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Theorem mbfmullem2 19484
Description: Lemma for mbfmul 19486. (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 5532 . . . 4  |-  ( F : A --> RR  ->  F  Fn  A )
31, 2syl 16 . . 3  |-  ( ph  ->  F  Fn  A )
4 mbfmul.4 . . . 4  |-  ( ph  ->  G : A --> RR )
5 ffn 5532 . . . 4  |-  ( G : A --> RR  ->  G  Fn  A )
64, 5syl 16 . . 3  |-  ( ph  ->  G  Fn  A )
7 fdm 5536 . . . . 5  |-  ( F : A --> RR  ->  dom 
F  =  A )
81, 7syl 16 . . . 4  |-  ( ph  ->  dom  F  =  A )
9 mbfmul.1 . . . . 5  |-  ( ph  ->  F  e. MblFn )
10 mbfdm 19388 . . . . 5  |-  ( F  e. MblFn  ->  dom  F  e.  dom  vol )
119, 10syl 16 . . . 4  |-  ( ph  ->  dom  F  e.  dom  vol )
128, 11eqeltrrd 2463 . . 3  |-  ( ph  ->  A  e.  dom  vol )
13 inidm 3494 . . 3  |-  ( A  i^i  A )  =  A
14 eqidd 2389 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  ( F `  x ) )
15 eqidd 2389 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  ( G `  x )  =  ( G `  x ) )
163, 6, 12, 12, 13, 14, 15offval 6252 . 2  |-  ( ph  ->  ( F  o F  x.  G )  =  ( x  e.  A  |->  ( ( F `  x )  x.  ( G `  x )
) ) )
17 nnuz 10454 . . 3  |-  NN  =  ( ZZ>= `  1 )
18 1z 10244 . . . 4  |-  1  e.  ZZ
1918a1i 11 . . 3  |-  ( ph  ->  1  e.  ZZ )
2018a1i 11 . . . 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 9939 . . . . . 6  |-  NN  e.  _V
2322mptex 5906 . . . . 5  |-  ( n  e.  NN  |->  ( ( ( P `  n
) `  x )  x.  ( ( Q `  n ) `  x
) ) )  e. 
_V
2423a1i 11 . . . 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 )
2726ffvelrnda 5810 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN )  ->  ( P `
 n )  e. 
dom  S.1 )
28 i1ff 19436 . . . . . . . . . 10  |-  ( ( P `  n )  e.  dom  S.1  ->  ( P `  n ) : RR --> RR )
2927, 28syl 16 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN )  ->  ( P `
 n ) : RR --> RR )
3029adantlr 696 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  A )  /\  n  e.  NN )  ->  ( P `  n ) : RR --> RR )
31 mblss 19295 . . . . . . . . . . 11  |-  ( A  e.  dom  vol  ->  A 
C_  RR )
3212, 31syl 16 . . . . . . . . . 10  |-  ( ph  ->  A  C_  RR )
3332sselda 3292 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  RR )
3433adantr 452 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  A )  /\  n  e.  NN )  ->  x  e.  RR )
3530, 34ffvelrnd 5811 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  A )  /\  n  e.  NN )  ->  (
( P `  n
) `  x )  e.  RR )
3635recnd 9048 . . . . . 6  |-  ( ( ( ph  /\  x  e.  A )  /\  n  e.  NN )  ->  (
( P `  n
) `  x )  e.  CC )
37 eqid 2388 . . . . . 6  |-  ( n  e.  NN  |->  ( ( P `  n ) `
 x ) )  =  ( n  e.  NN  |->  ( ( P `
 n ) `  x ) )
3836, 37fmptd 5833 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  (
n  e.  NN  |->  ( ( P `  n
) `  x )
) : NN --> CC )
3938ffvelrnda 5810 . . . 4  |-  ( ( ( ph  /\  x  e.  A )  /\  k  e.  NN )  ->  (
( n  e.  NN  |->  ( ( P `  n ) `  x
) ) `  k
)  e.  CC )
40 mbfmul.7 . . . . . . . . . . 11  |-  ( ph  ->  Q : NN --> dom  S.1 )
4140ffvelrnda 5810 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN )  ->  ( Q `
 n )  e. 
