Theorem ovoliun 19354

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 19334, 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 10477	. . . . . . . . . 10 |- NN = ( ZZ>= ` 1 )
2		1z 10267	. . . . . . . . . . 11 |- 1 e. ZZ
3	2	a1i 11	. . . . . . . . . 10 |- ( ph -> 1 e. ZZ )
4		ovoliun.v	. . . . . . . . . . . 12 |- ( ( ph /\ n e. NN ) -> ( vol * ` A ) e. RR )
5		ovoliun.g	. . . . . . . . . . . 12 |- G = ( n e. NN |-> ( vol * ` A ) )
6	4, 5	fmptd 5852	. . . . . . . . . . 11 |- ( ph -> G : NN --> RR )
7	6	ffvelrnda 5829	. . . . . . . . . 10 |- ( ( ph /\ k e. NN ) -> ( G ` k ) e. RR )
8	1, 3, 7	serfre 11307	. . . . . . . . 9 |- ( ph -> seq 1 ( + , G ) : NN --> RR )
9		ovoliun.t	. . . . . . . . . 10 |- T = seq 1 ( + , G )
10	9	feq1i 5544	. . . . . . . . 9 |- ( T : NN --> RR <-> seq 1 ( + , G ) : NN --> RR )
11	8, 10	sylibr 204	. . . . . . . 8 |- ( ph -> T : NN --> RR )
12		frn 5556	. . . . . . . 8 |- ( T : NN --> RR -> ran T C_ RR )
13	11, 12	syl 16	. . . . . . 7 |- ( ph -> ran T C_ RR )
14		ressxr 9085	. . . . . . 7 |- RR C_ RR*
15	13, 14	syl6ss 3320	. . . . . 6 |- ( ph -> ran T C_ RR* )
16		supxrcl 10849	. . . . . 6 |- ( ran T C_ RR* -> sup ( ran T , RR* , < ) e. RR* )
17	15, 16	syl 16	. . . . 5 |- ( ph -> sup ( ran T , RR* , < ) e. RR* )
18		xrrebnd 10712	. . . . 5 |- ( sup ( ran T , RR* , < ) e. RR* -> ( sup ( ran T , RR* , < ) e. RR <-> ( -oo < sup ( ran T , RR* , < ) /\ sup ( ran T , RR* , < ) < +oo ) ) )
19	17, 18	syl 16	. . . 4 |- ( ph -> ( sup ( ran T , RR* , < ) e. RR <-> ( -oo < sup ( ran T , RR* , < ) /\ sup ( ran T , RR* , < ) < +oo ) ) )
20		mnfxr 10670	. . . . . . 7 |- -oo e. RR*
21	20	a1i 11	. . . . . 6 |- ( ph -> -oo e. RR* )
22		1nn 9967	. . . . . . . 8 |- 1 e. NN
23		ffvelrn 5827	. . . . . . . 8 |- ( ( T : NN --> RR /\ 1 e. NN ) -> ( T ` 1 ) e. RR )
24	11, 22, 23	sylancl 644	. . . . . . 7 |- ( ph -> ( T ` 1 ) e. RR )
25	24	rexrd 9090	. . . . . 6 |- ( ph -> ( T ` 1 ) e. RR* )
26		mnflt 10678	. . . . . . 7 |- ( ( T ` 1 ) e. RR -> -oo < ( T ` 1 ) )
27	24, 26	syl 16	. . . . . 6 |- ( ph -> -oo < ( T ` 1 ) )
28		ffn 5550	. . . . . . . . 9 |- ( T : NN --> RR -> T Fn NN )
29	11, 28	syl 16	. . . . . . . 8 |- ( ph -> T Fn NN )
30		fnfvelrn 5826	. . . . . . . 8 |- ( ( T Fn NN /\ 1 e. NN ) -> ( T ` 1 ) e. ran T )
31	29, 22, 30	sylancl 644	. . . . . . 7 |- ( ph -> ( T ` 1 ) e. ran T )
32		supxrub 10859	. . . . . . 7 |- ( ( ran T C_ RR* /\ ( T ` 1 ) e. ran T ) -> ( T ` 1 ) <_ sup ( ran T , RR* , < ) )
33	15, 31, 32	syl2anc 643	. . . . . 6 |- ( ph -> ( T ` 1 ) <_ sup ( ran T , RR* , < ) )
34	21, 25, 17, 27, 33	xrltletrd 10707	. . . . 5 |- ( ph -> -oo < sup ( ran T , RR* , < ) )
35	34	biantrurd 495	. . . 4 |- ( ph -> ( sup ( ran T , RR* , < ) < +oo <-> ( -oo < sup ( ran T , RR* , < ) /\ sup ( ran T , RR* , < ) < +oo ) ) )
36	19, 35	bitr4d 248	. . 3 |- ( ph -> ( sup ( ran T , RR* , < ) e. RR <-> sup ( ran T , RR* , < ) < +oo ) )
37		nfcv 2540	. . . . . . . . 9 |- F/_ m A
38		nfcsb1v 3243	. . . . . . . . 9 |- F/_ n [_ m / n ]_ A
