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Theorem i1fima2 21849
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
StepHypRef Expression
1 i1fima 21848 . . . 4  |-  ( F  e.  dom  S.1  ->  ( `' F " A )  e.  dom  vol )
21adantr 465 . . 3  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  ( `' F " A )  e. 
dom  vol )
3 mblvol 21704 . . 3  |-  ( ( `' F " A )  e.  dom  vol  ->  ( vol `  ( `' F " A ) )  =  ( vol* `  ( `' F " A ) ) )
42, 3syl 16 . 2  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  ( vol `  ( `' F " A ) )  =  ( vol* `  ( `' F " A ) ) )
5 i1ff 21846 . . . . . . 7  |-  ( F  e.  dom  S.1  ->  F : RR --> RR )
65adantr 465 . . . . . 6  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  F : RR
--> RR )
7 ffun 5733 . . . . . 6  |-  ( F : RR --> RR  ->  Fun 
F )
8 inpreima 6008 . . . . . 6  |-  ( Fun 
F  ->  ( `' F " ( A  i^i  ran 
F ) )  =  ( ( `' F " A )  i^i  ( `' F " ran  F
) ) )
96, 7, 83syl 20 . . . . 5  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  ( `' F " ( A  i^i  ran 
F ) )  =  ( ( `' F " A )  i^i  ( `' F " ran  F
) ) )
10 cnvimass 5357 . . . . . . 7  |-  ( `' F " A ) 
C_  dom  F
11 cnvimarndm 5358 . . . . . . 7  |-  ( `' F " ran  F
)  =  dom  F
1210, 11sseqtr4i 3537 . . . . . 6  |-  ( `' F " A ) 
C_  ( `' F " ran  F )
13 df-ss 3490 . . . . . 6  |-  ( ( `' F " A ) 
C_  ( `' F " ran  F )  <->  ( ( `' F " A )  i^i  ( `' F " ran  F ) )  =  ( `' F " A ) )
1412, 13mpbi 208 . . . . 5  |-  ( ( `' F " A )  i^i  ( `' F " ran  F ) )  =  ( `' F " A )
159, 14syl6req 2525 . . . 4  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  ( `' F " A )  =  ( `' F "
( A  i^i  ran  F ) ) )
16 inss1 3718 . . . . . . . . . 10  |-  ( A  i^i  ran  F )  C_  A
1716sseli 3500 . . . . . . . . 9  |-  ( 0  e.  ( A  i^i  ran 
F )  ->  0  e.  A )
1817con3i 135 . . . . . . . 8  |-  ( -.  0  e.  A  ->  -.  0  e.  ( A  i^i  ran  F )
)
1918adantl 466 . . . . . . 7  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  -.  0  e.  ( A  i^i  ran  F ) )
20 disjsn 4088 . . . . . . 7  |-  ( ( ( A  i^i  ran  F )  i^i  { 0 } )  =  (/)  <->  -.  0  e.  ( A  i^i  ran  F ) )
2119, 20sylibr 212 . . . . . 6  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  ( ( A  i^i  ran  F )  i^i  { 0 } )  =  (/) )
22 inss2 3719 . . . . . . . . 9  |-  ( A  i^i  ran  F )  C_ 
ran  F
23 frn 5737 . . . . . . . . . 10  |-  ( F : RR --> RR  ->  ran 
F  C_  RR )
245, 23syl 16 . . . . . . . . 9  |-  ( F  e.  dom  S.1  ->  ran 
F  C_  RR )
2522, 24syl5ss 3515 . . . . . . . 8  |-  ( F  e.  dom  S.1  ->  ( A  i^i  ran  F
)  C_  RR )
2625adantr 465 . . . . . . 7  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  ( A  i^i  ran  F )  C_  RR )
27 reldisj 3870 . . . . . . 7  |-  ( ( A  i^i  ran  F
)  C_  RR  ->  ( ( ( A  i^i  ran 
F )  i^i  {
0 } )  =  (/) 
<->  ( A  i^i  ran  F )  C_  ( RR  \  { 0 } ) ) )
2826, 27syl 16 . . . . . 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 } ) ) )
2921, 28mpbid 210 . . . . 5  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  ( A  i^i  ran  F )  C_  ( RR  \  { 0 } ) )
30 imass2 5372 . . . . 5  |-  ( ( A  i^i  ran  F
)  C_  ( RR  \  { 0 } )  ->  ( `' F " ( A  i^i  ran  F ) )  C_  ( `' F " ( RR 
\  { 0 } ) ) )
