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Theorem i1fmulc 19455
Description: A nonnegative constant times a simple function gives another simple function. (Contributed by Mario Carneiro, 25-Jun-2014.)
Hypotheses
Ref Expression
i1fmulc.2  |-  ( ph  ->  F  e.  dom  S.1 )
i1fmulc.3  |-  ( ph  ->  A  e.  RR )
Assertion
Ref Expression
i1fmulc  |-  ( ph  ->  ( ( RR  X.  { A } )  o F  x.  F )  e.  dom  S.1 )

Proof of Theorem i1fmulc
Dummy variables  w  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 reex 9007 . . . . 5  |-  RR  e.  _V
21a1i 11 . . . 4  |-  ( (
ph  /\  A  = 
0 )  ->  RR  e.  _V )
3 i1fmulc.2 . . . . . 6  |-  ( ph  ->  F  e.  dom  S.1 )
4 i1ff 19428 . . . . . 6  |-  ( F  e.  dom  S.1  ->  F : RR --> RR )
53, 4syl 16 . . . . 5  |-  ( ph  ->  F : RR --> RR )
65adantr 452 . . . 4  |-  ( (
ph  /\  A  = 
0 )  ->  F : RR --> RR )
7 i1fmulc.3 . . . . 5  |-  ( ph  ->  A  e.  RR )
87adantr 452 . . . 4  |-  ( (
ph  /\  A  = 
0 )  ->  A  e.  RR )
9 0re 9017 . . . . 5  |-  0  e.  RR
109a1i 11 . . . 4  |-  ( (
ph  /\  A  = 
0 )  ->  0  e.  RR )
11 simplr 732 . . . . . 6  |-  ( ( ( ph  /\  A  =  0 )  /\  x  e.  RR )  ->  A  =  0 )
1211oveq1d 6028 . . . . 5  |-  ( ( ( ph  /\  A  =  0 )  /\  x  e.  RR )  ->  ( A  x.  x
)  =  ( 0  x.  x ) )
13 mul02lem2 9168 . . . . . 6  |-  ( x  e.  RR  ->  (
0  x.  x )  =  0 )
1413adantl 453 . . . . 5  |-  ( ( ( ph  /\  A  =  0 )  /\  x  e.  RR )  ->  ( 0  x.  x
)  =  0 )
1512, 14eqtrd 2412 . . . 4  |-  ( ( ( ph  /\  A  =  0 )  /\  x  e.  RR )  ->  ( A  x.  x
)  =  0 )
162, 6, 8, 10, 15caofid2 6267 . . 3  |-  ( (
ph  /\  A  = 
0 )  ->  (
( RR  X.  { A } )  o F  x.  F )  =  ( RR  X.  {
0 } ) )
17 i1f0 19439 . . 3  |-  ( RR 
X.  { 0 } )  e.  dom  S.1
1816, 17syl6eqel 2468 . 2  |-  ( (
ph  /\  A  = 
0 )  ->  (
( RR  X.  { A } )  o F  x.  F )  e. 
dom  S.1 )
19 remulcl 9001 . . . . . 6  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  x.  y
)  e.  RR )
2019adantl 453 . . . . 5  |-  ( (
ph  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
( x  x.  y
)  e.  RR )
21 fconst6g 5565 . . . . . 6  |-  ( A  e.  RR  ->  ( RR  X.  { A }
) : RR --> RR )
227, 21syl 16 . . . . 5  |-  ( ph  ->  ( RR  X.  { A } ) : RR --> RR )
231a1i 11 . . . . 5  |-  ( ph  ->  RR  e.  _V )
24 inidm 3486 . . . . 5  |-  ( RR 
i^i  RR )  =  RR
2520, 22, 5, 23, 23, 24off 6252 . . . 4  |-  ( ph  ->  ( ( RR  X.  { A } )  o F  x.  F ) : RR --> RR )
2625adantr 452 . . 3  |-  ( (
ph  /\  A  =/=  0 )  ->  (
( RR  X.  { A } )  o F  x.  F ) : RR --> RR )
27 i1frn 19429 . . . . . . 7  |-  ( F  e.  dom  S.1  ->  ran 
F  e.  Fin )
283, 27syl 16 . . . . . 6  |-  ( ph  ->  ran  F  e.  Fin )
29 ovex 6038 . . . . . . . 8  |-  ( A  x.  y )  e. 
