Theorem ovolfs2 23339

Description: Alternative expression for the interval length function. (Contributed by Mario Carneiro, 26-Mar-2015.)

Hypothesis

Ref	Expression
ovolfs2.1	|- G = ( ( abs o. - ) o. F )

Assertion

Ref	Expression
ovolfs2	|- ( F : NN --> ( <_ i^i ( RR X. RR ) ) -> G = ( ( vol* o. (,) ) o. F ) )

Proof of Theorem ovolfs2

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

Step	Hyp	Ref	Expression
1		ovolfcl 23235	. . . . 5 |- ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ n e. NN ) -> ( ( 1st ` ( F ` n ) ) e. RR /\ ( 2nd ` ( F ` n ) ) e. RR /\ ( 1st ` ( F ` n ) ) <_ ( 2nd ` ( F ` n ) ) ) )
2		ovolioo 23336	. . . . 5 |- ( ( ( 1st ` ( F ` n ) ) e. RR /\ ( 2nd ` ( F ` n ) ) e. RR /\ ( 1st ` ( F ` n ) ) <_ ( 2nd ` ( F ` n ) ) ) -> ( vol* ` ( ( 1st ` ( F ` n ) ) (,) ( 2nd ` ( F ` n ) ) ) ) = ( ( 2nd ` ( F ` n ) ) - ( 1st ` ( F ` n ) ) ) )
3	1, 2	syl 17	. . . 4 |- ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ n e. NN ) -> ( vol* ` ( ( 1st ` ( F ` n ) ) (,) ( 2nd ` ( F ` n ) ) ) ) = ( ( 2nd ` ( F ` n ) ) - ( 1st ` ( F ` n ) ) ) )
4		inss2 3834	. . . . . . . . . 10 |- ( <_ i^i ( RR X. RR ) ) C_ ( RR X. RR )
5		rexpssxrxp 10084	. . . . . . . . . 10 |- ( RR X. RR ) C_ ( RR* X. RR* )
6	4, 5	sstri 3612	. . . . . . . . 9 |- ( <_ i^i ( RR X. RR ) ) C_ ( RR* X. RR* )
7		ffvelrn 6357	. . . . . . . . 9 |- ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ n e. NN ) -> ( F ` n ) e. ( <_ i^i ( RR X. RR ) ) )
8	6, 7	sseldi 3601	. . . . . . . 8 |- ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ n e. NN ) -> ( F ` n ) e. ( RR* X. RR* ) )
9		1st2nd2 7205	. . . . . . . 8 |- ( ( F ` n ) e. ( RR* X. RR* ) -> ( F ` n ) = <. ( 1st ` ( F ` n ) ) , ( 2nd ` ( F ` n ) ) >. )
10	8, 9	syl 17	. . . . . . 7 |- ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ n e. NN ) -> ( F ` n ) = <. ( 1st ` ( F ` n ) ) , ( 2nd ` ( F ` n ) ) >. )
11	10	fveq2d 6195	. . . . . 6 |- ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ n e. NN ) -> ( (,) ` ( F ` n ) ) = ( (,) ` <. ( 1st ` ( F ` n ) ) , ( 2nd ` ( F ` n ) ) >. ) )
12		df-ov 6653	. . . . . 6 |- ( ( 1st ` ( F ` n ) ) (,) ( 2nd ` ( F ` n ) ) ) = ( (,) ` <. ( 1st ` ( F ` n ) ) , ( 2nd ` ( F ` n ) ) >. )
13	11, 12	syl6eqr 2674	. . . . 5 |- ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ n e. NN ) -> ( (,) ` ( F ` n ) ) = ( ( 1st ` ( F ` n ) ) (,) ( 2nd ` ( F ` n ) ) ) )
14	13	fveq2d 6195	. . . 4 |- ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ n e. NN ) -> ( vol* ` ( (,) ` ( F ` n ) ) ) = ( vol* ` ( ( 1st ` ( F ` n ) ) (,) ( 2nd ` ( F ` n ) ) ) ) )
15		ovolfs2.1	. . . . 5 |- G = ( ( abs o. - ) o. F )
16	15	ovolfsval 23239	. . . 4 |- ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ n e. NN ) -> ( G ` n ) = ( ( 2nd ` ( F ` n ) ) - ( 1st ` ( F ` n ) ) ) )
17	3, 14, 16	3eqtr4rd 2667	. . 3 |- ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ n e. NN ) -> ( G ` n ) = ( vol* ` ( (,) ` ( F ` n ) ) ) )
