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Theorem bj-finsumval0 31772
Description: Value of a finite sum. (Contributed by BJ, 9-Jun-2019.)
Hypotheses
Ref Expression
bj-finsumval0.1  |-  ( ph  ->  A  e. CMnd )
bj-finsumval0.2  |-  ( ph  ->  I  e.  Fin )
bj-finsumval0.3  |-  ( ph  ->  B : I --> ( Base `  A ) )
Assertion
Ref Expression
bj-finsumval0  |-  ( ph  ->  ( A FinSum  B )  =  ( iota s E. m  e.  NN0  E. f ( f : ( 1 ... m
)
-1-1-onto-> I  /\  s  =  (  seq 1 ( ( +g  `  A ) ,  ( n  e.  NN  |->  ( B `  ( f `  n
) ) ) ) `
 ( # `  I
) ) ) ) )
Distinct variable groups:    A, s,
f, m, n    B, f, m, n, s    f, I, n    ph, f, m, s
Allowed substitution hints:    ph( n)    I( m, s)

Proof of Theorem bj-finsumval0
Dummy variables  t  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ov 6311 . 2  |-  ( A FinSum  B )  =  ( FinSum  `  <. A ,  B >. )
2 df-bj-finsum 31771 . . . 4  |- FinSum  =  ( x  e.  { <. y ,  z >.  |  ( y  e. CMnd  /\  E. t  e.  Fin  z : t --> ( Base `  y
) ) }  |->  ( iota s E. m  e.  NN0  E. f ( f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x )  /\  s  =  (  seq 1 ( ( +g  `  ( 1st `  x
) ) ,  ( n  e.  NN  |->  ( ( 2nd `  x
) `  ( f `  n ) ) ) ) `  m ) ) ) )
32a1i 11 . . 3  |-  ( ph  -> FinSum 
=  ( x  e. 
{ <. y ,  z
>.  |  ( y  e. CMnd  /\  E. t  e. 
Fin  z : t --> ( Base `  y
) ) }  |->  ( iota s E. m  e.  NN0  E. f ( f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x )  /\  s  =  (  seq 1 ( ( +g  `  ( 1st `  x
) ) ,  ( n  e.  NN  |->  ( ( 2nd `  x
) `  ( f `  n ) ) ) ) `  m ) ) ) ) )
4 simpr 468 . . . . . . . . . 10  |-  ( (
ph  /\  x  =  <. A ,  B >. )  ->  x  =  <. A ,  B >. )
54fveq2d 5883 . . . . . . . . 9  |-  ( (
ph  /\  x  =  <. A ,  B >. )  ->  ( 1st `  x
)  =  ( 1st `  <. A ,  B >. ) )
6 bj-finsumval0.1 . . . . . . . . . . 11  |-  ( ph  ->  A  e. CMnd )
76adantr 472 . . . . . . . . . 10  |-  ( (
ph  /\  x  =  <. A ,  B >. )  ->  A  e. CMnd )
8 bj-finsumval0.3 . . . . . . . . . . . 12  |-  ( ph  ->  B : I --> ( Base `  A ) )
9 bj-finsumval0.2 . . . . . . . . . . . 12  |-  ( ph  ->  I  e.  Fin )
10 fex 6155 . . . . . . . . . . . 12  |-  ( ( B : I --> ( Base `  A )  /\  I  e.  Fin )  ->  B  e.  _V )
118, 9, 10syl2anc 673 . . . . . . . . . . 11  |-  ( ph  ->  B  e.  _V )
1211adantr 472 . . . . . . . . . 10  |-  ( (
ph  /\  x  =  <. A ,  B >. )  ->  B  e.  _V )
13 op1stg 6824 . . . . . . . . . 10  |-  ( ( A  e. CMnd  /\  B  e.  _V )  ->  ( 1st `  <. A ,  B >. )  =  A )
147, 12, 13syl2anc 673 . . . . . . . . 9  |-  ( (
ph  /\  x  =  <. A ,  B >. )  ->  ( 1st `  <. A ,  B >. )  =  A )
155, 14eqtrd 2505 . . . . . . . 8  |-  ( (
ph  /\  x  =  <. A ,  B >. )  ->  ( 1st `  x
)  =  A )
164fveq2d 5883 . . . . . . . . 9  |-  ( (
ph  /\  x  =  <. A ,  B >. )  ->  ( 2nd `  x
)  =  ( 2nd `  <. A ,  B >. ) )
17 op2ndg 6825 . . . . . . . . . 10  |-  ( ( A  e. CMnd  /\  B  e.  _V )  ->  ( 2nd `  <. A ,  B >. )  =  B )
187, 12, 17syl2anc 673 . . . . . . . . 9  |-  ( (
ph  /\  x  =  <. A ,  B >. )  ->  ( 2nd `  <. A ,  B >. )  =  B )
1916, 18eqtrd 2505 . . . . . . . 8  |-  ( (
ph  /\  x  =  <. A ,  B >. )  ->  ( 2nd `  x
)  =  B )
2019dmeqd 5042 . . . . . . . . 9  |-  ( (
ph  /\  x  =  <. A ,  B >. )  ->  dom  ( 2nd `  x )  =  dom  B )
21 fdm 5745 . . . . . . . . . . 11  |-  ( B : I --> ( Base `  A )  ->  dom  B  =  I )
228, 21syl 17 . . . . . . . . . 10  |-  ( ph  ->  dom  B  =  I )
2322adantr 472 . . . . . . . . 9  |-  ( (
ph  /\  x  =  <. A ,  B >. )  ->  dom  B  =  I )
2420, 23eqtrd 2505 . . . . . . . 8  |-  ( (
ph  /\  x  =  <. A ,  B >. )  ->  dom  ( 2nd `  x )  =  I )
25 f1oeq3 5820 . . . . . . . . . . . . . . 15  |-  ( dom  ( 2nd `  x
)  =  I  -> 
( f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x
)  <->  f : ( 1 ... m ) -1-1-onto-> I ) )
2625biimpd 212 . . . . . . . . . . . . . 14  |-  ( dom  ( 2nd `  x
)  =  I  -> 
( f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x
)  ->  f :
( 1 ... m
)
-1-1-onto-> I ) )
2726ad2antll 743 . . . . . . . . . . . . 13  |-  ( ( ( 1st `  x
)  =  A  /\  ( ( 2nd `  x
)  =  B  /\  dom  ( 2nd `  x
)  =  I ) )  ->  ( f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x
)  ->  f :
( 1 ... m
)
-1-1-onto-> I ) )
2827adantrd 475 . . . . . . . . . . . 12  |-  ( ( ( 1st `  x
)  =  A  /\  ( ( 2nd `  x
)  =  B  /\  dom  ( 2nd `  x
)  =  I ) )  ->  ( (
f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x )  /\  s  =  (  seq 1 ( ( +g  `  ( 1st `  x
) ) ,  ( n  e.  NN  |->  ( ( 2nd `  x
) `  ( f `  n ) ) ) ) `  m ) )  ->  f :
( 1 ... m
)
-1-1-onto-> I ) )
2928adantr 472 . . . . . . . . . . 11  |-  ( ( ( ( 1st `  x
)  =  A  /\  ( ( 2nd `  x
)  =  B  /\  dom  ( 2nd `  x
)  =  I ) )  /\  m  e. 
