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Theorem mbfmullem2 19095
Description: Lemma for mbfmul 19097. (Contributed by Mario Carneiro, 7-Sep-2014.)
Hypotheses
Ref Expression
mbfmul.1  |-  ( ph  ->  F  e. MblFn )
mbfmul.2  |-  ( ph  ->  G  e. MblFn )
mbfmul.3  |-  ( ph  ->  F : A --> RR )
mbfmul.4  |-  ( ph  ->  G : A --> RR )
mbfmul.5  |-  ( ph  ->  P : NN --> dom  S.1 )
mbfmul.6  |-  ( (
ph  /\  x  e.  A )  ->  (
n  e.  NN  |->  ( ( P `  n
) `  x )
)  ~~>  ( F `  x ) )
mbfmul.7  |-  ( ph  ->  Q : NN --> dom  S.1 )
mbfmul.8  |-  ( (
ph  /\  x  e.  A )  ->  (
n  e.  NN  |->  ( ( Q `  n
) `  x )
)  ~~>  ( G `  x ) )
Assertion
Ref Expression
mbfmullem2  |-  ( ph  ->  ( F  o F  x.  G )  e. MblFn
)
Distinct variable groups:    x, n, A    P, n, x    ph, n, x    Q, n, x    n, F, x    n, G, x

Proof of Theorem mbfmullem2
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 mbfmul.3 . . . 4  |-  ( ph  ->  F : A --> RR )
2 ffn 5405 . . . 4  |-  ( F : A --> RR  ->  F  Fn  A )
31, 2syl 15 . . 3  |-  ( ph  ->  F  Fn  A )
4 mbfmul.4 . . . 4  |-  ( ph  ->  G : A --> RR )
5 ffn 5405 . . . 4  |-  ( G : A --> RR  ->  G  Fn  A )
64, 5syl 15 . . 3  |-  ( ph  ->  G  Fn  A )
7 fdm 5409 . . . . 5  |-  ( F : A --> RR  ->  dom 
F  =  A )
81, 7syl 15 . . . 4  |-  ( ph  ->  dom  F  =  A )
9 mbfmul.1 . . . . 5  |-  ( ph  ->  F  e. MblFn )
10 mbfdm 18999 . . . . 5  |-  ( F  e. MblFn  ->  dom  F  e.  dom  vol )
119, 10syl 15 . . . 4  |-  ( ph  ->  dom  F  e.  dom  vol )
128, 11eqeltrrd 2371 . . 3  |-  ( ph  ->  A  e.  dom  vol )
13 inidm 3391 . . 3  |-  ( A  i^i  A )  =  A
14 eqidd 2297 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  ( F `  x ) )
15 eqidd 2297 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  ( G `  x )  =  ( G `  x ) )
163, 6, 12, 12, 13, 14, 15offval 6101 . 2  |-  ( ph  ->  ( F  o F  x.  G )  =  ( x  e.  A  |->  ( ( F `  x )  x.  ( G `  x )
) ) )
17 nnuz 10279 . . 3  |-  NN  =  ( ZZ>= `  1 )
18 1z 10069 . . . 4  |-  1  e.  ZZ
1918a1i 10 . . 3  |-  ( ph  ->  1  e.  ZZ )
2018a1i 10 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  1  e.  ZZ )
21 mbfmul.6 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  (
n  e.  NN  |->  ( ( P `  n
) `  x )
)  ~~>  ( F `  x ) )
22 nnex 9768 . . . . . 6  |-  NN  e.  _V
2322mptex 5762 . . . . 5  |-  ( n  e.  NN  |->  ( ( ( P `  n
) `  x )  x.  ( ( Q `  n ) `  x
) ) )  e. 
