . . . 11 |- ( ph -> G : NN --> RR )
6	5	ffvelrnda 6014	. . . . . . . . . 10 |- ( ( ph /\ k e. NN ) -> ( G ` k ) e. RR )
7	1, 2, 6	serfre 12094	. . . . . . . . 9 |- ( ph -> seq 1 ( + , G ) : NN --> RR )
8		ovoliun.t	. . . . . . . . . 10 |- T = seq 1 ( + , G )
9	8	feq1i 5716	. . . . . . . . 9 |- ( T : NN --> RR <-> seq 1 ( + , G ) : NN --> RR )
10	7, 9	sylibr 212	. . . . . . . 8 |- ( ph -> T : NN --> RR )
11		frn 5730	. . . . . . . 8 |- ( T : NN --> RR -> ran T C_ RR )
12	10, 11	syl 16	. . . . . . 7 |- ( ph -> ran T C_ RR )
13		ressxr 9628	. . . . . . 7 |- RR C_ RR*
14	12, 13	syl6ss 3511	. . . . . 6 |- ( ph -> ran T C_ RR* )
15		supxrcl 11497	. . . . . 6 |- ( ran T C_ RR* -> sup ( ran T , RR* , < ) e. RR* )
16	14, 15	syl 16	. . . . 5 |- ( ph -> sup ( ran T , RR* , < ) e. RR* )
17		xrrebnd 11360	. . . . 5 |- ( sup ( ran T , RR* , < ) e. RR* -> ( sup ( ran T , RR* , < ) e. RR <-> ( -oo < sup ( ran T , RR* , < ) /\ sup ( ran T , RR* , < ) < +oo ) ) )
18	16, 17	syl 16	. . . 4 |- ( ph -> ( sup ( ran T , RR* , < ) e. RR <-> ( -oo < sup ( ran T , RR* , < ) /\ sup ( ran T , RR* , < ) < +oo ) ) )
19		mnfxr 11314	. . . . . . 7 |- -oo e. RR*
20	19	a1i 11	. . . . . 6 |- ( ph -> -oo e. RR* )
21		1nn 10538	. . . . . . . 8 |- 1 e. NN
22		ffvelrn 6012	. . . . . . . 8 |- ( ( T : NN --> RR /\ 1 e. NN ) -> ( T ` 1 ) e. RR )
23	10, 21, 22	sylancl 662	. . . . . . 7 |- ( ph -> ( T ` 1 ) e. RR )
24	23	rexrd 9634	. . . . . 6 |- ( ph -> ( T ` 1 ) e. RR* )
25		mnflt 11324	. . . . . . 7 |- ( ( T ` 1 ) e. RR -> -oo < ( T ` 1 ) )
26	23, 25	syl 16	. . . . . 6 |- ( ph -> -oo < ( T ` 1 ) )
27		ffn 5724	. . . . . . . . 9 |- ( T : NN --> RR -> T Fn NN )
28	10, 27	syl 16	. . . . . . . 8 |- ( ph -> T Fn NN )
29		fnfvelrn 6011	. . . . . . . 8 |- ( ( T Fn NN /\ 1 e. NN ) -> ( T ` 1 ) e. ran T )
30	28, 21, 29	sylancl 662	. . . . . . 7 |- ( ph -> ( T ` 1 ) e. ran T )
31		supxrub 11507	. . . . . . 7 |- ( ( ran T C_ RR* /\ ( T ` 1 ) e. ran T ) -> ( T ` 1 ) <_ sup ( ran T , RR* , < ) )
32	14, 30, 31	syl2anc 661	. . . . . 6 |- ( ph -> ( T ` 1 ) <_ sup ( ran T , RR* , < ) )
33	20, 24, 16, 26, 32	xrltletrd 11355	. . . . 5 |- ( ph -> -oo < sup ( ran T , RR* , < ) )
34	33	biantrurd 508	. . . 4 |- ( ph -> ( sup ( ran T , RR* , < ) < +oo <-> ( -oo < sup ( ran T , RR* , < ) /\ sup ( ran T , RR* , < ) < +oo ) ) )
35	18, 34	bitr4d 256	. . 3 |- ( ph -> ( sup ( ran T , RR* , < ) e. RR <-> sup ( ran T , RR* , < ) < +oo ) )
36		nfcv 2624	. . . . . . . . 9 |- F/_ m A
37		nfcsb1v 3446	. . . . . . . . 9 |- F/_ n [_ m / n ]_ A
38		csbeq1a 3439	. . . . . . . . 9 |- ( n = m -> A = [_ m / n ]_ A )
39	36, 37, 38	cbviun 4357	. . . . . . . 8 |- U_ n e. NN A = U_ m e. NN [_ m / n ]_ A
40	39	fveq2i 5862	. . . . . . 7 |- ( vol* ` U_ n e. NN A ) = ( vol* ` U_ m e. NN [_ m / n ]_ A )
41		nfcv 2624	. . . . . . . . . 10 |- F/_ m ( vol* ` A )
42		nfcv 2624	. . . . . . . . . . 11 |- F/_ n vol*
43	42, 37	nffv 5866	. . . . . . . . . 10 |- F/_ n ( vol* ` [_ m / n ]_ A )
44	38	fveq2d 5863	. . . . . . . . . 10 |- ( n = m -> ( vol* ` A ) = ( vol* ` [_ m / n ]_ A ) )
