Theorem ovoliun 22536

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 22516, 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 11218	. . . . . . . . . 10 |- NN = ( ZZ>= ` 1 )
2		1zzd 10992	. . . . . . . . . 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 6061	. . . . . . . . . . 11 |- ( ph -> G : NN --> RR )
6	5	ffvelrnda 6037	. . . . . . . . . 10 |- ( ( ph /\ k e. NN ) -> ( G ` k ) e. RR )
7	1, 2, 6	serfre 12280	. . . . . . . . 9 |- ( ph -> seq 1 ( + , G ) : NN --> RR )
8		ovoliun.t	. . . . . . . . . 10 |- T = seq 1 ( + , G )
9	8	feq1i 5730	. . . . . . . . 9 |- ( T : NN --> RR <-> seq 1 ( + , G ) : NN --> RR )
10	7, 9	sylibr 217	. . . . . . . 8 |- ( ph -> T : NN --> RR )
11		frn 5747	. . . . . . . 8 |- ( T : NN --> RR -> ran T C_ RR )
12	10, 11	syl 17	. . . . . . 7 |- ( ph -> ran T C_ RR )
13		ressxr 9702	. . . . . . 7 |- RR C_ RR*
14	12, 13	syl6ss 3430	. . . . . 6 |- ( ph -> ran T C_ RR* )
15		supxrcl 11625	. . . . . 6 |- ( ran T C_ RR* -> sup ( ran T , RR* , < ) e. RR* )
16	14, 15	syl 17	. . . . 5 |- ( ph -> sup ( ran T , RR* , < ) e. RR* )
17		xrrebnd 11486	. . . . 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 17	. . . 4 |- ( ph -> ( sup ( ran T , RR* , < ) e. RR <-> ( -oo < sup ( ran T , RR* , < ) /\ sup ( ran T , RR* , < ) < +oo ) ) )
19		mnfxr 11437	. . . . . . 7 |- -oo e. RR*
20	19	a1i 11	. . . . . 6 |- ( ph -> -oo e. RR* )
21		1nn 10642	. . . . . . . 8 |- 1 e. NN
22		ffvelrn 6035	. . . . . . . 8 |- ( ( T : NN --> RR /\ 1 e. NN ) -> ( T ` 1 ) e. RR )
23	10, 21, 22	sylancl 675	. . . . . . 7 |- ( ph -> ( T ` 1 ) e. RR )
24	23	rexrd 9708	. . . . . 6 |- ( ph -> ( T ` 1 ) e. RR* )
25		mnflt 11448	. . . . . . 7 |- ( ( T ` 1 ) e. RR -> -oo < ( T ` 1 ) )
26	23, 25	syl 17	. . . . . 6 |- ( ph -> -oo < ( T ` 1 ) )
27		ffn 5739	. . . . . . . . 9 |- ( T : NN --> RR -> T Fn NN )
28	10, 27	syl 17	. . . . . . . 8 |- ( ph -> T Fn NN )
29		fnfvelrn 6034	. . . . . . . 8 |- ( ( T Fn NN /\ 1 e. NN ) -> ( T ` 1 ) e. ran T )
30	28, 21, 29	sylancl 675	. . . . . . 7 |- ( ph -> ( T ` 1 ) e. ran T )
31		supxrub 11635	. . . . . . 7 |- ( ( ran T C_ RR* /\ ( T ` 1 ) e. ran T ) -> ( T ` 1 ) <_ sup ( ran T , RR* , < ) )
32	14, 30, 31	syl2anc 673	. . . . . 6 |- ( ph -> ( T ` 1 ) <_ sup ( ran T , RR* , < ) )
33	20, 24, 16, 26, 32	xrltletrd 11481	. . . . 5 |- ( ph -> -oo < sup ( ran T , RR* , < ) )
34	33	biantrurd 516	. . . 4 |- ( ph -> ( sup ( ran T , RR* , < ) < +oo <-> ( -oo < sup ( ran T , RR* , < ) /\ sup ( ran T , RR* , < ) < +oo ) ) )
35	18, 34	bitr4d 264	. . 3 |- ( ph -> ( sup ( ran T , RR* , < ) e. RR <-> sup ( ran T , RR* , < ) < +oo ) )
36		nfcv 2612	. . . . . . . . 9 |- F/_ m A
37		nfcsb1v 3365	. . . . . . . . 9 |- F/_ n [_ m / n ]_ A