_V
2423a1i 10 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  (
n  e.  NN  |->  ( ( ( P `  n ) `  x
)  x.  ( ( Q `  n ) `
 x ) ) )  e.  _V )
25 mbfmul.8 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  (
n  e.  NN  |->  ( ( Q `  n
) `  x )
)  ~~>  ( G `  x ) )
26 mbfmul.5 . . . . . . . . . . 11  |-  ( ph  ->  P : NN --> dom  S.1 )
27 ffvelrn 5679 . . . . . . . . . . 11  |-  ( ( P : NN --> dom  S.1  /\  n  e.  NN )  ->  ( P `  n )  e.  dom  S.1 )
2826, 27sylan 457 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN )  ->  ( P `
 n )  e. 
dom  S.1 )
29 i1ff 19047 . . . . . . . . . 10  |-  ( ( P `  n )  e.  dom  S.1  ->  ( P `  n ) : RR --> RR )
3028, 29syl 15 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN )  ->  ( P `
 n ) : RR --> RR )
3130adantlr 695 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  A )  /\  n  e.  NN )  ->  ( P `  n ) : RR --> RR )
32 mblss 18906 . . . . . . . . . . 11  |-  ( A  e.  dom  vol  ->  A 
C_  RR )
3312, 32syl 15 . . . . . . . . . 10  |-  ( ph  ->  A  C_  RR )
3433sselda 3193 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  RR )
3534adantr 451 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  A )  /\  n  e.  NN )  ->  x  e.  RR )
36 ffvelrn 5679 . . . . . . . 8  |-  ( ( ( P `  n
) : RR --> RR  /\  x  e.  RR )  ->  ( ( P `  n ) `  x
)  e.  RR )
3731, 35, 36syl2anc 642 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  A )  /\  n  e.  NN )  ->  (
( P `  n
) `  x )  e.  RR )
3837recnd 8877 . . . . . 6  |-  ( ( ( ph  /\  x  e.  A )  /\  n  e.  NN )  ->  (
( P `  n
) `  x )  e.  CC )
39 eqid 2296 . . . . . 6  |-  ( n  e.  NN  |->  ( ( P `  n ) `
 x ) )  =  ( n  e.  NN  |->  ( ( P `
 n ) `  x ) )
4038, 39fmptd 5700 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  (
n  e.  NN  |->  ( ( P `  n
) `  x )
) : NN --> CC )
41 ffvelrn 5679 . . . . 5  |-  ( ( ( n  e.  NN  |->  ( ( P `  n ) `  x
) ) : NN --> CC  /\  k  e.  NN )  ->  ( ( n  e.  NN  |->  ( ( P `  n ) `
 x ) ) `
 k )  e.  CC )
4240, 41sylan 457 . . . 4  |-  ( ( ( ph  /\  x  e.  A )  /\  k  e.  NN )  ->  (
( n  e.  NN  |->  ( ( P `  n ) `  x
) ) `  k
)  e.  CC )
43 mbfmul.7 . . . . . . . . . . 11  |-  ( ph  ->  Q : NN --> dom  S.1 )
44 ffvelrn 5679 . . . . . . . . . . 11  |-  ( ( Q : NN --> dom  S.1  /\  n  e.  NN )  ->  ( Q `  n )  e.  dom  S.1 )
4543, 44sylan 457 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN )  ->  ( Q `
 n )  e. 
