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Theorem fmuldfeq 37525
Description: X and Z are two equivalent definitions of the finite product of real functions. Y is a set of real functions from a common domain T, Y is closed under function multiplication and U is a finite sequence of functions in Y. M is the number of functions multiplied together. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypotheses
Ref Expression
fmuldfeq.1  |-  F/ i
ph
fmuldfeq.2  |-  F/_ t Y
fmuldfeq.3  |-  P  =  ( f  e.  Y ,  g  e.  Y  |->  ( t  e.  T  |->  ( ( f `  t )  x.  (
g `  t )
) ) )
fmuldfeq.4  |-  X  =  (  seq 1 ( P ,  U ) `
 M )
fmuldfeq.5  |-  F  =  ( t  e.  T  |->  ( i  e.  ( 1 ... M ) 
|->  ( ( U `  i ) `  t
) ) )
fmuldfeq.6  |-  Z  =  ( t  e.  T  |->  (  seq 1 (  x.  ,  ( F `
 t ) ) `
 M ) )
fmuldfeq.7  |-  ( ph  ->  T  e.  _V )
fmuldfeq.8  |-  ( ph  ->  M  e.  NN )
fmuldfeq.9  |-  ( ph  ->  U : ( 1 ... M ) --> Y )
fmuldfeq.10  |-  ( (
ph  /\  f  e.  Y )  ->  f : T --> RR )
fmuldfeq.11  |-  ( (
ph  /\  f  e.  Y  /\  g  e.  Y
)  ->  ( t  e.  T  |->  ( ( f `  t )  x.  ( g `  t ) ) )  e.  Y )
Assertion
Ref Expression
fmuldfeq  |-  ( (
ph  /\  t  e.  T )  ->  ( X `  t )  =  ( Z `  t ) )
Distinct variable groups:    t, T    f, g, t, T    f,
i, t, T    f, F, g    f, M, g    U, f, g, t    f, Y, g    ph, f, g   
i, M    U, i
Allowed substitution hints:    ph( t, i)    P( t, f, g, i)    F( t, i)    M( t)    X( t, f, g, i)    Y( t, i)    Z( t, f, g, i)

Proof of Theorem fmuldfeq
Dummy variables  k 
b  n  m are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fmuldfeq.8 . . . . . 6  |-  ( ph  ->  M  e.  NN )
21nnge1d 10654 . . . . 5  |-  ( ph  ->  1  <_  M )
32adantr 467 . . . 4  |-  ( (
ph  /\  t  e.  T )  ->  1  <_  M )
4 nnre 10618 . . . . . 6  |-  ( M  e.  NN  ->  M  e.  RR )
5 leid 9731 . . . . . 6  |-  ( M  e.  RR  ->  M  <_  M )
61, 4, 53syl 18 . . . . 5  |-  ( ph  ->  M  <_  M )
76adantr 467 . . . 4  |-  ( (
ph  /\  t  e.  T )  ->  M  <_  M )
81nnzd 11041 . . . . . 6  |-  ( ph  ->  M  e.  ZZ )
98adantr 467 . . . . 5  |-  ( (
ph  /\  t  e.  T )  ->  M  e.  ZZ )
10 1zzd 10970 . . . . 5  |-  ( (
ph  /\  t  e.  T )  ->  1  e.  ZZ )
11 elfz 11792 . . . . 5  |-  ( ( M  e.  ZZ  /\  1  e.  ZZ  /\  M  e.  ZZ )  ->  ( M  e.  ( 1 ... M )  <->  ( 1  <_  M  /\  M  <_  M ) ) )
129, 10, 9, 11syl3anc 1265 . . . 4  |-  ( (
ph  /\  t  e.  T )  ->  ( M  e.  ( 1 ... M )  <->  ( 1  <_  M  /\  M  <_  M ) ) )
133, 7, 12mpbir2and 931 . . 3  |-  ( (
ph  /\  t  e.  T )  ->  M  e.  ( 1 ... M
) )
1413ad2ant1 1027 . . . 4  |-  ( (
ph  /\  t  e.  T  /\  M  e.  ( 1 ... M ) )  ->  M  e.  NN )
15 eleq1 2495 . . . . . . 7  |-  ( m  =  1  ->  (
m  e.  ( 1 ... M )  <->  1  e.  ( 1 ... M
) ) )
16153anbi3d 1342 . . . . . 6  |-  ( m  =  1  ->  (
( ph  /\  t  e.  T  /\  m  e.  ( 1 ... M
) )  <->  ( ph  /\  t  e.  T  /\  1  e.  ( 1 ... M ) ) ) )
17 fveq2 5879 . . . . . . . 8  |-  ( m  =  1  ->  (  seq 1 ( P ,  U ) `  m
)  =  (  seq 1 ( P ,  U ) `  1
) )
1817fveq1d 5881 . . . . . . 7  |-  ( m  =  1  ->  (
(  seq 1 ( P ,  U ) `  m ) `  t
)  =  ( (  seq 1 ( P ,  U ) ` 
1 ) `  t
) )
19 fveq2 5879 . . . . . . 7  |-  ( m  =  1  ->  (  seq 1 (  x.  , 
( F `  t
) ) `  m
)  =  (  seq 1 (  x.  , 
( F `  t
) ) `  1
) )
2018, 19eqeq12d 2445 . . . . . 6  |-  ( m  =  1  ->  (
( (  seq 1
( P ,  U
) `  m ) `  t )  =  (  seq 1 (  x.  ,  ( F `  t ) ) `  m )  <->  ( (  seq 1 ( P ,  U ) `  1
) `  t )  =  (  seq 1
(  x.  ,  ( F `  t ) ) `  1 ) ) )
