Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  meadjiunlem Structured version   Visualization version   Unicode version

Theorem meadjiunlem 38419
Description: The sum of nonnegative extended reals, restricted to the range of another function. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
meadjiunlem.f  |-  ( ph  ->  M  e. Meas )
meadjiunlem.3  |-  S  =  dom  M
meadjiunlem.x  |-  ( ph  ->  X  e.  V )
meadjiunlem.g  |-  ( ph  ->  G : X --> S )
meadjiunlem.y  |-  Y  =  { i  e.  X  |  ( G `  i )  =/=  (/) }
meadjiunlem.dj  |-  ( ph  -> Disj  i  e.  X  ( G `  i )
)
Assertion
Ref Expression
meadjiunlem  |-  ( ph  ->  (Σ^ `  ( M  |`  ran  G
) )  =  (Σ^ `  ( M  o.  G )
) )
Distinct variable groups:    i, G    i, X    i, Y    ph, i
Allowed substitution hints:    S( i)    M( i)    V( i)

Proof of Theorem meadjiunlem
Dummy variables  j 
k  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1769 . . . 4  |-  F/ k
ph
2 meadjiunlem.g . . . . . 6  |-  ( ph  ->  G : X --> S )
3 meadjiunlem.x . . . . . 6  |-  ( ph  ->  X  e.  V )
42, 3jca 541 . . . . 5  |-  ( ph  ->  ( G : X --> S  /\  X  e.  V
) )
5 fex 6155 . . . . 5  |-  ( ( G : X --> S  /\  X  e.  V )  ->  G  e.  _V )
6 rnexg 6744 . . . . 5  |-  ( G  e.  _V  ->  ran  G  e.  _V )
74, 5, 63syl 18 . . . 4  |-  ( ph  ->  ran  G  e.  _V )
8 difssd 3550 . . . 4  |-  ( ph  ->  ( ran  G  \  { (/) } )  C_  ran  G )
9 meadjiunlem.f . . . . . . 7  |-  ( ph  ->  M  e. Meas )
10 meadjiunlem.3 . . . . . . 7  |-  S  =  dom  M
119, 10meaf 38407 . . . . . 6  |-  ( ph  ->  M : S --> ( 0 [,] +oo ) )
1211adantr 472 . . . . 5  |-  ( (
ph  /\  k  e.  ( ran  G  \  { (/)
} ) )  ->  M : S --> ( 0 [,] +oo ) )
13 frn 5747 . . . . . . . 8  |-  ( G : X --> S  ->  ran  G  C_  S )
142, 13syl 17 . . . . . . 7  |-  ( ph  ->  ran  G  C_  S
)
1514adantr 472 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ran  G  \  { (/)
} ) )  ->  ran  G  C_  S )
168sselda 3418 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ran  G  \  { (/)
} ) )  -> 
k  e.  ran  G
)
1715, 16sseldd 3419 . . . . 5  |-  ( (
ph  /\  k  e.  ( ran  G  \  { (/)
} ) )  -> 
k  e.  S )
1812, 17ffvelrnd 6038 . . . 4  |-  ( (
ph  /\  k  e.  ( ran  G  \  { (/)
} ) )  -> 
( M `  k
)  e.  ( 0 [,] +oo ) )
19 simpl 464 . . . . 5  |-  ( (
ph  /\  k  e.  ( ran  G  \  ( ran  G  \  { (/) } ) ) )  ->  ph )
20 id 22 . . . . . . . 8  |-  ( k  e.  ( ran  G  \  ( ran  G  \  { (/) } ) )  ->  k  e.  ( ran  G  \  ( ran  G  \  { (/) } ) ) )
21 dfin4 3674 . . . . . . . . 9  |-  ( ran 
G  i^i  { (/) } )  =  ( ran  G  \  ( ran  G  \  { (/) } ) )
2221eqcomi 2480 . . . . . . . 8  |-  ( ran 
G  \  ( ran  G 
\  { (/) } ) )  =  ( ran 
G  i^i  { (/) } )
2320, 22syl6eleq 2559 . . . . . . 7  |-  ( k  e.  ( ran  G  \  ( ran  G  \  { (/) } ) )  ->  k  e.  ( ran  G  i^i  { (/)
} ) )
