Theorem voliooicof 40213

Description: The Lebesgue measure of open intervals is the same as the Lebesgue measure of left-closed right open intervals. (Contributed by Glauco Siliprandi, 3-Mar-2021.)

Hypothesis

Ref	Expression
voliooicof.1	|- ( ph -> F : A --> ( RR X. RR ) )

Assertion

Ref	Expression
voliooicof	|- ( ph -> ( ( vol o. (,) ) o. F ) = ( ( vol o. [,) ) o. F ) )

Proof of Theorem voliooicof

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		volioof 40204	. . . . 5 |- ( vol o. (,) ) : ( RR* X. RR* ) --> ( 0 [,] +oo )
2	1	a1i 11	. . . 4 |- ( ph -> ( vol o. (,) ) : ( RR* X. RR* ) --> ( 0 [,] +oo ) )
3		rexpssxrxp 10084	. . . . 5 |- ( RR X. RR ) C_ ( RR* X. RR* )
4	3	a1i 11	. . . 4 |- ( ph -> ( RR X. RR ) C_ ( RR* X. RR* ) )
5		voliooicof.1	. . . 4 |- ( ph -> F : A --> ( RR X. RR ) )
6	2, 4, 5	fcoss 39402	. . 3 |- ( ph -> ( ( vol o. (,) ) o. F ) : A --> ( 0 [,] +oo ) )
7		ffn 6045	. . 3 |- ( ( ( vol o. (,) ) o. F ) : A --> ( 0 [,] +oo ) -> ( ( vol o. (,) ) o. F ) Fn A )
8	6, 7	syl 17	. 2 |- ( ph -> ( ( vol o. (,) ) o. F ) Fn A )
9		volf 23297	. . . . . 6 |- vol : dom vol --> ( 0 [,] +oo )
10	9	a1i 11	. . . . 5 |- ( ph -> vol : dom vol --> ( 0 [,] +oo ) )
11		icof 39411	. . . . . . . . . 10 |- [,) : ( RR* X. RR* ) --> ~P RR*
12	11	a1i 11	. . . . . . . . 9 |- ( ph -> [,) : ( RR* X. RR* ) --> ~P RR* )
13	12, 4, 5	fcoss 39402	. . . . . . . 8 |- ( ph -> ( [,) o. F ) : A --> ~P RR* )
14		ffn 6045	. . . . . . . 8 |- ( ( [,) o. F ) : A --> ~P RR* -> ( [,) o. F ) Fn A )
15	13, 14	syl 17	. . . . . . 7 |- ( ph -> ( [,) o. F ) Fn A )
16	5	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ x e. A ) -> F : A --> ( RR X. RR ) )
17		simpr 477	. . . . . . . . . 10 |- ( ( ph /\ x e. A ) -> x e. A )
18	16, 17	fvovco 39381	. . . . . . . . 9 |- ( ( ph /\ x e. A ) -> ( ( [,) o. F ) ` x ) = ( ( 1st ` ( F ` x ) ) [,) ( 2nd ` ( F ` x ) ) ) )
19	5	ffvelrnda 6359	. . . . . . . . . . 11 |- ( ( ph /\ x e. A ) -> ( F ` x ) e. ( RR X. RR ) )
20		xp1st 7198	. . . . . . . . . . 11 |- ( ( F ` x ) e. ( RR X. RR ) -> ( 1st ` ( F ` x ) ) e. RR )
21	19, 20	syl 17	. . . . . . . . . 10 |- ( ( ph /\ x e. A ) -> ( 1st ` ( F ` x ) ) e. RR )
22		xp2nd 7199	. . . . . . . . . . . 12 |- ( ( F ` x ) e. ( RR X. RR ) -> ( 2nd ` ( F ` x ) ) e. RR )
23	19, 22	syl 17	. . . . . . . . . . 11 |- ( ( ph /\ x e. A ) -> ( 2nd ` ( F ` x ) ) e. RR )
24	23	rexrd 10089	. . . . . . . . . 10 |- ( ( ph /\ x e. A ) -> ( 2nd ` ( F ` x ) ) e. RR* )
25		icombl 23332	. . . . . . . . . 10 |- ( ( ( 1st ` ( F ` x ) ) e. RR /\ ( 2nd ` ( F ` x ) ) e. RR* ) -> ( ( 1st ` ( F ` x ) ) [,) ( 2nd ` ( F ` x ) ) ) e. dom vol )