dom  S.1 )
46 i1ff 19047 . . . . . . . . . 10  |-  ( ( Q `  n )  e.  dom  S.1  ->  ( Q `  n ) : RR --> RR )
4745, 46syl 15 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN )  ->  ( Q `
 n ) : RR --> RR )
4847adantlr 695 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  A )  /\  n  e.  NN )  ->  ( Q `  n ) : RR --> RR )
49 ffvelrn 5679 . . . . . . . 8  |-  ( ( ( Q `  n
) : RR --> RR  /\  x  e.  RR )  ->  ( ( Q `  n ) `  x
)  e.  RR )
5048, 35, 49syl2anc 642 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  A )  /\  n  e.  NN )  ->  (
( Q `  n
) `  x )  e.  RR )
5150recnd 8877 . . . . . 6  |-  ( ( ( ph  /\  x  e.  A )  /\  n  e.  NN )  ->  (
( Q `  n
) `  x )  e.  CC )
52 eqid 2296 . . . . . 6  |-  ( n  e.  NN  |->  ( ( Q `  n ) `
 x ) )  =  ( n  e.  NN  |->  ( ( Q `
 n ) `  x ) )
5351, 52fmptd 5700 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  (
n  e.  NN  |->  ( ( Q `  n
) `  x )
) : NN --> CC )
54 ffvelrn 5679 . . . . 5  |-  ( ( ( n  e.  NN  |->  ( ( Q `  n ) `  x
) ) : NN --> CC  /\  k  e.  NN )  ->  ( ( n  e.  NN  |->  ( ( Q `  n ) `
 x ) ) `
 k )  e.  CC )
5553, 54sylan 457 . . . 4  |-  ( ( ( ph  /\  x  e.  A )  /\  k  e.  NN )  ->  (
( n  e.  NN  |->  ( ( Q `  n ) `  x
) ) `  k
)  e.  CC )
56 fveq2 5541 . . . . . . . . 9  |-  ( n  =  k  ->  ( P `  n )  =  ( P `  k ) )
5756fveq1d 5543 . . . . . . . 8  |-  ( n  =  k  ->  (
( P `  n
) `  x )  =  ( ( P `
 k ) `  x ) )
58 fveq2 5541 . . . . . . . . 9  |-  ( n  =  k  ->  ( Q `  n )  =  ( Q `  k ) )
5958fveq1d 5543 . . . . . . . 8  |-  ( n  =  k  ->  (
( Q `  n
) `  x )  =  ( ( Q `
 k ) `  x ) )
6057, 59oveq12d 5892 . . . . . . 7  |-  ( n  =  k  ->  (
( ( P `  n ) `  x
)  x.  ( ( Q `  n ) `
 x ) )  =  ( ( ( P `  k ) `
 x )  x.  ( ( Q `  k ) `  x
) ) )
61 eqid 2296 . . . . . . 7  |-  ( n  e.  NN  |->  ( ( ( P `  n
) `  x )  x.  ( ( Q `  n ) `  x
) ) )  =  ( n  e.  NN  |->  ( ( ( P `
 n ) `  x )  x.  (
( Q `  n
) `  x )
) )
62 ovex 5899 . . . . . . 7  |-  ( ( ( P `  k
) `  x )  x.  ( ( Q `  k ) `  x
) )  e.  _V
6360, 61, 62fvmpt 5618 . . . . . 6  |-  ( k  e.  NN  ->  (
( n  e.  NN  |->  ( ( ( P `
 n ) `  x )  x.  (
( Q `  n
) `  x )
) ) `  k
)  =  ( ( ( P `  k
) `  x )  x.  ( ( Q `  k ) `  x
) ) )
6463adantl 452 . . . . 5  |-  ( ( ( ph  /\  x  e.  A )  /\  k  e.  NN )  ->  (
( n  e.  NN  |->  ( ( ( P `
 n ) `  x )  x.  (
( Q `  n
) `  x )
) ) `  k
)  =  ( ( ( P `  k
) `  x )  x.  ( ( Q `  k ) `  x
) ) )
65 fvex 5555 . . . . . . . 8  |-  ( ( P `  k ) `
 x )  e. 
_V
6657, 39, 65fvmpt 5618 . . . . . . 7  |-  ( k  e.  NN  ->  (
( n  e.  NN  |->  ( ( P `  n ) `  x
) ) `  k
)  =  ( ( P `  k ) `
 x ) )
67 fvex 5555 . . . . . . . 8  |-  ( ( Q `  k ) `
 x )  e. 