2116, 20imbi12d 322 . . . . 5  |-  ( m  =  1  ->  (
( ( ph  /\  t  e.  T  /\  m  e.  ( 1 ... M ) )  ->  ( (  seq 1 ( P ,  U ) `  m
) `  t )  =  (  seq 1
(  x.  ,  ( F `  t ) ) `  m ) )  <->  ( ( ph  /\  t  e.  T  /\  1  e.  ( 1 ... M ) )  ->  ( (  seq 1 ( P ,  U ) `  1
) `  t )  =  (  seq 1
(  x.  ,  ( F `  t ) ) `  1 ) ) ) )
22 eleq1 2495 . . . . . . 7  |-  ( m  =  n  ->  (
m  e.  ( 1 ... M )  <->  n  e.  ( 1 ... M
) ) )
23223anbi3d 1342 . . . . . 6  |-  ( m  =  n  ->  (
( ph  /\  t  e.  T  /\  m  e.  ( 1 ... M
) )  <->  ( ph  /\  t  e.  T  /\  n  e.  ( 1 ... M ) ) ) )
24 fveq2 5879 . . . . . . . 8  |-  ( m  =  n  ->  (  seq 1 ( P ,  U ) `  m
)  =  (  seq 1 ( P ,  U ) `  n
) )
2524fveq1d 5881 . . . . . . 7  |-  ( m  =  n  ->  (
(  seq 1 ( P ,  U ) `  m ) `  t
)  =  ( (  seq 1 ( P ,  U ) `  n ) `  t
) )
26 fveq2 5879 . . . . . . 7  |-  ( m  =  n  ->  (  seq 1 (  x.  , 
( F `  t
) ) `  m
)  =  (  seq 1 (  x.  , 
( F `  t
) ) `  n
) )
2725, 26eqeq12d 2445 . . . . . 6  |-  ( m  =  n  ->  (
( (  seq 1
( P ,  U
) `  m ) `  t )  =  (  seq 1 (  x.  ,  ( F `  t ) ) `  m )  <->  ( (  seq 1 ( P ,  U ) `  n
) `  t )  =  (  seq 1
(  x.  ,  ( F `  t ) ) `  n ) ) )
2823, 27imbi12d 322 . . . . 5  |-  ( m  =  n  ->  (
( ( ph  /\  t  e.  T  /\  m  e.  ( 1 ... M ) )  ->  ( (  seq 1 ( P ,  U ) `  m
) `  t )  =  (  seq 1
(  x.  ,  ( F `  t ) ) `  m ) )  <->  ( ( ph  /\  t  e.  T  /\  n  e.  ( 1 ... M ) )  ->  ( (  seq 1 ( P ,  U ) `  n
) `  t )  =  (  seq 1
(  x.  ,  ( F `  t ) ) `  n ) ) ) )
29 eleq1 2495 . . . . . . 7  |-  ( m  =  ( n  + 
1 )  ->  (
m  e.  ( 1 ... M )  <->  ( n  +  1 )  e.  ( 1 ... M
) ) )
30293anbi3d 1342 . . . . . 6  |-  ( m  =  ( n  + 
1 )  ->  (
( ph  /\  t  e.  T  /\  m  e.  ( 1 ... M
) )  <->  ( ph  /\  t  e.  T  /\  ( n  +  1
)  e.  ( 1 ... M ) ) ) )
31 fveq2 5879 . . . . . . . 8  |-  ( m  =  ( n  + 
1 )  ->  (  seq 1 ( P ,  U ) `  m
)  =  (  seq 1 ( P ,  U ) `  (
n  +  1 ) ) )
3231fveq1d 5881 . . . . . . 7  |-  ( m  =  ( n  + 
1 )  ->  (
(  seq 1 ( P ,  U ) `  m ) `  t
)  =  ( (  seq 1 ( P ,  U ) `  ( n  +  1
) ) `  t
) )
33 fveq2 5879 . . . . . . 7  |-  ( m  =  ( n  + 
1 )  ->  (  seq 1 (  x.  , 
( F `  t
) ) `  m
)  =  (  seq 1 (  x.  , 
( F `  t
) ) `  (
n  +  1 ) ) )
3432, 33eqeq12d 2445 . . . . . 6  |-  ( m  =  ( n  + 
1 )  ->  (
( (  seq 1
( P ,  U
) `  m ) `  t )  =  (  seq 1 (  x.  ,  ( F `  t ) ) `  m )  <->  ( (  seq 1 ( P ,  U ) `  (
n  +  1 ) ) `  t )  =  (  seq 1
(  x.  ,  ( F `  t ) ) `  ( n  +  1 ) ) ) )
3530, 34imbi12d 322 . . . . 5  |-  ( m  =  ( n  + 
1 )  ->  (
( ( ph  /\  t  e.  T  /\  m  e.  ( 1 ... M ) )  ->  ( (  seq 1 ( P ,  U ) `  m
) `  t )  =  (  seq 1
(  x.  ,  ( F `  t ) ) `  m ) )  <->  ( ( ph  /\  t  e.  T  /\  ( n  +  1
)  e.  ( 1 ... M ) )  ->  ( (  seq 1 ( P ,  U ) `  (
n  +  1 ) ) `  t )  =  (  seq 1
(  x.  ,  ( F `  t ) ) `  ( n  +  1 ) ) ) ) )
36 eleq1 2495 . . . . . . 7  |-  ( m  =  M  ->  (
m  e.  ( 1 ... M )  <->  M  e.  ( 1 ... M
) ) )
37363anbi3d 1342 . . . . . 6  |-  ( m  =  M  ->  (
( ph  /\  t  e.  T  /\  m  e.  ( 1 ... M
) )  <->  ( ph  /\  t  e.  T  /\  M  e.  ( 1 ... M ) ) ) )
38 fveq2 5879 . . . . . . . 8  |-  ( m  =  M  ->  (  seq 1 ( P ,  U ) `  m
)  =  (  seq 1 ( P ,  U ) `  M
) )
3938fveq1d 5881 . . . . . . 7  |-  ( m  =  M  ->  (
(  seq 1 ( P ,  U ) `  m ) `  t
)  =  ( (  seq 1 ( P ,  U ) `  M ) `  t
) )
40 fveq2 5879 . . . . . . 7  |-  ( m  =  M  ->  (  seq 1 (  x.  , 
( F `  t
) ) `  m
)  =  (  seq 1 (  x.  , 
( F `  t
) ) `  M
) )
