Theorem i1fima2 23446

Description: Any preimage of a simple function not containing zero has finite measure. (Contributed by Mario Carneiro, 26-Jun-2014.)

Assertion

Ref	Expression
i1fima2	|- ( ( F e. dom S.1 /\ -. 0 e. A ) -> ( vol ` ( `' F " A ) ) e. RR )

Proof of Theorem i1fima2

Step	Hyp	Ref	Expression
1		i1fima 23445	. . . 4 |- ( F e. dom S.1 -> ( `' F " A ) e. dom vol )
2	1	adantr 481	. . 3 |- ( ( F e. dom S.1 /\ -. 0 e. A ) -> ( `' F " A ) e. dom vol )
3		mblvol 23298	. . 3 |- ( ( `' F " A ) e. dom vol -> ( vol ` ( `' F " A ) ) = ( vol* ` ( `' F " A ) ) )
4	2, 3	syl 17	. 2 |- ( ( F e. dom S.1 /\ -. 0 e. A ) -> ( vol ` ( `' F " A ) ) = ( vol* ` ( `' F " A ) ) )
5		i1ff 23443	. . . . . . 7 |- ( F e. dom S.1 -> F : RR --> RR )
6	5	adantr 481	. . . . . 6 |- ( ( F e. dom S.1 /\ -. 0 e. A ) -> F : RR --> RR )
7		ffun 6048	. . . . . 6 |- ( F : RR --> RR -> Fun F )
8		inpreima 6342	. . . . . 6 |- ( Fun F -> ( `' F " ( A i^i ran F ) ) = ( ( `' F " A ) i^i ( `' F " ran F ) ) )
9	6, 7, 8	3syl 18	. . . . 5 |- ( ( F e. dom S.1 /\ -. 0 e. A ) -> ( `' F " ( A i^i ran F ) ) = ( ( `' F " A ) i^i ( `' F " ran F ) ) )
10		cnvimass 5485	. . . . . . 7 |- ( `' F " A ) C_ dom F
11		cnvimarndm 5486	. . . . . . 7 |- ( `' F " ran F ) = dom F
12	10, 11	sseqtr4i 3638	. . . . . 6 |- ( `' F " A ) C_ ( `' F " ran F )
13		df-ss 3588	. . . . . 6 |- ( ( `' F " A ) C_ ( `' F " ran F ) <-> ( ( `' F " A ) i^i ( `' F " ran F ) ) = ( `' F " A ) )
14	12, 13	mpbi 220	. . . . 5 |- ( ( `' F " A ) i^i ( `' F " ran F ) ) = ( `' F " A )
15	9, 14	syl6req 2673	. . . 4 |- ( ( F e. dom S.1 /\ -. 0 e. A ) -> ( `' F " A ) = ( `' F " ( A i^i ran F ) ) )
16		inss1 3833	. . . . . . . . . 10 |- ( A i^i ran F ) C_ A
17	16	sseli 3599	. . . . . . . . 9 |- ( 0 e. ( A i^i ran F ) -> 0 e. A )
18	17	con3i 150	. . . . . . . 8 |- ( -. 0 e. A -> -. 0 e. ( A i^i ran F ) )
19	18	adantl 482	. . . . . . 7 |- ( ( F e. dom S.1 /\ -. 0 e. A ) -> -. 0 e. ( A i^i ran F ) )
20		disjsn 4246	. . . . . . 7 |- ( ( ( A i^i ran F ) i^i { 0 } ) = (/) <-> -. 0 e. ( A i^i ran F ) )
21	19, 20	sylibr 224	. . . . . 6 |- ( ( F e. dom S.1 /\ -. 0 e. A ) -> ( ( A i^i ran F ) i^i { 0 } ) = (/) )
22		inss2 3834	. . . . . . . . 9 |- ( A i^i ran F ) C_ ran F
23		frn 6053	. . . . . . . . . 10 |- ( F : RR --> RR -> ran F C_ RR )
24	5, 23	syl 17	. . . . . . . . 9 |- ( F e. dom S.1 -> ran F C_ RR )
25	22, 24	syl5ss 3614	. . . . . . . 8 |- ( F e. dom S.1 -> ( A i^i ran F ) C_ RR )
26	25	adantr 481	. . . . . . 7 |- ( ( F e. dom S.1 /\ -. 0 e. A ) -> ( A i^i ran F ) C_ RR )
27		reldisj 4020	. . . . . . 7 |- ( ( A i^i ran F ) C_ RR -> ( ( ( A i^i ran F ) i^i { 0 } ) = (/) <-> ( A i^i ran F ) C_ ( RR \ { 0 } ) ) )
28	26, 27	syl 17	. . . . . 6 |- ( ( F e. dom S.1 /\ -. 0 e. A ) -> ( ( ( A i^i ran F ) i^i { 0 } ) = (/) <-> ( A i^i ran F ) C_ ( RR \ { 0 } ) ) )
