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Theorem meadjiunlem 38082
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 1751 . . . 4  |-  F/ k
ph
2 meadjiunlem.g . . . . . 6  |-  ( ph  ->  G : X --> S )
3 meadjiunlem.x . . . . . 6  |-  ( ph  ->  X  e.  V )
42, 3jca 534 . . . . 5  |-  ( ph  ->  ( G : X --> S  /\  X  e.  V
) )
5 fex 6150 . . . . 5  |-  ( ( G : X --> S  /\  X  e.  V )  ->  G  e.  _V )
6 rnexg 6736 . . . . 5  |-  ( G  e.  _V  ->  ran  G  e.  _V )
74, 5, 63syl 18 . . . 4  |-  ( ph  ->  ran  G  e.  _V )
8 difssd 3593 . . . 4  |-  ( ph  ->  ( ran  G  \  { (/) } )  C_  ran  G )
9 meadjiunlem.f . . . . . . 7  |-  ( ph  ->  M  e. Meas )
10 meadjiunlem.3 . . . . . . 7  |-  S  =  dom  M
119, 10meaf 38070 . . . . . 6  |-  ( ph  ->  M : S --> ( 0 [,] +oo ) )
1211adantr 466 . . . . 5  |-  ( (
ph  /\  k  e.  ( ran  G  \  { (/)
} ) )  ->  M : S --> ( 0 [,] +oo ) )
13 frn 5749 . . . . . . . 8  |-  ( G : X --> S  ->  ran  G  C_  S )
142, 13syl 17 . . . . . . 7  |-  ( ph  ->  ran  G  C_  S
)
1514adantr 466 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ran  G  \  { (/)
} ) )  ->  ran  G  C_  S )
168sselda 3464 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ran  G  \  { (/)
} ) )  -> 
k  e.  ran  G
)
1715, 16sseldd 3465 . . . . 5  |-  ( (
ph  /\  k  e.  ( ran  G  \  { (/)
} ) )  -> 
k  e.  S )
1812, 17ffvelrnd 6035 . . . 4  |-  ( (
ph  /\  k  e.  ( ran  G  \  { (/)
} ) )  -> 
( M `  k
)  e.  ( 0 [,] +oo ) )
19 simpl 458 . . . . 5  |-  ( (
ph  /\  k  e.  ( ran  G  \  ( ran  G  \  { (/) } ) ) )  ->  ph )
20 id 23 . . . . . . . 8  |-  ( k  e.  ( ran  G  \  ( ran  G  \  { (/) } ) )  ->  k  e.  ( ran  G  \  ( ran  G  \  { (/) } ) ) )
21 dfin4 3713 . . . . . . . . 9  |-  ( ran 
G  i^i  { (/) } )  =  ( ran  G  \  ( ran  G  \  { (/) } ) )
2221eqcomi 2435 . . . . . . . 8  |-  ( ran 
G  \  ( ran  G 
\  { (/) } ) )  =  ( ran 
G  i^i  { (/) } )
2320, 22syl6eleq 2520 . . . . . . 7  |-  ( k  e.  ( ran  G  \  ( ran  G  \  { (/) } ) )  ->  k  e.  ( ran  G  i^i  { (/)
} ) )
24 elinel2 3652 . . . . . . . 8  |-  ( k  e.  ( ran  G  i^i  { (/) } )  -> 
k  e.  { (/) } )
25 elsni 4021 . . . . . . . 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 467 . . . . 5  |-  ( (
ph  /\  k  e.  ( ran  G  \  ( ran  G  \  { (/) } ) ) )  -> 
k  =  (/) )
29 simpr 462 . . . . . . 7  |-  ( (
ph  /\  k  =  (/) )  ->  k  =  (/) )
3029fveq2d 5882 . . . . . 6  |-  ( (
ph  /\  k  =  (/) )  ->  ( M `  k )  =  ( M `  (/) ) )
319mea0 38071 . . . . . . 7  |-  ( ph  ->  ( M `  (/) )  =  0 )
3231adantr 466 . . . . . 6  |-  ( (
ph  /\  k  =  (/) )  ->  ( M `  (/) )  =  0 )
3330, 32eqtrd 2463 . . . . 5  |-  ( (
