Theorem ioorcl 23345

Description: The function F does not always return real numbers, but it does on intervals of finite volume. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by AV, 13-Sep-2020.)

Hypothesis

Ref	Expression
ioorf.1	|- F = ( x e. ran (,) |-> if ( x = (/) , <. 0 , 0 >. , <.inf ( x , RR* , < ) , sup ( x , RR* , < ) >. ) )

Assertion

Ref	Expression
ioorcl	|- ( ( A e. ran (,) /\ ( vol* ` A ) e. RR ) -> ( F ` A ) e. ( <_ i^i ( RR X. RR ) ) )

Distinct variable group:   x, A

Allowed substitution hint:   F( x)

Proof of Theorem ioorcl

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		inss1 3833	. . 3 |- ( <_ i^i ( RR* X. RR* ) ) C_ <_
2		ioorf.1	. . . . . 6 |- F = ( x e. ran (,) |-> if ( x = (/) , <. 0 , 0 >. , <.inf ( x , RR* , < ) , sup ( x , RR* , < ) >. ) )
3	2	ioorf 23341	. . . . 5 |- F : ran (,) --> ( <_ i^i ( RR* X. RR* ) )
4	3	ffvelrni 6358	. . . 4 |- ( A e. ran (,) -> ( F ` A ) e. ( <_ i^i ( RR* X. RR* ) ) )
5	4	adantr 481	. . 3 |- ( ( A e. ran (,) /\ ( vol* ` A ) e. RR ) -> ( F ` A ) e. ( <_ i^i ( RR* X. RR* ) ) )
6	1, 5	sseldi 3601	. 2 |- ( ( A e. ran (,) /\ ( vol* ` A ) e. RR ) -> ( F ` A ) e. <_ )
7	2	ioorval 23342	. . . . . 6 |- ( A e. ran (,) -> ( F ` A ) = if ( A = (/) , <. 0 , 0 >. , <.inf ( A , RR* , < ) , sup ( A , RR* , < ) >. ) )
8	7	adantr 481	. . . . 5 |- ( ( A e. ran (,) /\ ( vol* ` A ) e. RR ) -> ( F ` A ) = if ( A = (/) , <. 0 , 0 >. , <.inf ( A , RR* , < ) , sup ( A , RR* , < ) >. ) )
9		iftrue 4092	. . . . 5 |- ( A = (/) -> if ( A = (/) , <. 0 , 0 >. , <.inf ( A , RR* , < ) , sup ( A , RR* , < ) >. ) = <. 0 , 0 >. )
10	8, 9	sylan9eq 2676	. . . 4 |- ( ( ( A e. ran (,) /\ ( vol* ` A ) e. RR ) /\ A = (/) ) -> ( F ` A ) = <. 0 , 0 >. )
11		0re 10040	. . . . 5 |- 0 e. RR
12		opelxpi 5148	. . . . 5 |- ( ( 0 e. RR /\ 0 e. RR ) -> <. 0 , 0 >. e. ( RR X. RR ) )
13	11, 11, 12	mp2an 708	. . . 4 |- <. 0 , 0 >. e. ( RR X. RR )
14	10, 13	syl6eqel 2709	. . 3 |- ( ( ( A e. ran (,) /\ ( vol* ` A ) e. RR ) /\ A = (/) ) -> ( F ` A ) e. ( RR X. RR ) )
15		ioof 12271	. . . . . 6 |- (,) : ( RR* X. RR* ) --> ~P RR
16		ffn 6045	. . . . . 6 |- ( (,) : ( RR* X. RR* ) --> ~P RR -> (,) Fn ( RR* X. RR* ) )
17		ovelrn 6810	. . . . . 6 |- ( (,) Fn ( RR* X. RR* ) -> ( A e. ran (,) <-> E. a e. RR* E. b e. RR* A = ( a (,) b ) ) )
18	15, 16, 17	mp2b 10	. . . . 5 |- ( A e. ran (,) <-> E. a e. RR* E. b e. RR* A = ( a (,) b ) )
19	2	ioorinv2 23343	. . . . . . . . . 10 |- ( ( a (,) b ) =/= (/) -> ( F ` ( a (,) b ) ) = <. a , b >. )
20	19	adantl 482	. . . . . . . . 9 |- ( ( ( vol* ` ( a (,) b ) ) e. RR /\ ( a (,) b ) =/= (/) ) -> ( F ` ( a (,) b ) ) = <. a , b >. )
21		ioorcl2 23340	. . . . . . . . . . 11 |- ( ( ( a (,) b ) =/= (/) /\ ( vol* ` ( a (,) b ) ) e. RR ) -> ( a e. RR /\ b e. RR ) )