24 elinel2 3611 . . . . . . . 8  |-  ( k  e.  ( ran  G  i^i  { (/) } )  -> 
k  e.  { (/) } )
25 elsni 3985 . . . . . . . 8  |-  ( k  e.  { (/) }  ->  k  =  (/) )
2624, 25syl 17 . . . . . . 7  |-  ( k  e.  ( ran  G  i^i  { (/) } )  -> 
k  =  (/) )
2723, 26syl 17 . . . . . 6  |-  ( k  e.  ( ran  G  \  ( ran  G  \  { (/) } ) )  ->  k  =  (/) )
2827adantl 473 . . . . 5  |-  ( (
ph  /\  k  e.  ( ran  G  \  ( ran  G  \  { (/) } ) ) )  -> 
k  =  (/) )
29 simpr 468 . . . . . . 7  |-  ( (
ph  /\  k  =  (/) )  ->  k  =  (/) )
3029fveq2d 5883 . . . . . 6  |-  ( (
ph  /\  k  =  (/) )  ->  ( M `  k )  =  ( M `  (/) ) )
319mea0 38408 . . . . . . 7  |-  ( ph  ->  ( M `  (/) )  =  0 )
3231adantr 472 . . . . . 6  |-  ( (
ph  /\  k  =  (/) )  ->  ( M `  (/) )  =  0 )
3330, 32eqtrd 2505 . . . . 5  |-  ( (
ph  /\  k  =  (/) )  ->  ( M `  k )  =  0 )
3419, 28, 33syl2anc 673 . . . 4  |-  ( (
ph  /\  k  e.  ( ran  G  \  ( ran  G  \  { (/) } ) ) )  -> 
( M `  k
)  =  0 )
351, 7, 8, 18, 34sge0ss 38368 . . 3  |-  ( ph  ->  (Σ^ `  ( k  e.  ( ran  G  \  { (/)
} )  |->  ( M `
 k ) ) )  =  (Σ^ `  ( k  e.  ran  G 
|->  ( M `  k
) ) ) )
3635eqcomd 2477 . 2  |-  ( ph  ->  (Σ^ `  ( k  e.  ran  G 
|->  ( M `  k
) ) )  =  (Σ^ `  ( k  e.  ( ran  G  \  { (/)
} )  |->  ( M `
 k ) ) ) )
3711, 14feqresmpt 5933 . . 3  |-  ( ph  ->  ( M  |`  ran  G
)  =  ( k  e.  ran  G  |->  ( M `  k ) ) )
3837fveq2d 5883 . 2  |-  ( ph  ->  (Σ^ `  ( M  |`  ran  G
) )  =  (Σ^ `  (
k  e.  ran  G  |->  ( M `  k
) ) ) )
392ffvelrnda 6037 . . . . 5  |-  ( (
ph  /\  j  e.  X )  ->  ( G `  j )  e.  S )
402feqmptd 5932 . . . . 5  |-  ( ph  ->  G  =  ( j  e.  X  |->  ( G `
 j ) ) )
4111feqmptd 5932 . . . . 5  |-  ( ph  ->  M  =  ( k  e.  S  |->  ( M `
 k ) ) )
42 fveq2 5879 . . . . 5  |-  ( k  =  ( G `  j )  ->  ( M `  k )  =  ( M `  ( G `  j ) ) )
4339, 40, 41, 42fmptco 6072 . . . 4  |-  ( ph  ->  ( M  o.  G
)  =  ( j  e.  X  |->  ( M `
 ( G `  j ) ) ) )
4443fveq2d 5883 . . 3  |-  ( ph  ->  (Σ^ `  ( M  o.  G
) )  =  (Σ^ `  (
j  e.  X  |->  ( M `  ( G `
 j ) ) ) ) )
45 nfv 1769 . . . . 5  |-  F/ j
ph
46 meadjiunlem.y . . . . . 6  |-  Y  =  { i  e.  X  |  ( G `  i )  =/=  (/) }
47 ssrab2 3500 . . . . . . 7  |-  { i  e.  X  |  ( G `  i )  =/=  (/) }  C_  X
4847a1i 11 . . . . . 6  |-  ( ph  ->  { i  e.  X  |  ( G `  i )  =/=  (/) }  C_  X )
4946, 48syl5eqss 3462 . . . . 5  |-  ( ph  ->  Y  C_  X )
5011adantr 472 . . . . . 6  |-  ( (
ph  /\  j  e.  Y )  ->  M : S --> ( 0 [,] +oo ) )
512adantr 472 . . . . . . 7  |-  ( (
ph  /\  j  e.  Y )  ->  G : X --> S )
5249sselda 3418 . . . . . . 7  |-  ( (
ph  /\  j  e.  Y )  ->  j  e.  X )
5351, 52ffvelrnd 6038 . . . . . 6  |-  ( (
ph  /\  j  e.  Y )  ->  ( G `  j )  e.  S )
5450, 53ffvelrnd 6038 . . . . 5  |-  ( (
ph  /\  j  e.  Y )  ->  ( M `  ( G `  j ) )  e.  ( 0 [,] +oo ) )
