MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mbfmullem2 Structured version   Unicode version

Theorem mbfmullem2 19608
Description: Lemma for mbfmul 19610. (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 5583 . . . 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 5583 . . . 4  |-  ( G : A --> RR  ->  G  Fn  A )
64, 5syl 16 . . 3  |-  ( ph  ->  G  Fn  A )
7 fdm 5587 . . . . 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 19512 . . . . 5  |-  ( F  e. MblFn  ->  dom  F  e.  dom  vol )
119, 10syl 16 . . . 4  |-  ( ph  ->  dom  F  e.  dom  vol )
128, 11eqeltrrd 2510 . . 3  |-  ( ph  ->  A  e.  dom  vol )
13 inidm 3542 . . 3  |-  ( A  i^i  A )  =  A
14 eqidd 2436 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  ( F `  x ) )
15 eqidd 2436 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  ( G `  x )  =  ( G `  x ) )
163, 6, 12, 12, 13, 14, 15offval 6304 . 2  |-  ( ph  ->  ( F  o F  x.  G )  =  ( x  e.  A  |->  ( ( F `  x )  x.  ( G `  x )
) ) )
17 nnuz 10513 . . 3  |-  NN  =  ( ZZ>= `  1 )
18 1z 10303 . . . 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 9998 . . . . . 6  |-  NN  e.  _V
2322mptex 5958 . . . . 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 5862 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN )  ->  ( P `
 n )  e. 
dom  S.1 )
28 i1ff 19560 . . . . . . . . . 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 19419 . . . . . . . . . . 11  |-  ( A  e.  dom  vol  ->  A 
C_  RR )
3212, 31syl 16 . . . . . . . . . 10  |-  ( ph  ->  A  C_  RR )
3332sselda 3340 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  RR )
3433adantr 452 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  A )  /\  n  e.  NN )  ->  x  e.  RR )
3530, 34ffvelrnd 5863 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  A )  /\  n  e.  NN )  ->  (
( P `  n
) `  x )  e.  RR )
3635recnd 9106 . . . . . 6  |-  ( ( ( ph  /\  x  e.  A )  /\  n  e.  NN )  ->  (
( P `  n
) `  x )  e.  CC )
37 eqid 2435 . . . . . 6  |-  ( n  e.  NN  |->  ( ( P `  n ) `
 x ) )  =  ( n  e.  NN  |->  ( ( P `
 n ) `  x ) )
3836, 37fmptd 5885 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  (
n  e.  NN  |->  ( ( P `  n
) `  x )
) : NN --> CC )
3938ffvelrnda 5862 . . . 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 5862 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN )  ->  ( Q `
 n )  e. 
dom  S.1 )
42 i1ff 19560 . . . . . . . . . 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 5863 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  A )  /\  n  e.  NN )  ->  (
( Q `  n
) `  x )  e.  RR )
4645recnd 9106 . . . . . 6  |-  ( ( ( ph  /\  x  e.  A )  /\  n  e.  NN )  ->  (
( Q `  n
) `  x )  e.  CC )
47 eqid 2435 . . . . . 6  |-  ( n  e.  NN  |->  ( ( Q `  n ) `
 x ) )  =  ( n  e.  NN  |->  ( ( Q `
 n ) `  x ) )
4846, 47fmptd 5885 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  (
n  e.  NN  |->  ( ( Q `  n
) `  x )
) : NN --> CC )
4948ffvelrnda 5862 . . . 4  |-  ( ( ( ph  /\  x  e.  A )  /\  k  e.  NN )  ->  (
( n  e.  NN  |->  ( ( Q `  n ) `  x
) ) `  k
)  e.  CC )
50 fveq2 5720 . . . . . . . . 9  |-  ( n  =  k  ->  ( P `  n )  =  ( P `  k ) )
5150fveq1d 5722 . . . . . . . 8  |-  ( n  =  k  ->  (
( P `  n
) `  x )  =  ( ( P `
 k ) `  x ) )
52 fveq2 5720 . . . . . . . . 9  |-  ( n  =  k  ->  ( Q `  n )  =  ( Q `  k ) )
5352fveq1d 5722 . . . . . . . 8  |-  ( n  =  k  ->  (
( Q `  n
) `  x )  =  ( ( Q `
 k ) `  x ) )
5451, 53oveq12d 6091 . . . . . . 7  |-  ( n  =  k  ->  (
( ( P `  n ) `  x
)  x.  ( ( Q `  n ) `
 x ) )  =  ( ( ( P `  k ) `
 x )  x.  ( ( Q `  k ) `  x
) ) )
55 eqid 2435 . . . . . . 7  |-  ( n  e.  NN  |->  ( ( ( P `  n
) `  x )  x.  ( ( Q `  n ) `  x
) ) )  =  ( n  e.  NN  |->  ( ( ( P `
 n ) `  x )  x.  (
( Q `  n
) `  x )
) )
56 ovex 6098 . . . . . . 7  |-  ( ( ( P `  k
) `  x )  x.  ( ( Q `  k ) `  x
) )  e.  _V
5754, 55, 56fvmpt 5798 . . . . . 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 5734 . . . . . . . 8  |-  ( ( P `  k ) `
 x )  e. 