ph  /\  k  =  (/) )  ->  ( M `  k )  =  0 )
3419, 28, 33syl2anc 665 . . . 4  |-  ( (
ph  /\  k  e.  ( ran  G  \  ( ran  G  \  { (/) } ) ) )  -> 
( M `  k
)  =  0 )
351, 7, 8, 18, 34sge0ss 38042 . . 3  |-  ( ph  ->  (Σ^ `  ( k  e.  ( ran  G  \  { (/)
} )  |->  ( M `
 k ) ) )  =  (Σ^ `  ( k  e.  ran  G 
|->  ( M `  k
) ) ) )
3635eqcomd 2430 . 2  |-  ( ph  ->  (Σ^ `  ( k  e.  ran  G 
|->  ( M `  k
) ) )  =  (Σ^ `  ( k  e.  ( ran  G  \  { (/)
} )  |->  ( M `
 k ) ) ) )
3711, 14feqresmpt 5932 . . 3  |-  ( ph  ->  ( M  |`  ran  G
)  =  ( k  e.  ran  G  |->  ( M `  k ) ) )
3837fveq2d 5882 . 2  |-  ( ph  ->  (Σ^ `  ( M  |`  ran  G
) )  =  (Σ^ `  (
k  e.  ran  G  |->  ( M `  k
) ) ) )
392ffvelrnda 6034 . . . . 5  |-  ( (
ph  /\  j  e.  X )  ->  ( G `  j )  e.  S )
402feqmptd 5931 . . . . 5  |-  ( ph  ->  G  =  ( j  e.  X  |->  ( G `
 j ) ) )
4111feqmptd 5931 . . . . 5  |-  ( ph  ->  M  =  ( k  e.  S  |->  ( M `
 k ) ) )
42 fveq2 5878 . . . . 5  |-  ( k  =  ( G `  j )  ->  ( M `  k )  =  ( M `  ( G `  j ) ) )
4339, 40, 41, 42fmptco 6068 . . . 4  |-  ( ph  ->  ( M  o.  G
)  =  ( j  e.  X  |->  ( M `
 ( G `  j ) ) ) )
4443fveq2d 5882 . . 3  |-  ( ph  ->  (Σ^ `  ( M  o.  G
) )  =  (Σ^ `  (
j  e.  X  |->  ( M `  ( G `
 j ) ) ) ) )
45 nfv 1751 . . . . 5  |-  F/ j
ph
46 meadjiunlem.y . . . . . 6  |-  Y  =  { i  e.  X  |  ( G `  i )  =/=  (/) }
47 ssrab2 3546 . . . . . . 7  |-  { i  e.  X  |  ( G `  i )  =/=  (/) }  C_  X
4847a1i 11 . . . . . 6  |-  ( ph  ->  { i  e.  X  |  ( G `  i )  =/=  (/) }  C_  X )
4946, 48syl5eqss 3508 . . . . 5  |-  ( ph  ->  Y  C_  X )
5011adantr 466 . . . . . 6  |-  ( (
ph  /\  j  e.  Y )  ->  M : S --> ( 0 [,] +oo ) )
512adantr 466 . . . . . . 7  |-  ( (
ph  /\  j  e.  Y )  ->  G : X --> S )
5249sselda 3464 . . . . . . 7  |-  ( (
ph  /\  j  e.  Y )  ->  j  e.  X )
5351, 52ffvelrnd 6035 . . . . . 6  |-  ( (
ph  /\  j  e.  Y )  ->  ( G `  j )  e.  S )
5450, 53ffvelrnd 6035 . . . . 5  |-  ( (
ph  /\  j  e.  Y )  ->  ( M `  ( G `  j ) )  e.  ( 0 [,] +oo ) )
55 eldifi 3587 . . . . . . . . . . 11  |-  ( j  e.  ( X  \  Y )  ->  j  e.  X )
5655ad2antlr 731 . . . . . . . . . 10  |-  ( ( ( ph  /\  j  e.  ( X  \  Y
) )  /\  ( M `  ( G `  j ) )  =/=  0 )  ->  j  e.  X )
57 fveq2 5878 . . . . . . . . . . . . . . 15  |-  ( ( G `  j )  =  (/)  ->  ( M `
 ( G `  j ) )  =  ( M `  (/) ) )
5857adantl 467 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( G `  j )  =  (/) )  ->  ( M `  ( G `  j ) )  =  ( M `
 (/) ) )
599adantr 466 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( G `  j )  =  (/) )  ->  M  e. Meas )
6059mea0 38071 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( G `  j )  =  (/) )  ->  ( M `  (/) )  =  0 )