26	21, 24, 25	syl2anc 693	. . . . . . . . 9 |- ( ( ph /\ x e. A ) -> ( ( 1st ` ( F ` x ) ) [,) ( 2nd ` ( F ` x ) ) ) e. dom vol )
27	18, 26	eqeltrd 2701	. . . . . . . 8 |- ( ( ph /\ x e. A ) -> ( ( [,) o. F ) ` x ) e. dom vol )
28	27	ralrimiva 2966	. . . . . . 7 |- ( ph -> A. x e. A ( ( [,) o. F ) ` x ) e. dom vol )
29	15, 28	jca 554	. . . . . 6 |- ( ph -> ( ( [,) o. F ) Fn A /\ A. x e. A ( ( [,) o. F ) ` x ) e. dom vol ) )
30		ffnfv 6388	. . . . . 6 |- ( ( [,) o. F ) : A --> dom vol <-> ( ( [,) o. F ) Fn A /\ A. x e. A ( ( [,) o. F ) ` x ) e. dom vol ) )
31	29, 30	sylibr 224	. . . . 5 |- ( ph -> ( [,) o. F ) : A --> dom vol )
32		fco 6058	. . . . 5 |- ( ( vol : dom vol --> ( 0 [,] +oo ) /\ ( [,) o. F ) : A --> dom vol ) -> ( vol o. ( [,) o. F ) ) : A --> ( 0 [,] +oo ) )
33	10, 31, 32	syl2anc 693	. . . 4 |- ( ph -> ( vol o. ( [,) o. F ) ) : A --> ( 0 [,] +oo ) )
34		coass 5654	. . . . . 6 |- ( ( vol o. [,) ) o. F ) = ( vol o. ( [,) o. F ) )
35	34	a1i 11	. . . . 5 |- ( ph -> ( ( vol o. [,) ) o. F ) = ( vol o. ( [,) o. F ) ) )
36	35	feq1d 6030	. . . 4 |- ( ph -> ( ( ( vol o. [,) ) o. F ) : A --> ( 0 [,] +oo ) <-> ( vol o. ( [,) o. F ) ) : A --> ( 0 [,] +oo ) ) )
37	33, 36	mpbird 247	. . 3 |- ( ph -> ( ( vol o. [,) ) o. F ) : A --> ( 0 [,] +oo ) )
38		ffn 6045	. . 3 |- ( ( ( vol o. [,) ) o. F ) : A --> ( 0 [,] +oo ) -> ( ( vol o. [,) ) o. F ) Fn A )
39	37, 38	syl 17	. 2 |- ( ph -> ( ( vol o. [,) ) o. F ) Fn A )
40	21, 23	voliooico 40209	. . 3 |- ( ( ph /\ x e. A ) -> ( vol ` ( ( 1st ` ( F ` x ) ) (,) ( 2nd ` ( F ` x ) ) ) ) = ( vol ` ( ( 1st ` ( F ` x ) ) [,) ( 2nd ` ( F ` x ) ) ) ) )
41	5, 4	fssd 6057	. . . . 5 |- ( ph -> F : A --> ( RR* X. RR* ) )
42	41	adantr 481	. . . 4 |- ( ( ph /\ x e. A ) -> F : A --> ( RR* X. RR* ) )
43	42, 17	fvvolioof 40206	. . 3 |- ( ( ph /\ x e. A ) -> ( ( ( vol o. (,) ) o. F ) ` x ) = ( vol ` ( ( 1st ` ( F ` x ) ) (,) ( 2nd ` ( F ` x ) ) ) ) )
44	42, 17	fvvolicof 40208	. . 3 |- ( ( ph /\ x e. A ) -> ( ( ( vol o. [,) ) o. F ) ` x ) = ( vol ` ( ( 1st ` ( F ` x ) ) [,) ( 2nd ` ( F ` x ) ) ) ) )
45	40, 43, 44	3eqtr4d 2666	. 2 |- ( ( ph /\ x e. A ) -> ( ( ( vol o. (,) ) o. F ) ` x ) = ( ( ( vol o. [,) ) o. F ) ` x ) )
46	8, 39, 45	eqfnfvd 6314	1 |- ( ph -> ( ( vol o. (,) ) o. F ) = ( ( vol o. [,) ) o. F ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574   ~Pcpw 4158   X. cxp 5112   dom cdm 5114   o. ccom 5118   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   RRcr 9935   0cc0 9936   +oocpnf 10071   RR*cxr 10073   (,)cioo 12175   [,)cico 12177   [,]cicc 12178   volcvol 23232

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:  ovolval5lem3  40868