_V
6051, 37, 59fvmpt 5798 . . . . . . 7  |-  ( k  e.  NN  ->  (
( n  e.  NN  |->  ( ( P `  n ) `  x
) ) `  k
)  =  ( ( P `  k ) `
 x ) )
61 fvex 5734 . . . . . . . 8  |-  ( ( Q `  k ) `
 x )  e. 
_V
6253, 47, 61fvmpt 5798 . . . . . . 7  |-  ( k  e.  NN  ->  (
( n  e.  NN  |->  ( ( Q `  n ) `  x
) ) `  k
)  =  ( ( Q `  k ) `
 x ) )
6360, 62oveq12d 6091 . . . . . 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 2470 . . . 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 12418 . . 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 5183 . . . . 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 5583 . . . . . . . 8  |-  ( ( P `  n ) : RR --> RR  ->  ( P `  n )  Fn  RR )
7129, 70syl 16 . . . . . . 7  |-  ( (
ph  /\  n  e.  NN )  ->  ( P `
 n )  Fn  RR )
72 ffn 5583 . . . . . . . 8  |-  ( ( Q `  n ) : RR --> RR  ->  ( Q `  n )  Fn  RR )
7343, 72syl 16 . . . . . . 7  |-  ( (
ph  /\  n  e.  NN )  ->  ( Q `
 n )  Fn  RR )
74 reex 9073 . . . . . . . 8  |-  RR  e.  _V
7574a1i 11 . . . . . . 7  |-  ( (
ph  /\  n  e.  NN )  ->  RR  e.  _V )
76 inidm 3542 . . . . . . 7  |-  ( RR 
i^i  RR )  =  RR
77 eqidd 2436 . . . . . . 7  |-  ( ( ( ph  /\  n  e.  NN )  /\  x  e.  RR )  ->  (
( P `  n
) `  x )  =  ( ( P `
 n ) `  x ) )
78 eqidd 2436 . . . . . . 7  |-  ( ( ( ph  /\  n  e.  NN )  /\  x  e.  RR )  ->  (
( Q `  n
) `  x )  =  ( ( Q `
 n ) `  x ) )
7971, 73, 75, 75, 76, 77, 78offval 6304 . . . . . 6  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( P `  n )  o F  x.  ( Q `  n )
)  =  ( x  e.  RR  |->  ( ( ( P `  n
) `  x )  x.  ( ( Q `  n ) `  x
) ) ) )
8027, 41i1fmul 19580 . . . . . . 7  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( P `  n )  o F  x.  ( Q `  n )
)  e.  dom  S.1 )
81 i1fmbf 19559 . . . . . . 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 2510 . . . . 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 19528 . . . . 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 2510 . . 3  |-  ( (
ph  /\  n  e.  NN )  ->  ( x  e.  A  |->  ( ( ( P `  n
) `  x )  x.  ( ( Q `  n ) `  x
) ) )  e. MblFn
)
88 ovex 6098 . . . 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 19552 . 2  |-  ( ph  ->  ( x  e.  A  |->  ( ( F `  x )  x.  ( G `  x )
) )  e. MblFn )
9116, 90eqeltrd 2509 1  |-  ( ph  ->  ( F  o F  x.  G )  e. MblFn
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2948    C_ wss 3312   class class class wbr 4204    e. cmpt 4258   dom cdm 4870    |` cres 4872    Fn wfn 5441   -->wf 5442   ` cfv 5446  (class class class)co 6073    o Fcof 6295   CCcc 8980   RRcr 8981   1c1 8983    x. cmul 8987   NNcn 9992   ZZcz 10274    ~~> cli 12270   volcvol 19352  MblFncmbf 19498   S.1citg1 19499
This theorem is referenced by:  mbfmullem  19609
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cc 8307  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-disj 4175  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-omul 6721  df-er 6897  df-map 7012  df-pm 7013  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7471  df-card 7818  df-acn 7821  df-cda 8040  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-q 10567  df-rp 10605  df-xadd 10703  df-ioo 10912  df-ioc 10913  df-ico 10914  df-icc 10915  df-fz 11036  df-fzo 11128  df-fl 11194  df-seq 11316  df-exp 11375  df-hash 11611  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-limsup 12257  df-clim 12274  df-rlim 12275  df-sum 12472  df-xmet 16687  df-met 16688  df-ovol 19353  df-vol 19354  df-mbf 19504  df-itg1 19505
  Copyright terms: Public domain W3C validator