NN0 )  ->  (
( f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x
)  /\  s  =  (  seq 1 ( ( +g  `  ( 1st `  x ) ) ,  ( n  e.  NN  |->  ( ( 2nd `  x
) `  ( f `  n ) ) ) ) `  m ) )  ->  f :
( 1 ... m
)
-1-1-onto-> I ) )
30 eqidd 2472 . . . . . . . . . . . . . . . . 17  |-  ( ( f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x )  /\  ( ( ( 1st `  x )  =  A  /\  ( ( 2nd `  x )  =  B  /\  dom  ( 2nd `  x )  =  I ) )  /\  m  e.  NN0 ) )  -> 
1  =  1 )
31 simprl 772 . . . . . . . . . . . . . . . . . . 19  |-  ( ( f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x )  /\  ( ( 1st `  x
)  =  A  /\  ( ( 2nd `  x
)  =  B  /\  dom  ( 2nd `  x
)  =  I ) ) )  ->  ( 1st `  x )  =  A )
3231fveq2d 5883 . . . . . . . . . . . . . . . . . 18  |-  ( ( f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x )  /\  ( ( 1st `  x
)  =  A  /\  ( ( 2nd `  x
)  =  B  /\  dom  ( 2nd `  x
)  =  I ) ) )  ->  ( +g  `  ( 1st `  x
) )  =  ( +g  `  A ) )
3332adantrr 731 . . . . . . . . . . . . . . . . 17  |-  ( ( f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x )  /\  ( ( ( 1st `  x )  =  A  /\  ( ( 2nd `  x )  =  B  /\  dom  ( 2nd `  x )  =  I ) )  /\  m  e.  NN0 ) )  -> 
( +g  `  ( 1st `  x ) )  =  ( +g  `  A
) )
34 simprrl 782 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x )  /\  ( ( 1st `  x
)  =  A  /\  ( ( 2nd `  x
)  =  B  /\  dom  ( 2nd `  x
)  =  I ) ) )  ->  ( 2nd `  x )  =  B )
3534adantr 472 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x
)  /\  ( ( 1st `  x )  =  A  /\  ( ( 2nd `  x )  =  B  /\  dom  ( 2nd `  x )  =  I ) ) )  /\  n  e.  NN )  ->  ( 2nd `  x )  =  B )
3635fveq1d 5881 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x
)  /\  ( ( 1st `  x )  =  A  /\  ( ( 2nd `  x )  =  B  /\  dom  ( 2nd `  x )  =  I ) ) )  /\  n  e.  NN )  ->  (
( 2nd `  x
) `  ( f `  n ) )  =  ( B `  (
f `  n )
) )
3736mpteq2dva 4482 . . . . . . . . . . . . . . . . . 18  |-  ( ( f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x )  /\  ( ( 1st `  x
)  =  A  /\  ( ( 2nd `  x
)  =  B  /\  dom  ( 2nd `  x
)  =  I ) ) )  ->  (
n  e.  NN  |->  ( ( 2nd `  x
) `  ( f `  n ) ) )  =  ( n  e.  NN  |->  ( B `  ( f `  n
) ) ) )
3837adantrr 731 . . . . . . . . . . . . . . . . 17  |-  ( ( f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x )  /\  ( ( ( 1st `  x )  =  A  /\  ( ( 2nd `  x )  =  B  /\  dom  ( 2nd `  x )  =  I ) )  /\  m  e.  NN0 ) )  -> 
( n  e.  NN  |->  ( ( 2nd `  x
) `  ( f `  n ) ) )  =  ( n  e.  NN  |->  ( B `  ( f `  n
) ) ) )
3930, 33, 38seqeq123d 12260 . . . . . . . . . . . . . . . 16  |-  ( ( f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x )  /\  ( ( ( 1st `  x )  =  A  /\  ( ( 2nd `  x )  =  B  /\  dom  ( 2nd `  x )  =  I ) )  /\  m  e.  NN0 ) )  ->  seq 1 ( ( +g  `  ( 1st `  x
) ) ,  ( n  e.  NN  |->  ( ( 2nd `  x
) `  ( f `  n ) ) ) )  =  seq 1
( ( +g  `  A
) ,  ( n  e.  NN  |->  ( B `
 ( f `  n ) ) ) ) )
40 simpr 468 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( 1st `  x
)  =  A  /\  ( ( 2nd `  x
)  =  B  /\  dom  ( 2nd `  x
)  =  I ) )  /\  m  e. 