4139, 40eqeq12d 2445 . . . . . 6  |-  ( m  =  M  ->  (
( (  seq 1
( P ,  U
) `  m ) `  t )  =  (  seq 1 (  x.  ,  ( F `  t ) ) `  m )  <->  ( (  seq 1 ( P ,  U ) `  M
) `  t )  =  (  seq 1
(  x.  ,  ( F `  t ) ) `  M ) ) )
4237, 41imbi12d 322 . . . . 5  |-  ( m  =  M  ->  (
( ( ph  /\  t  e.  T  /\  m  e.  ( 1 ... M ) )  ->  ( (  seq 1 ( P ,  U ) `  m
) `  t )  =  (  seq 1
(  x.  ,  ( F `  t ) ) `  m ) )  <->  ( ( ph  /\  t  e.  T  /\  M  e.  ( 1 ... M ) )  ->  ( (  seq 1 ( P ,  U ) `  M
) `  t )  =  (  seq 1
(  x.  ,  ( F `  t ) ) `  M ) ) ) )
43 1z 10969 . . . . . . . 8  |-  1  e.  ZZ
44 seq1 12227 . . . . . . . 8  |-  ( 1  e.  ZZ  ->  (  seq 1 (  x.  , 
( F `  t
) ) `  1
)  =  ( ( F `  t ) `
 1 ) )
4543, 44ax-mp 5 . . . . . . 7  |-  (  seq 1 (  x.  , 
( F `  t
) ) `  1
)  =  ( ( F `  t ) `
 1 )
46 1le1 10242 . . . . . . . . . . . . 13  |-  1  <_  1
4746a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  1  <_  1 )
48 1zzd 10970 . . . . . . . . . . . . 13  |-  ( ph  ->  1  e.  ZZ )
49 elfz 11792 . . . . . . . . . . . . 13  |-  ( ( 1  e.  ZZ  /\  1  e.  ZZ  /\  M  e.  ZZ )  ->  (
1  e.  ( 1 ... M )  <->  ( 1  <_  1  /\  1  <_  M ) ) )
5048, 48, 8, 49syl3anc 1265 . . . . . . . . . . . 12  |-  ( ph  ->  ( 1  e.  ( 1 ... M )  <-> 
( 1  <_  1  /\  1  <_  M ) ) )
5147, 2, 50mpbir2and 931 . . . . . . . . . . 11  |-  ( ph  ->  1  e.  ( 1 ... M ) )
52 nfv 1752 . . . . . . . . . . . . 13  |-  F/ i  t  e.  T
53 fmuldfeq.5 . . . . . . . . . . . . . . . . 17  |-  F  =  ( t  e.  T  |->  ( i  e.  ( 1 ... M ) 
|->  ( ( U `  i ) `  t
) ) )
54 nfcv 2585 . . . . . . . . . . . . . . . . . 18  |-  F/_ i T
55 nfmpt1 4511 . . . . . . . . . . . . . . . . . 18  |-  F/_ i
( i  e.  ( 1 ... M ) 
|->  ( ( U `  i ) `  t
) )
5654, 55nfmpt 4510 . . . . . . . . . . . . . . . . 17  |-  F/_ i
( t  e.  T  |->  ( i  e.  ( 1 ... M ) 
|->  ( ( U `  i ) `  t
) ) )
5753, 56nfcxfr 2583 . . . . . . . . . . . . . . . 16  |-  F/_ i F
58 nfcv 2585 . . . . . . . . . . . . . . . 16  |-  F/_ i
t
5957, 58nffv 5886 . . . . . . . . . . . . . . 15  |-  F/_ i
( F `  t
)
60 nfcv 2585 . . . . . . . . . . . . . . 15  |-  F/_ i
1
6159, 60nffv 5886 . . . . . . . . . . . . . 14  |-  F/_ i
( ( F `  t ) `  1
)
62 nffvmpt1 5887 . . . . . . . . . . . . . 14  |-  F/_ i
( ( i  e.  ( 1 ... M
)  |->  ( ( U `
 i ) `  t ) ) ` 
1 )
6361, 62nfeq 2596 . . . . . . . . . . . . 13  |-  F/ i ( ( F `  t ) `  1
)  =  ( ( i  e.  ( 1 ... M )  |->  ( ( U `  i
) `  t )
) `  1 )
6452, 63nfim 1977 . . . . . . . . . . . 12  |-  F/ i ( t  e.  T  ->  ( ( F `  t ) `  1
)  =  ( ( i  e.  ( 1 ... M )  |->  ( ( U `  i
) `  t )
) `  1 )
)
65 fveq2 5879 . . . . . . . . . . . . . 14  |-  ( i  =  1  ->  (
( F `  t
) `  i )  =  ( ( F `
 t ) ` 
1 ) )
66 fveq2 5879 . . . . . . . . . . . . . 14  |-  ( i  =  1  ->  (
( i  e.  ( 1 ... M ) 
|->  ( ( U `  i ) `  t
) ) `  i
)  =  ( ( i  e.  ( 1 ... M )  |->  ( ( U `  i
) `  t )
) `  1 )
)
6765, 66eqeq12d 2445 . . . . . . . . . . . . 13  |-  ( i  =  1  ->  (
( ( F `  t ) `  i
)  =  ( ( i  e.  ( 1 ... M )  |->  ( ( U `  i
) `  t )
) `  i )  <->  ( ( F `  t
) `  1 )  =  ( ( i  e.  ( 1 ... M )  |->  ( ( U `  i ) `
 t ) ) `
 1 ) ) )
6867imbi2d 318 . . . . . . . . . . . 12  |-  ( i  =  1  ->  (
( t  e.  T  ->  ( ( F `  t ) `  i
)  =  ( ( i  e.  ( 1 ... M )  |->  ( ( U `  i
) `  t )
) `  i )
)  <->  ( t  e.  T  ->  ( ( F `  t ) `  1 )  =  ( ( i  e.  ( 1 ... M
)  |->  ( ( U `
 i ) `  t ) ) ` 
1 ) ) ) )
69 ovex 6331 . . . . . . . . . . . . . . 15  |-  ( 1 ... M )  e. 