29	21, 28	mpbid 222	. . . . 5 |- ( ( F e. dom S.1 /\ -. 0 e. A ) -> ( A i^i ran F ) C_ ( RR \ { 0 } ) )
30		imass2 5501	. . . . 5 |- ( ( A i^i ran F ) C_ ( RR \ { 0 } ) -> ( `' F " ( A i^i ran F ) ) C_ ( `' F " ( RR \ { 0 } ) ) )
31	29, 30	syl 17	. . . 4 |- ( ( F e. dom S.1 /\ -. 0 e. A ) -> ( `' F " ( A i^i ran F ) ) C_ ( `' F " ( RR \ { 0 } ) ) )
32	15, 31	eqsstrd 3639	. . 3 |- ( ( F e. dom S.1 /\ -. 0 e. A ) -> ( `' F " A ) C_ ( `' F " ( RR \ { 0 } ) ) )
33		i1fima 23445	. . . . 5 |- ( F e. dom S.1 -> ( `' F " ( RR \ { 0 } ) ) e. dom vol )
34	33	adantr 481	. . . 4 |- ( ( F e. dom S.1 /\ -. 0 e. A ) -> ( `' F " ( RR \ { 0 } ) ) e. dom vol )
35		mblss 23299	. . . 4 |- ( ( `' F " ( RR \ { 0 } ) ) e. dom vol -> ( `' F " ( RR \ { 0 } ) ) C_ RR )
36	34, 35	syl 17	. . 3 |- ( ( F e. dom S.1 /\ -. 0 e. A ) -> ( `' F " ( RR \ { 0 } ) ) C_ RR )
37		mblvol 23298	. . . . 5 |- ( ( `' F " ( RR \ { 0 } ) ) e. dom vol -> ( vol ` ( `' F " ( RR \ { 0 } ) ) ) = ( vol* ` ( `' F " ( RR \ { 0 } ) ) ) )
38	34, 37	syl 17	. . . 4 |- ( ( F e. dom S.1 /\ -. 0 e. A ) -> ( vol ` ( `' F " ( RR \ { 0 } ) ) ) = ( vol* ` ( `' F " ( RR \ { 0 } ) ) ) )
39		isi1f 23441	. . . . . . 7 |- ( F e. dom S.1 <-> ( F e. MblFn /\ ( F : RR --> RR /\ ran F e. Fin /\ ( vol ` ( `' F " ( RR \ { 0 } ) ) ) e. RR ) ) )
40	39	simprbi 480	. . . . . 6 |- ( F e. dom S.1 -> ( F : RR --> RR /\ ran F e. Fin /\ ( vol ` ( `' F " ( RR \ { 0 } ) ) ) e. RR ) )
41	40	simp3d 1075	. . . . 5 |- ( F e. dom S.1 -> ( vol ` ( `' F " ( RR \ { 0 } ) ) ) e. RR )
42	41	adantr 481	. . . 4 |- ( ( F e. dom S.1 /\ -. 0 e. A ) -> ( vol ` ( `' F " ( RR \ { 0 } ) ) ) e. RR )
43	38, 42	eqeltrrd 2702	. . 3 |- ( ( F e. dom S.1 /\ -. 0 e. A ) -> ( vol* ` ( `' F " ( RR \ { 0 } ) ) ) e. RR )
44		ovolsscl 23254	. . 3 |- ( ( ( `' F " A ) C_ ( `' F " ( RR \ { 0 } ) ) /\ ( `' F " ( RR \ { 0 } ) ) C_ RR /\ ( vol* ` ( `' F " ( RR \ { 0 } ) ) ) e. RR ) -> ( vol* ` ( `' F " A ) ) e. RR )
45	32, 36, 43, 44	syl3anc 1326	. 2 |- ( ( F e. dom S.1 /\ -. 0 e. A ) -> ( vol* ` ( `' F " A ) ) e. RR )
46	4, 45	eqeltrd 2701	1 |- ( ( F e. dom S.1 /\ -. 0 e. A ) -> ( vol ` ( `' F " A ) ) e. RR )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   \ cdif 3571   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   `'ccnv 5113   dom cdm 5114   ran crn 5115   "cima 5117   Fun wfun 5882   -->wf 5884   ` cfv 5888   Fincfn 7955   RRcr 9935   0cc0 9936   vol*covol 23231   volcvol 23232  MblFncmbf 23383   S.1citg1 23384

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-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-xadd 11947  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-sum 14417  df-xmet 19739  df-met 19740  df-ovol 23233  df-vol 23234  df-mbf 23388  df-itg1 23389

This theorem is referenced by:  i1fima2sn  23447  i1f0rn  23449  itg2addnclem  33461  itg2addnclem2  33462  ftc1anclem3  33487