55 eldifi 3544 . . . . . . . . . . 11  |-  ( j  e.  ( X  \  Y )  ->  j  e.  X )
5655ad2antlr 741 . . . . . . . . . 10  |-  ( ( ( ph  /\  j  e.  ( X  \  Y
) )  /\  ( M `  ( G `  j ) )  =/=  0 )  ->  j  e.  X )
57 fveq2 5879 . . . . . . . . . . . . . . 15  |-  ( ( G `  j )  =  (/)  ->  ( M `
 ( G `  j ) )  =  ( M `  (/) ) )
5857adantl 473 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( G `  j )  =  (/) )  ->  ( M `  ( G `  j ) )  =  ( M `
 (/) ) )
599adantr 472 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( G `  j )  =  (/) )  ->  M  e. Meas )
6059mea0 38408 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( G `  j )  =  (/) )  ->  ( M `  (/) )  =  0 )
6158, 60eqtrd 2505 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( G `  j )  =  (/) )  ->  ( M `  ( G `  j ) )  =  0 )
6261ad4ant14 1259 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  j  e.  ( X  \  Y ) )  /\  ( M `  ( G `
 j ) )  =/=  0 )  /\  ( G `  j )  =  (/) )  ->  ( M `  ( G `  j ) )  =  0 )
63 neneq 2649 . . . . . . . . . . . . 13  |-  ( ( M `  ( G `
 j ) )  =/=  0  ->  -.  ( M `  ( G `
 j ) )  =  0 )
6463ad2antlr 741 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  j  e.  ( X  \  Y ) )  /\  ( M `  ( G `
 j ) )  =/=  0 )  /\  ( G `  j )  =  (/) )  ->  -.  ( M `  ( G `
 j ) )  =  0 )
6562, 64pm2.65da 586 . . . . . . . . . . 11  |-  ( ( ( ph  /\  j  e.  ( X  \  Y
) )  /\  ( M `  ( G `  j ) )  =/=  0 )  ->  -.  ( G `  j )  =  (/) )
6665neqned 2650 . . . . . . . . . 10  |-  ( ( ( ph  /\  j  e.  ( X  \  Y
) )  /\  ( M `  ( G `  j ) )  =/=  0 )  ->  ( G `  j )  =/=  (/) )
6756, 66jca 541 . . . . . . . . 9  |-  ( ( ( ph  /\  j  e.  ( X  \  Y
) )  /\  ( M `  ( G `  j ) )  =/=  0 )  ->  (
j  e.  X  /\  ( G `  j )  =/=  (/) ) )
68 fveq2 5879 . . . . . . . . . . 11  |-  ( i  =  j  ->  ( G `  i )  =  ( G `  j ) )
6968neeq1d 2702 . . . . . . . . . 10  |-  ( i  =  j  ->  (
( G `  i
)  =/=  (/)  <->  ( G `  j )  =/=  (/) ) )
7069elrab 3184 . . . . . . . . 9  |-  ( j  e.  { i  e.  X  |  ( G `
 i )  =/=  (/) }  <->  ( j  e.  X  /\  ( G `
 j )  =/=  (/) ) )
7167, 70sylibr 217 . . . . . . . 8  |-  ( ( ( ph  /\  j  e.  ( X  \  Y
) )  /\  ( M `  ( G `  j ) )  =/=  0 )  ->  j  e.  { i  e.  X  |  ( G `  i )  =/=  (/) } )
7271, 46syl6eleqr 2560 . . . . . . 7  |-  ( ( ( ph  /\  j  e.  ( X  \  Y
) )  /\  ( M `  ( G `  j ) )  =/=  0 )  ->  j  e.  Y )
73 eldifn 3545 . . . . . . . 8  |-  ( j  e.  ( X  \  Y )  ->  -.  j  e.  Y )
7473ad2antlr 741 . . . . . . 7  |-  ( ( ( ph  /\  j  e.  ( X  \  Y
) )  /\  ( M `  ( G `  j ) )  =/=  0 )  ->  -.  j  e.  Y )
7572, 74pm2.65da 586 . . . . . 6  |-  ( (
ph  /\  j  e.  ( X  \  Y ) )  ->  -.  ( M `  ( G `  j ) )  =/=  0 )
76 nne 2647 . . . . . 6  |-  ( -.  ( M `  ( G `  j )
)  =/=  0  <->  ( M `  ( G `  j ) )  =  0 )
7775, 76sylib 201 . . . . 5  |-  ( (
ph  /\  j  e.  ( X  \  Y ) )  ->  ( M `  ( G `  j
