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Theorem i1fmulc 22402
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 9613 . . . . 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 22375 . . . . . 6  |-  ( F  e.  dom  S.1  ->  F : RR --> RR )
53, 4syl 17 . . . . 5  |-  ( ph  ->  F : RR --> RR )
65adantr 463 . . . 4  |-  ( (
ph  /\  A  = 
0 )  ->  F : RR --> RR )
7 i1fmulc.3 . . . . 5  |-  ( ph  ->  A  e.  RR )
87adantr 463 . . . 4  |-  ( (
ph  /\  A  = 
0 )  ->  A  e.  RR )
9 0red 9627 . . . 4  |-  ( (
ph  /\  A  = 
0 )  ->  0  e.  RR )
10 simplr 754 . . . . . 6  |-  ( ( ( ph  /\  A  =  0 )  /\  x  e.  RR )  ->  A  =  0 )
1110oveq1d 6293 . . . . 5  |-  ( ( ( ph  /\  A  =  0 )  /\  x  e.  RR )  ->  ( A  x.  x
)  =  ( 0  x.  x ) )
12 mul02lem2 9791 . . . . . 6  |-  ( x  e.  RR  ->  (
0  x.  x )  =  0 )
1312adantl 464 . . . . 5  |-  ( ( ( ph  /\  A  =  0 )  /\  x  e.  RR )  ->  ( 0  x.  x
)  =  0 )
1411, 13eqtrd 2443 . . . 4  |-  ( ( ( ph  /\  A  =  0 )  /\  x  e.  RR )  ->  ( A  x.  x
)  =  0 )
152, 6, 8, 9, 14caofid2 6553 . . 3  |-  ( (
ph  /\  A  = 
0 )  ->  (
( RR  X.  { A } )  oF  x.  F )  =  ( RR  X.  {
0 } ) )
16 i1f0 22386 . . 3  |-  ( RR 
X.  { 0 } )  e.  dom  S.1
1715, 16syl6eqel 2498 . 2  |-  ( (
ph  /\  A  = 
0 )  ->  (
( RR  X.  { A } )  oF  x.  F )  e. 
dom  S.1 )
18 remulcl 9607 . . . . . 6  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  x.  y
)  e.  RR )
1918adantl 464 . . . . 5  |-  ( (
ph  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
( x  x.  y
)  e.  RR )
20 fconst6g 5757 . . . . . 6  |-  ( A  e.  RR  ->  ( RR  X.  { A }
) : RR --> RR )
217, 20syl 17 . . . . 5  |-  ( ph  ->  ( RR  X.  { A } ) : RR --> RR )
221a1i 11 . . . . 5  |-  ( ph  ->  RR  e.  _V )
23 inidm 3648 . . . . 5  |-  ( RR 
i^i  RR )  =  RR
2419, 21, 5, 22, 22, 23off 6536 . . . 4  |-  ( ph  ->  ( ( RR  X.  { A } )  oF  x.  F ) : RR --> RR )
2524adantr 463 . . 3  |-  ( (
ph  /\  A  =/=  0 )  ->  (
( RR  X.  { A } )  oF  x.  F ) : RR --> RR )
26 i1frn 22376 . . . . . . 7  |-  ( F  e.  dom  S.1  ->  ran 
F  e.  Fin )
273, 26syl 17 . . . . . 6  |-  ( ph  ->  ran  F  e.  Fin )
28 ovex 6306 . . . . . . . 8  |-  ( A  x.  y )  e. 
