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Theorem i1fmulc 21156
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 } )  oF  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 9365 . . . . 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 21129 . . . . . 6  |-  ( F  e.  dom  S.1  ->  F : RR --> RR )
53, 4syl 16 . . . . 5  |-  ( ph  ->  F : RR --> RR )
65adantr 465 . . . 4  |-  ( (
ph  /\  A  = 
0 )  ->  F : RR --> RR )
7 i1fmulc.3 . . . . 5  |-  ( ph  ->  A  e.  RR )
87adantr 465 . . . 4  |-  ( (
ph  /\  A  = 
0 )  ->  A  e.  RR )
9 0red 9379 . . . 4  |-  ( (
ph  /\  A  = 
0 )  ->  0  e.  RR )
10 simplr 754 . . . . . 6  |-  ( ( ( ph  /\  A  =  0 )  /\  x  e.  RR )  ->  A  =  0 )
1110oveq1d 6101 . . . . 5  |-  ( ( ( ph  /\  A  =  0 )  /\  x  e.  RR )  ->  ( A  x.  x
)  =  ( 0  x.  x ) )
12 mul02lem2 9538 . . . . . 6  |-  ( x  e.  RR  ->  (
0  x.  x )  =  0 )
1312adantl 466 . . . . 5  |-  ( ( ( ph  /\  A  =  0 )  /\  x  e.  RR )  ->  ( 0  x.  x
)  =  0 )
1411, 13eqtrd 2470 . . . 4  |-  ( ( ( ph  /\  A  =  0 )  /\  x  e.  RR )  ->  ( A  x.  x
)  =  0 )
152, 6, 8, 9, 14caofid2 6346 . . 3  |-  ( (
ph  /\  A  = 
0 )  ->  (
( RR  X.  { A } )  oF  x.  F )  =  ( RR  X.  {
0 } ) )
16 i1f0 21140 . . 3  |-  ( RR 
X.  { 0 } )  e.  dom  S.1
1715, 16syl6eqel 2526 . 2  |-  ( (
ph  /\  A  = 
0 )  ->  (
( RR  X.  { A } )  oF  x.  F )  e. 
dom  S.1 )
18 remulcl 9359 . . . . . 6  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  x.  y
)  e.  RR )
1918adantl 466 . . . . 5  |-  ( (
ph  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
( x  x.  y
)  e.  RR )
20 fconst6g 5594 . . . . . 6  |-  ( A  e.  RR  ->  ( RR  X.  { A }
) : RR --> RR )
217, 20syl 16 . . . . 5  |-  ( ph  ->  ( RR  X.  { A } ) : RR --> RR )
221a1i 11 . . . . 5  |-  ( ph  ->  RR  e.  _V )
23 inidm 3554 . . . . 5  |-  ( RR 
i^i  RR )  =  RR
2419, 21, 5, 22, 22, 23off 6329 . . . 4  |-  ( ph  ->  ( ( RR  X.  { A } )  oF  x.  F ) : RR --> RR )
2524adantr 465 . . 3  |-  ( (
ph  /\  A  =/=  0 )  ->  (
( RR  X.  { A } )  oF  x.  F ) : RR --> RR )
26 i1frn 21130 . . . . . . 7  |-  ( F  e.  dom  S.1  ->  ran 
F  e.  Fin )
273, 26syl 16 . . . . . 6  |-  ( ph  ->  ran  F  e.  Fin )
28 ovex 6111 . . . . . . . 8  |-  ( A  x.  y )  e. 
_V
29 eqid 2438 . . . . . . . 8  |-  ( y  e.  ran  F  |->  ( A  x.  y ) )  =  ( y  e.  ran  F  |->  ( A  x.  y ) )
3028, 29fnmpti 5534 . . . . . . 7  |-  ( y  e.  ran  F  |->  ( A  x.  y ) )  Fn  ran  F
31 dffn4 5621 . . . . . . 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 ) ) )
3230, 31mpbi 208 . . . . . 6  |-  ( y  e.  ran  F  |->  ( A  x.  y ) ) : ran  F -onto-> ran  ( y  e.  ran  F 
|->  ( A  x.  y
) )
33 fofi 7589 . . . . . 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 )
3427, 32, 33sylancl 662 . . . . 5  |-  ( ph  ->  ran  ( y  e. 
ran  F  |->  ( A  x.  y ) )  e.  Fin )
35 id 22 . . . . . . . . . . 11  |-  ( w  e.  ran  F  ->  w  e.  ran  F )
36 elsni 3897 . . . . . . . . . . . 12  |-  ( x  e.  { A }  ->  x  =  A )
3736oveq1d 6101 . . . . . . . . . . 11  |-  ( x  e.  { A }  ->  ( x  x.  w
)  =  ( A  x.  w ) )
38 oveq2 6094 . . . . . . . . . . . . 13  |-  ( y  =  w  ->  ( A  x.  y )  =  ( A  x.  w ) )
3938eqeq2d 2449 . . . . . . . . . . . 12  |-  ( y  =  w  ->  (
( x  x.  w
)  =  ( A  x.  y )  <->  ( x  x.  w )  =  ( A  x.  w ) ) )
4039rspcev 3068 . . . . . . . . . . 11  |-  ( ( w  e.  ran  F  /\  ( x  x.  w
)  =  ( A  x.  w ) )  ->  E. y  e.  ran  F ( x  x.  w
)  =  ( A  x.  y ) )
4135, 37, 40syl2anr 478 . . . . . . . . . 10  |-  ( ( x  e.  { A }  /\  w  e.  ran  F )  ->  E. y  e.  ran  F ( x  x.  w )  =  ( A  x.  y
) )
42 ovex 6111 . . . . . . . . . . 11  |-  ( x  x.  w )  e. 