45	41, 43, 44	cbvmpt 4532	. . . . . . . . 9 |- ( n e. NN |-> ( vol* ` A ) ) = ( m e. NN |-> ( vol* ` [_ m / n ]_ A ) )
46	4, 45	eqtri 2491	. . . . . . . 8 |- G = ( m e. NN |-> ( vol* ` [_ m / n ]_ A ) )
47		ovoliun.a	. . . . . . . . . . . 12 |- ( ( ph /\ n e. NN ) -> A C_ RR )
48	47	ralrimiva 2873	. . . . . . . . . . 11 |- ( ph -> A. n e. NN A C_ RR )
49		nfv 1678	. . . . . . . . . . . 12 |- F/ m A C_ RR
50		nfcv 2624	. . . . . . . . . . . . 13 |- F/_ n RR
51	37, 50	nfss 3492	. . . . . . . . . . . 12 |- F/ n [_ m / n ]_ A C_ RR
52	38	sseq1d 3526	. . . . . . . . . . . 12 |- ( n = m -> ( A C_ RR <-> [_ m / n ]_ A C_ RR ) )
53	49, 51, 52	cbvral 3079	. . . . . . . . . . 11 |- ( A. n e. NN A C_ RR <-> A. m e. NN [_ m / n ]_ A C_ RR )
54	48, 53	sylib 196	. . . . . . . . . 10 |- ( ph -> A. m e. NN [_ m / n ]_ A C_ RR )
55	54	ad2antrr 725	. . . . . . . . 9 |- ( ( ( ph /\ sup ( ran T , RR* , < ) e. RR ) /\ x e. RR+ ) -> A. m e. NN [_ m / n ]_ A C_ RR )
56	55	r19.21bi 2828	. . . . . . . 8 |- ( ( ( ( ph /\ sup ( ran T , RR* , < ) e. RR ) /\ x e. RR+ ) /\ m e. NN ) -> [_ m / n ]_ A C_ RR )
57	3	ralrimiva 2873	. . . . . . . . . . 11 |- ( ph -> A. n e. NN ( vol* ` A ) e. RR )
58	41	nfel1 2640	. . . . . . . . . . . 12 |- F/ m ( vol* ` A ) e. RR
59	43	nfel1 2640	. . . . . . . . . . . 12 |- F/ n ( vol* ` [_ m / n ]_ A ) e. RR
60	44	eleq1d 2531	. . . . . . . . . . . 12 |- ( n = m -> ( ( vol* ` A ) e. RR <-> ( vol* ` [_ m / n ]_ A ) e. RR ) )
61	58, 59, 60	cbvral 3079	. . . . . . . . . . 11 |- ( A. n e. NN ( vol* ` A ) e. RR <-> A. m e. NN ( vol* ` [_ m / n ]_ A ) e. RR )
62	57, 61	sylib 196	. . . . . . . . . 10 |- ( ph -> A. m e. NN ( vol* ` [_ m / n ]_ A ) e. RR )
63	62	ad2antrr 725	. . . . . . . . 9 |- ( ( ( ph /\ sup ( ran T , RR* , < ) e. RR ) /\ x e. RR+ ) -> A. m e. NN ( vol* ` [_ m / n ]_ A ) e. RR )
64	63	r19.21bi 2828	. . . . . . . 8 |- ( ( ( ( ph /\ sup ( ran T , RR* , < ) e. RR ) /\ x e. RR+ ) /\ m e. NN ) -> ( vol* ` [_ m / n ]_ A ) e. RR )
65		simplr 754	. . . . . . . 8 |- ( ( ( ph /\ sup ( ran T , RR* , < ) e. RR ) /\ x e. RR+ ) -> sup ( ran T , RR* , < ) e. RR )
66		simpr 461	. . . . . . . 8 |- ( ( ( ph /\ sup ( ran T , RR* , < ) e. RR ) /\ x e. RR+ ) -> x e. RR+ )
67	8, 46, 56, 64, 65, 66	ovoliunlem3 21645	. . . . . . 7 |- ( ( ( ph /\ sup ( ran T , RR* , < ) e. RR ) /\ x e. RR+ ) -> ( vol* ` U_ m e. NN [_ m / n ]_ A ) <_ ( sup ( ran T , RR* , < ) + x ) )
68	40, 67	syl5eqbr 4475	. . . . . 6 |- ( ( ( ph /\ sup ( ran T , RR* , < ) e. RR ) /\ x e. RR+ ) -> ( vol* ` U_ n e. NN A ) <_ ( sup ( ran T , RR* , < ) + x ) )
69	68	ralrimiva 2873	. . . . 5 |- ( ( ph /\ sup ( ran T , RR* , < ) e. RR ) -> A. x e. RR+ ( vol* ` U_ n e. NN A ) <_ ( sup ( ran T , RR* , < ) + x ) )
70		iunss 4361	. . . . . . . 8 |- ( U_ n e. NN A C_ RR <-> A. n e. NN A C_ RR )
71	48, 70	sylibr 212	. . . . . . 7 |- ( ph -> U_ n e. NN A C_ RR )
72		ovolcl 21619	. . . . . . 7 |- ( U_ n e. NN A C_ RR -> ( vol* ` U_ n e. NN A ) e. RR* )
73	71, 72	syl 16	. . . . . 6 |- ( ph -> ( vol* ` U_ n e. NN A ) e. RR* )
