MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  i1fmulc Unicode version

Theorem i1fmulc 19058
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 8828 . . . . 5  |-  RR  e.  _V
21a1i 10 . . . 4  |-  ( (
ph  /\  A  = 
0 )  ->  RR  e.  _V )
3 i1fmulc.2 . . . . . 6  |-  ( ph  ->  F  e.  dom  S.1 )
4 i1ff 19031 . . . . . 6  |-  ( F  e.  dom  S.1  ->  F : RR --> RR )
53, 4syl 15 . . . . 5  |-  ( ph  ->  F : RR --> RR )
65adantr 451 . . . 4  |-  ( (
ph  /\  A  = 
0 )  ->  F : RR --> RR )
7 i1fmulc.3 . . . . 5  |-  ( ph  ->  A  e.  RR )
87adantr 451 . . . 4  |-  ( (
ph  /\  A  = 
0 )  ->  A  e.  RR )
9 0re 8838 . . . . 5  |-  0  e.  RR
109a1i 10 . . . 4  |-  ( (
ph  /\  A  = 
0 )  ->  0  e.  RR )
11 simplr 731 . . . . . 6  |-  ( ( ( ph  /\  A  =  0 )  /\  x  e.  RR )  ->  A  =  0 )
1211oveq1d 5873 . . . . 5  |-  ( ( ( ph  /\  A  =  0 )  /\  x  e.  RR )  ->  ( A  x.  x
)  =  ( 0  x.  x ) )
13 mul02lem2 8989 . . . . . 6  |-  ( x  e.  RR  ->  (
0  x.  x )  =  0 )
1413adantl 452 . . . . 5  |-  ( ( ( ph  /\  A  =  0 )  /\  x  e.  RR )  ->  ( 0  x.  x
)  =  0 )
1512, 14eqtrd 2315 . . . 4  |-  ( ( ( ph  /\  A  =  0 )  /\  x  e.  RR )  ->  ( A  x.  x
)  =  0 )
162, 6, 8, 10, 15caofid2 6108 . . 3  |-  ( (
ph  /\  A  = 
0 )  ->  (
( RR  X.  { A } )  o F  x.  F )  =  ( RR  X.  {
0 } ) )
17 i1f0 19042 . . 3  |-  ( RR 
X.  { 0 } )  e.  dom  S.1
1816, 17syl6eqel 2371 . 2  |-  ( (
ph  /\  A  = 
0 )  ->  (
( RR  X.  { A } )  o F  x.  F )  e. 
dom  S.1 )
19 remulcl 8822 . . . . . 6  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  x.  y
)  e.  RR )
2019adantl 452 . . . . 5  |-  ( (
ph  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
( x  x.  y
)  e.  RR )
21 fconst6g 5430 . . . . . 6  |-  ( A  e.  RR  ->  ( RR  X.  { A }
) : RR --> RR )
227, 21syl 15 . . . . 5  |-  ( ph  ->  ( RR  X.  { A } ) : RR --> RR )
231a1i 10 . . . . 5  |-  ( ph  ->  RR  e.  _V )
24 inidm 3378 . . . . 5  |-  ( RR 
i^i  RR )  =  RR
2520, 22, 5, 23, 23, 24off 6093 . . . 4  |-  ( ph  ->  ( ( RR  X.  { A } )  o F  x.  F ) : RR --> RR )
2625adantr 451 . . 3  |-  ( (
ph  /\  A  =/=  0 )  ->  (
( RR  X.  { A } )  o F  x.  F ) : RR --> RR )
27 i1frn 19032 . . . . . . 7  |-  ( F  e.  dom  S.1  ->  ran 
F  e.  Fin )
283, 27syl 15 . . . . . 6  |-  ( ph  ->  ran  F  e.  Fin )
29 ovex 5883 . . . . . . . 8  |-  ( A  x.  y )  e. 