_V
7069mptex 6149 . . . . . . . . . . . . . 14  |-  ( i  e.  ( 1 ... M )  |->  ( ( U `  i ) `
 t ) )  e.  _V
7153fvmpt2 5971 . . . . . . . . . . . . . 14  |-  ( ( t  e.  T  /\  ( i  e.  ( 1 ... M ) 
|->  ( ( U `  i ) `  t
) )  e.  _V )  ->  ( F `  t )  =  ( i  e.  ( 1 ... M )  |->  ( ( U `  i
) `  t )
) )
7270, 71mpan2 676 . . . . . . . . . . . . 13  |-  ( t  e.  T  ->  ( F `  t )  =  ( i  e.  ( 1 ... M
)  |->  ( ( U `
 i ) `  t ) ) )
7372fveq1d 5881 . . . . . . . . . . . 12  |-  ( t  e.  T  ->  (
( F `  t
) `  i )  =  ( ( i  e.  ( 1 ... M )  |->  ( ( U `  i ) `
 t ) ) `
 i ) )
7464, 68, 73vtoclg1f 3139 . . . . . . . . . . 11  |-  ( 1  e.  ( 1 ... M )  ->  (
t  e.  T  -> 
( ( F `  t ) `  1
)  =  ( ( i  e.  ( 1 ... M )  |->  ( ( U `  i
) `  t )
) `  1 )
) )
7551, 74syl 17 . . . . . . . . . 10  |-  ( ph  ->  ( t  e.  T  ->  ( ( F `  t ) `  1
)  =  ( ( i  e.  ( 1 ... M )  |->  ( ( U `  i
) `  t )
) `  1 )
) )
7675imp 431 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  T )  ->  (
( F `  t
) `  1 )  =  ( ( i  e.  ( 1 ... M )  |->  ( ( U `  i ) `
 t ) ) `
 1 ) )
7751adantr 467 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  T )  ->  1  e.  ( 1 ... M
) )
78 fmuldfeq.9 . . . . . . . . . . . . 13  |-  ( ph  ->  U : ( 1 ... M ) --> Y )
7978, 51ffvelrnd 6036 . . . . . . . . . . . 12  |-  ( ph  ->  ( U `  1
)  e.  Y )
8079ancli 554 . . . . . . . . . . . 12  |-  ( ph  ->  ( ph  /\  ( U `  1 )  e.  Y ) )
81 eleq1 2495 . . . . . . . . . . . . . . 15  |-  ( f  =  ( U ` 
1 )  ->  (
f  e.  Y  <->  ( U `  1 )  e.  Y ) )
8281anbi2d 709 . . . . . . . . . . . . . 14  |-  ( f  =  ( U ` 
1 )  ->  (
( ph  /\  f  e.  Y )  <->  ( ph  /\  ( U `  1
)  e.  Y ) ) )
83 feq1 5726 . . . . . . . . . . . . . 14  |-  ( f  =  ( U ` 
1 )  ->  (
f : T --> RR  <->  ( U `  1 ) : T --> RR ) )
8482, 83imbi12d 322 . . . . . . . . . . . . 13  |-  ( f  =  ( U ` 
1 )  ->  (
( ( ph  /\  f  e.  Y )  ->  f : T --> RR )  <-> 
( ( ph  /\  ( U `  1 )  e.  Y )  -> 
( U `  1
) : T --> RR ) ) )
85 fmuldfeq.10 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  f  e.  Y )  ->  f : T --> RR )
8685a1i 11 . . . . . . . . . . . . 13  |-  ( f  e.  Y  ->  (
( ph  /\  f  e.  Y )  ->  f : T --> RR ) )
8784, 86vtoclga 3146 . . . . . . . . . . . 12  |-  ( ( U `  1 )  e.  Y  ->  (
( ph  /\  ( U `  1 )  e.  Y )  ->  ( U `  1 ) : T --> RR ) )
8879, 80, 87sylc 63 . . . . . . . . . . 11  |-  ( ph  ->  ( U `  1
) : T --> RR )
8988ffvelrnda 6035 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  T )  ->  (
( U `  1
) `  t )  e.  RR )
90 fveq2 5879 . . . . . . . . . . . 12  |-  ( i  =  1  ->  ( U `  i )  =  ( U ` 
1 ) )
9190fveq1d 5881 . . . . . . . . . . 11  |-  ( i  =  1  ->  (
( U `  i
) `  t )  =  ( ( U `
 1 ) `  t ) )
92 eqid 2423 . . . . . . . . . . 11  |-  ( i  e.  ( 1 ... M )  |->  ( ( U `  i ) `
 t ) )  =  ( i  e.  ( 1 ... M
)  |->  ( ( U `
 i ) `  t ) )
9391, 92fvmptg 5960 . . . . . . . . . 10  |-  ( ( 1  e.  ( 1 ... M )  /\  ( ( U ` 
1 ) `  t
)  e.  RR )  ->  ( ( i  e.  ( 1 ... M )  |->  ( ( U `  i ) `
 t ) ) `
 1 )  =  ( ( U ` 
1 ) `  t
) )
9477, 89, 93syl2anc 666 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  T )  ->  (
( i  e.  ( 1 ... M ) 
|->  ( ( U `  i ) `  t
) ) `  1
)  =  ( ( U `  1 ) `
 t ) )
9576, 94eqtrd 2464 . . . . . . . 8  |-  ( (
ph  /\  t  e.  T )  ->  (
( F `  t
) `  1 )  =  ( ( U `
 1 ) `  t ) )
96 seq1 12227 . . . . . . . . . 10  |-  ( 1  e.  ZZ  ->  (  seq 1 ( P ,  U ) `  1
)  =  ( U `
 1 ) )
9743, 96ax-mp 5 . . . . . . . . 9  |-  (  seq 1 ( P ,  U ) `  1
)  =  ( U `
 1 )
9897fveq1i 5880 . . . . . . . 8  |-  ( (  seq 1 ( P ,  U ) ` 
1 ) `  t
)  =  ( ( U `  1 ) `
 t )
9995, 98syl6eqr 2482 . . . . . . 7  |-  ( (
ph  /\  t  e.  T )  ->  (
( F `  t
) `  1 )  =  ( (  seq 1 ( P ,  U ) `  1
) `  t )
)
10045, 99syl5req 2477 . . . . . 6  |-  ( (
ph  /\  t  e.  T )  ->  (
(  seq 1 ( P ,  U ) ` 
1 ) `  t
)  =  (  seq 1 (  x.  , 
( F `  t
) ) `  1
) )
1011003adant3 1026 . . . . 5  |-  ( (
ph  /\  t  e.  T  /\  1  e.  ( 1 ... M ) )  ->  ( (  seq 1 ( P ,  U ) `  1
) `  t )  =  (  seq 1
(  x.  ,  ( F `  t ) ) `  1 ) )
102 simp31 1042 . . . . . . 7  |-  ( ( n  e.  NN  /\  ( ( ph  /\  t  e.  T  /\  n  e.  ( 1 ... M ) )  ->  ( (  seq 1 ( P ,  U ) `  n