38		csbeq1a 3358	. . . . . . . . 9 |- ( n = m -> A = [_ m / n ]_ A )
39	36, 37, 38	cbviun 4306	. . . . . . . 8 |- U_ n e. NN A = U_ m e. NN [_ m / n ]_ A
40	39	fveq2i 5882	. . . . . . 7 |- ( vol* ` U_ n e. NN A ) = ( vol* ` U_ m e. NN [_ m / n ]_ A )
41		nfcv 2612	. . . . . . . . . 10 |- F/_ m ( vol* ` A )
42		nfcv 2612	. . . . . . . . . . 11 |- F/_ n vol*
43	42, 37	nffv 5886	. . . . . . . . . 10 |- F/_ n ( vol* ` [_ m / n ]_ A )
44	38	fveq2d 5883	. . . . . . . . . 10 |- ( n = m -> ( vol* ` A ) = ( vol* ` [_ m / n ]_ A ) )
45	41, 43, 44	cbvmpt 4487	. . . . . . . . 9 |- ( n e. NN |-> ( vol* ` A ) ) = ( m e. NN |-> ( vol* ` [_ m / n ]_ A ) )
46	4, 45	eqtri 2493	. . . . . . . 8 |- G = ( m e. NN |-> ( vol* ` [_ m / n ]_ A ) )
47		ovoliun.a	. . . . . . . . . . . 12 |- ( ( ph /\ n e. NN ) -> A C_ RR )
48	47	ralrimiva 2809	. . . . . . . . . . 11 |- ( ph -> A. n e. NN A C_ RR )
49		nfv 1769	. . . . . . . . . . . 12 |- F/ m A C_ RR
50		nfcv 2612	. . . . . . . . . . . . 13 |- F/_ n RR
51	37, 50	nfss 3411	. . . . . . . . . . . 12 |- F/ n [_ m / n ]_ A C_ RR
52	38	sseq1d 3445	. . . . . . . . . . . 12 |- ( n = m -> ( A C_ RR <-> [_ m / n ]_ A C_ RR ) )
53	49, 51, 52	cbvral 3001	. . . . . . . . . . 11 |- ( A. n e. NN A C_ RR <-> A. m e. NN [_ m / n ]_ A C_ RR )
54	48, 53	sylib 201	. . . . . . . . . 10 |- ( ph -> A. m e. NN [_ m / n ]_ A C_ RR )
55	54	ad2antrr 740	. . . . . . . . 9 |- ( ( ( ph /\ sup ( ran T , RR* , < ) e. RR ) /\ x e. RR+ ) -> A. m e. NN [_ m / n ]_ A C_ RR )
56	55	r19.21bi 2776	. . . . . . . 8 |- ( ( ( ( ph /\ sup ( ran T , RR* , < ) e. RR ) /\ x e. RR+ ) /\ m e. NN ) -> [_ m / n ]_ A C_ RR )
57	3	ralrimiva 2809	. . . . . . . . . . 11 |- ( ph -> A. n e. NN ( vol* ` A ) e. RR )
58	41	nfel1 2626	. . . . . . . . . . . 12 |- F/ m ( vol* ` A ) e. RR
59	43	nfel1 2626	. . . . . . . . . . . 12 |- F/ n ( vol* ` [_ m / n ]_ A ) e. RR
60	44	eleq1d 2533	. . . . . . . . . . . 12 |- ( n = m -> ( ( vol* ` A ) e. RR <-> ( vol* ` [_ m / n ]_ A ) e. RR ) )
61	58, 59, 60	cbvral 3001	. . . . . . . . . . 11 |- ( A. n e. NN ( vol* ` A ) e. RR <-> A. m e. NN ( vol* ` [_ m / n ]_ A ) e. RR )
62	57, 61	sylib 201	. . . . . . . . . 10 |- ( ph -> A. m e. NN ( vol* ` [_ m / n ]_ A ) e. RR )
63	62	ad2antrr 740	. . . . . . . . 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 2776	. . . . . . . 8 |- ( ( ( ( ph /\ sup ( ran T , RR* , < ) e. RR ) /\ x e. RR+ ) /\ m e. NN ) -> ( vol* ` [_ m / n ]_ A ) e. RR )
65		simplr 770	. . . . . . . 8 |- ( ( ( ph /\ sup ( ran T , RR* , < ) e. RR ) /\ x e. RR+ ) -> sup ( ran T , RR* , < ) e. RR )
66		simpr 468	. . . . . . . 8 |- ( ( ( ph /\ sup ( ran T , RR* , < ) e. RR ) /\ x e. RR+ ) -> x e. RR+ )