_V
29 eqid 2402 . . . . . . . 8  |-  ( y  e.  ran  F  |->  ( A  x.  y ) )  =  ( y  e.  ran  F  |->  ( A  x.  y ) )
3028, 29fnmpti 5692 . . . . . . 7  |-  ( y  e.  ran  F  |->  ( A  x.  y ) )  Fn  ran  F
31 dffn4 5784 . . . . . . 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 7840 . . . . . 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 660 . . . . 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 3997 . . . . . . . . . . . 12  |-  ( x  e.  { A }  ->  x  =  A )
3736oveq1d 6293 . . . . . . . . . . 11  |-  ( x  e.  { A }  ->  ( x  x.  w
)  =  ( A  x.  w ) )
38 oveq2 6286 . . . . . . . . . . . . 13  |-  ( y  =  w  ->  ( A  x.  y )  =  ( A  x.  w ) )
3938eqeq2d 2416 . . . . . . . . . . . 12  |-  ( y  =  w  ->  (
( x  x.  w
)  =  ( A  x.  y )  <->  ( x  x.  w )  =  ( A  x.  w ) ) )
4039rspcev 3160 . . . . . . . . . . 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 476 . . . . . . . . . 10  |-  ( ( x  e.  { A }  /\  w  e.  ran  F )  ->  E. y  e.  ran  F ( x  x.  w )  =  ( A  x.  y
) )
42 ovex 6306 . . . . . . . . . . 11  |-  ( x  x.  w )  e. 
_V
43 eqeq1 2406 . . . . . . . . . . . 12  |-  ( z  =  ( x  x.  w )  ->  (
z  =  ( A  x.  y )  <->  ( x  x.  w )  =  ( A  x.  y ) ) )
4443rexbidv 2918 . . . . . . . . . . 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 3196 . . . . . . . . . 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 464 . . . . . . . 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 5755 . . . . . . . . 9  |-  ( A  e.  RR  ->  ( RR  X.  { A }
) : RR --> { A } )
497, 48syl 17 . . . . . . . 8  |-  ( ph  ->  ( RR  X.  { A } ) : RR --> { A } )
50 ffn 5714 . . . . . . . . . 10  |-  ( F : RR --> RR  ->  F  Fn  RR )
515, 50syl 17 . . . . . . . . 9  |-  ( ph  ->  F  Fn  RR )
52 dffn3 5721 . . . . . . . . 9  |-  ( F  Fn  RR  <->  F : RR
--> ran  F )
5351, 52sylib 196 . . . . . . . 8  |-  ( ph  ->  F : RR --> ran  F
)
5447, 49, 53, 22, 22, 23off 6536 . . . . . . 7  |-  ( ph  ->  ( ( RR  X.  { A } )  oF  x.  F ) : RR --> { z  |  E. y  e. 
ran  F  z  =  ( A  x.  y
) } )
55 frn 5720 . . . . . . 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 17 . . . . . 6  |-  ( ph  ->  ran  ( ( RR 
X.  { A }
)  oF  x.  F )  C_  { z  |  E. y  e. 
ran  F  z  =  ( A  x.  y
) } )
5729rnmpt 5069 . . . . . 6  |-  ran  (
y  e.  ran  F  |->  ( A  x.  y
) )  =  {
z  |  E. y  e.  ran  F  z  =  ( A  x.  y
) }
5856, 57syl6sseqr 3489 . . . . 5  |-  ( ph  ->  ran  ( ( RR 
X.  { A }
)  oF  x.  F )  C_  ran  ( y  e.  ran  F 
|->  ( A  x.  y
) ) )
59 ssfi 7775 . . . . 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 659 . . . 4  |-  ( ph  ->  ran  ( ( RR 
X.  { A }
)  oF  x.  F )  e.  Fin )
6160adantr 463 . . 3  |-  ( (
ph  /\  A  =/=  0 )  ->  ran  ( ( RR  X.  { A } )  oF  x.  F )  e.  Fin )
62 frn 5720 . . . . . . . . 9  |-  ( ( ( RR  X.  { A } )  oF  x.  F ) : RR --> RR  ->  ran  ( ( RR  X.  { A } )  oF  x.  F ) 
C_  RR )
6324, 62syl 17 . . . . . . . 8  |-  ( ph  ->  ran  ( ( RR 
X.  { A }
)  oF  x.  F )  C_  RR )
6463ssdifssd 3581 . . . . . . 7  |-  ( ph  ->  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) 
C_  RR )
6564adantr 463 . . . . . 6  |-  ( (
ph  /\  A  =/=  0 )  ->  ( ran  ( ( RR  X.  { A } )  oF  x.  F ) 
\  { 0 } )  C_  RR )
6665sselda 3442 . . . . 5  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  y  e.  RR )
673, 7i1fmulclem 22401 . . . . 5  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  RR )  ->  ( `' ( ( RR 
X.  { A }
)  oF  x.  F ) " {
y } )  =  ( `' F " { ( y  /  A ) } ) )
6866, 67syldan 468 . . . 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 22377 . . . . . 6  |-  ( F  e.  dom  S.1  ->  ( `' F " { ( y  /  A ) } )  e.  dom  vol )