) `  t )  =  (  seq 1
(  x.  ,  ( F `  t ) ) `  n ) )  /\  ( ph  /\  t  e.  T  /\  ( n  +  1
)  e.  ( 1 ... M ) ) )  ->  ph )
103 simp1 1006 . . . . . . . . . 10  |-  ( ( n  e.  NN  /\  ( ( ph  /\  t  e.  T  /\  n  e.  ( 1 ... M ) )  ->  ( (  seq 1 ( P ,  U ) `  n
) `  t )  =  (  seq 1
(  x.  ,  ( F `  t ) ) `  n ) )  /\  ( ph  /\  t  e.  T  /\  ( n  +  1
)  e.  ( 1 ... M ) ) )  ->  n  e.  NN )
104 simp33 1044 . . . . . . . . . 10  |-  ( ( n  e.  NN  /\  ( ( ph  /\  t  e.  T  /\  n  e.  ( 1 ... M ) )  ->  ( (  seq 1 ( P ,  U ) `  n
) `  t )  =  (  seq 1
(  x.  ,  ( F `  t ) ) `  n ) )  /\  ( ph  /\  t  e.  T  /\  ( n  +  1
)  e.  ( 1 ... M ) ) )  ->  ( n  +  1 )  e.  ( 1 ... M
) )
105103, 104jca 535 . . . . . . . . 9  |-  ( ( n  e.  NN  /\  ( ( ph  /\  t  e.  T  /\  n  e.  ( 1 ... M ) )  ->  ( (  seq 1 ( P ,  U ) `  n
) `  t )  =  (  seq 1
(  x.  ,  ( F `  t ) ) `  n ) )  /\  ( ph  /\  t  e.  T  /\  ( n  +  1
)  e.  ( 1 ... M ) ) )  ->  ( n  e.  NN  /\  ( n  +  1 )  e.  ( 1 ... M
) ) )
106 elnnuz 11197 . . . . . . . . . . 11  |-  ( n  e.  NN  <->  n  e.  ( ZZ>= `  1 )
)
107106biimpi 198 . . . . . . . . . 10  |-  ( n  e.  NN  ->  n  e.  ( ZZ>= `  1 )
)
108107anim1i 571 . . . . . . . . 9  |-  ( ( n  e.  NN  /\  ( n  +  1
)  e.  ( 1 ... M ) )  ->  ( n  e.  ( ZZ>= `  1 )  /\  ( n  +  1 )  e.  ( 1 ... M ) ) )
109 peano2fzr 11814 . . . . . . . . 9  |-  ( ( n  e.  ( ZZ>= ` 
1 )  /\  (
n  +  1 )  e.  ( 1 ... M ) )  ->  n  e.  ( 1 ... M ) )
110105, 108, 1093syl 18 . . . . . . . 8  |-  ( ( n  e.  NN  /\  ( ( ph  /\  t  e.  T  /\  n  e.  ( 1 ... M ) )  ->  ( (  seq 1 ( P ,  U ) `  n
) `  t )  =  (  seq 1
(  x.  ,  ( F `  t ) ) `  n ) )  /\  ( ph  /\  t  e.  T  /\  ( n  +  1
)  e.  ( 1 ... M ) ) )  ->  n  e.  ( 1 ... M
) )
111 simp32 1043 . . . . . . . . 9  |-  ( ( n  e.  NN  /\  ( ( ph  /\  t  e.  T  /\  n  e.  ( 1 ... M ) )  ->  ( (  seq 1 ( P ,  U ) `  n
) `  t )  =  (  seq 1
(  x.  ,  ( F `  t ) ) `  n ) )  /\  ( ph  /\  t  e.  T  /\  ( n  +  1
)  e.  ( 1 ... M ) ) )  ->  t  e.  T )
112 simp2 1007 . . . . . . . . 9  |-  ( ( n  e.  NN  /\  ( ( ph  /\  t  e.  T  /\  n  e.  ( 1 ... M ) )  ->  ( (  seq 1 ( P ,  U ) `  n
) `  t )  =  (  seq 1
(  x.  ,  ( F `  t ) ) `  n ) )  /\  ( ph  /\  t  e.  T  /\  ( n  +  1
)  e.  ( 1 ... M ) ) )  ->  ( ( ph  /\  t  e.  T  /\  n  e.  (
1 ... M ) )  ->  ( (  seq 1 ( P ,  U ) `  n
) `  t )  =  (  seq 1
(  x.  ,  ( F `  t ) ) `  n ) ) )
113102, 111, 110, 112mp3and 1364 . . . . . . . 8  |-  ( ( n  e.  NN  /\  ( ( ph  /\  t  e.  T  /\  n  e.  ( 1 ... M ) )  ->  ( (  seq 1 ( P ,  U ) `  n
) `  t )  =  (  seq 1
(  x.  ,  ( F `  t ) ) `  n ) )  /\  ( ph  /\  t  e.  T  /\  ( n  +  1
)  e.  ( 1 ... M ) ) )  ->  ( (  seq 1 ( P ,  U ) `  n
) `  t )  =  (  seq 1
(  x.  ,  ( F `  t ) ) `  n ) )
114110, 104, 1133jca 1186 . . . . . . 7  |-  ( ( n  e.  NN  /\  ( ( ph  /\  t  e.  T  /\  n  e.  ( 1 ... M ) )  ->  ( (  seq 1 ( P ,  U ) `  n
) `  t )  =  (  seq 1
(  x.  ,  ( F `  t ) ) `  n ) )  /\  ( ph  /\  t  e.  T  /\  ( n  +  1
)  e.  ( 1 ... M ) ) )  ->  ( n  e.  ( 1 ... M
)  /\  ( n  +  1 )  e.  ( 1 ... M
)  /\  ( (  seq 1 ( P ,  U ) `  n
) `  t )  =  (  seq 1
(  x.  ,  ( F `  t ) ) `  n ) ) )
115 nfv 1752 . . . . . . . . 9  |-  F/ f
ph
116 nfv 1752 . . . . . . . . . 10  |-  F/ f  n  e.  ( 1 ... M )
117 nfv 1752 . . . . . . . . . 10  |-  F/ f ( n  +  1 )  e.  ( 1 ... M )
118 nfcv 2585 . . . . . . . . . . . . . 14  |-  F/_ f
1
119 fmuldfeq.3 . . . . . . . . . . . . . . 15  |-  P  =  ( f  e.  Y ,  g  e.  Y  |->  ( t  e.  T  |->  ( ( f `  t )  x.  (
g `  t )
) ) )
120 nfmpt21 6370 . . . . . . . . . . . . . . 15  |-  F/_ f
( f  e.  Y ,  g  e.  Y  |->  ( t  e.  T  |->  ( ( f `  t )  x.  (
g `  t )
) ) )
121119, 120nfcxfr 2583 . . . . . . . . . . . . . 14  |-  F/_ f P
122 nfcv 2585 . . . . . . . . . . . . . 14  |-  F/_ f U
123118, 121, 122nfseq 12224 . . . . . . . . . . . . 13  |-  F/_ f  seq 1 ( P ,  U )
124 nfcv 2585 . . . . . . . . . . . . 13  |-  F/_ f
n
125123, 124nffv 5886 . . . . . . . . . . . 12  |-  F/_ f
(  seq 1 ( P ,  U ) `  n )
126 nfcv 2585 . . . . . . . . . . . 12  |-  F/_ f
t
127125, 126nffv 5886 . . . . . . . . . . 11  |-  F/_ f