_V
30 eqid 2380 . . . . . . . 8  |-  ( y  e.  ran  F  |->  ( A  x.  y ) )  =  ( y  e.  ran  F  |->  ( A  x.  y ) )
3129, 30fnmpti 5506 . . . . . . 7  |-  ( y  e.  ran  F  |->  ( A  x.  y ) )  Fn  ran  F
32 dffn4 5592 . . . . . . 7  |-  ( ( y  e.  ran  F  |->  ( A  x.  y
) )  Fn  ran  F  <-> 
( y  e.  ran  F 
|->  ( A  x.  y
) ) : ran  F
-onto->
ran  ( y  e. 
ran  F  |->  ( A  x.  y ) ) )
3331, 32mpbi 200 . . . . . 6  |-  ( y  e.  ran  F  |->  ( A  x.  y ) ) : ran  F -onto-> ran  ( y  e.  ran  F 
|->  ( A  x.  y
) )
34 fofi 7321 . . . . . 6  |-  ( ( ran  F  e.  Fin  /\  ( y  e.  ran  F 
|->  ( A  x.  y
) ) : ran  F
-onto->
ran  ( y  e. 
ran  F  |->  ( A  x.  y ) ) )  ->  ran  ( y  e.  ran  F  |->  ( A  x.  y ) )  e.  Fin )
3528, 33, 34sylancl 644 . . . . 5  |-  ( ph  ->  ran  ( y  e. 
ran  F  |->  ( A  x.  y ) )  e.  Fin )
36 id 20 . . . . . . . . . . 11  |-  ( w  e.  ran  F  ->  w  e.  ran  F )
37 elsni 3774 . . . . . . . . . . . 12  |-  ( x  e.  { A }  ->  x  =  A )
3837oveq1d 6028 . . . . . . . . . . 11  |-  ( x  e.  { A }  ->  ( x  x.  w
)  =  ( A  x.  w ) )
39 oveq2 6021 . . . . . . . . . . . . 13  |-  ( y  =  w  ->  ( A  x.  y )  =  ( A  x.  w ) )
4039eqeq2d 2391 . . . . . . . . . . . 12  |-  ( y  =  w  ->  (
( x  x.  w
)  =  ( A  x.  y )  <->  ( x  x.  w )  =  ( A  x.  w ) ) )
4140rspcev 2988 . . . . . . . . . . 11  |-  ( ( w  e.  ran  F  /\  ( x  x.  w
)  =  ( A  x.  w ) )  ->  E. y  e.  ran  F ( x  x.  w
)  =  ( A  x.  y ) )
4236, 38, 41syl2anr 465 . . . . . . . . . 10  |-  ( ( x  e.  { A }  /\  w  e.  ran  F )  ->  E. y  e.  ran  F ( x  x.  w )  =  ( A  x.  y
) )
43 ovex 6038 . . . . . . . . . . 11  |-  ( x  x.  w )  e. 
_V
44 eqeq1 2386 . . . . . . . . . . . 12  |-  ( z  =  ( x  x.  w )  ->  (
z  =  ( A  x.  y )  <->  ( x  x.  w )  =  ( A  x.  y ) ) )
4544rexbidv 2663 . . . . . . . . . . 11  |-  ( z  =  ( x  x.  w )  ->  ( E. y  e.  ran  F  z  =  ( A  x.  y )  <->  E. y  e.  ran  F ( x  x.  w )  =  ( A  x.  y
) ) )
4643, 45elab 3018 . . . . . . . . . 10  |-  ( ( x  x.  w )  e.  { z  |  E. y  e.  ran  F  z  =  ( A  x.  y ) }  <->  E. y  e.  ran  F ( x  x.  w
)  =  ( A  x.  y ) )
4742, 46sylibr 204 . . . . . . . . 9  |-  ( ( x  e.  { A }  /\  w  e.  ran  F )  ->  ( x  x.  w )  e.  {
z  |  E. y  e.  ran  F  z  =  ( A  x.  y
) } )
4847adantl 453 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  { A }  /\  w  e.  ran  F ) )  ->  ( x  x.  w )  e.  {
z  |  E. y  e.  ran  F  z  =  ( A  x.  y
) } )
49 fconstg 5563 . . . . . . . . 9  |-  ( A  e.  RR  ->  ( RR  X.  { A }
) : RR --> { A } )
507, 49syl 16 . . . . . . . 8  |-  ( ph  ->  ( RR  X.  { A } ) : RR --> { A } )
51 ffn 5524 . . . . . . . . . 10  |-  ( F : RR --> RR  ->  F  Fn  RR )
525, 51syl 16 . . . . . . . . 9  |-  ( ph  ->  F  Fn  RR )
53 dffn3 5531 . . . . . . . . 9  |-  ( F  Fn  RR  <->  F : RR
--> ran  F )
5452, 53sylib 189 . . . . . . . 8  |-  ( ph  ->  F : RR --> ran  F
)
5548, 50, 54, 23, 23, 24off 6252 . . . . . . 7  |-  ( ph  ->  ( ( RR  X.  { A } )  o F  x.  F ) : RR --> { z  |  E. y  e. 