3129, 30syl 16 . . . 4  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  ( `' F " ( A  i^i  ran 
F ) )  C_  ( `' F " ( RR 
\  { 0 } ) ) )
3215, 31eqsstrd 3538 . . 3  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  ( `' F " A )  C_  ( `' F " ( RR 
\  { 0 } ) ) )
33 i1fima 21848 . . . . 5  |-  ( F  e.  dom  S.1  ->  ( `' F " ( RR 
\  { 0 } ) )  e.  dom  vol )
3433adantr 465 . . . 4  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  ( `' F " ( RR  \  { 0 } ) )  e.  dom  vol )
35 mblss 21705 . . . 4  |-  ( ( `' F " ( RR 
\  { 0 } ) )  e.  dom  vol 
->  ( `' F "
( RR  \  {
0 } ) ) 
C_  RR )
3634, 35syl 16 . . 3  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  ( `' F " ( RR  \  { 0 } ) )  C_  RR )
37 mblvol 21704 . . . . 5  |-  ( ( `' F " ( RR 
\  { 0 } ) )  e.  dom  vol 
->  ( vol `  ( `' F " ( RR 
\  { 0 } ) ) )  =  ( vol* `  ( `' F " ( RR 
\  { 0 } ) ) ) )
3834, 37syl 16 . . . 4  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  ( vol `  ( `' F "
( RR  \  {
0 } ) ) )  =  ( vol* `  ( `' F " ( RR  \  { 0 } ) ) ) )
39 isi1f 21844 . . . . . . 7  |-  ( F  e.  dom  S.1  <->  ( F  e. MblFn  /\  ( F : RR
--> RR  /\  ran  F  e.  Fin  /\  ( vol `  ( `' F "
( RR  \  {
0 } ) ) )  e.  RR ) ) )
4039simprbi 464 . . . . . 6  |-  ( F  e.  dom  S.1  ->  ( F : RR --> RR  /\  ran  F  e.  Fin  /\  ( vol `  ( `' F " ( RR 
\  { 0 } ) ) )  e.  RR ) )
4140simp3d 1010 . . . . 5  |-  ( F  e.  dom  S.1  ->  ( vol `  ( `' F " ( RR 
\  { 0 } ) ) )  e.  RR )
4241adantr 465 . . . 4  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  ( vol `  ( `' F "
( RR  \  {
0 } ) ) )  e.  RR )
4338, 42eqeltrrd 2556 . . 3  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  ( vol* `  ( `' F " ( RR  \  {
0 } ) ) )  e.  RR )
44 ovolsscl 21660 . . 3  |-  ( ( ( `' F " A )  C_  ( `' F " ( RR 
\  { 0 } ) )  /\  ( `' F " ( RR 
\  { 0 } ) )  C_  RR  /\  ( vol* `  ( `' F " ( RR 
\  { 0 } ) ) )  e.  RR )  ->  ( vol* `  ( `' F " A ) )  e.  RR )
4532, 36, 43, 44syl3anc 1228 . 2  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  ( vol* `  ( `' F " A ) )  e.  RR )
464, 45eqeltrd 2555 1  |-  ( ( F  e.  dom  S.1  /\ 
-.  0  e.  A
)  ->  ( vol `  ( `' F " A ) )  e.  RR )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    \ cdif 3473    i^i cin 3475    C_ wss 3476   (/)c0 3785   {csn 4027   `'ccnv 4998   dom cdm 4999   ran crn 5000   "cima 5002   Fun wfun 5582   -->wf 5584   ` cfv 5588   Fincfn 7516   RRcr 9491   0cc0 9492   vol*covol 21637   volcvol 21638  MblFncmbf 21786   S.1citg1 21787
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-of 6524  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-sup 7901  df-oi 7935  df-card 8320  df-cda 8548  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-n0 10796  df-z 10865  df-uz 11083  df-q 11183  df-rp 11221  df-xadd 11319  df-ioo 11533  df-ico 11535  df-icc 11536  df-fz 11673  df-fzo 11793  df-fl 11897  df-seq 12076  df-exp 12135  df-hash 12374  df-cj 12895  df-re 12896  df-im 12897  df-sqrt 13031  df-abs 13032  df-clim 13274  df-sum 13472  df-xmet 18211  df-met 18212  df-ovol 21639  df-vol 21640  df-mbf 21791  df-itg1 21792
This theorem is referenced by:  i1fima2sn  21850  i1f0rn  21852  itg2addnclem  29671  itg2addnclem2  29672  ftc1anclem3  29697
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