39		csbeq1a 3219	. . . . . . . . 9 |- ( n = m -> A = [_ m / n ]_ A )
40	37, 38, 39	cbviun 4088	. . . . . . . 8 |- U_ n e. NN A = U_ m e. NN [_ m / n ]_ A
41	40	fveq2i 5690	. . . . . . 7 |- ( vol * ` U_ n e. NN A ) = ( vol * ` U_ m e. NN [_ m / n ]_ A )
42		nfcv 2540	. . . . . . . . . 10 |- F/_ m ( vol * ` A )
43		nfcv 2540	. . . . . . . . . . 11 |- F/_ n vol *
44	43, 38	nffv 5694	. . . . . . . . . 10 |- F/_ n ( vol * ` [_ m / n ]_ A )
45	39	fveq2d 5691	. . . . . . . . . 10 |- ( n = m -> ( vol * ` A ) = ( vol * ` [_ m / n ]_ A ) )
46	42, 44, 45	cbvmpt 4259	. . . . . . . . 9 |- ( n e. NN |-> ( vol * ` A ) ) = ( m e. NN |-> ( vol * ` [_ m / n ]_ A ) )
47	5, 46	eqtri 2424	. . . . . . . 8 |- G = ( m e. NN |-> ( vol * ` [_ m / n ]_ A ) )
48		ovoliun.a	. . . . . . . . . . . 12 |- ( ( ph /\ n e. NN ) -> A C_ RR )
49	48	ralrimiva 2749	. . . . . . . . . . 11 |- ( ph -> A. n e. NN A C_ RR )
50		nfv 1626	. . . . . . . . . . . 12 |- F/ m A C_ RR
51		nfcv 2540	. . . . . . . . . . . . 13 |- F/_ n RR
52	38, 51	nfss 3301	. . . . . . . . . . . 12 |- F/ n [_ m / n ]_ A C_ RR
53	39	sseq1d 3335	. . . . . . . . . . . 12 |- ( n = m -> ( A C_ RR <-> [_ m / n ]_ A C_ RR ) )
54	50, 52, 53	cbvral 2888	. . . . . . . . . . 11 |- ( A. n e. NN A C_ RR <-> A. m e. NN [_ m / n ]_ A C_ RR )
55	49, 54	sylib 189	. . . . . . . . . 10 |- ( ph -> A. m e. NN [_ m / n ]_ A C_ RR )
56	55	ad2antrr 707	. . . . . . . . 9 |- ( ( ( ph /\ sup ( ran T , RR* , < ) e. RR ) /\ x e. RR+ ) -> A. m e. NN [_ m / n ]_ A C_ RR )
57	56	r19.21bi 2764	. . . . . . . 8 |- ( ( ( ( ph /\ sup ( ran T , RR* , < ) e. RR ) /\ x e. RR+ ) /\ m e. NN ) -> [_ m / n ]_ A C_ RR )
58	4	ralrimiva 2749	. . . . . . . . . . 11 |- ( ph -> A. n e. NN ( vol * ` A ) e. RR )
59	42	nfel1 2550	. . . . . . . . . . . 12 |- F/ m ( vol * ` A ) e. RR
60	44	nfel1 2550	. . . . . . . . . . . 12 |- F/ n ( vol * ` [_ m / n ]_ A ) e. RR
61	45	eleq1d 2470	. . . . . . . . . . . 12 |- ( n = m -> ( ( vol * ` A ) e. RR <-> ( vol * ` [_ m / n ]_ A ) e. RR ) )
62	59, 60, 61	cbvral 2888	. . . . . . . . . . 11 |- ( A. n e. NN ( vol * ` A ) e. RR <-> A. m e. NN ( vol * ` [_ m / n ]_ A ) e. RR )
63	58, 62	sylib 189	. . . . . . . . . 10 |- ( ph -> A. m e. NN ( vol * ` [_ m / n ]_ A ) e. RR )
64	63	ad2antrr 707	. . . . . . . . 9 |- ( ( ( ph /\ sup ( ran T , RR* , < ) e. RR ) /\ x e. RR+ ) -> A. m e. NN ( vol * ` [_ m / n ]_ A ) e. RR )
65	64	r19.21bi 2764	. . . . . . . 8 |- ( ( ( ( ph /\ sup ( ran T , RR* , < ) e. RR ) /\ x e. RR+ ) /\ m e. NN ) -> ( vol * ` [_ m / n ]_ A ) e. RR )
66		simplr 732	. . . . . . . 8 |- ( ( ( ph /\ sup ( ran T , RR* , < ) e. RR ) /\ x e. RR+ ) -> sup ( ran T , RR* , < ) e. RR )
67		simpr 448	. . . . . . . 8 |- ( ( ( ph /\ sup ( ran T , RR* , < ) e. RR ) /\ x e. RR+ ) -> x e. RR+ )
68	9, 47, 57, 65, 66, 67	ovoliunlem3 19353	. . . . . . 7 |- ( ( ( ph /\ sup ( ran T , RR* , < ) e. RR ) /\ x e. RR+ ) -> ( vol * ` U_ m e. NN [_ m / n ]_ A ) <_ ( sup ( ran T , RR* , < ) + x ) )