ran  F  z  =  ( A  x.  y
) } )
56 frn 5530 . . . . . . 7  |-  ( ( ( RR  X.  { A } )  o F  x.  F ) : RR --> { z  |  E. y  e.  ran  F  z  =  ( A  x.  y ) }  ->  ran  ( ( RR  X.  { A }
)  o F  x.  F )  C_  { z  |  E. y  e. 
ran  F  z  =  ( A  x.  y
) } )
5755, 56syl 16 . . . . . 6  |-  ( ph  ->  ran  ( ( RR 
X.  { A }
)  o F  x.  F )  C_  { z  |  E. y  e. 
ran  F  z  =  ( A  x.  y
) } )
5830rnmpt 5049 . . . . . 6  |-  ran  (
y  e.  ran  F  |->  ( A  x.  y
) )  =  {
z  |  E. y  e.  ran  F  z  =  ( A  x.  y
) }
5957, 58syl6sseqr 3331 . . . . 5  |-  ( ph  ->  ran  ( ( RR 
X.  { A }
)  o F  x.  F )  C_  ran  ( y  e.  ran  F 
|->  ( A  x.  y
) ) )
60 ssfi 7258 . . . . 5  |-  ( ( ran  ( y  e. 
ran  F  |->  ( A  x.  y ) )  e.  Fin  /\  ran  ( ( RR  X.  { A } )  o F  x.  F ) 
C_  ran  ( y  e.  ran  F  |->  ( A  x.  y ) ) )  ->  ran  ( ( RR  X.  { A } )  o F  x.  F )  e. 
Fin )
6135, 59, 60syl2anc 643 . . . 4  |-  ( ph  ->  ran  ( ( RR 
X.  { A }
)  o F  x.  F )  e.  Fin )
6261adantr 452 . . 3  |-  ( (
ph  /\  A  =/=  0 )  ->  ran  ( ( RR  X.  { A } )  o F  x.  F )  e.  Fin )
63 frn 5530 . . . . . . . . 9  |-  ( ( ( RR  X.  { A } )  o F  x.  F ) : RR --> RR  ->  ran  ( ( RR  X.  { A } )  o F  x.  F ) 
C_  RR )
6425, 63syl 16 . . . . . . . 8  |-  ( ph  ->  ran  ( ( RR 
X.  { A }
)  o F  x.  F )  C_  RR )
6564ssdifssd 3421 . . . . . . 7  |-  ( ph  ->  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) 
C_  RR )
6665adantr 452 . . . . . 6  |-  ( (
ph  /\  A  =/=  0 )  ->  ( ran  ( ( RR  X.  { A } )  o F  x.  F ) 
\  { 0 } )  C_  RR )
6766sselda 3284 . . . . 5  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  y  e.  RR )
683, 7i1fmulclem 19454 . . . . 5  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  RR )  ->  ( `' ( ( RR 
X.  { A }
)  o F  x.  F ) " {
y } )  =  ( `' F " { ( y  /  A ) } ) )
6967, 68syldan 457 . . . 4  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  ( `' ( ( RR  X.  { A } )  o F  x.  F )
" { y } )  =  ( `' F " { ( y  /  A ) } ) )
70 i1fima 19430 . . . . . 6  |-  ( F  e.  dom  S.1  ->  ( `' F " { ( y  /  A ) } )  e.  dom  vol )
713, 70syl 16 . . . . 5  |-  ( ph  ->  ( `' F " { ( y  /  A ) } )  e.  dom  vol )
7271ad2antrr 707 . . . 4  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  ( `' F " { ( y  /  A ) } )  e.  dom  vol )
7369, 72eqeltrd 2454 . . 3  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  ( `' ( ( RR  X.  { A } )  o F  x.  F )
" { y } )  e.  dom  vol )
7469fveq2d 5665 . . . 4  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  ( vol `  ( `' ( ( RR  X.  { A } )  o F  x.  F ) " { y } ) )  =  ( vol `  ( `' F " { ( y  /  A ) } ) ) )
753ad2antrr 707 . . . . 5  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  F  e.  dom  S.1 )
767ad2antrr 707 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  A  e.  RR )
77 simplr 732 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  A  =/=  0 )
7867, 76, 77redivcld 9767 . . . . . 6  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  ( y  /  A )  e.  RR )
7967recnd 9040 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  y  e.  CC )
