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Theorem bj-finsumval0 33735
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 6285 . 2  |-  ( A FinSum  B )  =  ( FinSum  `  <. A ,  B >. )
2 df-bj-finsum 33734 . . . 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 461 . . . . . . . . . 10  |-  ( (
ph  /\  x  =  <. A ,  B >. )  ->  x  =  <. A ,  B >. )
54fveq2d 5868 . . . . . . . . 9  |-  ( (
ph  /\  x  =  <. A ,  B >. )  ->  ( 1st `  x
)  =  ( 1st `  <. A ,  B >. ) )
6 bj-finsumval0.1 . . . . . . . . . . 11  |-  ( ph  ->  A  e. CMnd )
76adantr 465 . . . . . . . . . 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 6131 . . . . . . . . . . . 12  |-  ( ( B : I --> ( Base `  A )  /\  I  e.  Fin )  ->  B  e.  _V )
118, 9, 10syl2anc 661 . . . . . . . . . . 11  |-  ( ph  ->  B  e.  _V )
1211adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  x  =  <. A ,  B >. )  ->  B  e.  _V )
13 op1stg 6793 . . . . . . . . . 10  |-  ( ( A  e. CMnd  /\  B  e.  _V )  ->  ( 1st `  <. A ,  B >. )  =  A )
147, 12, 13syl2anc 661 . . . . . . . . 9  |-  ( (
ph  /\  x  =  <. A ,  B >. )  ->  ( 1st `  <. A ,  B >. )  =  A )
155, 14eqtrd 2508 . . . . . . . 8  |-  ( (
ph  /\  x  =  <. A ,  B >. )  ->  ( 1st `  x
)  =  A )
164fveq2d 5868 . . . . . . . . 9  |-  ( (
ph  /\  x  =  <. A ,  B >. )  ->  ( 2nd `  x
)  =  ( 2nd `  <. A ,  B >. ) )
17 op2ndg 6794 . . . . . . . . . 10  |-  ( ( A  e. CMnd  /\  B  e.  _V )  ->  ( 2nd `  <. A ,  B >. )  =  B )
187, 12, 17syl2anc 661 . . . . . . . . 9  |-  ( (
ph  /\  x  =  <. A ,  B >. )  ->  ( 2nd `  <. A ,  B >. )  =  B )
1916, 18eqtrd 2508 . . . . . . . 8  |-  ( (
ph  /\  x  =  <. A ,  B >. )  ->  ( 2nd `  x
)  =  B )
2019dmeqd 5203 . . . . . . . . 9  |-  ( (
ph  /\  x  =  <. A ,  B >. )  ->  dom  ( 2nd `  x )  =  dom  B )
21 fdm 5733 . . . . . . . . . . 11  |-  ( B : I --> ( Base `  A )  ->  dom  B  =  I )
228, 21syl 16 . . . . . . . . . 10  |-  ( ph  ->  dom  B  =  I )
2322adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  x  =  <. A ,  B >. )  ->  dom  B  =  I )
2420, 23eqtrd 2508 . . . . . . . 8  |-  ( (
ph  /\  x  =  <. A ,  B >. )  ->  dom  ( 2nd `  x )  =  I )
25 f1oeq3 5807 . . . . . . . . . . . . . . 15  |-  ( dom  ( 2nd `  x
)  =  I  -> 
( f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x
)  <->  f : ( 1 ... m ) -1-1-onto-> I ) )
2625biimpd 207 . . . . . . . . . . . . . 14  |-  ( dom  ( 2nd `  x
)  =  I  -> 
( f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x
)  ->  f :
( 1 ... m
)
-1-1-onto-> I ) )
2726ad2antll 728 . . . . . . . . . . . . 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 468 . . . . . . . . . . . 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 465 . . . . . . . . . . 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 2468 . . . . . . . . . . . . . . . . 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 755 . . . . . . . . . . . . . . . . . . 19  |-  ( ( f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x )  /\  ( ( 1st `  x
)  =  A  /\  ( ( 2nd `  x
)  =  B  /\  dom  ( 2nd `  x
)  =  I ) ) )  ->  ( 1st `  x )  =  A )
3231fveq2d 5868 . . . . . . . . . . . . . . . . . 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 716 . . . . . . . . . . . . . . . . 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 763 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x )  /\  ( ( 1st `  x
)  =  A  /\  ( ( 2nd `  x
)  =  B  /\  dom  ( 2nd `  x
)  =  I ) ) )  ->  ( 2nd `  x )  =  B )
3534adantr 465 . . . . . . . . . . . . . . . . . . . 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 5866 . . . . . . . . . . . . . . . . . . 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 4533 . . . . . . . . . . . . . . . . . 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 716 . . . . . . . . . . . . . . . . 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 12080 . . . . . . . . . . . . . . . 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 461 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( 1st `  x
)  =  A  /\  ( ( 2nd `  x
)  =  B  /\  dom  ( 2nd `  x
)  =  I ) )  /\  m  e. 