_V
30 eqid 2283 . . . . . . . 8  |-  ( y  e.  ran  F  |->  ( A  x.  y ) )  =  ( y  e.  ran  F  |->  ( A  x.  y ) )
3129, 30fnmpti 5372 . . . . . . 7  |-  ( y  e.  ran  F  |->  ( A  x.  y ) )  Fn  ran  F
32 dffn4 5457 . . . . . . 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 199 . . . . . 6  |-  ( y  e.  ran  F  |->  ( A  x.  y ) ) : ran  F -onto-> ran  ( y  e.  ran  F 
|->  ( A  x.  y
) )
34 fofi 7142 . . . . . 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 643 . . . . 5  |-  ( ph  ->  ran  ( y  e. 
ran  F  |->  ( A  x.  y ) )  e.  Fin )
36 id 19 . . . . . . . . . . 11  |-  ( w  e.  ran  F  ->  w  e.  ran  F )
37 elsni 3664 . . . . . . . . . . . 12  |-  ( x  e.  { A }  ->  x  =  A )
3837oveq1d 5873 . . . . . . . . . . 11  |-  ( x  e.  { A }  ->  ( x  x.  w
)  =  ( A  x.  w ) )
39 oveq2 5866 . . . . . . . . . . . . 13  |-  ( y  =  w  ->  ( A  x.  y )  =  ( A  x.  w ) )
4039eqeq2d 2294 . . . . . . . . . . . 12  |-  ( y  =  w  ->  (
( x  x.  w
)  =  ( A  x.  y )  <->  ( x  x.  w )  =  ( A  x.  w ) ) )
4140rspcev 2884 . . . . . . . . . . 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 464 . . . . . . . . . 10  |-  ( ( x  e.  { A }  /\  w  e.  ran  F )  ->  E. y  e.  ran  F ( x  x.  w )  =  ( A  x.  y
) )
43 ovex 5883 . . . . . . . . . . 11  |-  ( x  x.  w )  e. 
_V
44 eqeq1 2289 . . . . . . . . . . . 12  |-  ( z  =  ( x  x.  w )  ->  (
z  =  ( A  x.  y )  <->  ( x  x.  w )  =  ( A  x.  y ) ) )
4544rexbidv 2564 . . . . . . . . . . 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 2914 . . . . . . . . . 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 203 . . . . . . . . 9  |-  ( ( x  e.  { A }  /\  w  e.  ran  F )  ->  ( x  x.  w )  e.  {
z  |  E. y  e.  ran  F  z  =  ( A  x.  y
) } )
4847adantl 452 . . . . . . . 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 5428 . . . . . . . . 9  |-  ( A  e.  RR  ->  ( RR  X.  { A }
) : RR --> { A } )
507, 49syl 15 . . . . . . . 8  |-  ( ph  ->  ( RR  X.  { A } ) : RR --> { A } )
51 ffn 5389 . . . . . . . . . 10  |-  ( F : RR --> RR  ->  F  Fn  RR )
525, 51syl 15 . . . . . . . . 9  |-  ( ph  ->  F  Fn  RR )
53 dffn3 5396 . . . . . . . . 9  |-  ( F  Fn  RR  <->  F : RR
--> ran  F )
5452, 53sylib 188 . . . . . . . 8  |-  ( ph  ->  F : RR --> ran  F
)
5548, 50, 54, 23, 23, 24off 6093 . . . . . . 7  |-  ( ph  ->  ( ( RR  X.  { A } )  o F  x.  F ) : RR --> { z  |  E. y  e. 
ran  F  z  =  ( A  x.  y
) } )
56 frn 5395 . . . . . . 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 15 . . . . . 6  |-  ( ph  ->  ran  ( ( RR 
X.  { A }
)  o F  x.  F )  C_  { z  |  E. y  e. 