dom  S.1 )
42 i1ff 19436 . . . . . . . . . 10  |-  ( ( Q `  n )  e.  dom  S.1  ->  ( Q `  n ) : RR --> RR )
4341, 42syl 16 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN )  ->  ( Q `
 n ) : RR --> RR )
4443adantlr 696 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  A )  /\  n  e.  NN )  ->  ( Q `  n ) : RR --> RR )
4544, 34ffvelrnd 5811 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  A )  /\  n  e.  NN )  ->  (
( Q `  n
) `  x )  e.  RR )
4645recnd 9048 . . . . . 6  |-  ( ( ( ph  /\  x  e.  A )  /\  n  e.  NN )  ->  (
( Q `  n
) `  x )  e.  CC )
47 eqid 2388 . . . . . 6  |-  ( n  e.  NN  |->  ( ( Q `  n ) `
 x ) )  =  ( n  e.  NN  |->  ( ( Q `
 n ) `  x ) )
4846, 47fmptd 5833 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  (
n  e.  NN  |->  ( ( Q `  n
) `  x )
) : NN --> CC )
4948ffvelrnda 5810 . . . 4  |-  ( ( ( ph  /\  x  e.  A )  /\  k  e.  NN )  ->  (
( n  e.  NN  |->  ( ( Q `  n ) `  x
) ) `  k
)  e.  CC )
50 fveq2 5669 . . . . . . . . 9  |-  ( n  =  k  ->  ( P `  n )  =  ( P `  k ) )
5150fveq1d 5671 . . . . . . . 8  |-  ( n  =  k  ->  (
( P `  n
) `  x )  =  ( ( P `
 k ) `  x ) )
52 fveq2 5669 . . . . . . . . 9  |-  ( n  =  k  ->  ( Q `  n )  =  ( Q `  k ) )
5352fveq1d 5671 . . . . . . . 8  |-  ( n  =  k  ->  (
( Q `  n
) `  x )  =  ( ( Q `
 k ) `  x ) )
5451, 53oveq12d 6039 . . . . . . 7  |-  ( n  =  k  ->  (
( ( P `  n ) `  x
)  x.  ( ( Q `  n ) `
 x ) )  =  ( ( ( P `  k ) `
 x )  x.  ( ( Q `  k ) `  x
) ) )
55 eqid 2388 . . . . . . 7  |-  ( n  e.  NN  |->  ( ( ( P `  n
) `  x )  x.  ( ( Q `  n ) `  x
) ) )  =  ( n  e.  NN  |->  ( ( ( P `
 n ) `  x )  x.  (
( Q `  n
) `  x )
) )
56 ovex 6046 . . . . . . 7  |-  ( ( ( P `  k
) `  x )  x.  ( ( Q `  k ) `  x
) )  e.  _V
5754, 55, 56fvmpt 5746 . . . . . 6  |-  ( k  e.  NN  ->  (
( n  e.  NN  |->  ( ( ( P `
 n ) `  x )  x.  (
( Q `  n
) `  x )
) ) `  k
)  =  ( ( ( P `  k
) `  x )  x.  ( ( Q `  k ) `  x
) ) )
5857adantl 453 . . . . 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
) ) )
59 fvex 5683 . . . . . . . 8  |-  ( ( P `  k ) `
 x )  e. 
_V
6051, 37, 59fvmpt 5746 . . . . . . 7  |-  ( k  e.  NN  ->  (
( n  e.  NN  |->  ( ( P `  n ) `  x
) ) `  k
)  =  ( ( P `  k ) `
 x ) )
61 fvex 5683 . . . . . . . 8  |-  ( ( Q `  k ) `
 x )  e. 
_V
6253, 47, 61fvmpt 5746 . . . . . . 7  |-  ( k  e.  NN  ->  (
( n  e.  NN  |->  ( ( Q `  n ) `  x
) ) `  k
)  =  ( ( Q `  k ) `
 x ) )
6360, 62oveq12d 6039 . . . . . 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 ) ) )
6463adantl 453 . . . . 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 ) ) )
6558, 64eqtr4d 2423 . . . 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 )
) )
6617, 20, 21, 24, 25, 39, 49, 65climmul 12354 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  (
n  e.  NN  |->  ( ( ( P `  n ) `  x
)  x.  ( ( Q `  n ) `
 x ) ) )  ~~>  ( ( F `
 x )  x.  ( G `  x
) ) )
6732adantr 452 . . . . 5  |-  ( (
ph  /\  n  e.  NN )  ->  A  C_  RR )
68 resmpt 5132 . . . . 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
) ) ) )
6967, 68syl 16 . . . 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
) ) ) )
70 ffn 5532 . . . . . . . 8  |-  ( ( P `  n ) : RR --> RR  ->  ( P `  n )  Fn  RR )
7129, 70syl 16 . . . . . . 7  |-  ( (
ph  /\  n  e.  NN )  ->  ( P `
 n )  Fn  RR )
72 ffn 5532 . . . . . . . 8  |-  ( ( Q `  n ) : RR --> RR  ->  ( Q `  n )  Fn  RR )
7343, 72syl 16 . . . . . . 7  |-  ( (
ph  /\  n  e.  NN )  ->  ( Q `
 n )  Fn  RR )
74 reex 9015 . . . . . . . 8  |-  RR  e.  _V
7574a1i 11 . . . . . . 7  |-  ( (
ph  /\  n  e.  NN )  ->  RR  e.  _V )
76 inidm 3494 . . . . . . 7  |-  ( RR 