NN0 )  ->  m  e.  NN0 )
41 simprr 774 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( 1st `  x
)  =  A  /\  ( ( 2nd `  x
)  =  B  /\  dom  ( 2nd `  x
)  =  I ) )  ->  dom  ( 2nd `  x )  =  I )
4241adantr 472 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( 1st `  x
)  =  A  /\  ( ( 2nd `  x
)  =  B  /\  dom  ( 2nd `  x
)  =  I ) )  /\  m  e. 
NN0 )  ->  dom  ( 2nd `  x )  =  I )
4340, 42jca 541 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( 1st `  x
)  =  A  /\  ( ( 2nd `  x
)  =  B  /\  dom  ( 2nd `  x
)  =  I ) )  /\  m  e. 
NN0 )  ->  (
m  e.  NN0  /\  dom  ( 2nd `  x
)  =  I ) )
44 hashfz1 12567 . . . . . . . . . . . . . . . . . . . . 21  |-  ( m  e.  NN0  ->  ( # `  ( 1 ... m
) )  =  m )
4544eqcomd 2477 . . . . . . . . . . . . . . . . . . . 20  |-  ( m  e.  NN0  ->  m  =  ( # `  (
1 ... m ) ) )
4645ad2antrl 742 . . . . . . . . . . . . . . . . . . 19  |-  ( ( f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x )  /\  ( m  e.  NN0  /\ 
dom  ( 2nd `  x
)  =  I ) )  ->  m  =  ( # `  ( 1 ... m ) ) )
47 fzfid 12224 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x )  /\  m  e.  NN0 )  -> 
( 1 ... m
)  e.  Fin )
48 f1ofo 5835 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x
)  ->  f :
( 1 ... m
) -onto-> dom  ( 2nd `  x
) )
4948adantr 472 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x )  /\  m  e.  NN0 )  -> 
f : ( 1 ... m ) -onto-> dom  ( 2nd `  x
) )
50 fornex 6781 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( 1 ... m )  e.  Fin  ->  (
f : ( 1 ... m ) -onto-> dom  ( 2nd `  x
)  ->  dom  ( 2nd `  x )  e.  _V ) )
5147, 49, 50sylc 61 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x )  /\  m  e.  NN0 )  ->  dom  ( 2nd `  x
)  e.  _V )
5247, 51jca 541 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x )  /\  m  e.  NN0 )  -> 
( ( 1 ... m )  e.  Fin  /\ 
dom  ( 2nd `  x
)  e.  _V )
)
5352adantrr 731 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x )  /\  ( m  e.  NN0  /\ 
dom  ( 2nd `  x
)  =  I ) )  ->  ( (
1 ... m )  e. 
Fin  /\  dom  ( 2nd `  x )  e.  _V ) )
54 19.8a 1955 . . . . . . . . . . . . . . . . . . . . 21  |-  ( f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x
)  ->  E. f 
f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x ) )
5554adantr 472 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x )  /\  ( m  e.  NN0  /\ 
dom  ( 2nd `  x
)  =  I ) )  ->  E. f 
f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x ) )
56 hasheqf1oi 12572 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( 1 ... m
)  e.  Fin  /\  dom  ( 2nd `  x
)  e.  _V )  ->  ( E. f  f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x
)  ->  ( # `  (
1 ... m ) )  =  ( # `  dom  ( 2nd `  x ) ) ) )
5753, 55, 56sylc 61 . . . . . . . . . . . . . . . . . . 19  |-  ( ( f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x )  /\  ( m  e.  NN0  /\ 
dom  ( 2nd `  x
)  =  I ) )  ->  ( # `  (
1 ... m ) )  =  ( # `  dom  ( 2nd `  x ) ) )
5846, 57eqtrd 2505 . . . . . . . . . . . . . . . . . 18  |-  ( ( f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x )  /\  ( m  e.  NN0  /\ 
dom  ( 2nd `  x
)  =  I ) )  ->  m  =  ( # `  dom  ( 2nd `  x ) ) )
59 simprr 774 . . . . . . . . . . . . . . . . . . 19  |-  ( ( f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x )  /\  ( m  e.  NN0  /\ 
dom  ( 2nd `  x
)  =  I ) )  ->  dom  ( 2nd `  x )  =  I )
6059fveq2d 5883 . . . . . . . . . . . . . . . . . 18  |-  ( ( f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x )  /\  ( m  e.  NN0  /\ 
dom  ( 2nd `  x
)  =  I ) )  ->  ( # `  dom  ( 2nd `  x ) )  =  ( # `  I ) )
6158, 60eqtrd 2505 . . . . . . . . . . . . . . . . 17  |-  ( ( f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x )  /\  ( m  e.  NN0  /\ 
dom  ( 2nd `  x
)  =  I ) )  ->  m  =  ( # `  I ) )
6243, 61sylan2 482 . . . . . . . . . . . . . . . 16  |-  ( ( f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x )  /\  ( ( ( 1st `  x )  =  A  /\  ( ( 2nd `  x )  =  B  /\  dom  ( 2nd `  x )  =  I ) )  /\  m  e.  NN0 ) )  ->  m  =  ( # `  I
) )
6339, 62fveq12d 5885 . . . . . . . . . . . . . . 15  |-  ( ( f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x )  /\  ( ( ( 1st `  x )  =  A  /\  ( ( 2nd `  x )  =  B  /\  dom  ( 2nd `  x )  =  I ) )  /\  m  e.  NN0 ) )  -> 
(  seq 1 ( ( +g  `  ( 1st `  x ) ) ,  ( n  e.  NN  |->  ( ( 2nd `  x
) `  ( f `  n ) ) ) ) `  m )  =  (  seq 1
( ( +g  `  A
) ,  ( n  e.  NN  |->  ( B `
 ( f `  n ) ) ) ) `  ( # `  I ) ) )
6463eqeq2d 2481 . . . . . . . . . . . . . 14  |-  ( ( f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x )  /\  ( ( ( 1st `  x )  =  A  /\  ( ( 2nd `  x )  =  B  /\  dom  ( 2nd `  x )  =  I ) )  /\  m  e.  NN0 ) )  -> 
( s  =  (  seq 1 ( ( +g  `  ( 1st `  x ) ) ,  ( n  e.  NN  |->  ( ( 2nd `  x
) `  ( f `  n ) ) ) ) `  m )  <-> 
s  =  (  seq 1 ( ( +g  `  A ) ,  ( n  e.  NN  |->  ( B `  ( f `
 n ) ) ) ) `  ( # `
 I ) ) ) )
6564biimpd 212 . . . . . . . . . . . . 13  |-  ( ( f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x )  /\  ( ( ( 1st `  x )  =  A  /\  ( ( 2nd `  x )  =  B  /\  dom  ( 2nd `  x )  =  I ) )  /\  m  e.  NN0 ) )  -> 
( s  =  (  seq 1 ( ( +g  `  ( 1st `  x ) ) ,  ( n  e.  NN  |->  ( ( 2nd `  x
) `  ( f `  n ) ) ) ) `  m )  ->  s  =  (  seq 1 ( ( +g  `  A ) ,  ( n  e.  NN  |->  ( B `  ( f `  n
) ) ) ) `
 ( # `  I
) ) ) )
6665impancom 447 . . . . . . . . . . . 12  |-  ( ( f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x )  /\  s  =  (  seq 1 ( ( +g  `  ( 1st `  x
) ) ,  ( n  e.  NN  |->  ( ( 2nd `  x
) `  ( f `  n ) ) ) ) `  m ) )  ->  ( (
( ( 1st `  x
)  =  A  /\  ( ( 2nd `  x
)  =  B  /\  dom  ( 2nd `  x
)  =  I ) )  /\  m  e. 
