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Theorem ovolfsf 23240
Description: Closure for the interval length function. (Contributed by Mario Carneiro, 16-Mar-2014.)
Hypothesis
Ref Expression
ovolfs.1 |- G = ( ( abs o. - ) o. F )
Assertion
Ref Expression
ovolfsf |- ( F : NN --> ( <_ i^i ( RR X. RR ) ) -> G : NN --> ( 0 [,) +oo ) )
Proof of Theorem ovolfsf
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 absf 14077 . . . . . 6 |- abs : CC --> RR
2 subf 10283 . . . . . 6 |- - : ( CC X. CC ) --> CC
3 fco 6058 . . . . . 6 |- ( ( abs : CC --> RR /\ - : ( CC X. CC ) --> CC ) -> ( abs o. - ) : ( CC X. CC ) --> RR )
41, 2, 3mp2an 708 . . . . 5 |- ( abs o. - ) : ( CC X. CC ) --> RR
5 inss2 3834 . . . . . . 7 |- ( <_ i^i ( RR X. RR ) ) C_ ( RR X. RR )
6 ax-resscn 9993 . . . . . . . 8 |- RR C_ CC
7 xpss12 5225 . . . . . . . 8 |- ( ( RR C_ CC /\ RR C_ CC ) -> ( RR X. RR ) C_ ( CC X. CC ) )
86, 6, 7mp2an 708 . . . . . . 7 |- ( RR X. RR ) C_ ( CC X. CC )
95, 8sstri 3612 . . . . . 6 |- ( <_ i^i ( RR X. RR ) ) C_ ( CC X. CC )
10 fss 6056 . . . . . 6 |- ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ ( <_ i^i ( RR X. RR ) ) C_ ( CC X. CC ) ) -> F : NN --> ( CC X. CC ) )
119, 10mpan2 707 . . . . 5 |- ( F : NN --> ( <_ i^i ( RR X. RR ) ) -> F : NN --> ( CC X. CC ) )
12 fco 6058 . . . . 5 |- ( ( ( abs o. - ) : ( CC X. CC ) --> RR /\ F : NN --> ( CC X. CC ) ) -> ( ( abs o. - ) o. F ) : NN --> RR )
134, 11, 12sylancr 695 . . . 4 |- ( F : NN --> ( <_ i^i ( RR X. RR ) ) -> ( ( abs o. - ) o. F ) : NN --> RR )
14 ovolfs.1 . . . . 5 |- G = ( ( abs o. - ) o. F )
1514feq1i 6036 . . . 4 |- ( G : NN --> RR <-> ( ( abs o. - ) o. F ) : NN --> RR )
1613, 15sylibr 224 . . 3 |- ( F : NN --> ( <_ i^i ( RR X. RR ) ) -> G : NN --> RR )
17 ffn 6045 . . 3 |- ( G : NN --> RR -> G Fn NN )
1816, 17syl 17 . 2 |- ( F : NN --> ( <_ i^i ( RR X. RR ) ) -> G Fn NN )
1916ffvelrnda 6359 . . . 4 |- ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ x e. NN ) -> ( G ` x ) e. RR )
20 ovolfcl 23235 . . . . . 6 |- ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ x e. NN ) -> ( ( 1st ` ( F ` x ) ) e. RR /\ ( 2nd ` ( F ` x ) ) e. RR /\ ( 1st ` ( F ` x ) ) <_ ( 2nd ` ( F ` x ) ) ) )
21 subge0 10541 . . . . . . . 8 |- ( ( ( 2nd ` ( F ` x ) ) e. RR /\ ( 1st ` ( F ` x ) ) e. RR ) -> ( 0 <_ ( ( 2nd ` ( F ` x ) ) - ( 1st ` ( F ` x ) ) ) <-> ( 1st ` ( F ` x ) ) <_ ( 2nd ` ( F ` x ) ) ) )
2221ancoms 469 . . . . . . 7 |- ( ( ( 1st ` ( F ` x ) ) e. RR /\ ( 2nd ` ( F ` x ) ) e. RR ) -> ( 0 <_ ( ( 2nd ` ( F ` x ) ) - ( 1st ` ( F ` x ) ) ) <-> ( 1st ` ( F ` x ) ) <_ ( 2nd ` ( F ` x ) ) ) )
2322biimp3ar 1433 . . . . . 6 |- ( ( ( 1st ` ( F ` x ) ) e. RR /\ ( 2nd ` ( F ` x ) ) e. RR /\ ( 1st ` ( F ` x ) ) <_ ( 2nd ` ( F ` x ) ) ) -> 0 <_ ( ( 2nd ` ( F ` x ) ) - ( 1st ` ( F ` x ) ) ) )
2420, 23syl 17 . . . . 5 |- ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ x e. NN ) -> 0 <_ ( ( 2nd ` ( F ` x ) ) - ( 1st ` ( F ` x ) ) ) )
2514ovolfsval 23239 . . . . 5 |- ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ x e. NN ) -> ( G ` x ) = ( ( 2nd ` ( F ` x ) ) - ( 1st ` ( F ` x ) ) ) )
2624, 25breqtrrd 4681 . . . 4 |- ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ x e. NN ) -> 0 <_ ( G ` x ) )
27 elrege0 12278 . . . 4 |- ( ( G ` x ) e. ( 0 [,) +oo ) <-> ( ( G ` x ) e. RR /\ 0 <_ ( G ` x ) ) )
2819, 26, 27sylanbrc 698 . . 3 |- ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ x e. NN ) -> ( G ` x ) e. ( 0 [,) +oo ) )
2928ralrimiva 2966 . 2 |- ( F : NN --> ( <_ i^i ( RR X. RR ) ) -> A. x e. NN ( G ` x ) e. ( 0 [,) +oo ) )
30 ffnfv 6388 . 2 |- ( G : NN --> ( 0 [,) +oo ) <-> ( G Fn NN /\ A. x e. NN ( G ` x ) e. ( 0 [,) +oo ) ) )
3118, 29, 30sylanbrc 698 1 |- ( F : NN --> ( <_ i^i ( RR X. RR ) ) -> G : NN --> ( 0 [,) +oo ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   i^i cin 3573   C_ wss 3574   class class class wbr 4653   X. cxp 5112   o. ccom 5118   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   CCcc 9934   RRcr 9935   0cc0 9936   +oocpnf 10071   <_ cle 10075   - cmin 10266   NNcn 11020   [,)cico 12177   abscabs 13974
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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  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-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-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-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-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-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  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-rp 11833  df-ico 12181  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976
This theorem is referenced by:  ovolsf  23241  ovollb2lem  23256  ovolunlem1a  23264  ovoliunlem1  23270  ovolshftlem1  23277  ovolicc2lem4  23288  ioombl1lem4  23329  ovolfs2  23339  uniioombllem2  23351  uniioombllem6  23356
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