( (  seq 1
( P ,  U
) `  n ) `  t )
128 nfcv 2585 . . . . . . . . . . 11  |-  F/_ f
(  seq 1 (  x.  ,  ( F `  t ) ) `  n )
129127, 128nfeq 2596 . . . . . . . . . 10  |-  F/ f ( (  seq 1
( P ,  U
) `  n ) `  t )  =  (  seq 1 (  x.  ,  ( F `  t ) ) `  n )
130116, 117, 129nf3an 1987 . . . . . . . . 9  |-  F/ f ( n  e.  ( 1 ... M )  /\  ( n  + 
1 )  e.  ( 1 ... M )  /\  ( (  seq 1 ( P ,  U ) `  n
) `  t )  =  (  seq 1
(  x.  ,  ( F `  t ) ) `  n ) )
131115, 130nfan 1985 . . . . . . . 8  |-  F/ f ( ph  /\  (
n  e.  ( 1 ... M )  /\  ( n  +  1
)  e.  ( 1 ... M )  /\  ( (  seq 1
( P ,  U
) `  n ) `  t )  =  (  seq 1 (  x.  ,  ( F `  t ) ) `  n ) ) )
132 nfv 1752 . . . . . . . . 9  |-  F/ g
ph
133 nfv 1752 . . . . . . . . . 10  |-  F/ g  n  e.  ( 1 ... M )
134 nfv 1752 . . . . . . . . . 10  |-  F/ g ( n  +  1 )  e.  ( 1 ... M )
135 nfcv 2585 . . . . . . . . . . . . . 14  |-  F/_ g
1
136 nfmpt22 6371 . . . . . . . . . . . . . . 15  |-  F/_ g
( f  e.  Y ,  g  e.  Y  |->  ( t  e.  T  |->  ( ( f `  t )  x.  (
g `  t )
) ) )
137119, 136nfcxfr 2583 . . . . . . . . . . . . . 14  |-  F/_ g P
138 nfcv 2585 . . . . . . . . . . . . . 14  |-  F/_ g U
139135, 137, 138nfseq 12224 . . . . . . . . . . . . 13  |-  F/_ g  seq 1 ( P ,  U )
140 nfcv 2585 . . . . . . . . . . . . 13  |-  F/_ g
n
141139, 140nffv 5886 . . . . . . . . . . . 12  |-  F/_ g
(  seq 1 ( P ,  U ) `  n )
142 nfcv 2585 . . . . . . . . . . . 12  |-  F/_ g
t
143141, 142nffv 5886 . . . . . . . . . . 11  |-  F/_ g
( (  seq 1
( P ,  U
) `  n ) `  t )
144 nfcv 2585 . . . . . . . . . . 11  |-  F/_ g
(  seq 1 (  x.  ,  ( F `  t ) ) `  n )
145143, 144nfeq 2596 . . . . . . . . . 10  |-  F/ g ( (  seq 1
( P ,  U
) `  n ) `  t )  =  (  seq 1 (  x.  ,  ( F `  t ) ) `  n )
146133, 134, 145nf3an 1987 . . . . . . . . 9  |-  F/ g ( n  e.  ( 1 ... M )  /\  ( n  + 
1 )  e.  ( 1 ... M )  /\  ( (  seq 1 ( P ,  U ) `  n
) `  t )  =  (  seq 1
(  x.  ,  ( F `  t ) ) `  n ) )
147132, 146nfan 1985 . . . . . . . 8  |-  F/ g ( ph  /\  (
n  e.  ( 1 ... M )  /\  ( n  +  1
)  e.  ( 1 ... M )  /\  ( (  seq 1
( P ,  U
) `  n ) `  t )  =  (  seq 1 (  x.  ,  ( F `  t ) ) `  n ) ) )
148 fmuldfeq.2 . . . . . . . 8  |-  F/_ t Y
149 fmuldfeq.7 . . . . . . . . 9  |-  ( ph  ->  T  e.  _V )
150149adantr 467 . . . . . . . 8  |-  ( (
ph  /\  ( n  e.  ( 1 ... M
)  /\  ( n  +  1 )  e.  ( 1 ... M
)  /\  ( (  seq 1 ( P ,  U ) `  n
) `  t )  =  (  seq 1
(  x.  ,  ( F `  t ) ) `  n ) ) )  ->  T  e.  _V )
15178adantr 467 . . . . . . . 8  |-  ( (
ph  /\  ( n  e.  ( 1 ... M
)  /\  ( n  +  1 )  e.  ( 1 ... M
)  /\  ( (  seq 1 ( P ,  U ) `  n
) `  t )  =  (  seq 1
(  x.  ,  ( F `  t ) ) `  n ) ) )  ->  U : ( 1 ... M ) --> Y )
152 fmuldfeq.11 . . . . . . . . 9  |-  ( (
ph  /\  f  e.  Y  /\  g  e.  Y
)  ->  ( t  e.  T  |->  ( ( f `  t )  x.  ( g `  t ) ) )  e.  Y )
1531523adant1r 1258 . . . . . . . 8  |-  ( ( ( ph  /\  (
n  e.  ( 1 ... M )  /\  ( n  +  1
)  e.  ( 1 ... M )  /\  ( (  seq 1
( P ,  U
) `  n ) `  t )  =  (  seq 1 (  x.  ,  ( F `  t ) ) `  n ) ) )  /\  f  e.  Y  /\  g  e.  Y
)  ->  ( t  e.  T  |->  ( ( f `  t )  x.  ( g `  t ) ) )  e.  Y )
154 simpr1 1012 . . . . . . . 8  |-  ( (
ph  /\  ( n  e.  ( 1 ... M
)  /\  ( n  +  1 )  e.  ( 1 ... M
)  /\  ( (  seq 1 ( P ,  U ) `  n
) `  t )  =  (  seq 1
(  x.  ,  ( F `  t ) ) `  n ) ) )  ->  n  e.  ( 1 ... M
) )
155 simpr2 1013 . . . . . . . 8  |-  ( (
ph  /\  ( n  e.  ( 1 ... M
)  /\  ( n  +  1 )  e.  ( 1 ... M
)  /\  ( (  seq 1 ( P ,  U ) `  n
) `  t )  =  (  seq 1
(  x.  ,  ( F `  t ) ) `  n ) ) )  ->  (
n  +  1 )  e.  ( 1 ... M ) )
156 simpr3 1014 . . . . . . . 8  |-  ( (
ph  /\  ( n  e.  ( 1 ... M
)  /\  ( n  +  1 )  e.  ( 1 ... M
)  /\  ( (  seq 1 ( P ,  U ) `  n
) `  t )  =  (  seq 1
(  x.  ,  ( F `  t ) ) `  n ) ) )  ->  (
(  seq 1 ( P ,  U ) `  n ) `  t
)  =  (  seq 1 (  x.  , 
( F `  t
) ) `  n
) )
15785adantlr 720 . . . . . . . 8  |-  ( ( ( ph  /\  (
n  e.  ( 1 ... M )  /\  ( n  +  1
)  e.  ( 1 ... M )  /\  ( (  seq 1
( P ,  U
) `  n ) `  t )  =  (  seq 1 (  x.  ,  ( F `  t ) ) `  n ) ) )  /\  f  e.  Y
)  ->  f : T
--> RR )
158131, 147, 148, 119, 53, 150, 151, 153, 154, 155, 156, 157fmuldfeqlem1 37524 . . . . . . 7  |-  ( ( ( ph  /\  (
