Theorem ovollb2 23257

Description: It is often more convenient to do calculations with *closed* coverings rather than open ones; here we show that it makes no difference (compare ovollb 23247). (Contributed by Mario Carneiro, 24-Mar-2015.)

Hypothesis

Ref	Expression
ovollb2.1	|- S = seq 1 ( + , ( ( abs o. - ) o. F ) )

Assertion

Ref	Expression
ovollb2	|- ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ A C_ U. ran ( [,] o. F ) ) -> ( vol* ` A ) <_ sup ( ran S , RR* , < ) )

Proof of Theorem ovollb2

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

Step	Hyp	Ref	Expression
1		simpr 477	. . . . . . 7 |- ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ A C_ U. ran ( [,] o. F ) ) -> A C_ U. ran ( [,] o. F ) )
2		ovolficcss 23238	. . . . . . . 8 |- ( F : NN --> ( <_ i^i ( RR X. RR ) ) -> U. ran ( [,] o. F ) C_ RR )
3	2	adantr 481	. . . . . . 7 |- ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ A C_ U. ran ( [,] o. F ) ) -> U. ran ( [,] o. F ) C_ RR )
4	1, 3	sstrd 3613	. . . . . 6 |- ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ A C_ U. ran ( [,] o. F ) ) -> A C_ RR )
5		ovolcl 23246	. . . . . 6 |- ( A C_ RR -> ( vol* ` A ) e. RR* )
6	4, 5	syl 17	. . . . 5 |- ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ A C_ U. ran ( [,] o. F ) ) -> ( vol* ` A ) e. RR* )
7	6	adantr 481	. . . 4 |- ( ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ A C_ U. ran ( [,] o. F ) ) /\ sup ( ran S , RR* , < ) = +oo ) -> ( vol* ` A ) e. RR* )
8		pnfge 11964	. . . 4 |- ( ( vol* ` A ) e. RR* -> ( vol* ` A ) <_ +oo )
9	7, 8	syl 17	. . 3 |- ( ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ A C_ U. ran ( [,] o. F ) ) /\ sup ( ran S , RR* , < ) = +oo ) -> ( vol* ` A ) <_ +oo )
10		simpr 477	. . 3 |- ( ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ A C_ U. ran ( [,] o. F ) ) /\ sup ( ran S , RR* , < ) = +oo ) -> sup ( ran S , RR* , < ) = +oo )
11	9, 10	breqtrrd 4681	. 2 |- ( ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ A C_ U. ran ( [,] o. F ) ) /\ sup ( ran S , RR* , < ) = +oo ) -> ( vol* ` A ) <_ sup ( ran S , RR* , < ) )
12		eqid 2622	. . . . . . . . 9 |- ( ( abs o. - ) o. F ) = ( ( abs o. - ) o. F )
13		ovollb2.1	. . . . . . . . 9 |- S = seq 1 ( + , ( ( abs o. - ) o. F ) )
14	12, 13	ovolsf 23241	. . . . . . . 8 |- ( F : NN --> ( <_ i^i ( RR X. RR ) ) -> S : NN --> ( 0 [,) +oo ) )
15	14	adantr 481	. . . . . . 7 |- ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ A C_ U. ran ( [,] o. F ) ) -> S : NN --> ( 0 [,) +oo ) )
16		frn 6053	. . . . . . 7 |- ( S : NN --> ( 0 [,) +oo ) -> ran S C_ ( 0 [,) +oo ) )
17	15, 16	syl 17	. . . . . 6 |- ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ A C_ U. ran ( [,] o. F ) ) -> ran S C_ ( 0 [,) +oo ) )
18		rge0ssre 12280	. . . . . 6 |- ( 0 [,) +oo ) C_ RR
19	17, 18	syl6ss 3615	. . . . 5 |- ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ A C_ U. ran ( [,] o. F ) ) -> ran S C_ RR )
20		1nn 11031	. . . . . . . 8 |- 1 e. NN
21		fdm 6051	. . . . . . . . 9 |- ( S : NN --> ( 0 [,) +oo ) -> dom S = NN )