NN0 )  ->  s  =  (  seq 1
( ( +g  `  A
) ,  ( n  e.  NN  |->  ( B `
 ( f `  n ) ) ) ) `  ( # `  I ) ) ) )
6766com12 31 . . . . . . . . . . 11  |-  ( ( ( ( 1st `  x
)  =  A  /\  ( ( 2nd `  x
)  =  B  /\  dom  ( 2nd `  x
)  =  I ) )  /\  m  e. 
NN0 )  ->  (
( f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x
)  /\  s  =  (  seq 1 ( ( +g  `  ( 1st `  x ) ) ,  ( n  e.  NN  |->  ( ( 2nd `  x
) `  ( f `  n ) ) ) ) `  m ) )  ->  s  =  (  seq 1 ( ( +g  `  A ) ,  ( n  e.  NN  |->  ( B `  ( f `  n
) ) ) ) `
 ( # `  I
) ) ) )
6829, 67jcad 542 . . . . . . . . . 10  |-  ( ( ( ( 1st `  x
)  =  A  /\  ( ( 2nd `  x
)  =  B  /\  dom  ( 2nd `  x
)  =  I ) )  /\  m  e. 
NN0 )  ->  (
( f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x
)  /\  s  =  (  seq 1 ( ( +g  `  ( 1st `  x ) ) ,  ( n  e.  NN  |->  ( ( 2nd `  x
) `  ( f `  n ) ) ) ) `  m ) )  ->  ( f : ( 1 ... m ) -1-1-onto-> I  /\  s  =  (  seq 1
( ( +g  `  A
) ,  ( n  e.  NN  |->  ( B `
 ( f `  n ) ) ) ) `  ( # `  I ) ) ) ) )
6925biimprd 231 . . . . . . . . . . . . . 14  |-  ( dom  ( 2nd `  x
)  =  I  -> 
( f : ( 1 ... m ) -1-1-onto-> I  ->  f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x
) ) )
7069ad2antll 743 . . . . . . . . . . . . 13  |-  ( ( ( 1st `  x
)  =  A  /\  ( ( 2nd `  x
)  =  B  /\  dom  ( 2nd `  x
)  =  I ) )  ->  ( f : ( 1 ... m ) -1-1-onto-> I  ->  f : ( 1 ... m
)
-1-1-onto-> dom  ( 2nd `  x
) ) )
7170adantr 472 . . . . . . . . . . . 12  |-  ( ( ( ( 1st `  x
)  =  A  /\  ( ( 2nd `  x
)  =  B  /\  dom  ( 2nd `  x
)  =  I ) )  /\  m  e. 
NN0 )  ->  (
f : ( 1 ... m ) -1-1-onto-> I  -> 
f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x ) ) )
7271adantrd 475 . . . . . . . . . . 11  |-  ( ( ( ( 1st `  x
)  =  A  /\  ( ( 2nd `  x
)  =  B  /\  dom  ( 2nd `  x
)  =  I ) )  /\  m  e. 
NN0 )  ->  (
( f : ( 1 ... m ) -1-1-onto-> I  /\  s  =  (  seq 1 ( ( +g  `  A ) ,  ( n  e.  NN  |->  ( B `  ( f `  n
) ) ) ) `
 ( # `  I
) ) )  -> 
f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x ) ) )
73 eqidd 2472 . . . . . . . . . . . . . . . . 17  |-  ( ( f : ( 1 ... m ) -1-1-onto-> I  /\  ( ( ( 1st `  x )  =  A  /\  ( ( 2nd `  x )  =  B  /\  dom  ( 2nd `  x )  =  I ) )  /\  m  e.  NN0 ) )  -> 
1  =  1 )
74 simpl 464 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( 1st `  x
)  =  A  /\  ( ( 2nd `  x
)  =  B  /\  dom  ( 2nd `  x
)  =  I ) )  ->  ( 1st `  x )  =  A )
75 tru 1456 . . . . . . . . . . . . . . . . . . . . 21  |- T.