6158, 60eqtrd 2463 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( G `  j )  =  (/) )  ->  ( M `  ( G `  j ) )  =  0 )
6261ad4ant14 1231 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  j  e.  ( X  \  Y ) )  /\  ( M `  ( G `
 j ) )  =/=  0 )  /\  ( G `  j )  =  (/) )  ->  ( M `  ( G `  j ) )  =  0 )
63 neneq 2626 . . . . . . . . . . . . 13  |-  ( ( M `  ( G `
 j ) )  =/=  0  ->  -.  ( M `  ( G `
 j ) )  =  0 )
6463ad2antlr 731 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  j  e.  ( X  \  Y ) )  /\  ( M `  ( G `
 j ) )  =/=  0 )  /\  ( G `  j )  =  (/) )  ->  -.  ( M `  ( G `
 j ) )  =  0 )
6562, 64pm2.65da 578 . . . . . . . . . . 11  |-  ( ( ( ph  /\  j  e.  ( X  \  Y
) )  /\  ( M `  ( G `  j ) )  =/=  0 )  ->  -.  ( G `  j )  =  (/) )
6665neqned 2627 . . . . . . . . . 10  |-  ( ( ( ph  /\  j  e.  ( X  \  Y
) )  /\  ( M `  ( G `  j ) )  =/=  0 )  ->  ( G `  j )  =/=  (/) )
6756, 66jca 534 . . . . . . . . 9  |-  ( ( ( ph  /\  j  e.  ( X  \  Y
) )  /\  ( M `  ( G `  j ) )  =/=  0 )  ->  (
j  e.  X  /\  ( G `  j )  =/=  (/) ) )
68 fveq2 5878 . . . . . . . . . . 11  |-  ( i  =  j  ->  ( G `  i )  =  ( G `  j ) )
6968neeq1d 2701 . . . . . . . . . 10  |-  ( i  =  j  ->  (
( G `  i
)  =/=  (/)  <->  ( G `  j )  =/=  (/) ) )
7069elrab 3229 . . . . . . . . 9  |-  ( j  e.  { i  e.  X  |  ( G `
 i )  =/=  (/) }  <->  ( j  e.  X  /\  ( G `
 j )  =/=  (/) ) )
7167, 70sylibr 215 . . . . . . . 8  |-  ( ( ( ph  /\  j  e.  ( X  \  Y
) )  /\  ( M `  ( G `  j ) )  =/=  0 )  ->  j  e.  { i  e.  X  |  ( G `  i )  =/=  (/) } )
7271, 46syl6eleqr 2521 . . . . . . 7  |-  ( ( ( ph  /\  j  e.  ( X  \  Y
) )  /\  ( M `  ( G `  j ) )  =/=  0 )  ->  j  e.  Y )
73 eldifn 3588 . . . . . . . 8  |-  ( j  e.  ( X  \  Y )  ->  -.  j  e.  Y )
7473ad2antlr 731 . . . . . . 7  |-  ( ( ( ph  /\  j  e.  ( X  \  Y
) )  /\  ( M `  ( G `  j ) )  =/=  0 )  ->  -.  j  e.  Y )
7572, 74pm2.65da 578 . . . . . 6  |-  ( (
ph  /\  j  e.  ( X  \  Y ) )  ->  -.  ( M `  ( G `  j ) )  =/=  0 )
76 nne 2624 . . . . . 6  |-  ( -.  ( M `  ( G `  j )
)  =/=  0  <->  ( M `  ( G `  j ) )  =  0 )
7775, 76sylib 199 . . . . 5  |-  ( (
ph  /\  j  e.  ( X  \  Y ) )  ->  ( M `  ( G `  j
) )  =  0 )
7845, 3, 49, 54, 77sge0ss 38042 . . . 4  |-  ( ph  ->  (Σ^ `  ( j  e.  Y  |->  ( M `  ( G `  j )
) ) )  =  (Σ^ `  ( j  e.  X  |->  ( M `  ( G `  j )
) ) ) )
7978eqcomd 2430 . . 3  |-  ( ph  ->  (Σ^ `  ( j  e.  X  |->  ( M `  ( G `  j )
) ) )  =  (Σ^ `  ( j  e.  Y  |->  ( M `  ( G `  j )
) ) ) )
803, 49ssexd 4568 . . . . 5  |-  ( ph  ->  Y  e.  _V )
81 nfv 1751 . . . . . . . . 9  |-  F/ i
ph