NN0 )  ->  m  e.  NN0 )
41 simprr 756 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( 1st `  x
)  =  A  /\  ( ( 2nd `  x
)  =  B  /\  dom  ( 2nd `  x
)  =  I ) )  ->  dom  ( 2nd `  x )  =  I )
4241adantr 465 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( 1st `  x
)  =  A  /\  ( ( 2nd `  x
)  =  B  /\  dom  ( 2nd `  x
)  =  I ) )  /\  m  e. 
NN0 )  ->  dom  ( 2nd `  x )  =  I )
4340, 42jca 532 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( 1st `  x
)  =  A  /\  ( ( 2nd `  x
)  =  B  /\  dom  ( 2nd `  x
)  =  I ) )  /\  m  e. 
NN0 )  ->  (
m  e.  NN0  /\  dom  ( 2nd `  x
)  =  I ) )
44 hashfz1 12383 . . . . . . . . . . . . . . . . . . . . 21  |-  ( m  e.  NN0  ->  ( # `  ( 1 ... m
) )  =  m )
4544eqcomd 2475 . . . . . . . . . . . . . . . . . . . 20  |-  ( m  e.  NN0  ->  m  =  ( # `  (
1 ... m ) ) )
4645ad2antrl 727 . . . . . . . . . . . . . . . . . . 19  |-  ( ( f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x )  /\  ( m  e.  NN0  /\ 
dom  ( 2nd `  x
)  =  I ) )  ->  m  =  ( # `  ( 1 ... m ) ) )
47 fzfid 12047 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x )  /\  m  e.  NN0 )  -> 
( 1 ... m
)  e.  Fin )
48 f1ofo 5821 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x
)  ->  f :
( 1 ... m
) -onto-> dom  ( 2nd `  x
) )
4948adantr 465 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x )  /\  m  e.  NN0 )  -> 
f : ( 1 ... m ) -onto-> dom  ( 2nd `  x
) )
50 fornex 6750 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( 1 ... m )  e.  Fin  ->  (
f : ( 1 ... m ) -onto-> dom  ( 2nd `  x
)  ->  dom  ( 2nd `  x )  e.  _V ) )
5147, 49, 50sylc 60 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x )  /\  m  e.  NN0 )  ->  dom  ( 2nd `  x
)  e.  _V )
5247, 51jca 532 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x )  /\  m  e.  NN0 )  -> 
( ( 1 ... m )  e.  Fin  /\ 
dom  ( 2nd `  x
)  e.  _V )
)
5352adantrr 716 . . . . . . . . . . . . . . . . . . . 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 1806 . . . . . . . . . . . . . . . . . . . . 21  |-  ( f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x
)  ->  E. f 
f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x ) )
5554adantr 465 . . . . . . . . . . . . . . . . . . . 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 12388 . . . . . . . . . . . . . . . . . . . 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 60 . . . . . . . . . . . . . . . . . . 19  |-  ( ( f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x )  /\  ( m  e.  NN0  /\ 
dom  ( 2nd `  x
)  =  I ) )  ->  ( # `  (
1 ... m ) )  =  ( # `  dom  ( 2nd `  x ) ) )
5846, 57eqtrd 2508 . . . . . . . . . . . . . . . . . 18  |-  ( ( f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x )  /\  ( m  e.  NN0  /\ 
dom  ( 2nd `  x
)  =  I ) )  ->  m  =  ( # `  dom  ( 2nd `  x ) ) )
59 simprr 756 . . . . . . . . . . . . . . . . . . 19  |-  ( ( f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x )  /\  ( m  e.  NN0  /\ 
dom  ( 2nd `  x
)  =  I ) )  ->  dom  ( 2nd `  x )  =  I )
6059fveq2d 5868 . . . . . . . . . . . . . . . . . 18  |-  ( ( f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x )  /\  ( m  e.  NN0  /\ 
dom  ( 2nd `  x
)  =  I ) )  ->  ( # `  dom  ( 2nd `  x ) )  =  ( # `  I ) )
6158, 60eqtrd 2508 . . . . . . . . . . . . . . . . 17  |-  ( ( f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x )  /\  ( m  e.  NN0  /\ 
dom  ( 2nd `  x
)  =  I ) )  ->  m  =  ( # `  I ) )
6243, 61sylan2 474 . . . . . . . . . . . . . . . 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 5870 . . . . . . . . . . . . . . 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 207 . . . . . . . . . . . . 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 440 . . . . . . . . . . . 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 533 . . . . . . . . . 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 223 . . . . . . . . . . . . . 14  |-  ( dom  ( 2nd `  x
)  =  I  -> 
( f : ( 1 ... m ) -1-1-onto-> I  ->  f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x