) )  =  0 )
7845, 3, 49, 54, 77sge0ss 38368 . . . 4  |-  ( ph  ->  (Σ^ `  ( j  e.  Y  |->  ( M `  ( G `  j )
) ) )  =  (Σ^ `  ( j  e.  X  |->  ( M `  ( G `  j )
) ) ) )
7978eqcomd 2477 . . 3  |-  ( ph  ->  (Σ^ `  ( j  e.  X  |->  ( M `  ( G `  j )
) ) )  =  (Σ^ `  ( j  e.  Y  |->  ( M `  ( G `  j )
) ) ) )
803, 49ssexd 4543 . . . . 5  |-  ( ph  ->  Y  e.  _V )
81 nfv 1769 . . . . . . . . 9  |-  F/ i
ph
82 eqid 2471 . . . . . . . . 9  |-  ( i  e.  Y  |->  ( G `
 i ) )  =  ( i  e.  Y  |->  ( G `  i ) )
832ffnd 5740 . . . . . . . . . . . . 13  |-  ( ph  ->  G  Fn  X )
84 dffn3 5748 . . . . . . . . . . . . 13  |-  ( G  Fn  X  <->  G : X
--> ran  G )
8583, 84sylib 201 . . . . . . . . . . . 12  |-  ( ph  ->  G : X --> ran  G
)
8685adantr 472 . . . . . . . . . . 11  |-  ( (
ph  /\  i  e.  Y )  ->  G : X --> ran  G )
8749sselda 3418 . . . . . . . . . . 11  |-  ( (
ph  /\  i  e.  Y )  ->  i  e.  X )
8886, 87ffvelrnd 6038 . . . . . . . . . 10  |-  ( (
ph  /\  i  e.  Y )  ->  ( G `  i )  e.  ran  G )
8946eleq2i 2541 . . . . . . . . . . . . . . 15  |-  ( i  e.  Y  <->  i  e.  { i  e.  X  | 
( G `  i
)  =/=  (/) } )
90 rabid 2953 . . . . . . . . . . . . . . 15  |-  ( i  e.  { i  e.  X  |  ( G `
 i )  =/=  (/) }  <->  ( i  e.  X  /\  ( G `
 i )  =/=  (/) ) )
9189, 90bitri 257 . . . . . . . . . . . . . 14  |-  ( i  e.  Y  <->  ( i  e.  X  /\  ( G `  i )  =/=  (/) ) )
9291biimpi 199 . . . . . . . . . . . . 13  |-  ( i  e.  Y  ->  (
i  e.  X  /\  ( G `  i )  =/=  (/) ) )
9392simprd 470 . . . . . . . . . . . 12  |-  ( i  e.  Y  ->  ( G `  i )  =/=  (/) )
9493adantl 473 . . . . . . . . . . 11  |-  ( (
ph  /\  i  e.  Y )  ->  ( G `  i )  =/=  (/) )
95 nelsn 3992 . . . . . . . . . . 11  |-  ( ( G `  i )  =/=  (/)  ->  -.  ( G `  i )  e.  { (/) } )
9694, 95syl 17 . . . . . . . . . 10  |-  ( (
ph  /\  i  e.  Y )  ->  -.  ( G `  i )  e.  { (/) } )
9788, 96eldifd 3401 . . . . . . . . 9  |-  ( (
ph  /\  i  e.  Y )  ->  ( G `  i )  e.  ( ran  G  \  { (/) } ) )
98 meadjiunlem.dj . . . . . . . . . 10  |-  ( ph  -> Disj  i  e.  X  ( G `  i )
)
99 disjss1 4372 . . . . . . . . . 10  |-  ( Y 
C_  X  ->  (Disj  i  e.  X  ( G `
 i )  -> Disj  i  e.  Y  ( G `
 i ) ) )
10049, 98, 99sylc 61 . . . . . . . . 9  |-  ( ph  -> Disj  i  e.  Y  ( G `  i )
)
10181, 82, 97, 94, 100disjf1 37528 . . . . . . . 8  |-  ( ph  ->  ( i  e.  Y  |->  ( G `  i
) ) : Y -1-1-> ( ran  G  \  { (/)
} ) )
1022, 49feqresmpt 5933 . . . . . . . . 9  |-  ( ph  ->  ( G  |`  Y )  =  ( i  e.  Y  |->  ( G `  i ) ) )
103 f1eq1 5787 . . . . . . . . 9  |-  ( ( G  |`  Y )  =  ( i  e.  Y  |->  ( G `  i ) )  -> 
( ( G  |`  Y ) : Y -1-1-> ( ran  G  \  { (/)
} )  <->  ( i  e.  Y  |->  ( G `
 i ) ) : Y -1-1-> ( ran 
G  \  { (/) } ) ) )
104102, 103syl 17 . . . . . . . 8  |-  ( ph  ->  ( ( G  |`  Y ) : Y -1-1-> ( ran  G  \  { (/)