_V
6859, 52, 67fvmpt 5618 . . . . . . 7  |-  ( k  e.  NN  ->  (
( n  e.  NN  |->  ( ( Q `  n ) `  x
) ) `  k
)  =  ( ( Q `  k ) `
 x ) )
6966, 68oveq12d 5892 . . . . . 6  |-  ( k  e.  NN  ->  (
( ( n  e.  NN  |->  ( ( P `
 n ) `  x ) ) `  k )  x.  (
( n  e.  NN  |->  ( ( Q `  n ) `  x
) ) `  k
) )  =  ( ( ( P `  k ) `  x
)  x.  ( ( Q `  k ) `
 x ) ) )
7069adantl 452 . . . . 5  |-  ( ( ( ph  /\  x  e.  A )  /\  k  e.  NN )  ->  (
( ( n  e.  NN  |->  ( ( P `
 n ) `  x ) ) `  k )  x.  (
( n  e.  NN  |->  ( ( Q `  n ) `  x
) ) `  k
) )  =  ( ( ( P `  k ) `  x
)  x.  ( ( Q `  k ) `
 x ) ) )
7164, 70eqtr4d 2331 . . . 4  |-  ( ( ( ph  /\  x  e.  A )  /\  k  e.  NN )  ->  (
( n  e.  NN  |->  ( ( ( P `
 n ) `  x )  x.  (
( Q `  n
) `  x )
) ) `  k
)  =  ( ( ( n  e.  NN  |->  ( ( P `  n ) `  x
) ) `  k
)  x.  ( ( n  e.  NN  |->  ( ( Q `  n
) `  x )
) `  k )
) )
7217, 20, 21, 24, 25, 42, 55, 71climmul 12122 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  (
n  e.  NN  |->  ( ( ( P `  n ) `  x
)  x.  ( ( Q `  n ) `
 x ) ) )  ~~>  ( ( F `
 x )  x.  ( G `  x
) ) )
7333adantr 451 . . . . 5  |-  ( (
ph  /\  n  e.  NN )  ->  A  C_  RR )
74 resmpt 5016 . . . . 5  |-  ( A 
C_  RR  ->  ( ( x  e.  RR  |->  ( ( ( P `  n ) `  x
)  x.  ( ( Q `  n ) `
 x ) ) )  |`  A )  =  ( x  e.  A  |->  ( ( ( P `  n ) `
 x )  x.  ( ( Q `  n ) `  x
) ) ) )
7573, 74syl 15 . . . 4  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( x  e.  RR  |->  ( ( ( P `  n ) `  x
)  x.  ( ( Q `  n ) `
 x ) ) )  |`  A )  =  ( x  e.  A  |->  ( ( ( P `  n ) `
 x )  x.  ( ( Q `  n ) `  x
) ) ) )
76 ffn 5405 . . . . . . . 8  |-  ( ( P `  n ) : RR --> RR  ->  ( P `  n )  Fn  RR )
7730, 76syl 15 . . . . . . 7  |-  ( (
ph  /\  n  e.  NN )  ->  ( P `
 n )  Fn  RR )
78 ffn 5405 . . . . . . . 8  |-  ( ( Q `  n ) : RR --> RR  ->  ( Q `  n )  Fn  RR )
7947, 78syl 15 . . . . . . 7  |-  ( (
ph  /\  n  e.  NN )  ->  ( Q `
 n )  Fn  RR )
80 reex 8844 . . . . . . . 8  |-  RR  e.  _V
8180a1i 10 . . . . . . 7  |-  ( (
ph  /\  n  e.  NN )  ->  RR  e.  _V )
82 inidm 3391 . . . . . . 7  |-  ( RR 
i^i  RR )  =  RR
83 eqidd 2297 . . . . . . 7  |-  ( ( ( ph  /\  n  e.  NN )  /\  x  e.  RR )  ->  (
( P `  n
) `  x )  =  ( ( P `
 n ) `  x ) )
84 eqidd 2297 . . . . . . 7  |-  ( ( ( ph  /\  n  e.  NN )  /\  x  e.  RR )  ->  (
( Q `  n
) `  x )  =  ( ( Q `
 n ) `  x ) )
8577, 79, 81, 81, 82, 83, 84offval 6101 . . . . . 6  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( P `  n )  o F  x.  ( Q `  n )
)  =  ( x  e.  RR  |->  ( ( ( P `  n
) `  x )  x.  ( ( Q `  n ) `  x
) ) ) )