_V
43 eqeq1 2444 . . . . . . . . . . . 12  |-  ( z  =  ( x  x.  w )  ->  (
z  =  ( A  x.  y )  <->  ( x  x.  w )  =  ( A  x.  y ) ) )
4443rexbidv 2731 . . . . . . . . . . 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
) ) )
4542, 44elab 3101 . . . . . . . . . 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 ) )
4641, 45sylibr 212 . . . . . . . . 9  |-  ( ( x  e.  { A }  /\  w  e.  ran  F )  ->  ( x  x.  w )  e.  {
z  |  E. y  e.  ran  F  z  =  ( A  x.  y
) } )
4746adantl 466 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  { A }  /\  w  e.  ran  F ) )  ->  ( x  x.  w )  e.  {
z  |  E. y  e.  ran  F  z  =  ( A  x.  y
) } )
48 fconstg 5592 . . . . . . . . 9  |-  ( A  e.  RR  ->  ( RR  X.  { A }
) : RR --> { A } )
497, 48syl 16 . . . . . . . 8  |-  ( ph  ->  ( RR  X.  { A } ) : RR --> { A } )
50 ffn 5554 . . . . . . . . . 10  |-  ( F : RR --> RR  ->  F  Fn  RR )
515, 50syl 16 . . . . . . . . 9  |-  ( ph  ->  F  Fn  RR )
52 dffn3 5561 . . . . . . . . 9  |-  ( F  Fn  RR  <->  F : RR
--> ran  F )
5351, 52sylib 196 . . . . . . . 8  |-  ( ph  ->  F : RR --> ran  F
)
5447, 49, 53, 22, 22, 23off 6329 . . . . . . 7  |-  ( ph  ->  ( ( RR  X.  { A } )  oF  x.  F ) : RR --> { z  |  E. y  e. 
ran  F  z  =  ( A  x.  y
) } )
55 frn 5560 . . . . . . 7  |-  ( ( ( RR  X.  { A } )  oF  x.  F ) : RR --> { z  |  E. y  e.  ran  F  z  =  ( A  x.  y ) }  ->  ran  ( ( RR  X.  { A }
)  oF  x.  F )  C_  { z  |  E. y  e. 
ran  F  z  =  ( A  x.  y
) } )
5654, 55syl 16 . . . . . 6  |-  ( ph  ->  ran  ( ( RR 
X.  { A }
)  oF  x.  F )  C_  { z  |  E. y  e. 
ran  F  z  =  ( A  x.  y
) } )
5729rnmpt 5080 . . . . . 6  |-  ran  (
y  e.  ran  F  |->  ( A  x.  y
) )  =  {
z  |  E. y  e.  ran  F  z  =  ( A  x.  y
) }
5856, 57syl6sseqr 3398 . . . . 5  |-  ( ph  ->  ran  ( ( RR 
X.  { A }
)  oF  x.  F )  C_  ran  ( y  e.  ran  F 
|->  ( A  x.  y
) ) )
59 ssfi 7525 . . . . 5  |-  ( ( ran  ( y  e. 
ran  F  |->  ( A  x.  y ) )  e.  Fin  /\  ran  ( ( RR  X.  { A } )  oF  x.  F ) 
C_  ran  ( y  e.  ran  F  |->  ( A  x.  y ) ) )  ->  ran  ( ( RR  X.  { A } )  oF  x.  F )  e. 