18	17	mpteq2dva 4744	. 2 |- ( F : NN --> ( <_ i^i ( RR X. RR ) ) -> ( n e. NN |-> ( G ` n ) ) = ( n e. NN |-> ( vol* ` ( (,) ` ( F ` n ) ) ) ) )
19	15	ovolfsf 23240	. . 3 |- ( F : NN --> ( <_ i^i ( RR X. RR ) ) -> G : NN --> ( 0 [,) +oo ) )
20	19	feqmptd 6249	. 2 |- ( F : NN --> ( <_ i^i ( RR X. RR ) ) -> G = ( n e. NN |-> ( G ` n ) ) )
21		id 22	. . . 4 |- ( F : NN --> ( <_ i^i ( RR X. RR ) ) -> F : NN --> ( <_ i^i ( RR X. RR ) ) )
22	21	feqmptd 6249	. . 3 |- ( F : NN --> ( <_ i^i ( RR X. RR ) ) -> F = ( n e. NN |-> ( F ` n ) ) )
23		ioof 12271	. . . . . 6 |- (,) : ( RR* X. RR* ) --> ~P RR
24	23	a1i 11	. . . . 5 |- ( F : NN --> ( <_ i^i ( RR X. RR ) ) -> (,) : ( RR* X. RR* ) --> ~P RR )
25	24	ffvelrnda 6359	. . . 4 |- ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ x e. ( RR* X. RR* ) ) -> ( (,) ` x ) e. ~P RR )
26	24	feqmptd 6249	. . . 4 |- ( F : NN --> ( <_ i^i ( RR X. RR ) ) -> (,) = ( x e. ( RR* X. RR* ) |-> ( (,) ` x ) ) )
27		ovolf 23250	. . . . . 6 |- vol* : ~P RR --> ( 0 [,] +oo )
28	27	a1i 11	. . . . 5 |- ( F : NN --> ( <_ i^i ( RR X. RR ) ) -> vol* : ~P RR --> ( 0 [,] +oo ) )
29	28	feqmptd 6249	. . . 4 |- ( F : NN --> ( <_ i^i ( RR X. RR ) ) -> vol* = ( y e. ~P RR |-> ( vol* ` y ) ) )
30		fveq2 6191	. . . 4 |- ( y = ( (,) ` x ) -> ( vol* ` y ) = ( vol* ` ( (,) ` x ) ) )
31	25, 26, 29, 30	fmptco 6396	. . 3 |- ( F : NN --> ( <_ i^i ( RR X. RR ) ) -> ( vol* o. (,) ) = ( x e. ( RR* X. RR* ) |-> ( vol* ` ( (,) ` x ) ) ) )
32		fveq2 6191	. . . 4 |- ( x = ( F ` n ) -> ( (,) ` x ) = ( (,) ` ( F ` n ) ) )
33	32	fveq2d 6195	. . 3 |- ( x = ( F ` n ) -> ( vol* ` ( (,) ` x ) ) = ( vol* ` ( (,) ` ( F ` n ) ) ) )
34	8, 22, 31, 33	fmptco 6396	. 2 |- ( F : NN --> ( <_ i^i ( RR X. RR ) ) -> ( ( vol* o. (,) ) o. F ) = ( n e. NN |-> ( vol* ` ( (,) ` ( F ` n ) ) ) ) )
35	18, 20, 34	3eqtr4d 2666	1 |- ( F : NN --> ( <_ i^i ( RR X. RR ) ) -> G = ( ( vol* o. (,) ) o. F ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   i^i cin 3573   ~Pcpw 4158   <.cop 4183   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   o. ccom 5118   -->wf 5884   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   RRcr 9935   0cc0 9936   +oocpnf 10071   RR*cxr 10073   <_ cle 10075   - cmin 10266   NNcn 11020   (,)cioo 12175   [,)cico 12177   [,]cicc 12178   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-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  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-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  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-rlim 14220  df-sum 14417  df-rest 16083  df-topgen 16104  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-top 20699  df-topon 20716  df-bases 20750  df-cmp 21190  df-ovol 23233  df-vol 23234

This theorem is referenced by:  uniioombllem2  23351