8628, 45i1fmul 19067 . . . . . . 7  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( P `  n )  o F  x.  ( Q `  n )
)  e.  dom  S.1 )
87 i1fmbf 19046 . . . . . . 7  |-  ( ( ( P `  n
)  o F  x.  ( Q `  n ) )  e.  dom  S.1  ->  ( ( P `  n )  o F  x.  ( Q `  n ) )  e. MblFn
)
8886, 87syl 15 . . . . . 6  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( P `  n )  o F  x.  ( Q `  n )
)  e. MblFn )
8985, 88eqeltrrd 2371 . . . . 5  |-  ( (
ph  /\  n  e.  NN )  ->  ( x  e.  RR  |->  ( ( ( P `  n
) `  x )  x.  ( ( Q `  n ) `  x
) ) )  e. MblFn
)
9012adantr 451 . . . . 5  |-  ( (
ph  /\  n  e.  NN )  ->  A  e. 
dom  vol )
91 mbfres 19015 . . . . 5  |-  ( ( ( x  e.  RR  |->  ( ( ( P `
 n ) `  x )  x.  (
( Q `  n
) `  x )
) )  e. MblFn  /\  A  e.  dom  vol )  -> 
( ( x  e.  RR  |->  ( ( ( P `  n ) `
 x )  x.  ( ( Q `  n ) `  x
) ) )  |`  A )  e. MblFn )
9289, 90, 91syl2anc 642 . . . 4  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( x  e.  RR  |->  ( ( ( P `  n ) `  x
)  x.  ( ( Q `  n ) `
 x ) ) )  |`  A )  e. MblFn )
9375, 92eqeltrrd 2371 . . 3  |-  ( (
ph  /\  n  e.  NN )  ->  ( x  e.  A  |->  ( ( ( P `  n
) `  x )  x.  ( ( Q `  n ) `  x
) ) )  e. MblFn
)
94 ovex 5899 . . . 4  |-  ( ( ( P `  n
) `  x )  x.  ( ( Q `  n ) `  x
) )  e.  _V
9594a1i 10 . . 3  |-  ( (
ph  /\  ( n  e.  NN  /\  x  e.  A ) )  -> 
( ( ( P `
 n ) `  x )  x.  (
( Q `  n
) `  x )
)  e.  _V )
9617, 19, 72, 93, 95mbflim 19039 . 2  |-  ( ph  ->  ( x  e.  A  |->  ( ( F `  x )  x.  ( G `  x )
) )  e. MblFn )
9716, 96eqeltrd 2370 1  |-  ( ph  ->  ( F  o F  x.  G )  e. MblFn
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801    C_ wss 3165   class class class wbr 4039    e. cmpt 4093   dom cdm 4705    |` cres 4707    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874    o Fcof 6092   CCcc 8751   RRcr 8752   1c1 8754    x. cmul 8758   NNcn 9762   ZZcz 10040    ~~> cli 11974   volcvol 18839  MblFncmbf 18985   S.1citg1 18986
This theorem is referenced by:  mbfmullem  19096
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cc 8077  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-disj 4010  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-omul 6500  df-er 6676  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-acn 7591  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-xadd 10469  df-ioo 10676  df-ioc 10677  df-ico 10678  df-icc 10679  df-fz 10799  df-fzo 10887  df-fl 10941  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-limsup 11961  df-clim 11978  df-rlim 11979  df-sum 12175  df-xmet 16389  df-met 16390  df-ovol 18840  df-vol 18841  df-mbf 18991  df-itg1 18992
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