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Theorem i1fmulc 21313
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 9483 . . . . 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 21286 . . . . . 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 9497 . . . 4  |-  ( (
ph  /\  A  = 
0 )  ->  0  e.  RR )
10 simplr 754 . . . . . 6  |-  ( ( ( ph  /\  A  =  0 )  /\  x  e.  RR )  ->  A  =  0 )
1110oveq1d 6214 . . . . 5  |-  ( ( ( ph  /\  A  =  0 )  /\  x  e.  RR )  ->  ( A  x.  x
)  =  ( 0  x.  x ) )
12 mul02lem2 9656 . . . . . 6  |-  ( x  e.  RR  ->  (
0  x.  x )  =  0 )
1312adantl 466 . . . . 5  |-  ( ( ( ph  /\  A  =  0 )  /\  x  e.  RR )  ->  ( 0  x.  x
)  =  0 )
1411, 13eqtrd 2495 . . . 4  |-  ( ( ( ph  /\  A  =  0 )  /\  x  e.  RR )  ->  ( A  x.  x
)  =  0 )
152, 6, 8, 9, 14caofid2 6460 . . 3  |-  ( (
ph  /\  A  = 
0 )  ->  (
( RR  X.  { A } )  oF  x.  F )  =  ( RR  X.  {
0 } ) )
16 i1f0 21297 . . 3  |-  ( RR 
X.  { 0 } )  e.  dom  S.1
1715, 16syl6eqel 2550 . 2  |-  ( (
ph  /\  A  = 
0 )  ->  (
( RR  X.  { A } )  oF  x.  F )  e. 
dom  S.1 )
18 remulcl 9477 . . . . . 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 5706 . . . . . 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 3666 . . . . 5  |-  ( RR 
i^i  RR )  =  RR
2419, 21, 5, 22, 22, 23off 6443 . . . 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 21287 . . . . . . 7  |-  ( F  e.  dom  S.1  ->  ran 
F  e.  Fin )
273, 26syl 16 . . . . . 6  |-  ( ph  ->  ran  F  e.  Fin )
28 ovex 6224 . . . . . . . 8  |-  ( A  x.  y )  e. 
_V
29 eqid 2454 . . . . . . . 8  |-  ( y  e.  ran  F  |->  ( A  x.  y ) )  =  ( y  e.  ran  F  |->  ( A  x.  y ) )
3028, 29fnmpti 5646 . . . . . . 7  |-  ( y  e.  ran  F  |->  ( A  x.  y ) )  Fn  ran  F
31 dffn4 5733 . . . . . . 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 7707 . . . . . 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 4009 . . . . . . . . . . . 12  |-  ( x  e.  { A }  ->  x  =  A )
3736oveq1d 6214 . . . . . . . . . . 11  |-  ( x  e.  { A }  ->  ( x  x.  w
)  =  ( A  x.  w ) )
38 oveq2 6207 . . . . . . . . . . . . 13  |-  ( y  =  w  ->  ( A  x.  y )  =  ( A  x.  w ) )
3938eqeq2d 2468 . . . . . . . . . . . 12  |-  ( y  =  w  ->  (
( x  x.  w
)  =  ( A  x.  y )  <->  ( x  x.  w )  =  ( A  x.  w ) ) )
4039rspcev 3177 . . . . . . . . . . 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 6224 . . . . . . . . . . 11  |-  ( x  x.  w )  e. 
_V
43 eqeq1 2458 . . . . . . . . . . . 12  |-  ( z  =  ( x  x.  w )  ->  (
z  =  ( A  x.  y )  <->  ( x  x.  w )  =  ( A  x.  y ) ) )
4443rexbidv 2864 . . . . . . . . . . 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 3211 . . . . . . . . . 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 5704 . . . . . . . . 9  |-  ( A  e.  RR  ->  ( RR  X.  { A }
) : RR --> { A } )
497, 48syl 16 . . . . . . . 8  |-  ( ph  ->  ( RR  X.  { A } ) : RR --> { A } )
50 ffn 5666 . . . . . . . . . 10  |-  ( F : RR --> RR  ->  F  Fn  RR )
515, 50syl 16 . . . . . . . . 9  |-  ( ph  ->  F  Fn  RR )
52 dffn3 5673 . . . . . . . . 9  |-  ( F  Fn  RR  <->  F : RR
--> ran  F )
5351, 52sylib 196 . . . . . . . 8  |-  ( ph  ->  F : RR --> ran  F
)
5447, 49, 53, 22, 22, 23off 6443 . . . . . . 7  |-  ( ph  ->  ( ( RR  X.  { A } )  oF  x.  F ) : RR --> { z  |  E. y  e. 