i^i  RR )  =  RR
77 eqidd 2389 . . . . . . 7  |-  ( ( ( ph  /\  n  e.  NN )  /\  x  e.  RR )  ->  (
( P `  n
) `  x )  =  ( ( P `
 n ) `  x ) )
78 eqidd 2389 . . . . . . 7  |-  ( ( ( ph  /\  n  e.  NN )  /\  x  e.  RR )  ->  (
( Q `  n
) `  x )  =  ( ( Q `
 n ) `  x ) )
7971, 73, 75, 75, 76, 77, 78offval 6252 . . . . . 6  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( P `  n )  o F  x.  ( Q `  n )
)  =  ( x  e.  RR  |->  ( ( ( P `  n
) `  x )  x.  ( ( Q `  n ) `  x
) ) ) )
8027, 41i1fmul 19456 . . . . . . 7  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( P `  n )  o F  x.  ( Q `  n )
)  e.  dom  S.1 )
81 i1fmbf 19435 . . . . . . 7  |-  ( ( ( P `  n
)  o F  x.  ( Q `  n ) )  e.  dom  S.1  ->  ( ( P `  n )  o F  x.  ( Q `  n ) )  e. MblFn
)
8280, 81syl 16 . . . . . 6  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( P `  n )  o F  x.  ( Q `  n )
)  e. MblFn )
8379, 82eqeltrrd 2463 . . . . 5  |-  ( (
ph  /\  n  e.  NN )  ->  ( x  e.  RR  |->  ( ( ( P `  n
) `  x )  x.  ( ( Q `  n ) `  x
) ) )  e. MblFn
)
8412adantr 452 . . . . 5  |-  ( (
ph  /\  n  e.  NN )  ->  A  e. 
dom  vol )
85 mbfres 19404 . . . . 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 )
8683, 84, 85syl2anc 643 . . . 4  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( x  e.  RR  |->  ( ( ( P `  n ) `  x
)  x.  ( ( Q `  n ) `
 x ) ) )  |`  A )  e. MblFn )
8769, 86eqeltrrd 2463 . . 3  |-  ( (
ph  /\  n  e.  NN )  ->  ( x  e.  A  |->  ( ( ( P `  n
) `  x )  x.  ( ( Q `  n ) `  x
) ) )  e. MblFn
)
88 ovex 6046 . . . 4  |-  ( ( ( P `  n
) `  x )  x.  ( ( Q `  n ) `  x
) )  e.  _V
8988a1i 11 . . 3  |-  ( (
ph  /\  ( n  e.  NN  /\  x  e.  A ) )  -> 
( ( ( P `
 n ) `  x )  x.  (
( Q `  n
) `  x )
)  e.  _V )
9017, 19, 66, 87, 89mbflim 19428 . 2  |-  ( ph  ->  ( x  e.  A  |->  ( ( F `  x )  x.  ( G `  x )
) )  e. MblFn )
9116, 90eqeltrd 2462 1  |-  ( ph  ->  ( F  o F  x.  G )  e. MblFn
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   _Vcvv 2900    C_ wss 3264   class class class wbr 4154    e. cmpt 4208   dom cdm 4819    |` cres 4821    Fn wfn 5390   -->wf 5391   ` cfv 5395  (class class class)co 6021    o Fcof 6243   CCcc 8922   RRcr 8923   1c1 8925    x. cmul 8929   NNcn 9933   ZZcz 10215    ~~> cli 12206   volcvol 19228  MblFncmbf 19374   S.1citg1 19375
This theorem is referenced by:  mbfmullem  19485
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-inf2 7530  ax-cc 8249  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001  ax-pre-sup 9002
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-disj 4125  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-se 4484  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-isom 5404  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-of 6245  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-1o 6661  df-2o 6662  df-oadd 6665  df-omul 6666  df-er 6842  df-map 6957  df-pm 6958  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-sup 7382  df-oi 7413  df-card 7760  df-acn 7763  df-cda 7982  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-div 9611  df-nn 9934  df-2 9991  df-3 9992  df-n0 10155  df-z 10216  df-uz 10422  df-q 10508  df-rp 10546  df-xadd 10644  df-ioo 10853  df-ioc 10854  df-ico 10855  df-icc 10856  df-fz 10977  df-fzo 11067  df-fl 11130  df-seq 11252  df-exp 11311  df-hash 11547  df-cj 11832  df-re 11833  df-im 11834  df-sqr 11968  df-abs 11969  df-limsup 12193  df-clim 12210  df-rlim 12211  df-sum 12408  df-xmet 16620  df-met 16621  df-ovol 19229  df-vol 19230  df-mbf 19380  df-itg1 19381
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