n  e.  ( 1 ... M )  /\  ( n  +  1
)  e.  ( 1 ... M )  /\  ( (  seq 1
( P ,  U
) `  n ) `  t )  =  (  seq 1 (  x.  ,  ( F `  t ) ) `  n ) ) )  /\  t  e.  T
)  ->  ( (  seq 1 ( P ,  U ) `  (
n  +  1 ) ) `  t )  =  (  seq 1
(  x.  ,  ( F `  t ) ) `  ( n  +  1 ) ) )
159102, 114, 111, 158syl21anc 1264 . . . . . 6  |-  ( ( n  e.  NN  /\  ( ( ph  /\  t  e.  T  /\  n  e.  ( 1 ... M ) )  ->  ( (  seq 1 ( P ,  U ) `  n
) `  t )  =  (  seq 1
(  x.  ,  ( F `  t ) ) `  n ) )  /\  ( ph  /\  t  e.  T  /\  ( n  +  1
)  e.  ( 1 ... M ) ) )  ->  ( (  seq 1 ( P ,  U ) `  (
n  +  1 ) ) `  t )  =  (  seq 1
(  x.  ,  ( F `  t ) ) `  ( n  +  1 ) ) )
1601593exp 1205 . . . . 5  |-  ( n  e.  NN  ->  (
( ( ph  /\  t  e.  T  /\  n  e.  ( 1 ... M ) )  ->  ( (  seq 1 ( P ,  U ) `  n
) `  t )  =  (  seq 1
(  x.  ,  ( F `  t ) ) `  n ) )  ->  ( ( ph  /\  t  e.  T  /\  ( n  +  1 )  e.  ( 1 ... M ) )  ->  ( (  seq 1 ( P ,  U ) `  (
n  +  1 ) ) `  t )  =  (  seq 1
(  x.  ,  ( F `  t ) ) `  ( n  +  1 ) ) ) ) )
16121, 28, 35, 42, 101, 160nnind 10629 . . . 4  |-  ( M  e.  NN  ->  (
( ph  /\  t  e.  T  /\  M  e.  ( 1 ... M
) )  ->  (
(  seq 1 ( P ,  U ) `  M ) `  t
)  =  (  seq 1 (  x.  , 
( F `  t
) ) `  M
) ) )
16214, 161mpcom 38 . . 3  |-  ( (
ph  /\  t  e.  T  /\  M  e.  ( 1 ... M ) )  ->  ( (  seq 1 ( P ,  U ) `  M
) `  t )  =  (  seq 1
(  x.  ,  ( F `  t ) ) `  M ) )
16313, 162mpd3an3 1362 . 2  |-  ( (
ph  /\  t  e.  T )  ->  (
(  seq 1 ( P ,  U ) `  M ) `  t
)  =  (  seq 1 (  x.  , 
( F `  t
) ) `  M
) )
164 fmuldfeq.4 . . . 4  |-  X  =  (  seq 1 ( P ,  U ) `
 M )
165164fveq1i 5880 . . 3  |-  ( X `
 t )  =  ( (  seq 1
( P ,  U
) `  M ) `  t )
166165a1i 11 . 2  |-  ( (
ph  /\  t  e.  T )  ->  ( X `  t )  =  ( (  seq 1 ( P ,  U ) `  M
) `  t )
)
167 simpr 463 . . 3  |-  ( (
ph  /\  t  e.  T )  ->  t  e.  T )
168 elnnuz 11197 . . . . . 6  |-  ( M  e.  NN  <->  M  e.  ( ZZ>= `  1 )
)
1691, 168sylib 200 . . . . 5  |-  ( ph  ->  M  e.  ( ZZ>= ` 
1 ) )
170169adantr 467 . . . 4  |-  ( (
ph  /\  t  e.  T )  ->  M  e.  ( ZZ>= `  1 )
)
171 fmuldfeq.1 . . . . . . . 8  |-  F/ i
ph
172171, 52nfan 1985 . . . . . . 7  |-  F/ i ( ph  /\  t  e.  T )
173 nfv 1752 . . . . . . 7  |-  F/ i  k  e.  ( 1 ... M )
174172, 173nfan 1985 . . . . . 6  |-  F/ i ( ( ph  /\  t  e.  T )  /\  k  e.  (
1 ... M ) )
175 nfcv 2585 . . . . . . . 8  |-  F/_ i
k
17659, 175nffv 5886 . . . . . . 7  |-  F/_ i
( ( F `  t ) `  k
)
177176nfel1 2601 . . . . . 6  |-  F/ i ( ( F `  t ) `  k
)  e.  RR
178174, 177nfim 1977 . . . . 5  |-  F/ i ( ( ( ph  /\  t  e.  T )  /\  k  e.  ( 1 ... M ) )  ->  ( ( F `  t ) `  k )  e.  RR )
179 eleq1 2495 . . . . . . 7  |-  ( i  =  k  ->  (
i  e.  ( 1 ... M )  <->  k  e.  ( 1 ... M
) ) )
180179anbi2d 709 . . . . . 6  |-  ( i  =  k  ->  (
( ( ph  /\  t  e.  T )  /\  i  e.  (
1 ... M ) )  <-> 
( ( ph  /\  t  e.  T )  /\  k  e.  (
1 ... M ) ) ) )
181 fveq2 5879 . . . . . . 7  |-  ( i  =  k  ->  (
( F `  t
) `  i )  =  ( ( F `
 t ) `  k ) )
182181eleq1d 2492 . . . . . 6  |-  ( i  =  k  ->  (
( ( F `  t ) `  i
)  e.  RR  <->  ( ( F `  t ) `  k )  e.  RR ) )
183180, 182imbi12d 322 . . . . 5  |-  ( i  =  k  ->  (
( ( ( ph  /\  t  e.  T )  /\  i  e.  ( 1 ... M ) )  ->  ( ( F `  t ) `  i )  e.  RR ) 
<->  ( ( ( ph  /\  t  e.  T )  /\  k  e.  ( 1 ... M ) )  ->  ( ( F `  t ) `  k )  e.  RR ) ) )
18473ad2antlr 732 . . . . . 6  |-  ( ( ( ph  /\  t  e.  T )  /\  i  e.  ( 1 ... M
) )  ->  (
( F `  t
) `  i )  =  ( ( i  e.  ( 1 ... M )  |->  ( ( U `  i ) `
 t ) ) `
 i ) )
185 simpr 463 . . . . . . . 8  |-  ( ( ( ph  /\  t  e.  T )  /\  i  e.  ( 1 ... M
) )  ->  i  e.  ( 1 ... M
) )
18678ffvelrnda 6035 . . . . . . . . . . 11  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( U `  i )  e.  Y )
187 simpl 459 . . . . . . . . . . . 12  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ph )
188187, 186jca 535 . . . . . . . . . . 11  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( ph  /\  ( U `  i )  e.  Y
) )
189 eleq1 2495 . . . . . . . . . . . . . 14  |-  ( f  =  ( U `  i )  ->  (
f  e.  Y  <->  ( U `  i )  e.  Y
) )
190189anbi2d 709 . . . . . . . . . . . . 13  |-  ( f  =  ( U `  i )  ->  (
( ph  /\  f  e.  Y )  <->  ( ph  /\  ( U `  i