74		xralrple 11395	. . . . . 6 |- ( ( ( vol* ` U_ n e. NN A ) e. RR* /\ sup ( ran T , RR* , < ) e. RR ) -> ( ( vol* ` U_ n e. NN A ) <_ sup ( ran T , RR* , < ) <-> A. x e. RR+ ( vol* ` U_ n e. NN A ) <_ ( sup ( ran T , RR* , < ) + x ) ) )
75	73, 74	sylan 471	. . . . 5 |- ( ( ph /\ sup ( ran T , RR* , < ) e. RR ) -> ( ( vol* ` U_ n e. NN A ) <_ sup ( ran T , RR* , < ) <-> A. x e. RR+ ( vol* ` U_ n e. NN A ) <_ ( sup ( ran T , RR* , < ) + x ) ) )
76	69, 75	mpbird 232	. . . 4 |- ( ( ph /\ sup ( ran T , RR* , < ) e. RR ) -> ( vol* ` U_ n e. NN A ) <_ sup ( ran T , RR* , < ) )
77	76	ex 434	. . 3 |- ( ph -> ( sup ( ran T , RR* , < ) e. RR -> ( vol* ` U_ n e. NN A ) <_ sup ( ran T , RR* , < ) ) )
78	35, 77	sylbird 235	. 2 |- ( ph -> ( sup ( ran T , RR* , < ) < +oo -> ( vol* ` U_ n e. NN A ) <_ sup ( ran T , RR* , < ) ) )
79		nltpnft 11358	. . . 4 |- ( sup ( ran T , RR* , < ) e. RR* -> ( sup ( ran T , RR* , < ) = +oo <-> -. sup ( ran T , RR* , < ) < +oo ) )
80	16, 79	syl 16	. . 3 |- ( ph -> ( sup ( ran T , RR* , < ) = +oo <-> -. sup ( ran T , RR* , < ) < +oo ) )
81		pnfge 11330	. . . . 5 |- ( ( vol* ` U_ n e. NN A ) e. RR* -> ( vol* ` U_ n e. NN A ) <_ +oo )
82	73, 81	syl 16	. . . 4 |- ( ph -> ( vol* ` U_ n e. NN A ) <_ +oo )
83		breq2 4446	. . . 4 |- ( sup ( ran T , RR* , < ) = +oo -> ( ( vol* ` U_ n e. NN A ) <_ sup ( ran T , RR* , < ) <-> ( vol* ` U_ n e. NN A ) <_ +oo ) )
84	82, 83	syl5ibrcom 222	. . 3 |- ( ph -> ( sup ( ran T , RR* , < ) = +oo -> ( vol* ` U_ n e. NN A ) <_ sup ( ran T , RR* , < ) ) )
85	80, 84	sylbird 235	. 2 |- ( ph -> ( -. sup ( ran T , RR* , < ) < +oo -> ( vol* ` U_ n e. NN A ) <_ sup ( ran T , RR* , < ) ) )
86	78, 85	pm2.61d 158	1 |- ( ph -> ( vol* ` U_ n e. NN A ) <_ sup ( ran T , RR* , < ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1374   e. wcel 1762   A.wral 2809   [_csb 3430   C_ wss 3471   U_ciun 4320   class class class wbr 4442   |-> cmpt 4500   ran crn 4995   Fn wfn 5576   -->wf 5577   ` cfv 5581  (class class class)co 6277   supcsup 7891   RRcr 9482   1c1 9484   + caddc 9486   +oocpnf 9616   -oocmnf 9617   RR*cxr 9618   < clt 9619   <_ cle 9620   NNcn 10527   RR+crp 11211   seqcseq 12065   vol*covol 21604

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-inf2 8049  ax-cc 8806  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560  ax-pre-sup 9561

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-se 4834  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-1st 6776  df-2nd 6777  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-en 7509  df-dom 7510  df-sdom 7511  df-fin 7512  df-sup 7892  df-oi 7926  df-card 8311  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-div 10198  df-nn 10528  df-2 10585  df-3 10586  df-n0 10787  df-z 10856  df-uz 11074  df-q 11174  df-rp 11212  df-ioo 11524  df-ico 11526  df-fz 11664  df-fzo 11784  df-fl 11888  df-seq 12066  df-exp 12125  df-hash 12363  df-cj 12884  df-re 12885  df-im 12886  df-sqr 13020  df-abs 13021  df-clim 13262  df-rlim 13263  df-sum 13460  df-ovol 21606

This theorem is referenced by:  ovoliun2  21647  voliunlem2  21691  voliunlem3  21692  ex-ovoliunnfl  29623