8076recnd 9040 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  A  e.  CC )
81 eldifsni 3864 . . . . . . . 8  |-  ( y  e.  ( ran  (
( RR  X.  { A } )  o F  x.  F )  \  { 0 } )  ->  y  =/=  0
)
8281adantl 453 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  y  =/=  0 )
8379, 80, 82, 77divne0d 9731 . . . . . 6  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  ( y  /  A )  =/=  0
)
84 eldifsn 3863 . . . . . 6  |-  ( ( y  /  A )  e.  ( RR  \  { 0 } )  <-> 
( ( y  /  A )  e.  RR  /\  ( y  /  A
)  =/=  0 ) )
8578, 83, 84sylanbrc 646 . . . . 5  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  ( y  /  A )  e.  ( RR  \  { 0 } ) )
86 i1fima2sn 19432 . . . . 5  |-  ( ( F  e.  dom  S.1  /\  ( y  /  A
)  e.  ( RR 
\  { 0 } ) )  ->  ( vol `  ( `' F " { ( y  /  A ) } ) )  e.  RR )
8775, 85, 86syl2anc 643 . . . 4  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  ( vol `  ( `' F " { ( y  /  A ) } ) )  e.  RR )
8874, 87eqeltrd 2454 . . 3  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  ( vol `  ( `' ( ( RR  X.  { A } )  o F  x.  F ) " { y } ) )  e.  RR )
8926, 62, 73, 88i1fd 19433 . 2  |-  ( (
ph  /\  A  =/=  0 )  ->  (
( RR  X.  { A } )  o F  x.  F )  e. 
dom  S.1 )
9018, 89pm2.61dane 2621 1  |-  ( ph  ->  ( ( RR  X.  { A } )  o F  x.  F )  e.  dom  S.1 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   {cab 2366    =/= wne 2543   E.wrex 2643   _Vcvv 2892    \ cdif 3253    C_ wss 3256   {csn 3750    e. cmpt 4200    X. cxp 4809   `'ccnv 4810   dom cdm 4811   ran crn 4812   "cima 4814    Fn wfn 5382   -->wf 5383   -onto->wfo 5385   ` cfv 5387  (class class class)co 6013    o Fcof 6235   Fincfn 7038   RRcr 8915   0cc0 8916    x. cmul 8921    / cdiv 9602   volcvol 19220   S.1citg1 19367
This theorem is referenced by:  itg1mulc  19456  i1fsub  19460  itg1sub  19461  itg2const  19492  itg2mulclem  19498  itg2monolem1  19502  i1fibl  19559  itgitg1  19560  itg2addnclem  25950
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-inf2 7522  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993  ax-pre-sup 8994
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-se 4476  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-isom 5396  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-of 6237  df-1st 6281  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-1o 6653  df-2o 6654  df-oadd 6657  df-er 6834  df-map 6949  df-pm 6950  df-en 7039  df-dom 7040  df-sdom 7041  df-fin 7042  df-sup 7374  df-oi 7405  df-card 7752  df-cda 7974  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-div 9603  df-nn 9926  df-2 9983  df-3 9984  df-n0 10147  df-z 10208  df-uz 10414  df-q 10500  df-rp 10538  df-xadd 10636  df-ioo 10845  df-ico 10847  df-icc 10848  df-fz 10969  df-fzo 11059  df-fl 11122  df-seq 11244  df-exp 11303  df-hash 11539  df-cj 11824  df-re 11825  df-im 11826  df-sqr 11960  df-abs 11961  df-clim 12202  df-sum 12400  df-xmet 16612  df-met 16613  df-ovol 19221  df-vol 19222  df-mbf 19372  df-itg1 19373
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