7674, 75jctir 547 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( 1st `  x
)  =  A  /\  ( ( 2nd `  x
)  =  B  /\  dom  ( 2nd `  x
)  =  I ) )  ->  ( ( 1st `  x )  =  A  /\ T.  ) )
7776ad2antrl 742 . . . . . . . . . . . . . . . . . . 19  |-  ( ( f : ( 1 ... m ) -1-1-onto-> I  /\  ( ( ( 1st `  x )  =  A  /\  ( ( 2nd `  x )  =  B  /\  dom  ( 2nd `  x )  =  I ) )  /\  m  e.  NN0 ) )  -> 
( ( 1st `  x
)  =  A  /\ T.  ) )
78 simpl 464 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( 1st `  x
)  =  A  /\ T.  )  ->  ( 1st `  x )  =  A )
7978eqcomd 2477 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( 1st `  x
)  =  A  /\ T.  )  ->  A  =  ( 1st `  x
) )
8077, 79syl 17 . . . . . . . . . . . . . . . . . 18  |-  ( ( f : ( 1 ... m ) -1-1-onto-> I  /\  ( ( ( 1st `  x )  =  A  /\  ( ( 2nd `  x )  =  B  /\  dom  ( 2nd `  x )  =  I ) )  /\  m  e.  NN0 ) )  ->  A  =  ( 1st `  x ) )
8180fveq2d 5883 . . . . . . . . . . . . . . . . 17  |-  ( ( f : ( 1 ... m ) -1-1-onto-> I  /\  ( ( ( 1st `  x )  =  A  /\  ( ( 2nd `  x )  =  B  /\  dom  ( 2nd `  x )  =  I ) )  /\  m  e.  NN0 ) )  -> 
( +g  `  A )  =  ( +g  `  ( 1st `  x ) ) )
82 simpl 464 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( 2nd `  x
)  =  B  /\  dom  ( 2nd `  x
)  =  I )  ->  ( 2nd `  x
)  =  B )
8382eqcomd 2477 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( 2nd `  x
)  =  B  /\  dom  ( 2nd `  x
)  =  I )  ->  B  =  ( 2nd `  x ) )
8483ad2antll 743 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( f : ( 1 ... m ) -1-1-onto-> I  /\  ( ( 1st `  x
)  =  A  /\  ( ( 2nd `  x
)  =  B  /\  dom  ( 2nd `  x
)  =  I ) ) )  ->  B  =  ( 2nd `  x
) )
8584adantr 472 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( f : ( 1 ... m ) -1-1-onto-> I  /\  ( ( 1st `  x )  =  A  /\  ( ( 2nd `  x )  =  B  /\  dom  ( 2nd `  x )  =  I ) ) )  /\  n  e.  NN )  ->  B  =  ( 2nd `  x ) )
8685fveq1d 5881 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( f : ( 1 ... m ) -1-1-onto-> I  /\  ( ( 1st `  x )  =  A  /\  ( ( 2nd `  x )  =  B  /\  dom  ( 2nd `  x )  =  I ) ) )  /\  n  e.  NN )  ->  ( B `  (
f `  n )
)  =  ( ( 2nd `  x ) `
 ( f `  n ) ) )
8786adantlrr 735 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( f : ( 1 ... m ) -1-1-onto-> I  /\  ( ( ( 1st `  x )  =  A  /\  (
( 2nd `  x
)  =  B  /\  dom  ( 2nd `  x
)  =  I ) )  /\  m  e. 
NN0 ) )  /\  n  e.  NN )  ->  ( B `  (
f `  n )
)  =  ( ( 2nd `  x ) `
 ( f `  n ) ) )
8887mpteq2dva 4482 . . . . . . . . . . . . . . . . 17  |-  ( ( f : ( 1 ... m ) -1-1-onto-> I  /\  ( ( ( 1st `  x )  =  A  /\  ( ( 2nd `  x )  =  B  /\  dom  ( 2nd `  x )  =  I ) )  /\  m  e.  NN0 ) )  -> 
( n  e.  NN  |->  ( B `  ( f `
 n ) ) )  =  ( n  e.  NN  |->  ( ( 2nd `  x ) `
 ( f `  n ) ) ) )
8973, 81, 88seqeq123d 12260 . . . . . . . . . . . . . . . 16  |-  ( ( f : ( 1 ... m ) -1-1-onto-> I  /\  ( ( ( 1st `  x )  =  A  /\  ( ( 2nd `  x )  =  B  /\  dom  ( 2nd `  x )  =  I ) )  /\  m  e.  NN0 ) )  ->  seq 1 ( ( +g  `  A ) ,  ( n  e.  NN  |->  ( B `  ( f `
 n ) ) ) )  =  seq 1 ( ( +g  `  ( 1st `  x
) ) ,  ( n  e.  NN  |->  ( ( 2nd `  x
) `  ( f `  n ) ) ) ) )
9071com12 31 . . . . . . . . . . . . . . . . . . 19  |-  ( f : ( 1 ... m ) -1-1-onto-> I  ->  ( ( ( ( 1st `  x
)  =  A  /\  ( ( 2nd `  x
)  =  B  /\  dom  ( 2nd `  x
)  =  I ) )  /\  m  e. 