82 eqid 2422 . . . . . . . . 9  |-  ( i  e.  Y  |->  ( G `
 i ) )  =  ( i  e.  Y  |->  ( G `  i ) )
832ffnd 37294 . . . . . . . . . . . . 13  |-  ( ph  ->  G  Fn  X )
84 dffn3 5750 . . . . . . . . . . . . 13  |-  ( G  Fn  X  <->  G : X
--> ran  G )
8583, 84sylib 199 . . . . . . . . . . . 12  |-  ( ph  ->  G : X --> ran  G
)
8685adantr 466 . . . . . . . . . . 11  |-  ( (
ph  /\  i  e.  Y )  ->  G : X --> ran  G )
8749sselda 3464 . . . . . . . . . . 11  |-  ( (
ph  /\  i  e.  Y )  ->  i  e.  X )
8886, 87ffvelrnd 6035 . . . . . . . . . 10  |-  ( (
ph  /\  i  e.  Y )  ->  ( G `  i )  e.  ran  G )
8946eleq2i 2500 . . . . . . . . . . . . . . 15  |-  ( i  e.  Y  <->  i  e.  { i  e.  X  | 
( G `  i
)  =/=  (/) } )
90 rabid 3005 . . . . . . . . . . . . . . 15  |-  ( i  e.  { i  e.  X  |  ( G `
 i )  =/=  (/) }  <->  ( i  e.  X  /\  ( G `
 i )  =/=  (/) ) )
9189, 90bitri 252 . . . . . . . . . . . . . 14  |-  ( i  e.  Y  <->  ( i  e.  X  /\  ( G `  i )  =/=  (/) ) )
9291biimpi 197 . . . . . . . . . . . . 13  |-  ( i  e.  Y  ->  (
i  e.  X  /\  ( G `  i )  =/=  (/) ) )
9392simprd 464 . . . . . . . . . . . 12  |-  ( i  e.  Y  ->  ( G `  i )  =/=  (/) )
9493adantl 467 . . . . . . . . . . 11  |-  ( (
ph  /\  i  e.  Y )  ->  ( G `  i )  =/=  (/) )
95 nelsn 37263 . . . . . . . . . . 11  |-  ( ( G `  i )  =/=  (/)  ->  -.  ( G `  i )  e.  { (/) } )
9694, 95syl 17 . . . . . . . . . 10  |-  ( (
ph  /\  i  e.  Y )  ->  -.  ( G `  i )  e.  { (/) } )
9788, 96eldifd 3447 . . . . . . . . 9  |-  ( (
ph  /\  i  e.  Y )  ->  ( G `  i )  e.  ( ran  G  \  { (/) } ) )
98 meadjiunlem.dj . . . . . . . . . 10  |-  ( ph  -> Disj  i  e.  X  ( G `  i )
)
99 disjss1 4397 . . . . . . . . . 10  |-  ( Y 
C_  X  ->  (Disj  i  e.  X  ( G `
 i )  -> Disj  i  e.  Y  ( G `
 i ) ) )
10049, 98, 99sylc 62 . . . . . . . . 9  |-  ( ph  -> Disj  i  e.  Y  ( G `  i )
)
10181, 82, 97, 94, 100disjf1 37307 . . . . . . . 8  |-  ( ph  ->  ( i  e.  Y  |->  ( G `  i
) ) : Y -1-1-> ( ran  G  \  { (/)
} ) )
1022, 49feqresmpt 5932 . . . . . . . . 9  |-  ( ph  ->  ( G  |`  Y )  =  ( i  e.  Y  |->  ( G `  i ) ) )
103 f1eq1 5788 . . . . . . . . 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 235 . . . . . . 7  |-  ( ph  ->  ( G  |`  Y ) : Y -1-1-> ( ran 
G  \  { (/) } ) )
106102rneqd 5078 . . . . . . . . 9  |-  ( ph  ->  ran  ( G  |`  Y )  =  ran  ( i  e.  Y  |->  ( G `  i
) ) )
10797ralrimiva 2839 . . . . . . . . . 10  |-  ( ph  ->  A. i  e.  Y  ( G `  i )  e.  ( ran  G  \  { (/) } ) )
10882rnmptss 6064 . . . . . . . . . 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 3498 . . . . . . . 8  |-  ( ph  ->  ran  ( G  |`  Y )  C_  ( ran  G  \  { (/) } ) )