67	8, 46, 56, 64, 65, 66	ovoliunlem3 22535	. . . . . . 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 4429	. . . . . 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 2809	. . . . 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 4310	. . . . . . . 8 |- ( U_ n e. NN A C_ RR <-> A. n e. NN A C_ RR )
71	48, 70	sylibr 217	. . . . . . 7 |- ( ph -> U_ n e. NN A C_ RR )
72		ovolcl 22509	. . . . . . 7 |- ( U_ n e. NN A C_ RR -> ( vol* ` U_ n e. NN A ) e. RR* )
73	71, 72	syl 17	. . . . . 6 |- ( ph -> ( vol* ` U_ n e. NN A ) e. RR* )
74		xralrple 11521	. . . . . 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 479	. . . . 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 240	. . . 4 |- ( ( ph /\ sup ( ran T , RR* , < ) e. RR ) -> ( vol* ` U_ n e. NN A ) <_ sup ( ran T , RR* , < ) )
77	76	ex 441	. . 3 |- ( ph -> ( sup ( ran T , RR* , < ) e. RR -> ( vol* ` U_ n e. NN A ) <_ sup ( ran T , RR* , < ) ) )
78	35, 77	sylbird 243	. 2 |- ( ph -> ( sup ( ran T , RR* , < ) < +oo -> ( vol* ` U_ n e. NN A ) <_ sup ( ran T , RR* , < ) ) )
79		nltpnft 11484	. . . 4 |- ( sup ( ran T , RR* , < ) e. RR* -> ( sup ( ran T , RR* , < ) = +oo <-> -. sup ( ran T , RR* , < ) < +oo ) )
80	16, 79	syl 17	. . 3 |- ( ph -> ( sup ( ran T , RR* , < ) = +oo <-> -. sup ( ran T , RR* , < ) < +oo ) )
81		pnfge 11455	. . . . 5 |- ( ( vol* ` U_ n e. NN A ) e. RR* -> ( vol* ` U_ n e. NN A ) <_ +oo )
82	73, 81	syl 17	. . . 4 |- ( ph -> ( vol* ` U_ n e. NN A ) <_ +oo )
83		breq2 4399	. . . 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 230	. . 3 |- ( ph -> ( sup ( ran T , RR* , < ) = +oo -> ( vol* ` U_ n e. NN A ) <_ sup ( ran T , RR* , < ) ) )
85	80, 84	sylbird 243	. 2 |- ( ph -> ( -. sup ( ran T , RR* , < ) < +oo -> ( vol* ` U_ n e. NN A ) <_ sup ( ran T , RR* , < ) ) )
86	78, 85	pm2.61d 163	1 |- ( ph -> ( vol* ` U_ n e. NN A ) <_ sup ( ran T , RR* , < ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 189   /\ wa 376   = wceq 1452   e. wcel 1904   A.wral 2756   [_csb 3349   C_ wss 3390   U_ciun 4269   class class class wbr 4395   |-> cmpt 4454   ran crn 4840   Fn wfn 5584   -->wf 5585   ` cfv 5589  (class class class)co 6308   supcsup 7972   RRcr 9556   1c1 9558   + caddc 9560   +oocpnf 9690   -oocmnf 9691   RR*cxr 9692   < clt 9693   <_ cle 9694   NNcn 10631   RR+crp 11325   seqcseq 12251   vol*covol 22491

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cc 8883  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-q 11288  df-rp 11326  df-ioo 11664  df-ico 11666  df-fz 11811  df-fzo 11943  df-fl 12061  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-clim 13629  df-rlim 13630  df-sum 13830  df-ovol 22494

This theorem is referenced by:  ovoliun2  22537  voliunlem2  22583  voliunlem3  22584  ex-ovoliunnfl  32047