22	15, 21	syl 17	. . . . . . . 8 |- ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ A C_ U. ran ( [,] o. F ) ) -> dom S = NN )
23	20, 22	syl5eleqr 2708	. . . . . . 7 |- ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ A C_ U. ran ( [,] o. F ) ) -> 1 e. dom S )
24		ne0i 3921	. . . . . . 7 |- ( 1 e. dom S -> dom S =/= (/) )
25	23, 24	syl 17	. . . . . 6 |- ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ A C_ U. ran ( [,] o. F ) ) -> dom S =/= (/) )
26		dm0rn0 5342	. . . . . . 7 |- ( dom S = (/) <-> ran S = (/) )
27	26	necon3bii 2846	. . . . . 6 |- ( dom S =/= (/) <-> ran S =/= (/) )
28	25, 27	sylib 208	. . . . 5 |- ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ A C_ U. ran ( [,] o. F ) ) -> ran S =/= (/) )
29		supxrre2 12161	. . . . 5 |- ( ( ran S C_ RR /\ ran S =/= (/) ) -> ( sup ( ran S , RR* , < ) e. RR <-> sup ( ran S , RR* , < ) =/= +oo ) )
30	19, 28, 29	syl2anc 693	. . . 4 |- ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ A C_ U. ran ( [,] o. F ) ) -> ( sup ( ran S , RR* , < ) e. RR <-> sup ( ran S , RR* , < ) =/= +oo ) )
31	30	biimpar 502	. . 3 |- ( ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ A C_ U. ran ( [,] o. F ) ) /\ sup ( ran S , RR* , < ) =/= +oo ) -> sup ( ran S , RR* , < ) e. RR )
32		fveq2 6191	. . . . . . . . . 10 |- ( m = n -> ( F ` m ) = ( F ` n ) )
33	32	fveq2d 6195	. . . . . . . . 9 |- ( m = n -> ( 1st ` ( F ` m ) ) = ( 1st ` ( F ` n ) ) )
34		oveq2 6658	. . . . . . . . . 10 |- ( m = n -> ( 2 ^ m ) = ( 2 ^ n ) )
35	34	oveq2d 6666	. . . . . . . . 9 |- ( m = n -> ( ( x / 2 ) / ( 2 ^ m ) ) = ( ( x / 2 ) / ( 2 ^ n ) ) )
36	33, 35	oveq12d 6668	. . . . . . . 8 |- ( m = n -> ( ( 1st ` ( F ` m ) ) - ( ( x / 2 ) / ( 2 ^ m ) ) ) = ( ( 1st ` ( F ` n ) ) - ( ( x / 2 ) / ( 2 ^ n ) ) ) )
37	32	fveq2d 6195	. . . . . . . . 9 |- ( m = n -> ( 2nd ` ( F ` m ) ) = ( 2nd ` ( F ` n ) ) )
38	37, 35	oveq12d 6668	. . . . . . . 8 |- ( m = n -> ( ( 2nd ` ( F ` m ) ) + ( ( x / 2 ) / ( 2 ^ m ) ) ) = ( ( 2nd ` ( F ` n ) ) + ( ( x / 2 ) / ( 2 ^ n ) ) ) )
39	36, 38	opeq12d 4410	. . . . . . 7 |- ( m = n -> <. ( ( 1st ` ( F ` m ) ) - ( ( x / 2 ) / ( 2 ^ m ) ) ) , ( ( 2nd ` ( F ` m ) ) + ( ( x / 2 ) / ( 2 ^ m ) ) ) >. = <. ( ( 1st ` ( F ` n ) ) - ( ( x / 2 ) / ( 2 ^ n ) ) ) , ( ( 2nd ` ( F ` n ) ) + ( ( x / 2 ) / ( 2 ^ n ) ) ) >. )
40	39	cbvmptv 4750	. . . . . 6 |- ( m e. NN |-> <. ( ( 1st ` ( F ` m ) ) - ( ( x / 2 ) / ( 2 ^ m ) ) ) , ( ( 2nd ` ( F ` m ) ) + ( ( x / 2 ) / ( 2 ^ m ) ) ) >. ) = ( n e. NN |-> <. ( ( 1st ` ( F ` n ) ) - ( ( x / 2 ) / ( 2 ^ n ) ) ) , ( ( 2nd ` ( F ` n ) ) + ( ( x / 2 ) / ( 2 ^ n ) ) ) >. )
41		eqid 2622	. . . . . 6 |- seq 1 ( + , ( ( abs o. - ) o. ( m e. NN |-> <. ( ( 1st ` ( F ` m ) ) - ( ( x / 2 ) / ( 2 ^ m ) ) ) , ( ( 2nd ` ( F ` m ) ) + ( ( x / 2 ) / ( 2 ^ m ) ) ) >. ) ) ) = seq 1 ( + , ( ( abs o. - ) o. ( m e. NN |-> <. ( ( 1st ` ( F ` m ) ) - ( ( x / 2 ) / ( 2 ^ m ) ) ) , ( ( 2nd ` ( F ` m ) ) + ( ( x / 2 ) / ( 2 ^ m ) ) ) >. ) ) )
42		simplll 798	. . . . . 6 |- ( ( ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ A C_ U. ran ( [,] o. F ) ) /\ sup ( ran S , RR* , < ) e. RR ) /\ x e. RR+ ) -> F : NN --> ( <_ i^i ( RR X. RR ) ) )