111 simpl 458 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( ran  G  \  { (/)
} ) )  ->  ph )
112 eldifi 3587 . . . . . . . . . . . 12  |-  ( x  e.  ( ran  G  \  { (/) } )  ->  x  e.  ran  G )
113112adantl 467 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( ran  G  \  { (/)
} ) )  ->  x  e.  ran  G )
114 eldifsni 4123 . . . . . . . . . . . 12  |-  ( x  e.  ( ran  G  \  { (/) } )  ->  x  =/=  (/) )
115114adantl 467 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( ran  G  \  { (/)
} ) )  ->  x  =/=  (/) )
116 simpr 462 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  ran  G )  ->  x  e.  ran  G )
117 fvelrnb 5925 . . . . . . . . . . . . . . . 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 466 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  ran  G )  ->  (
x  e.  ran  G  <->  E. i  e.  X  ( G `  i )  =  x ) )
120116, 119mpbid 213 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ran  G )  ->  E. i  e.  X  ( G `  i )  =  x )
1211203adant3 1025 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ran  G  /\  x  =/=  (/) )  ->  E. i  e.  X  ( G `  i )  =  x )
122 id 23 . . . . . . . . . . . . . . . . . 18  |-  ( ( G `  i )  =  x  ->  ( G `  i )  =  x )
123122eqcomd 2430 . . . . . . . . . . . . . . . . 17  |-  ( ( G `  i )  =  x  ->  x  =  ( G `  i ) )
1241233ad2ant3 1028 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  =/=  (/) )  /\  i  e.  X  /\  ( G `  i )  =  x )  ->  x  =  ( G `  i ) )
125 simp1l 1029 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  x  =/=  (/) )  /\  i  e.  X  /\  ( G `  i )  =  x )  ->  ph )
126 simp2 1006 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  x  =/=  (/) )  /\  i  e.  X  /\  ( G `  i )  =  x )  ->  i  e.  X )
127 simpr 462 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( x  =/=  (/)  /\  ( G `  i )  =  x )  ->  ( G `  i )  =  x )
128 simpl 458 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( x  =/=  (/)  /\  ( G `  i )  =  x )  ->  x  =/=  (/) )
129127, 128eqnetrd 2717 . . . . . . . . . . . . . . . . . . 19  |-  ( ( x  =/=  (/)  /\  ( G `  i )  =  x )  ->  ( G `  i )  =/=  (/) )
130129adantll 718 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  x  =/=  (/) )  /\  ( G `  i )  =  x )  ->  ( G `  i )  =/=  (/) )
1311303adant2 1024 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  x  =/=  (/) )  /\  i  e.  X  /\  ( G `  i )  =  x )  ->  ( G `  i )  =/=  (/) )
13291biimpri 209 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( i  e.  X  /\  ( G `  i )  =/=  (/) )  ->  i  e.  Y )
133 fvex 5888 . . . . . . . . . . . . . . . . . . . . 21  |-  ( G `
 i )  e. 