ran  F  z  =  ( A  x.  y
) } )
55 frn 5672 . . . . . . 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 5192 . . . . . 6  |-  ran  (
y  e.  ran  F  |->  ( A  x.  y
) )  =  {
z  |  E. y  e.  ran  F  z  =  ( A  x.  y
) }
5856, 57syl6sseqr 3510 . . . . 5  |-  ( ph  ->  ran  ( ( RR 
X.  { A }
)  oF  x.  F )  C_  ran  ( y  e.  ran  F 
|->  ( A  x.  y
) ) )
59 ssfi 7643 . . . . 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 5672 . . . . . . . . 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 3601 . . . . . . 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 3463 . . . . 5  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  y  e.  RR )
673, 7i1fmulclem 21312 . . . . 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 21288 . . . . . 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 2542 . . 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 5802 . . . 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 10269 . . . . . 6  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  ( y  /  A )  e.  RR )
7866recnd 9522 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  y  e.  CC )
7975recnd 9522 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  A  e.  CC )
80 eldifsni 4108 . . . . . . . 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 10233 . . . . . 6  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  ( y  /  A )  =/=  0
)
83 eldifsn 4107 . . . . . 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 21290 . . . . 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 2542 . . 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 21291 . 2  |-  ( (
ph  /\  A  =/=  0 )  ->  (
( RR  X.  { A } )  oF  x.  F )  e. 
dom  S.1 )
8917, 88pm2.61dane 2769 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 1370    e. wcel 1758   {cab 2439    =/= wne 2647   E.wrex 2799   _Vcvv 3076    \ cdif 3432    C_ wss 3435   {csn 3984    |-> cmpt 4457    X. cxp 4945   `'ccnv 4946   dom cdm 4947   ran crn 4948   "cima 4950    Fn wfn 5520   -->wf 5521   -onto->wfo 5523   ` cfv 5525  (class class class)co 6199    oFcof 6427   Fincfn 7419   RRcr 9391   0cc0 9392    x. cmul 9397    / cdiv 10103   volcvol 21078   S.1citg1 21227
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-inf2 7957  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469  ax-pre-sup 9470
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-int 4236  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-se 4787  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-isom 5534  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-of 6429  df-om 6586  df-1st 6686  df-2nd 6687  df-recs 6941  df-rdg 6975  df-1o 7029  df-2o 7030  df-oadd 7033  df-er 7210  df-map 7325  df-pm 7326  df-en 7420  df-dom 7421  df-sdom 7422  df-fin 7423  df-sup 7801  df-oi 7834  df-card 8219  df-cda 8447  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-div 10104  df-nn 10433  df-2 10490  df-3 10491  df-n0 10690  df-z 10757  df-uz 10972  df-q 11064  df-rp 11102  df-xadd 11200  df-ioo 11414  df-ico 11416  df-icc 11417  df-fz 11554  df-fzo 11665  df-fl 11758  df-seq 11923  df-exp 11982  df-hash 12220  df-cj 12705  df-re 12706  df-im 12707  df-sqr 12841  df-abs 12842  df-clim 13083  df-sum 13281  df-xmet 17934  df-met 17935  df-ovol 21079  df-vol 21080  df-mbf 21231  df-itg1 21232
This theorem is referenced by:  itg1mulc  21314  i1fsub  21318  itg1sub  21319  itg2const  21350  itg2mulclem  21356  itg2monolem1  21360  i1fibl  21417  itgitg1  21418  itg2addnclem  28590  ftc1anclem5  28618
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