NN0 )  ->  f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x
) ) )
9190imp 436 . . . . . . . . . . . . . . . . . 18  |-  ( ( f : ( 1 ... m ) -1-1-onto-> I  /\  ( ( ( 1st `  x )  =  A  /\  ( ( 2nd `  x )  =  B  /\  dom  ( 2nd `  x )  =  I ) )  /\  m  e.  NN0 ) )  -> 
f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x ) )
92 simprr 774 . . . . . . . . . . . . . . . . . 18  |-  ( ( f : ( 1 ... m ) -1-1-onto-> I  /\  ( ( ( 1st `  x )  =  A  /\  ( ( 2nd `  x )  =  B  /\  dom  ( 2nd `  x )  =  I ) )  /\  m  e.  NN0 ) )  ->  m  e.  NN0 )
9341ad2antrl 742 . . . . . . . . . . . . . . . . . 18  |-  ( ( f : ( 1 ... m ) -1-1-onto-> I  /\  ( ( ( 1st `  x )  =  A  /\  ( ( 2nd `  x )  =  B  /\  dom  ( 2nd `  x )  =  I ) )  /\  m  e.  NN0 ) )  ->  dom  ( 2nd `  x
)  =  I )
9491, 92, 93, 61syl12anc 1290 . . . . . . . . . . . . . . . . 17  |-  ( ( f : ( 1 ... m ) -1-1-onto-> I  /\  ( ( ( 1st `  x )  =  A  /\  ( ( 2nd `  x )  =  B  /\  dom  ( 2nd `  x )  =  I ) )  /\  m  e.  NN0 ) )  ->  m  =  ( # `  I
) )
9594eqcomd 2477 . . . . . . . . . . . . . . . 16  |-  ( ( f : ( 1 ... m ) -1-1-onto-> I  /\  ( ( ( 1st `  x )  =  A  /\  ( ( 2nd `  x )  =  B  /\  dom  ( 2nd `  x )  =  I ) )  /\  m  e.  NN0 ) )  -> 
( # `  I )  =  m )
9689, 95fveq12d 5885 . . . . . . . . . . . . . . 15  |-  ( ( f : ( 1 ... m ) -1-1-onto-> I  /\  ( ( ( 1st `  x )  =  A  /\  ( ( 2nd `  x )  =  B  /\  dom  ( 2nd `  x )  =  I ) )  /\  m  e.  NN0 ) )  -> 
(  seq 1 ( ( +g  `  A ) ,  ( n  e.  NN  |->  ( B `  ( f `  n
) ) ) ) `
 ( # `  I
) )  =  (  seq 1 ( ( +g  `  ( 1st `  x ) ) ,  ( n  e.  NN  |->  ( ( 2nd `  x
) `  ( f `  n ) ) ) ) `  m ) )
9796eqeq2d 2481 . . . . . . . . . . . . . 14  |-  ( ( f : ( 1 ... m ) -1-1-onto-> I  /\  ( ( ( 1st `  x )  =  A  /\  ( ( 2nd `  x )  =  B  /\  dom  ( 2nd `  x )  =  I ) )  /\  m  e.  NN0 ) )  -> 
( s  =  (  seq 1 ( ( +g  `  A ) ,  ( n  e.  NN  |->  ( B `  ( f `  n
) ) ) ) `
 ( # `  I
) )  <->  s  =  (  seq 1 ( ( +g  `  ( 1st `  x ) ) ,  ( n  e.  NN  |->  ( ( 2nd `  x
) `  ( f `  n ) ) ) ) `  m ) ) )
9897biimpd 212 . . . . . . . . . . . . 13  |-  ( ( f : ( 1 ... m ) -1-1-onto-> I  /\  ( ( ( 1st `  x )  =  A  /\  ( ( 2nd `  x )  =  B  /\  dom  ( 2nd `  x )  =  I ) )  /\  m  e.  NN0 ) )  -> 
( s  =  (  seq 1 ( ( +g  `  A ) ,  ( n  e.  NN  |->  ( B `  ( f `  n
) ) ) ) `
 ( # `  I
) )  ->  s  =  (  seq 1
( ( +g  `  ( 1st `  x ) ) ,  ( n  e.  NN  |->  ( ( 2nd `  x ) `  (
f `  n )
) ) ) `  m ) ) )
9998impancom 447 . . . . . . . . . . . 12  |-  ( ( f : ( 1 ... m ) -1-1-onto-> I  /\  s  =  (  seq 1 ( ( +g  `  A ) ,  ( n  e.  NN  |->  ( B `  ( f `
 n ) ) ) ) `  ( # `
 I ) ) )  ->  ( (
( ( 1st `  x
)  =  A  /\  ( ( 2nd `  x
)  =  B  /\  dom  ( 2nd `  x
)  =  I ) )  /\  m  e. 
NN0 )  ->  s  =  (  seq 1
( ( +g  `  ( 1st `  x ) ) ,  ( n  e.  NN  |->  ( ( 2nd `  x ) `  (
f `  n )
) ) ) `  m ) ) )
10099com12 31 . . . . . . . . . . 11  |-  ( ( ( ( 1st `  x
)  =  A  /\  ( ( 2nd `  x
)  =  B  /\  dom  ( 2nd `  x
)  =  I ) )  /\  m  e. 
NN0 )  ->  (
( f : ( 1 ... m ) -1-1-onto-> I  /\  s  =  (  seq 1 ( ( +g  `  A ) ,  ( n  e.  NN  |->  ( B `  ( f `  n
) ) ) ) `
 ( # `  I
) ) )  -> 
s  =  (  seq 1 ( ( +g  `  ( 1st `  x
) ) ,  ( n  e.  NN  |->  ( ( 2nd `  x
) `  ( f `  n ) ) ) ) `  m ) ) )
10172, 100jcad 542 . . . . . . . . . 10  |-  ( ( ( ( 1st `  x
)  =  A  /\  ( ( 2nd `  x
)  =  B  /\  dom  ( 2nd `  x
)  =  I ) )  /\  m  e. 
NN0 )  ->  (
( f : ( 1 ... m ) -1-1-onto-> I  /\  s  =  (  seq 1 ( ( +g  `  A ) ,  ( n  e.  NN  |->  ( B `  ( f `  n
) ) ) ) `
 ( # `  I
) ) )  -> 
( f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x
)  /\  s  =  (  seq 1 ( ( +g  `  ( 1st `  x ) ) ,  ( n  e.  NN  |->  ( ( 2nd `  x
) `  ( f `  n ) ) ) ) `  m ) ) ) )
10268, 101impbid 195 . . . . . . . . 9  |-  ( ( ( ( 1st `  x
)  =  A  /\  ( ( 2nd `  x
)  =  B  /\  dom  ( 2nd `  x
)  =  I ) )  /\  m  e. 