Fin )
6034, 58, 59syl2anc 661 . . . 4  |-  ( ph  ->  ran  ( ( RR 
X.  { A }
)  oF  x.  F )  e.  Fin )
6160adantr 465 . . 3  |-  ( (
ph  /\  A  =/=  0 )  ->  ran  ( ( RR  X.  { A } )  oF  x.  F )  e.  Fin )
62 frn 5560 . . . . . . . . 9  |-  ( ( ( RR  X.  { A } )  oF  x.  F ) : RR --> RR  ->  ran  ( ( RR  X.  { A } )  oF  x.  F ) 
C_  RR )
6324, 62syl 16 . . . . . . . 8  |-  ( ph  ->  ran  ( ( RR 
X.  { A }
)  oF  x.  F )  C_  RR )
6463ssdifssd 3489 . . . . . . 7  |-  ( ph  ->  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) 
C_  RR )
6564adantr 465 . . . . . 6  |-  ( (
ph  /\  A  =/=  0 )  ->  ( ran  ( ( RR  X.  { A } )  oF  x.  F ) 
\  { 0 } )  C_  RR )
6665sselda 3351 . . . . 5  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  y  e.  RR )
673, 7i1fmulclem 21155 . . . . 5  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  RR )  ->  ( `' ( ( RR 
X.  { A }
)  oF  x.  F ) " {
y } )  =  ( `' F " { ( y  /  A ) } ) )
6866, 67syldan 470 . . . 4  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  ( `' ( ( RR  X.  { A } )  oF  x.  F )
" { y } )  =  ( `' F " { ( y  /  A ) } ) )
69 i1fima 21131 . . . . . 6  |-  ( F  e.  dom  S.1  ->  ( `' F " { ( y  /  A ) } )  e.  dom  vol )
703, 69syl 16 . . . . 5  |-  ( ph  ->  ( `' F " { ( y  /  A ) } )  e.  dom  vol )
7170ad2antrr 725 . . . 4  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  ( `' F " { ( y  /  A ) } )  e.  dom  vol )
7268, 71eqeltrd 2512 . . 3  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  ( `' ( ( RR  X.  { A } )  oF  x.  F )
" { y } )  e.  dom  vol )
7368fveq2d 5690 . . . 4  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  ( vol `  ( `' ( ( RR  X.  { A } )  oF  x.  F ) " { y } ) )  =  ( vol `  ( `' F " { ( y  /  A ) } ) ) )
743ad2antrr 725 . . . . 5  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  F  e.  dom  S.1 )
757ad2antrr 725 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  A  e.  RR )
76 simplr 754 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  A  =/=  0 )
7766, 75, 76redivcld 10151 . . . . . 6  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  ( y  /  A )  e.  RR )
7866recnd 9404 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  y  e.  CC )
7975recnd 9404 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  A  e.  CC )
80 eldifsni 3996 . . . . . . . 8  |-  ( y  e.  ( ran  (
( RR  X.  { A } )  oF  x.  F )  \  { 0 } )  ->  y  =/=  0
)
8180adantl 466 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  y  =/=  0 )
8278, 79, 81, 76divne0d 10115 . . . . . 6  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  ( y  /  A )  =/=  0
)
83 eldifsn 3995 . . . . . 6  |-  ( ( y  /  A )  e.  ( RR  \  { 0 } )  <-> 
( ( y  /  A )  e.  RR  /\  ( y  /  A
)  =/=  0 ) )
8477, 82, 83sylanbrc 664 . . . . 5  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  ( y  /  A )  e.  ( RR  \  { 0 } ) )
85 i1fima2sn 21133 . . . . 5  |-  ( ( F  e.  dom  S.1  /\  ( y  /  A
)  e.  ( RR 
\  { 0 } ) )  ->  ( vol `  ( `' F " { ( y  /  A ) } ) )  e.  RR )
8674, 84, 85syl2anc 661 . . . 4  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  ( vol `  ( `' F " { ( y  /  A ) } ) )  e.  RR )
8773, 86eqeltrd 2512 . . 3  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  ( vol `  ( `' ( ( RR  X.  { A } )  oF  x.  F ) " { y } ) )  e.  RR )
8825, 61, 72, 87i1fd 21134 . 2  |-  ( (
ph  /\  A  =/=  0 )  ->  (
( RR  X.  { A } )  oF  x.  F )  e. 
dom  S.1 )
8917, 88pm2.61dane 2684 1  |-  ( ph  ->  ( ( RR  X.  { A } )  oF  x.  F )  e.  dom  S.1 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   {cab 2424    =/= wne 2601   E.wrex 2711   _Vcvv 2967    \ cdif 3320    C_ wss 3323   {csn 3872    e. cmpt 4345    X. cxp 4833   `'ccnv 4834   dom cdm 4835   ran crn 4836   "cima 4838    Fn wfn 5408   -->wf 5409   -onto->wfo 5411   ` cfv 5413  (class class class)co 6086    oFcof 6313   Fincfn 7302   RRcr 9273   0cc0 9274    x. cmul 9279    / cdiv 9985   volcvol 20922   S.1citg1 21070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  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-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-of 6315  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-2o 6913  df-oadd 6916  df-er 7093  df-map 7208  df-pm 7209  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-sup 7683  df-oi 7716  df-card 8101  df-cda 8329  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-n0 10572  df-z 10639  df-uz 10854  df-q 10946  df-rp 10984  df-xadd 11082  df-ioo 11296  df-ico 11298  df-icc 11299  df-fz 11430  df-fzo 11541  df-fl 11634  df-seq 11799  df-exp 11858  df-hash 12096  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-clim 12958  df-sum 13156  df-xmet 17785  df-met 17786  df-ovol 20923  df-vol 20924  df-mbf 21074  df-itg1 21075
This theorem is referenced by:  itg1mulc  21157  i1fsub  21161  itg1sub  21162  itg2const  21193  itg2mulclem  21199  itg2monolem1  21203  i1fibl  21260  itgitg1  21261  itg2addnclem  28396  ftc1anclem5  28424
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