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Theorem i1fmulc 19548
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 9037 . . . . 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 19521 . . . . . 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 9047 . . . . 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 6055 . . . . 5  |-  ( ( ( ph  /\  A  =  0 )  /\  x  e.  RR )  ->  ( A  x.  x
)  =  ( 0  x.  x ) )
13 mul02lem2 9199 . . . . . 6  |-  ( x  e.  RR  ->  (
0  x.  x )  =  0 )
1413adantl 453 . . . . 5  |-  ( ( ( ph  /\  A  =  0 )  /\  x  e.  RR )  ->  ( 0  x.  x
)  =  0 )
1512, 14eqtrd 2436 . . . 4  |-  ( ( ( ph  /\  A  =  0 )  /\  x  e.  RR )  ->  ( A  x.  x
)  =  0 )
162, 6, 8, 10, 15caofid2 6294 . . 3  |-  ( (
ph  /\  A  = 
0 )  ->  (
( RR  X.  { A } )  o F  x.  F )  =  ( RR  X.  {
0 } ) )
17 i1f0 19532 . . 3  |-  ( RR 
X.  { 0 } )  e.  dom  S.1
1816, 17syl6eqel 2492 . 2  |-  ( (
ph  /\  A  = 
0 )  ->  (
( RR  X.  { A } )  o F  x.  F )  e. 
dom  S.1 )
19 remulcl 9031 . . . . . 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 5591 . . . . . 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 3510 . . . . 5  |-  ( RR 
i^i  RR )  =  RR
2520, 22, 5, 23, 23, 24off 6279 . . . 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 19522 . . . . . . 7  |-  ( F  e.  dom  S.1  ->  ran 
F  e.  Fin )
283, 27syl 16 . . . . . 6  |-  ( ph  ->  ran  F  e.  Fin )
29 ovex 6065 . . . . . . . 8  |-  ( A  x.  y )  e. 
_V
30 eqid 2404 . . . . . . . 8  |-  ( y  e.  ran  F  |->  ( A  x.  y ) )  =  ( y  e.  ran  F  |->  ( A  x.  y ) )
3129, 30fnmpti 5532 . . . . . . 7  |-  ( y  e.  ran  F  |->  ( A  x.  y ) )  Fn  ran  F
32 dffn4 5618 . . . . . . 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 7351 . . . . . 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 3798 . . . . . . . . . . . 12  |-  ( x  e.  { A }  ->  x  =  A )
3837oveq1d 6055 . . . . . . . . . . 11  |-  ( x  e.  { A }  ->  ( x  x.  w
)  =  ( A  x.  w ) )
39 oveq2 6048 . . . . . . . . . . . . 13  |-  ( y  =  w  ->  ( A  x.  y )  =  ( A  x.  w ) )
4039eqeq2d 2415 . . . . . . . . . . . 12  |-  ( y  =  w  ->  (
( x  x.  w
)  =  ( A  x.  y )  <->  ( x  x.  w )  =  ( A  x.  w ) ) )
4140rspcev 3012 . . . . . . . . . . 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 6065 . . . . . . . . . . 11  |-  ( x  x.  w )  e. 
_V
44 eqeq1 2410 . . . . . . . . . . . 12  |-  ( z  =  ( x  x.  w )  ->  (
z  =  ( A  x.  y )  <->  ( x  x.  w )  =  ( A  x.  y ) ) )
4544rexbidv 2687 . . . . . . . . . . 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 3042 . . . . . . . . . 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 5589 . . . . . . . . 9  |-  ( A  e.  RR  ->  ( RR  X.  { A }
) : RR --> { A } )
507, 49syl 16 . . . . . . . 8  |-  ( ph  ->  ( RR  X.  { A } ) : RR --> { A } )
51 ffn 5550 . . . . . . . . . 10  |-  ( F : RR --> RR  ->  F  Fn  RR )
525, 51syl 16 . . . . . . . . 9  |-  ( ph  ->  F  Fn  RR )
53 dffn3 5557 . . . . . . . . 9  |-  ( F  Fn  RR  <->  F : RR
--> ran  F )
5452, 53sylib 189 . . . . . . . 8  |-  ( ph  ->  F : RR --> ran  F
)
5548, 50, 54, 23, 23, 24off 6279 . . . . . . 7  |-  ( ph  ->  ( ( RR  X.  { A } )  o F  x.  F ) : RR --> { z  |  E. y  e. 
ran  F  z  =  ( A  x.  y
) } )
56 frn 5556 . . . . . . 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 5075 . . . . . 6  |-  ran  (
y  e.  ran  F  |->  ( A  x.  y
) )  =  {
z  |  E. y  e.  ran  F  z  =  ( A  x.  y
) }
5957, 58syl6sseqr 3355 . . . . 5  |-  ( ph  ->  ran  ( ( RR 
X.  { A }
)  o F  x.  F )  C_  ran  ( y  e.  ran  F 
|->  ( A  x.  y
) ) )
60 ssfi 7288 . . . . 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 5556 . . . . . . . . 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 3445 . . . . . . 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 3308 . . . . 5  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  y  e.  RR )
683, 7i1fmulclem 19547 . . . . 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 19523 . . . . . 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 2478 . . 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 5691 . . . 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 9798 . . . . . 6  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  ( y  /  A )  e.  RR )
7967recnd 9070 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  y  e.  CC )
8076recnd 9070 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  A  e.  CC )
81 eldifsni 3888 . . . . . . . 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 9762 . . . . . 6  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  ( y  /  A )  =/=  0
)
84 eldifsn 3887 . . . . . 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 19525 . . . . 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 2478 . . 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 19526 . 2  |-  ( (
ph  /\  A  =/=  0 )  ->  (
( RR  X.  { A } )  o F  x.  F )  e. 
dom  S.1 )
9018, 89pm2.61dane 2645 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 1721   {cab 2390    =/= wne 2567   E.wrex 2667   _Vcvv 2916    \ cdif 3277    C_ wss 3280   {csn 3774    e. cmpt 4226    X. cxp 4835   `'ccnv 4836   dom cdm 4837   ran crn 4838   "cima 4840    Fn wfn 5408   -->wf 5409   -onto->wfo 5411   ` cfv 5413  (class class class)co 6040    o Fcof 6262   Fincfn 7068   RRcr 8945   0cc0 8946    x. cmul 8951    / cdiv 9633   volcvol 19313   S.1citg1 19460
This theorem is referenced by:  itg1mulc  19549  i1fsub  19553  itg1sub  19554  itg2const  19585  itg2mulclem  19591  itg2monolem1  19595  i1fibl  19652  itgitg1  19653  itg2addnclem  26155
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-q 10531  df-rp 10569  df-xadd 10667  df-ioo 10876  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-fl 11157  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-sum 12435  df-xmet 16650  df-met 16651  df-ovol 19314  df-vol 19315  df-mbf 19465  df-itg1 19466
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