NN0 )  ->  (
( f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x
)  /\  s  =  (  seq 1 ( ( +g  `  ( 1st `  x ) ) ,  ( n  e.  NN  |->  ( ( 2nd `  x
) `  ( f `  n ) ) ) ) `  m ) )  <->  ( f : ( 1 ... m
)
-1-1-onto-> I  /\  s  =  (  seq 1 ( ( +g  `  A ) ,  ( n  e.  NN  |->  ( B `  ( f `  n
) ) ) ) `
 ( # `  I
) ) ) ) )
103102ex 441 . . . . . . . 8  |-  ( ( ( 1st `  x
)  =  A  /\  ( ( 2nd `  x
)  =  B  /\  dom  ( 2nd `  x
)  =  I ) )  ->  ( m  e.  NN0  ->  ( (
f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x )  /\  s  =  (  seq 1 ( ( +g  `  ( 1st `  x
) ) ,  ( n  e.  NN  |->  ( ( 2nd `  x
) `  ( f `  n ) ) ) ) `  m ) )  <->  ( f : ( 1 ... m
)
-1-1-onto-> I  /\  s  =  (  seq 1 ( ( +g  `  A ) ,  ( n  e.  NN  |->  ( B `  ( f `  n
) ) ) ) `
 ( # `  I
) ) ) ) ) )
10415, 19, 24, 103syl12anc 1290 . . . . . . 7  |-  ( (
ph  /\  x  =  <. A ,  B >. )  ->  ( m  e. 
NN0  ->  ( ( f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x
)  /\  s  =  (  seq 1 ( ( +g  `  ( 1st `  x ) ) ,  ( n  e.  NN  |->  ( ( 2nd `  x
) `  ( f `  n ) ) ) ) `  m ) )  <->  ( f : ( 1 ... m
)
-1-1-onto-> I  /\  s  =  (  seq 1 ( ( +g  `  A ) ,  ( n  e.  NN  |->  ( B `  ( f `  n
) ) ) ) `
 ( # `  I
) ) ) ) ) )
105104imp 436 . . . . . 6  |-  ( ( ( ph  /\  x  =  <. A ,  B >. )  /\  m  e. 
NN0 )  ->  (
( f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x
)  /\  s  =  (  seq 1 ( ( +g  `  ( 1st `  x ) ) ,  ( n  e.  NN  |->  ( ( 2nd `  x
) `  ( f `  n ) ) ) ) `  m ) )  <->  ( f : ( 1 ... m
)
-1-1-onto-> I  /\  s  =  (  seq 1 ( ( +g  `  A ) ,  ( n  e.  NN  |->  ( B `  ( f `  n
) ) ) ) `
 ( # `  I
) ) ) ) )
106105exbidv 1776 . . . . 5  |-  ( ( ( ph  /\  x  =  <. A ,  B >. )  /\  m  e. 
NN0 )  ->  ( E. f ( f : ( 1 ... m
)
-1-1-onto-> dom  ( 2nd `  x
)  /\  s  =  (  seq 1 ( ( +g  `  ( 1st `  x ) ) ,  ( n  e.  NN  |->  ( ( 2nd `  x
) `  ( f `  n ) ) ) ) `  m ) )  <->  E. f ( f : ( 1 ... m ) -1-1-onto-> I  /\  s  =  (  seq 1
( ( +g  `  A
) ,  ( n  e.  NN  |->  ( B `
 ( f `  n ) ) ) ) `  ( # `  I ) ) ) ) )
107106rexbidva 2889 . . . 4  |-  ( (
ph  /\  x  =  <. A ,  B >. )  ->  ( E. m  e.  NN0  E. f ( f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x )  /\  s  =  (  seq 1 ( ( +g  `  ( 1st `  x
) ) ,  ( n  e.  NN  |->  ( ( 2nd `  x
) `  ( f `  n ) ) ) ) `  m ) )  <->  E. m  e.  NN0  E. f ( f : ( 1 ... m
)
-1-1-onto-> I  /\  s  =  (  seq 1 ( ( +g  `  A ) ,  ( n  e.  NN  |->  ( B `  ( f `  n
) ) ) ) `
 ( # `  I
) ) ) ) )
108107iotabidv 5574 . . 3  |-  ( (
ph  /\  x  =  <. A ,  B >. )  ->  ( iota s E. m  e.  NN0  E. f ( f : ( 1 ... m
)
-1-1-onto-> dom  ( 2nd `  x
)  /\  s  =  (  seq 1 ( ( +g  `  ( 1st `  x ) ) ,  ( n  e.  NN  |->  ( ( 2nd `  x
) `  ( f `  n ) ) ) ) `  m ) ) )  =  ( iota s E. m  e.  NN0  E. f ( f : ( 1 ... m ) -1-1-onto-> I  /\  s  =  (  seq 1 ( ( +g  `  A ) ,  ( n  e.  NN  |->  ( B `  ( f `
 n ) ) ) ) `  ( # `
 I ) ) ) ) )
109 eleq1 2537 . . . . . . . . . 10  |-  ( t  =  I  ->  (
t  e.  Fin  <->  I  e.  Fin ) )
110 feq2 5721 . . . . . . . . . 10  |-  ( t  =  I  ->  ( B : t --> ( Base `  A )  <->  B :
I --> ( Base `  A
) ) )
111109, 110anbi12d 725 . . . . . . . . 9  |-  ( t  =  I  ->  (
( t  e.  Fin  /\  B : t --> (
Base `  A )
)  <->  ( I  e. 