} )  <->  ( i  e.  Y  |->  ( G `
 i ) ) : Y -1-1-> ( ran 
G  \  { (/) } ) ) )
105101, 104mpbird 240 . . . . . . 7  |-  ( ph  ->  ( G  |`  Y ) : Y -1-1-> ( ran 
G  \  { (/) } ) )
106102rneqd 5068 . . . . . . . . 9  |-  ( ph  ->  ran  ( G  |`  Y )  =  ran  ( i  e.  Y  |->  ( G `  i
) ) )
10797ralrimiva 2809 . . . . . . . . . 10  |-  ( ph  ->  A. i  e.  Y  ( G `  i )  e.  ( ran  G  \  { (/) } ) )
10882rnmptss 6068 . . . . . . . . . 10  |-  ( A. i  e.  Y  ( G `  i )  e.  ( ran  G  \  { (/) } )  ->  ran  ( i  e.  Y  |->  ( G `  i
) )  C_  ( ran  G  \  { (/) } ) )
109107, 108syl 17 . . . . . . . . 9  |-  ( ph  ->  ran  ( i  e.  Y  |->  ( G `  i ) )  C_  ( ran  G  \  { (/)
} ) )
110106, 109eqsstrd 3452 . . . . . . . 8  |-  ( ph  ->  ran  ( G  |`  Y )  C_  ( ran  G  \  { (/) } ) )
111 simpl 464 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( ran  G  \  { (/)
} ) )  ->  ph )
112 eldifi 3544 . . . . . . . . . . . 12  |-  ( x  e.  ( ran  G  \  { (/) } )  ->  x  e.  ran  G )
113112adantl 473 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( ran  G  \  { (/)
} ) )  ->  x  e.  ran  G )
114 eldifsni 4089 . . . . . . . . . . . 12  |-  ( x  e.  ( ran  G  \  { (/) } )  ->  x  =/=  (/) )
115114adantl 473 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( ran  G  \  { (/)
} ) )  ->  x  =/=  (/) )
116 simpr 468 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  ran  G )  ->  x  e.  ran  G )
117 fvelrnb 5926 . . . . . . . . . . . . . . . 16  |-  ( G  Fn  X  ->  (
x  e.  ran  G  <->  E. i  e.  X  ( G `  i )  =  x ) )
11883, 117syl 17 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( x  e.  ran  G  <->  E. i  e.  X  ( G `  i )  =  x ) )
119118adantr 472 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  ran  G )  ->  (
x  e.  ran  G  <->  E. i  e.  X  ( G `  i )  =  x ) )
120116, 119mpbid 215 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ran  G )  ->  E. i  e.  X  ( G `  i )  =  x )
1211203adant3 1050 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ran  G  /\  x  =/=  (/) )  ->  E. i  e.  X  ( G `  i )  =  x )
122 id 22 . . . . . . . . . . . . . . . . . 18  |-  ( ( G `  i )  =  x  ->  ( G `  i )  =  x )
123122eqcomd 2477 . . . . . . . . . . . . . . . . 17  |-  ( ( G `  i )  =  x  ->  x  =  ( G `  i ) )
1241233ad2ant3 1053 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  =/=  (/) )  /\  i  e.  X  /\  ( G `  i )  =  x )  ->  x  =  ( G `  i ) )
125 simp1l 1054 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  x  =/=  (/) )  /\  i  e.  X  /\  ( G `  i )  =  x )  ->  ph )
126 simp2 1031 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  x  =/=  (/) )  /\  i  e.  X  /\  ( G `  i )  =  x )  ->  i  e.  X )
127 simpr 468 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( x  =/=  (/)  /\  ( G `  i )  =  x )  ->  ( G `  i )  =  x )