) ) )
7069ad2antll 728 . . . . . . . . . . . . 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 465 . . . . . . . . . . . 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 468 . . . . . . . . . . 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 2468 . . . . . . . . . . . . . . . . 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 457 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( 1st `  x
)  =  A  /\  ( ( 2nd `  x
)  =  B  /\  dom  ( 2nd `  x
)  =  I ) )  ->  ( 1st `  x )  =  A )
75 tru 1383 . . . . . . . . . . . . . . . . . . . . 21  |- T.
7674, 75jctir 538 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( 1st `  x
)  =  A  /\  ( ( 2nd `  x
)  =  B  /\  dom  ( 2nd `  x
)  =  I ) )  ->  ( ( 1st `  x )  =  A  /\ T.  ) )
7776ad2antrl 727 . . . . . . . . . . . . . . . . . . 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 457 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( 1st `  x
)  =  A  /\ T.  )  ->  ( 1st `  x )  =  A )
7978eqcomd 2475 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( 1st `  x
)  =  A  /\ T.  )  ->  A  =  ( 1st `  x
) )
8077, 79syl 16 . . . . . . . . . . . . . . . . . 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 5868 . . . . . . . . . . . . . . . . 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 457 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( 2nd `  x
)  =  B  /\  dom  ( 2nd `  x
)  =  I )  ->  ( 2nd `  x
)  =  B )
8382eqcomd 2475 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( 2nd `  x
)  =  B  /\  dom  ( 2nd `  x
)  =  I )  ->  B  =  ( 2nd `  x ) )
8483ad2antll 728 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( f : ( 1 ... m ) -1-1-onto-> I  /\  ( ( 1st `  x
)  =  A  /\  ( ( 2nd `  x
)  =  B  /\  dom  ( 2nd `  x
)  =  I ) ) )  ->  B  =  ( 2nd `  x
) )
8584adantr 465 . . . . . . . . . . . . . . . . . . . 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 5866 . . . . . . . . . . . . . . . . . . 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 720 . . . . . . . . . . . . . . . . . 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 4533 . . . . . . . . . . . . . . . . 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 12080 . . . . . . . . . . . . . . . 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 429 . . . . . . . . . . . . . . . . . 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 756 . . . . . . . . . . . . . . . . . 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 727 . . . . . . . . . . . . . . . . . 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 1226 . . . . . . . . . . . . . . . . 17  |-  ( ( f : ( 1 ... m ) -1-1-onto-> I  /\  ( ( ( 1st `  x )  =  A  /\  ( ( 2nd `  x )  =  B  /\  dom  ( 2nd `  x )  =  I ) )  /\  m  e.  NN0 ) )  ->  m  =  ( # `  I
) )
9594eqcomd 2475 . . . . . . . . . . . . . . . 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 5870 . . . . . . . . . . . . . . 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 207 . . . . . . . . . . . . 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 440 . . . . . . . . . . . 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 533 . . . . . . . . . 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 191 . . . . . . . . 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 434 . . . . . . . 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 1226 . . . . . . 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 429 . . . . . 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 1690 . . . . 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 2970 . . . 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 5570 . . 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 2539 . . . . . . . . . 10  |-  ( t  =  I  ->  (
t  e.  Fin  <->  I  e.  Fin ) )
110 feq2 5712 . . . . . . . . . 10  |-  ( t  =  I  ->  ( B : t --> ( Base `  A )  <->  B :
I --> ( Base `  A
) ) )
111109, 110anbi12d 710 . . . . . . . . 9  |-  ( t  =  I  ->  (
( t  e.  Fin  /\  B : t --> (
Base `  A )
)  <->  ( I  e. 