43		simpllr 799	. . . . . 6 |- ( ( ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ A C_ U. ran ( [,] o. F ) ) /\ sup ( ran S , RR* , < ) e. RR ) /\ x e. RR+ ) -> A C_ U. ran ( [,] o. F ) )
44		simpr 477	. . . . . 6 |- ( ( ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ A C_ U. ran ( [,] o. F ) ) /\ sup ( ran S , RR* , < ) e. RR ) /\ x e. RR+ ) -> x e. RR+ )
45		simplr 792	. . . . . 6 |- ( ( ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ A C_ U. ran ( [,] o. F ) ) /\ sup ( ran S , RR* , < ) e. RR ) /\ x e. RR+ ) -> sup ( ran S , RR* , < ) e. RR )
46	13, 40, 41, 42, 43, 44, 45	ovollb2lem 23256	. . . . 5 |- ( ( ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ A C_ U. ran ( [,] o. F ) ) /\ sup ( ran S , RR* , < ) e. RR ) /\ x e. RR+ ) -> ( vol* ` A ) <_ ( sup ( ran S , RR* , < ) + x ) )
47	46	ralrimiva 2966	. . . 4 |- ( ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ A C_ U. ran ( [,] o. F ) ) /\ sup ( ran S , RR* , < ) e. RR ) -> A. x e. RR+ ( vol* ` A ) <_ ( sup ( ran S , RR* , < ) + x ) )
48		xralrple 12036	. . . . 5 |- ( ( ( vol* ` A ) e. RR* /\ sup ( ran S , RR* , < ) e. RR ) -> ( ( vol* ` A ) <_ sup ( ran S , RR* , < ) <-> A. x e. RR+ ( vol* ` A ) <_ ( sup ( ran S , RR* , < ) + x ) ) )
49	6, 48	sylan 488	. . . 4 |- ( ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ A C_ U. ran ( [,] o. F ) ) /\ sup ( ran S , RR* , < ) e. RR ) -> ( ( vol* ` A ) <_ sup ( ran S , RR* , < ) <-> A. x e. RR+ ( vol* ` A ) <_ ( sup ( ran S , RR* , < ) + x ) ) )
50	47, 49	mpbird 247	. . 3 |- ( ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ A C_ U. ran ( [,] o. F ) ) /\ sup ( ran S , RR* , < ) e. RR ) -> ( vol* ` A ) <_ sup ( ran S , RR* , < ) )
51	31, 50	syldan 487	. 2 |- ( ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ A C_ U. ran ( [,] o. F ) ) /\ sup ( ran S , RR* , < ) =/= +oo ) -> ( vol* ` A ) <_ sup ( ran S , RR* , < ) )
52	11, 51	pm2.61dane 2881	1 |- ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ A C_ U. ran ( [,] o. F ) ) -> ( vol* ` A ) <_ sup ( ran S , RR* , < ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   i^i cin 3573   C_ wss 3574   (/)c0 3915   <.cop 4183   U.cuni 4436   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   dom cdm 5114   ran crn 5115   o. ccom 5118   -->wf 5884   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   supcsup 8346   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   +oocpnf 10071   RR*cxr 10073   < clt 10074   <_ cle 10075   - cmin 10266   / cdiv 10684   NNcn 11020   2c2 11070   RR+crp 11832   [,)cico 12177   [,]cicc 12178   seqcseq 12801   ^cexp 12860   abscabs 13974   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-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-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-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-ovol 23233

This theorem is referenced by:  ovolctb  23258  ovolicc1  23284  ioombl1lem4  23329  uniiccvol  23348