_V
134133a1i 11 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( i  e.  X  /\  ( G `  i )  =/=  (/) )  ->  ( G `  i )  e.  _V )
13582elrnmpt1 5099 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( i  e.  Y  /\  ( G `  i )  e.  _V )  -> 
( G `  i
)  e.  ran  (
i  e.  Y  |->  ( G `  i ) ) )
136132, 134, 135syl2anc 665 . . . . . . . . . . . . . . . . . . 19  |-  ( ( i  e.  X  /\  ( G `  i )  =/=  (/) )  ->  ( G `  i )  e.  ran  ( i  e.  Y  |->  ( G `  i ) ) )
1371363adant1 1023 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  i  e.  X  /\  ( G `  i )  =/=  (/) )  -> 
( G `  i
)  e.  ran  (
i  e.  Y  |->  ( G `  i ) ) )
138106eqcomd 2430 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ran  ( i  e.  Y  |->  ( G `  i ) )  =  ran  ( G  |`  Y ) )
1391383ad2ant1 1026 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  i  e.  X  /\  ( G `  i )  =/=  (/) )  ->  ran  ( i  e.  Y  |->  ( G `  i
) )  =  ran  ( G  |`  Y ) )
140137, 139eleqtrd 2512 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  i  e.  X  /\  ( G `  i )  =/=  (/) )  -> 
( G `  i
)  e.  ran  ( G  |`  Y ) )
141125, 126, 131, 140syl3anc 1264 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  =/=  (/) )  /\  i  e.  X  /\  ( G `  i )  =  x )  ->  ( G `  i )  e.  ran  ( G  |`  Y ) )
142124, 141eqeltrd 2510 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  =/=  (/) )  /\  i  e.  X  /\  ( G `  i )  =  x )  ->  x  e.  ran  ( G  |`  Y ) )
1431423exp 1204 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  =/=  (/) )  ->  ( i  e.  X  ->  ( ( G `  i )  =  x  ->  x  e.  ran  ( G  |`  Y ) ) ) )
144143rexlimdv 2915 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  =/=  (/) )  ->  ( E. i  e.  X  ( G `  i )  =  x  ->  x  e. 
ran  ( G  |`  Y ) ) )
1451443adant2 1024 . . . . . . . . . . . 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 1264 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( ran  G  \  { (/)
} ) )  ->  x  e.  ran  ( G  |`  Y ) )
148147ralrimiva 2839 . . . . . . . . 9  |-  ( ph  ->  A. x  e.  ( ran  G  \  { (/)
} ) x  e. 