69	41, 68	syl5eqbr 4205	. . . . . 6 |- ( ( ( ph /\ sup ( ran T , RR* , < ) e. RR ) /\ x e. RR+ ) -> ( vol * ` U_ n e. NN A ) <_ ( sup ( ran T , RR* , < ) + x ) )
70	69	ralrimiva 2749	. . . . 5 |- ( ( ph /\ sup ( ran T , RR* , < ) e. RR ) -> A. x e. RR+ ( vol * ` U_ n e. NN A ) <_ ( sup ( ran T , RR* , < ) + x ) )
71		iunss 4092	. . . . . . . 8 |- ( U_ n e. NN A C_ RR <-> A. n e. NN A C_ RR )
72	49, 71	sylibr 204	. . . . . . 7 |- ( ph -> U_ n e. NN A C_ RR )
73		ovolcl 19327	. . . . . . 7 |- ( U_ n e. NN A C_ RR -> ( vol * ` U_ n e. NN A ) e. RR* )
74	72, 73	syl 16	. . . . . 6 |- ( ph -> ( vol * ` U_ n e. NN A ) e. RR* )
75		xralrple 10747	. . . . . 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 ) ) )
76	74, 75	sylan 458	. . . . 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 ) ) )
77	70, 76	mpbird 224	. . . 4 |- ( ( ph /\ sup ( ran T , RR* , < ) e. RR ) -> ( vol * ` U_ n e. NN A ) <_ sup ( ran T , RR* , < ) )
78	77	ex 424	. . 3 |- ( ph -> ( sup ( ran T , RR* , < ) e. RR -> ( vol * ` U_ n e. NN A ) <_ sup ( ran T , RR* , < ) ) )
79	36, 78	sylbird 227	. 2 |- ( ph -> ( sup ( ran T , RR* , < ) < +oo -> ( vol * ` U_ n e. NN A ) <_ sup ( ran T , RR* , < ) ) )
80		nltpnft 10710	. . . 4 |- ( sup ( ran T , RR* , < ) e. RR* -> ( sup ( ran T , RR* , < ) = +oo <-> -. sup ( ran T , RR* , < ) < +oo ) )
81	17, 80	syl 16	. . 3 |- ( ph -> ( sup ( ran T , RR* , < ) = +oo <-> -. sup ( ran T , RR* , < ) < +oo ) )
82		pnfge 10683	. . . . 5 |- ( ( vol * ` U_ n e. NN A ) e. RR* -> ( vol * ` U_ n e. NN A ) <_ +oo )
83	74, 82	syl 16	. . . 4 |- ( ph -> ( vol * ` U_ n e. NN A ) <_ +oo )
84		breq2 4176	. . . 4 |- ( sup ( ran T , RR* , < ) = +oo -> ( ( vol * ` U_ n e. NN A ) <_ sup ( ran T , RR* , < ) <-> ( vol * ` U_ n e. NN A ) <_ +oo ) )
85	83, 84	syl5ibrcom 214	. . 3 |- ( ph -> ( sup ( ran T , RR* , < ) = +oo -> ( vol * ` U_ n e. NN A ) <_ sup ( ran T , RR* , < ) ) )
86	81, 85	sylbird 227	. 2 |- ( ph -> ( -. sup ( ran T , RR* , < ) < +oo -> ( vol * ` U_ n e. NN A ) <_ sup ( ran T , RR* , < ) ) )
87	79, 86	pm2.61d 152	1 |- ( ph -> ( vol * ` U_ n e. NN A ) <_ sup ( ran T , RR* , < ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 177   /\ wa 359   = wceq 1649   e. wcel 1721   A.wral 2666   [_csb 3211   C_ wss 3280   U_ciun 4053   class class class wbr 4172   e. cmpt 4226   ran crn 4838   Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040   supcsup 7403   RRcr 8945   1c1 8947   + caddc 8949   +oocpnf 9073   -oocmnf 9074   RR*cxr 9075   < clt 9076   <_ cle 9077   NNcn 9956   ZZcz 10238   RR+crp 10568   seq cseq 11278   vol *covol 19312

This theorem is referenced by:  ovoliun2  19355  voliunlem2  19398  voliunlem3  19399  ex-ovoliunnfl  26148

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cc 8271  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-q 10531  df-rp 10569  df-ioo 10876  df-ico 10878  df-fz 11000  df-fzo 11091  df-fl 11157  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-rlim 12238  df-sum 12435  df-ovol 19314