)  e.  Y ) ) )
191 feq1 5726 . . . . . . . . . . . . 13  |-  ( f  =  ( U `  i )  ->  (
f : T --> RR  <->  ( U `  i ) : T --> RR ) )
192190, 191imbi12d 322 . . . . . . . . . . . 12  |-  ( f  =  ( U `  i )  ->  (
( ( ph  /\  f  e.  Y )  ->  f : T --> RR )  <-> 
( ( ph  /\  ( U `  i )  e.  Y )  -> 
( U `  i
) : T --> RR ) ) )
193192, 86vtoclga 3146 . . . . . . . . . . 11  |-  ( ( U `  i )  e.  Y  ->  (
( ph  /\  ( U `  i )  e.  Y )  ->  ( U `  i ) : T --> RR ) )
194186, 188, 193sylc 63 . . . . . . . . . 10  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( U `  i ) : T --> RR )
195194adantlr 720 . . . . . . . . 9  |-  ( ( ( ph  /\  t  e.  T )  /\  i  e.  ( 1 ... M
) )  ->  ( U `  i ) : T --> RR )
196 simplr 761 . . . . . . . . 9  |-  ( ( ( ph  /\  t  e.  T )  /\  i  e.  ( 1 ... M
) )  ->  t  e.  T )
197195, 196ffvelrnd 6036 . . . . . . . 8  |-  ( ( ( ph  /\  t  e.  T )  /\  i  e.  ( 1 ... M
) )  ->  (
( U `  i
) `  t )  e.  RR )
19892fvmpt2 5971 . . . . . . . 8  |-  ( ( i  e.  ( 1 ... M )  /\  ( ( U `  i ) `  t
)  e.  RR )  ->  ( ( i  e.  ( 1 ... M )  |->  ( ( U `  i ) `
 t ) ) `
 i )  =  ( ( U `  i ) `  t
) )
199185, 197, 198syl2anc 666 . . . . . . 7  |-  ( ( ( ph  /\  t  e.  T )  /\  i  e.  ( 1 ... M
) )  ->  (
( i  e.  ( 1 ... M ) 
|->  ( ( U `  i ) `  t
) ) `  i
)  =  ( ( U `  i ) `
 t ) )
200199, 197eqeltrd 2511 . . . . . 6  |-  ( ( ( ph  /\  t  e.  T )  /\  i  e.  ( 1 ... M
) )  ->  (
( i  e.  ( 1 ... M ) 
|->  ( ( U `  i ) `  t
) ) `  i
)  e.  RR )
201184, 200eqeltrd 2511 . . . . 5  |-  ( ( ( ph  /\  t  e.  T )  /\  i  e.  ( 1 ... M
) )  ->  (
( F `  t
) `  i )  e.  RR )
202178, 183, 201chvar 2068 . . . 4  |-  ( ( ( ph  /\  t  e.  T )  /\  k  e.  ( 1 ... M
) )  ->  (
( F `  t
) `  k )  e.  RR )
203 remulcl 9626 . . . . 5  |-  ( ( k  e.  RR  /\  b  e.  RR )  ->  ( k  x.  b
)  e.  RR )
204203adantl 468 . . . 4  |-  ( ( ( ph  /\  t  e.  T )  /\  (
k  e.  RR  /\  b  e.  RR )
)  ->  ( k  x.  b )  e.  RR )
205170, 202, 204seqcl 12234 . . 3  |-  ( (
ph  /\  t  e.  T )  ->  (  seq 1 (  x.  , 
( F `  t
) ) `  M
)  e.  RR )
206 fmuldfeq.6 . . . 4  |-  Z  =  ( t  e.  T  |->  (  seq 1 (  x.  ,  ( F `
 t ) ) `
 M ) )
207206fvmpt2 5971 . . 3  |-  ( ( t  e.  T  /\  (  seq 1 (  x.  ,  ( F `  t ) ) `  M )  e.  RR )  ->  ( Z `  t )  =  (  seq 1 (  x.  ,  ( F `  t ) ) `  M ) )
208167, 205, 207syl2anc 666 . 2  |-  ( (
ph  /\  t  e.  T )  ->  ( Z `  t )  =  (  seq 1
(  x.  ,  ( F `  t ) ) `  M ) )
209163, 166, 2083eqtr4d 2474 1  |-  ( (
ph  /\  t  e.  T )  ->  ( X `  t )  =  ( Z `  t ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 983    = wceq 1438   F/wnf 1664    e. wcel 1869   F/_wnfc 2571   _Vcvv 3082   class class class wbr 4421    |-> cmpt 4480   -->wf 5595   ` cfv 5599  (class class class)co 6303    |-> cmpt2 6305   RRcr 9540   1c1 9542    + caddc 9544    x. cmul 9546    <_ cle 9678   NNcn 10611   ZZcz 10939   ZZ>=cuz 11161   ...cfz 11786    seqcseq 12214
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595  ax-cnex 9597  ax-resscn 9598  ax-1cn 9599  ax-icn 9600  ax-addcl 9601  ax-addrcl 9602  ax-mulcl 9603  ax-mulrcl 9604  ax-mulcom 9605  ax-addass 9606  ax-mulass 9607  ax-distr 9608  ax-i2m1 9609  ax-1ne0 9610  ax-1rid 9611  ax-rnegex 9612  ax-rrecex 9613  ax-cnre 9614  ax-pre-lttri 9615  ax-pre-lttrn 9616  ax-pre-ltadd 9617  ax-pre-mulgt0 9618
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-uni 4218  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-tr 4517  df-eprel 4762  df-id 4766  df-po 4772  df-so 4773  df-fr 4810  df-we 4812  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-pred 5397  df-ord 5443  df-on 5444  df-lim 5445  df-suc 5446  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-riota 6265  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-om 6705  df-1st 6805  df-2nd 6806  df-wrecs 7034  df-recs 7096  df-rdg 7134  df-er 7369  df-en 7576  df-dom 7577  df-sdom 7578  df-pnf 9679  df-mnf 9680  df-xr 9681  df-ltxr 9682  df-le 9683  df-sub 9864  df-neg 9865  df-nn 10612  df-n0 10872  df-z 10940  df-uz 11162  df-fz 11787  df-seq 12215
This theorem is referenced by:  stoweidlem42  37767  stoweidlem48  37773
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