128 simpl 464 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( x  =/=  (/)  /\  ( G `  i )  =  x )  ->  x  =/=  (/) )
129127, 128eqnetrd 2710 . . . . . . . . . . . . . . . . . . 19  |-  ( ( x  =/=  (/)  /\  ( G `  i )  =  x )  ->  ( G `  i )  =/=  (/) )
130129adantll 728 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  x  =/=  (/) )  /\  ( G `  i )  =  x )  ->  ( G `  i )  =/=  (/) )
1311303adant2 1049 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  x  =/=  (/) )  /\  i  e.  X  /\  ( G `  i )  =  x )  ->  ( G `  i )  =/=  (/) )
13291biimpri 211 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( i  e.  X  /\  ( G `  i )  =/=  (/) )  ->  i  e.  Y )
133 fvex 5889 . . . . . . . . . . . . . . . . . . . . 21  |-  ( G `
 i )  e. 
_V
134133a1i 11 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( i  e.  X  /\  ( G `  i )  =/=  (/) )  ->  ( G `  i )  e.  _V )
13582elrnmpt1 5089 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( i  e.  Y  /\  ( G `  i )  e.  _V )  -> 
( G `  i
)  e.  ran  (
i  e.  Y  |->  ( G `  i ) ) )
136132, 134, 135syl2anc 673 . . . . . . . . . . . . . . . . . . 19  |-  ( ( i  e.  X  /\  ( G `  i )  =/=  (/) )  ->  ( G `  i )  e.  ran  ( i  e.  Y  |->  ( G `  i ) ) )
1371363adant1 1048 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  i  e.  X  /\  ( G `  i )  =/=  (/) )  -> 
( G `  i
)  e.  ran  (
i  e.  Y  |->  ( G `  i ) ) )
138106eqcomd 2477 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ran  ( i  e.  Y  |->  ( G `  i ) )  =  ran  ( G  |`  Y ) )
1391383ad2ant1 1051 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  i  e.  X  /\  ( G `  i )  =/=  (/) )  ->  ran  ( i  e.  Y  |->  ( G `  i
) )  =  ran  ( G  |`  Y ) )
140137, 139eleqtrd 2551 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  i  e.  X  /\  ( G `  i )  =/=  (/) )  -> 
( G `  i
)  e.  ran  ( G  |`  Y ) )
141125, 126, 131, 140syl3anc 1292 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  =/=  (/) )  /\  i  e.  X  /\  ( G `  i )  =  x )  ->  ( G `  i )  e.  ran  ( G  |`  Y ) )
142124, 141eqeltrd 2549 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  =/=  (/) )  /\  i  e.  X  /\  ( G `  i )  =  x )  ->  x  e.  ran  ( G  |`  Y ) )
1431423exp 1230 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  =/=  (/) )  ->  ( i  e.  X  ->  ( ( G `  i )  =  x  ->  x  e.  ran  ( G  |`  Y ) ) ) )
144143rexlimdv 2870 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  =/=  (/) )  ->  ( E. i  e.  X  ( G `  i )  =  x  ->  x  e. 
ran  ( G  |`  Y ) ) )
1451443adant2 1049 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ran  G  /\  x  =/=  (/) )  ->  ( E. i  e.  X  ( G `  i )  =  x  ->  x  e.  ran  ( G  |`  Y ) ) )
146121, 145mpd 15 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ran  G  /\  x  =/=  (/) )  ->  x  e. 