703, 69syl 17 . . . . 5  |-  ( ph  ->  ( `' F " { ( y  /  A ) } )  e.  dom  vol )
7170ad2antrr 724 . . . 4  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  ( `' F " { ( y  /  A ) } )  e.  dom  vol )
7268, 71eqeltrd 2490 . . 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 5853 . . . 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 724 . . . . 5  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  F  e.  dom  S.1 )
757ad2antrr 724 . . . . . . 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 10413 . . . . . 6  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  ( y  /  A )  e.  RR )
7866recnd 9652 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  y  e.  CC )
7975recnd 9652 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  A  e.  CC )
80 eldifsni 4098 . . . . . . . 8  |-  ( y  e.  ( ran  (
( RR  X.  { A } )  oF  x.  F )  \  { 0 } )  ->  y  =/=  0
)
8180adantl 464 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  y  =/=  0 )
8278, 79, 81, 76divne0d 10377 . . . . . 6  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  ( y  /  A )  =/=  0
)
83 eldifsn 4097 . . . . . 6  |-  ( ( y  /  A )  e.  ( RR  \  { 0 } )  <-> 
( ( y  /  A )  e.  RR  /\  ( y  /  A
)  =/=  0 ) )
8477, 82, 83sylanbrc 662 . . . . 5  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  ( y  /  A )  e.  ( RR  \  { 0 } ) )
85 i1fima2sn 22379 . . . . 5  |-  ( ( F  e.  dom  S.1  /\  ( y  /  A
)  e.  ( RR 
\  { 0 } ) )  ->  ( vol `  ( `' F " { ( y  /  A ) } ) )  e.  RR )
8674, 84, 85syl2anc 659 . . . 4  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  ( vol `  ( `' F " { ( y  /  A ) } ) )  e.  RR )
8773, 86eqeltrd 2490 . . 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 22380 . 2  |-  ( (
ph  /\  A  =/=  0 )  ->  (
( RR  X.  { A } )  oF  x.  F )  e. 
dom  S.1 )
8917, 88pm2.61dane 2721 1  |-  ( ph  ->  ( ( RR  X.  { A } )  oF  x.  F )  e.  dom  S.1 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842   {cab 2387    =/= wne 2598   E.wrex 2755   _Vcvv 3059    \ cdif 3411    C_ wss 3414   {csn 3972    |-> cmpt 4453    X. cxp 4821   `'ccnv 4822   dom cdm 4823   ran crn 4824   "cima 4826    Fn wfn 5564   -->wf 5565   -onto->wfo 5567   ` cfv 5569  (class class class)co 6278    oFcof 6519   Fincfn 7554   RRcr 9521   0cc0 9522    x. cmul 9527    / cdiv 10247   volcvol 22167   S.1citg1 22316
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-inf2 8091  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-pre-sup 9600
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-isom 5578  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-of 6521  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-2o 7168  df-oadd 7171  df-er 7348  df-map 7459  df-pm 7460  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-sup 7935  df-oi 7969  df-card 8352  df-cda 8580  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-nn 10577  df-2 10635  df-3 10636  df-n0 10837  df-z 10906  df-uz 11128  df-q 11228  df-rp 11266  df-xadd 11372  df-ioo 11586  df-ico 11588  df-icc 11589  df-fz 11727  df-fzo 11855  df-fl 11966  df-seq 12152  df-exp 12211  df-hash 12453  df-cj 13081  df-re 13082  df-im 13083  df-sqrt 13217  df-abs 13218  df-clim 13460  df-sum 13658  df-xmet 18732  df-met 18733  df-ovol 22168  df-vol 22169  df-mbf 22320  df-itg1 22321
This theorem is referenced by:  itg1mulc  22403  i1fsub  22407  itg1sub  22408  itg2const  22439  itg2mulclem  22445  itg2monolem1  22449  i1fibl  22506  itgitg1  22507  itg2addnclem  31439  ftc1anclem5  31467
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