ran  F  z  =  ( A  x.  y
) } )
5830rnmpt 4925 . . . . . 6  |-  ran  (
y  e.  ran  F  |->  ( A  x.  y
) )  =  {
z  |  E. y  e.  ran  F  z  =  ( A  x.  y
) }
5957, 58syl6sseqr 3225 . . . . 5  |-  ( ph  ->  ran  ( ( RR 
X.  { A }
)  o F  x.  F )  C_  ran  ( y  e.  ran  F 
|->  ( A  x.  y
) ) )
60 ssfi 7083 . . . . 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 642 . . . 4  |-  ( ph  ->  ran  ( ( RR 
X.  { A }
)  o F  x.  F )  e.  Fin )
6261adantr 451 . . 3  |-  ( (
ph  /\  A  =/=  0 )  ->  ran  ( ( RR  X.  { A } )  o F  x.  F )  e.  Fin )
63 difss 3303 . . . . . . . 8  |-  ( ran  ( ( RR  X.  { A } )  o F  x.  F ) 
\  { 0 } )  C_  ran  ( ( RR  X.  { A } )  o F  x.  F )
64 frn 5395 . . . . . . . . 9  |-  ( ( ( RR  X.  { A } )  o F  x.  F ) : RR --> RR  ->  ran  ( ( RR  X.  { A } )  o F  x.  F ) 
C_  RR )
6525, 64syl 15 . . . . . . . 8  |-  ( ph  ->  ran  ( ( RR 
X.  { A }
)  o F  x.  F )  C_  RR )
6663, 65syl5ss 3190 . . . . . . 7  |-  ( ph  ->  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) 
C_  RR )
6766adantr 451 . . . . . 6  |-  ( (
ph  /\  A  =/=  0 )  ->  ( ran  ( ( RR  X.  { A } )  o F  x.  F ) 
\  { 0 } )  C_  RR )
6867sselda 3180 . . . . 5  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  y  e.  RR )
693, 7i1fmulclem 19057 . . . . 5  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  RR )  ->  ( `' ( ( RR 
X.  { A }
)  o F  x.  F ) " {
y } )  =  ( `' F " { ( y  /  A ) } ) )
7068, 69syldan 456 . . . 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 ) } ) )
71 i1fima 19033 . . . . . 6  |-  ( F  e.  dom  S.1  ->  ( `' F " { ( y  /  A ) } )  e.  dom  vol )
723, 71syl 15 . . . . 5  |-  ( ph  ->  ( `' F " { ( y  /  A ) } )  e.  dom  vol )
7372ad2antrr 706 . . . 4  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  ( `' F " { ( y  /  A ) } )  e.  dom  vol )
7470, 73eqeltrd 2357 . . 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 )
7570fveq2d 5529 . . . 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 ) } ) ) )
763ad2antrr 706 . . . . 5  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  F  e.  dom  S.1 )
777ad2antrr 706 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  A  e.  RR )
78 simplr 731 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  A  =/=  0 )
7968, 77, 78redivcld 9588 . . . . . 6  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  ( y  /  A )  e.  RR )
8068recnd 8861 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  y  e.  CC )
8177recnd 8861 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  A  e.  CC )
82 eldifsni 3750 . . . . . . . 8  |-  ( y  e.  ( ran  (
( RR  X.  { A } )  o F  x.  F )  \  { 0 } )  ->  y  =/=  0
)
8382adantl 452 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  y  =/=  0 )
8480, 81, 83, 78divne0d 9552 . . . . . 6  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  ( y  /  A )  =/=  0
)
85 eldifsn 3749 . . . . . 6  |-  ( ( y  /  A )  e.  ( RR  \  { 0 } )  <-> 
( ( y  /  A )  e.  RR  /\  ( y  /  A
)  =/=  0 ) )
8679, 84, 85sylanbrc 645 . . . . 5  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  ( y  /  A )  e.  ( RR  \  { 0 } ) )
87 i1fima2sn 19035 . . . . 5  |-  ( ( F  e.  dom  S.1  /\  ( y  /  A
)  e.  ( RR 
\  { 0 } ) )  ->  ( vol `  ( `' F " { ( y  /  A ) } ) )  e.  RR )
8876, 86, 87syl2anc 642 . . . 4  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  ( vol `  ( `' F " { ( y  /  A ) } ) )  e.  RR )
8975, 88eqeltrd 2357 . . 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 )
9026, 62, 74, 89i1fd 19036 . 2  |-  ( (
ph  /\  A  =/=  0 )  ->  (
( RR  X.  { A } )  o F  x.  F )  e. 
dom  S.1 )
9118, 90pm2.61dane 2524 1  |-  ( ph  ->  ( ( RR  X.  { A } )  o F  x.  F )  e.  dom  S.1 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269    =/= wne 2446   E.wrex 2544   _Vcvv 2788    \ cdif 3149    C_ wss 3152   {csn 3640    e. cmpt 4077    X. cxp 4687   `'ccnv 4688   dom cdm 4689   ran crn 4690   "cima 4692    Fn wfn 5250   -->wf 5251   -onto->wfo 5253   ` cfv 5255  (class class class)co 5858    o Fcof 6076   Fincfn 6863   RRcr 8736   0cc0 8737    x. cmul 8742    / cdiv 9423   volcvol 18823   S.1citg1 18970
This theorem is referenced by:  itg1mulc  19059  i1fsub  19063  itg1sub  19064  itg2const  19095  itg2mulclem  19101  itg2monolem1  19105  i1fibl  19162  itgitg1  19163
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-xadd 10453  df-ioo 10660  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-sum 12159  df-xmet 16373  df-met 16374  df-ovol 18824  df-vol 18825  df-mbf 18975  df-itg1 18976
  Copyright terms: Public domain W3C validator