Fin  /\  B :
I --> ( Base `  A
) ) ) )
112111ceqsexgv 3159 . . . . . . . 8  |-  ( I  e.  Fin  ->  ( E. t ( t  =  I  /\  ( t  e.  Fin  /\  B : t --> ( Base `  A ) ) )  <-> 
( I  e.  Fin  /\  B : I --> ( Base `  A ) ) ) )
1139, 112syl 17 . . . . . . 7  |-  ( ph  ->  ( E. t ( t  =  I  /\  ( t  e.  Fin  /\  B : t --> (
Base `  A )
) )  <->  ( I  e.  Fin  /\  B :
I --> ( Base `  A
) ) ) )
1149, 8, 113mpbir2and 936 . . . . . 6  |-  ( ph  ->  E. t ( t  =  I  /\  (
t  e.  Fin  /\  B : t --> ( Base `  A ) ) ) )
115 exsimpr 1738 . . . . . 6  |-  ( E. t ( t  =  I  /\  ( t  e.  Fin  /\  B : t --> ( Base `  A ) ) )  ->  E. t ( t  e.  Fin  /\  B : t --> ( Base `  A ) ) )
116114, 115syl 17 . . . . 5  |-  ( ph  ->  E. t ( t  e.  Fin  /\  B : t --> ( Base `  A ) ) )
117 df-rex 2762 . . . . 5  |-  ( E. t  e.  Fin  B : t --> ( Base `  A )  <->  E. t
( t  e.  Fin  /\  B : t --> (
Base `  A )
) )
118116, 117sylibr 217 . . . 4  |-  ( ph  ->  E. t  e.  Fin  B : t --> ( Base `  A ) )
119 eleq1 2537 . . . . . . 7  |-  ( y  =  A  ->  (
y  e. CMnd  <->  A  e. CMnd ) )
120 fveq2 5879 . . . . . . . . 9  |-  ( y  =  A  ->  ( Base `  y )  =  ( Base `  A
) )
121120feq3d 5726 . . . . . . . 8  |-  ( y  =  A  ->  (
z : t --> (
Base `  y )  <->  z : t --> ( Base `  A ) ) )
122121rexbidv 2892 . . . . . . 7  |-  ( y  =  A  ->  ( E. t  e.  Fin  z : t --> ( Base `  y )  <->  E. t  e.  Fin  z : t --> ( Base `  A
) ) )
123119, 122anbi12d 725 . . . . . 6  |-  ( y  =  A  ->  (
( y  e. CMnd  /\  E. t  e.  Fin  z : t --> ( Base `  y ) )  <->  ( A  e. CMnd  /\  E. t  e. 
Fin  z : t --> ( Base `  A
) ) ) )
124 feq1 5720 . . . . . . . 8  |-  ( z  =  B  ->  (
z : t --> (
Base `  A )  <->  B : t --> ( Base `  A ) ) )
125124rexbidv 2892 . . . . . . 7  |-  ( z  =  B  ->  ( E. t  e.  Fin  z : t --> ( Base `  A )  <->  E. t  e.  Fin  B : t --> ( Base `  A
) ) )
126125anbi2d 718 . . . . . 6  |-  ( z  =  B  ->  (
( A  e. CMnd  /\  E. t  e.  Fin  z : t --> ( Base `  A ) )  <->  ( A  e. CMnd  /\  E. t  e. 
Fin  B : t --> ( Base `  A
) ) ) )
127123, 126opelopabg 4719 . . . . 5  |-  ( ( A  e. CMnd  /\  B  e.  _V )  ->  ( <. A ,  B >.  e. 
{ <. y ,  z
>.  |  ( y  e. CMnd  /\  E. t  e. 
Fin  z : t --> ( Base `  y
) ) }  <->  ( A  e. CMnd  /\  E. t  e. 
Fin  B : t --> ( Base `  A
) ) ) )
1286, 11, 127syl2anc 673 . . . 4  |-  ( ph  ->  ( <. A ,  B >.  e.  { <. y ,  z >.  |  ( y  e. CMnd  /\  E. t  e.  Fin  z : t --> ( Base `  y
) ) }  <->  ( A  e. CMnd  /\  E. t  e. 
Fin  B : t --> ( Base `  A
) ) ) )
1296, 118, 128mpbir2and 936 . . 3  |-  ( ph  -> 
<. A ,  B >.  e. 
{ <. y ,  z
>.  |  ( y  e. CMnd  /\  E. t  e. 
Fin  z : t --> ( Base `  y
) ) } )
130 iotaex 5570 . . . 4  |-  ( iota s E. m  e. 
NN0  E. f ( f : ( 1 ... m ) -1-1-onto-> I  /\  s  =  (  seq 1
( ( +g  `  A
) ,  ( n  e.  NN  |->  ( B `
 ( f `  n ) ) ) ) `  ( # `  I ) ) ) )  e.  _V
131130a1i 11 . . 3  |-  ( ph  ->  ( iota s E. m  e.  NN0  E. f
( f : ( 1 ... m ) -1-1-onto-> I  /\  s  =  (  seq 1 ( ( +g  `  A ) ,  ( n  e.  NN  |->  ( B `  ( f `  n
) ) ) ) `
 ( # `  I
) ) ) )  e.  _V )
1323, 108, 129, 131fvmptd 5969 . 2  |-  ( ph  ->  ( FinSum  `  <. A ,  B >. )  =  ( iota s E. m  e.  NN0  E. f ( f : ( 1 ... m ) -1-1-onto-> I  /\  s  =  (  seq 1 ( ( +g  `  A ) ,  ( n  e.  NN  |->  ( B `  ( f `
 n ) ) ) ) `  ( # `
 I ) ) ) ) )
1331, 132syl5eq 2517 1  |-  ( ph  ->  ( A FinSum  B )  =  ( iota s E. m  e.  NN0  E. f ( f : ( 1 ... m
)
-1-1-onto-> I  /\  s  =  (  seq 1 ( ( +g  `  A ) ,  ( n  e.  NN  |->  ( B `  ( f `  n
) ) ) ) `
 ( # `  I
) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452   T. wtru 1453   E.wex 1671    e. wcel 1904   E.wrex 2757   _Vcvv 3031   <.cop 3965   {copab 4453    |-> cmpt 4454   dom cdm 4839   iotacio 5551   -->wf 5585   -onto->wfo 5587   -1-1-onto->wf1o 5588   ` cfv 5589  (class class class)co 6308   1stc1st 6810   2ndc2nd 6811   Fincfn 7587   1c1 9558   NNcn 10631   NN0cn0 10893   ...cfz 11810    seqcseq 12251   #chash 12553   Basecbs 15199   +g cplusg 15268  CMndccmn 17508   FinSum cfinsum 31770
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-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
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  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-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-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-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-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-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  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-nn 10632  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-seq 12252  df-hash 12554  df-bj-finsum 31771
This theorem is referenced by: (None)
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