ran  ( G  |`  Y ) )
147111, 113, 115, 146syl3anc 1292 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( ran  G  \  { (/)
} ) )  ->  x  e.  ran  ( G  |`  Y ) )
148147ralrimiva 2809 . . . . . . . . 9  |-  ( ph  ->  A. x  e.  ( ran  G  \  { (/)
} ) x  e. 
ran  ( G  |`  Y ) )
149 dfss3 3408 . . . . . . . . 9  |-  ( ( ran  G  \  { (/)
} )  C_  ran  ( G  |`  Y )  <->  A. x  e.  ( ran  G  \  { (/) } ) x  e.  ran  ( G  |`  Y ) )
150148, 149sylibr 217 . . . . . . . 8  |-  ( ph  ->  ( ran  G  \  { (/) } )  C_  ran  ( G  |`  Y ) )
151110, 150eqssd 3435 . . . . . . 7  |-  ( ph  ->  ran  ( G  |`  Y )  =  ( ran  G  \  { (/)
} ) )
152105, 151jca 541 . . . . . 6  |-  ( ph  ->  ( ( G  |`  Y ) : Y -1-1-> ( ran  G  \  { (/)
} )  /\  ran  ( G  |`  Y )  =  ( ran  G  \  { (/) } ) ) )
153 dff1o5 5837 . . . . . 6  |-  ( ( G  |`  Y ) : Y -1-1-onto-> ( ran  G  \  { (/) } )  <->  ( ( G  |`  Y ) : Y -1-1-> ( ran  G  \  { (/) } )  /\  ran  ( G  |`  Y )  =  ( ran  G  \  { (/) } ) ) )
154152, 153sylibr 217 . . . . 5  |-  ( ph  ->  ( G  |`  Y ) : Y -1-1-onto-> ( ran  G  \  { (/) } ) )
155 fvres 5893 . . . . . 6  |-  ( j  e.  Y  ->  (
( G  |`  Y ) `
 j )  =  ( G `  j
) )
156155adantl 473 . . . . 5  |-  ( (
ph  /\  j  e.  Y )  ->  (
( G  |`  Y ) `
 j )  =  ( G `  j
) )
1571, 45, 42, 80, 154, 156, 18sge0f1o 38338 . . . 4  |-  ( ph  ->  (Σ^ `  ( k  e.  ( ran  G  \  { (/)
} )  |->  ( M `
 k ) ) )  =  (Σ^ `  ( j  e.  Y  |->  ( M `  ( G `  j )
) ) ) )
158157eqcomd 2477 . . 3  |-  ( ph  ->  (Σ^ `  ( j  e.  Y  |->  ( M `  ( G `  j )
) ) )  =  (Σ^ `  ( k  e.  ( ran  G  \  { (/)
} )  |->  ( M `
 k ) ) ) )
15944, 79, 1583eqtrd 2509 . 2  |-  ( ph  ->  (Σ^ `  ( M  o.  G
) )  =  (Σ^ `  (
k  e.  ( ran 
G  \  { (/) } ) 
|->  ( M `  k
) ) ) )
16036, 38, 1593eqtr4d 2515 1  |-  ( ph  ->  (Σ^ `  ( M  |`  ran  G
) )  =  (Σ^ `  ( M  o.  G )
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   A.wral 2756   E.wrex 2757   {crab 2760   _Vcvv 3031    \ cdif 3387    i^i cin 3389    C_ wss 3390   (/)c0 3722   {csn 3959  Disj wdisj 4366    |-> cmpt 4454   dom cdm 4839   ran crn 4840    |` cres 4841    o. ccom 4843    Fn wfn 5584   -->wf 5585   -1-1->wf1 5586   -1-1-onto->wf1o 5588   ` cfv 5589  (class class class)co 6308   0cc0 9557   +oocpnf 9690   [,]cicc 11663  Σ^csumge0 38318  Meascmea 38403
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-disj 4367  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-oi 8043  df-card 8391  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-rp 11326  df-xadd 11433  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-clim 13629  df-sum 13830  df-sumge0 38319  df-mea 38404
This theorem is referenced by:  meadjiun  38420
  Copyright terms: Public domain W3C validator