22	21	ancoms 469	. . . . . . . . . 10 |- ( ( ( vol* ` ( a (,) b ) ) e. RR /\ ( a (,) b ) =/= (/) ) -> ( a e. RR /\ b e. RR ) )
23		opelxpi 5148	. . . . . . . . . 10 |- ( ( a e. RR /\ b e. RR ) -> <. a , b >. e. ( RR X. RR ) )
24	22, 23	syl 17	. . . . . . . . 9 |- ( ( ( vol* ` ( a (,) b ) ) e. RR /\ ( a (,) b ) =/= (/) ) -> <. a , b >. e. ( RR X. RR ) )
25	20, 24	eqeltrd 2701	. . . . . . . 8 |- ( ( ( vol* ` ( a (,) b ) ) e. RR /\ ( a (,) b ) =/= (/) ) -> ( F ` ( a (,) b ) ) e. ( RR X. RR ) )
26		fveq2 6191	. . . . . . . . . . 11 |- ( A = ( a (,) b ) -> ( vol* ` A ) = ( vol* ` ( a (,) b ) ) )
27	26	eleq1d 2686	. . . . . . . . . 10 |- ( A = ( a (,) b ) -> ( ( vol* ` A ) e. RR <-> ( vol* ` ( a (,) b ) ) e. RR ) )
28		neeq1 2856	. . . . . . . . . 10 |- ( A = ( a (,) b ) -> ( A =/= (/) <-> ( a (,) b ) =/= (/) ) )
29	27, 28	anbi12d 747	. . . . . . . . 9 |- ( A = ( a (,) b ) -> ( ( ( vol* ` A ) e. RR /\ A =/= (/) ) <-> ( ( vol* ` ( a (,) b ) ) e. RR /\ ( a (,) b ) =/= (/) ) ) )
30		fveq2 6191	. . . . . . . . . 10 |- ( A = ( a (,) b ) -> ( F ` A ) = ( F ` ( a (,) b ) ) )
31	30	eleq1d 2686	. . . . . . . . 9 |- ( A = ( a (,) b ) -> ( ( F ` A ) e. ( RR X. RR ) <-> ( F ` ( a (,) b ) ) e. ( RR X. RR ) ) )
32	29, 31	imbi12d 334	. . . . . . . 8 |- ( A = ( a (,) b ) -> ( ( ( ( vol* ` A ) e. RR /\ A =/= (/) ) -> ( F ` A ) e. ( RR X. RR ) ) <-> ( ( ( vol* ` ( a (,) b ) ) e. RR /\ ( a (,) b ) =/= (/) ) -> ( F ` ( a (,) b ) ) e. ( RR X. RR ) ) ) )
33	25, 32	mpbiri 248	. . . . . . 7 |- ( A = ( a (,) b ) -> ( ( ( vol* ` A ) e. RR /\ A =/= (/) ) -> ( F ` A ) e. ( RR X. RR ) ) )
34	33	a1i 11	. . . . . 6 |- ( ( a e. RR* /\ b e. RR* ) -> ( A = ( a (,) b ) -> ( ( ( vol* ` A ) e. RR /\ A =/= (/) ) -> ( F ` A ) e. ( RR X. RR ) ) ) )
35	34	rexlimivv 3036	. . . . 5 |- ( E. a e. RR* E. b e. RR* A = ( a (,) b ) -> ( ( ( vol* ` A ) e. RR /\ A =/= (/) ) -> ( F ` A ) e. ( RR X. RR ) ) )
36	18, 35	sylbi 207	. . . 4 |- ( A e. ran (,) -> ( ( ( vol* ` A ) e. RR /\ A =/= (/) ) -> ( F ` A ) e. ( RR X. RR ) ) )
37	36	impl 650	. . 3 |- ( ( ( A e. ran (,) /\ ( vol* ` A ) e. RR ) /\ A =/= (/) ) -> ( F ` A ) e. ( RR X. RR ) )
38	14, 37	pm2.61dane 2881	. 2 |- ( ( A e. ran (,) /\ ( vol* ` A ) e. RR ) -> ( F ` A ) e. ( RR X. RR ) )
39	6, 38	elind 3798	1 |- ( ( A e. ran (,) /\ ( vol* ` A ) e. RR ) -> ( F ` A ) e. ( <_ i^i ( RR X. RR ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   i^i cin 3573   (/)c0 3915   ifcif 4086   ~Pcpw 4158   <.cop 4183   |-> cmpt 4729   X. cxp 5112   ran crn 5115   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   supcsup 8346  infcinf 8347   RRcr 9935   0cc0 9936   RR*cxr 10073   < clt 10074   <_ cle 10075   (,)cioo 12175   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