ran  ( G  |`  Y ) )
149 dfss3 3454 . . . . . . . . 9  |-  ( ( ran  G  \  { (/)
} )  C_  ran  ( G  |`  Y )  <->  A. x  e.  ( ran  G  \  { (/) } ) x  e.  ran  ( G  |`  Y ) )
150148, 149sylibr 215 . . . . . . . 8  |-  ( ph  ->  ( ran  G  \  { (/) } )  C_  ran  ( G  |`  Y ) )
151110, 150eqssd 3481 . . . . . . 7  |-  ( ph  ->  ran  ( G  |`  Y )  =  ( ran  G  \  { (/)
} ) )
152105, 151jca 534 . . . . . 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 215 . . . . 5  |-  ( ph  ->  ( G  |`  Y ) : Y -1-1-onto-> ( ran  G  \  { (/) } ) )
155 fvres 5892 . . . . . 6  |-  ( j  e.  Y  ->  (
( G  |`  Y ) `
 j )  =  ( G `  j
) )
156155adantl 467 . . . . 5  |-  ( (
ph  /\  j  e.  Y )  ->  (
( G  |`  Y ) `
 j )  =  ( G `  j
) )
1571, 45, 42, 80, 154, 156, 18sge0f1o 38012 . . . 4  |-  ( ph  ->  (Σ^ `  ( k  e.  ( ran  G  \  { (/)
} )  |->  ( M `
 k ) ) )  =  (Σ^ `  ( j  e.  Y  |->  ( M `  ( G `  j )
) ) ) )
158157eqcomd 2430 . . 3  |-  ( ph  ->  (Σ^ `  ( j  e.  Y  |->  ( M `  ( G `  j )
) ) )  =  (Σ^ `  ( k  e.  ( ran  G  \  { (/)
} )  |->  ( M `
 k ) ) ) )
15944, 79, 1583eqtrd 2467 . 2  |-  ( ph  ->  (Σ^ `  ( M  o.  G
) )  =  (Σ^ `  (
k  e.  ( ran 
G  \  { (/) } ) 
|->  ( M `  k
) ) ) )
16036, 38, 1593eqtr4d 2473 1  |-  ( ph  ->  (Σ^ `  ( M  |`  ran  G
) )  =  (Σ^ `  ( M  o.  G )
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1868    =/= wne 2618   A.wral 2775   E.wrex 2776   {crab 2779   _Vcvv 3081    \ cdif 3433    i^i cin 3435    C_ wss 3436   (/)c0 3761   {csn 3996  Disj wdisj 4391    |-> cmpt 4479   dom cdm 4850   ran crn 4851    |` cres 4852    o. ccom 4854    Fn wfn 5593   -->wf 5594   -1-1->wf1 5595   -1-1-onto->wf1o 5597   ` cfv 5598  (class class class)co 6302   0cc0 9540   +oocpnf 9673   [,]cicc 11639  Σ^csumge0 37992  Meascmea 38066
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594  ax-inf2 8149  ax-cnex 9596  ax-resscn 9597  ax-1cn 9598  ax-icn 9599  ax-addcl 9600  ax-addrcl 9601  ax-mulcl 9602  ax-mulrcl 9603  ax-mulcom 9604  ax-addass 9605  ax-mulass 9606  ax-distr 9607  ax-i2m1 9608  ax-1ne0 9609  ax-1rid 9610  ax-rnegex 9611  ax-rrecex 9612  ax-cnre 9613  ax-pre-lttri 9614  ax-pre-lttrn 9615  ax-pre-ltadd 9616  ax-pre-mulgt0 9617  ax-pre-sup 9618
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-disj 4392  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4761  df-id 4765  df-po 4771  df-so 4772  df-fr 4809  df-se 4810  df-we 4811  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-pred 5396  df-ord 5442  df-on 5443  df-lim 5444  df-suc 5445  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-f1 5603  df-fo 5604  df-f1o 5605  df-fv 5606  df-isom 5607  df-riota 6264  df-ov 6305  df-oprab 6306  df-mpt2 6307  df-om 6704  df-1st 6804  df-2nd 6805  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-oadd 7191  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-sup 7959  df-oi 8028  df-card 8375  df-pnf 9678  df-mnf 9679  df-xr 9680  df-ltxr 9681  df-le 9682  df-sub 9863  df-neg 9864  df-div 10271  df-nn 10611  df-2 10669  df-3 10670  df-n0 10871  df-z 10939  df-uz 11161  df-rp 11304  df-xadd 11411  df-ico 11642  df-icc 11643  df-fz 11786  df-fzo 11917  df-seq 12214  df-exp 12273  df-hash 12516  df-cj 13151  df-re 13152  df-im 13153  df-sqrt 13287  df-abs 13288  df-clim 13540  df-sum 13741  df-sumge0 37993  df-mea 38067
This theorem is referenced by:  meadjiun  38083
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