Fin  /\  B :
I --> ( Base `  A
) ) ) )
112111ceqsexgv 3236 . . . . . . . 8  |-  ( I  e.  Fin  ->  ( E. t ( t  =  I  /\  ( t  e.  Fin  /\  B : t --> ( Base `  A ) ) )  <-> 
( I  e.  Fin  /\  B : I --> ( Base `  A ) ) ) )
1139, 112syl 16 . . . . . . 7  |-  ( ph  ->  ( E. t ( t  =  I  /\  ( t  e.  Fin  /\  B : t --> (
Base `  A )
) )  <->  ( I  e.  Fin  /\  B :
I --> ( Base `  A
) ) ) )
1149, 8, 113mpbir2and 920 . . . . . 6  |-  ( ph  ->  E. t ( t  =  I  /\  (
t  e.  Fin  /\  B : t --> ( Base `  A ) ) ) )
115 exsimpr 1655 . . . . . 6  |-  ( E. t ( t  =  I  /\  ( t  e.  Fin  /\  B : t --> ( Base `  A ) ) )  ->  E. t ( t  e.  Fin  /\  B : t --> ( Base `  A ) ) )
116114, 115syl 16 . . . . 5  |-  ( ph  ->  E. t ( t  e.  Fin  /\  B : t --> ( Base `  A ) ) )
117 df-rex 2820 . . . . 5  |-  ( E. t  e.  Fin  B : t --> ( Base `  A )  <->  E. t
( t  e.  Fin  /\  B : t --> (
Base `  A )
) )
118116, 117sylibr 212 . . . 4  |-  ( ph  ->  E. t  e.  Fin  B : t --> ( Base `  A ) )
119 eleq1 2539 . . . . . . 7  |-  ( y  =  A  ->  (
y  e. CMnd  <->  A  e. CMnd ) )
120 eqidd 2468 . . . . . . . . 9  |-  ( y  =  A  ->  t  =  t )
121 fveq2 5864 . . . . . . . . 9  |-  ( y  =  A  ->  ( Base `  y )  =  ( Base `  A
) )
122120, 121feq23d 5724 . . . . . . . 8  |-  ( y  =  A  ->  (
z : t --> (
Base `  y )  <->  z : t --> ( Base `  A ) ) )
123122rexbidv 2973 . . . . . . 7  |-  ( y  =  A  ->  ( E. t  e.  Fin  z : t --> ( Base `  y )  <->  E. t  e.  Fin  z : t --> ( Base `  A
) ) )
124119, 123anbi12d 710 . . . . . 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
) ) ) )
125 feq1 5711 . . . . . . . 8  |-  ( z  =  B  ->  (
z : t --> (
Base `  A )  <->  B : t --> ( Base `  A ) ) )
126125rexbidv 2973 . . . . . . 7  |-  ( z  =  B  ->  ( E. t  e.  Fin  z : t --> ( Base `  A )  <->  E. t  e.  Fin  B : t --> ( Base `  A
) ) )
127126anbi2d 703 . . . . . 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
) ) ) )
128124, 127opelopabg 4765 . . . . 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
) ) ) )
1296, 11, 128syl2anc 661 . . . 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
) ) ) )
1306, 118, 129mpbir2and 920 . . 3  |-  ( ph  -> 
<. A ,  B >.  e. 
{ <. y ,  z
>.  |  ( y  e. CMnd  /\  E. t  e. 
Fin  z : t --> ( Base `  y
) ) } )
131 iotaex 5566 . . . 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
132131a1i 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 )
1333, 108, 130, 132fvmptd 5953 . 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 ) ) ) ) )
1341, 133syl5eq 2520 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 184    /\ wa 369    = wceq 1379   T. wtru 1380   E.wex 1596    e. wcel 1767   E.wrex 2815   _Vcvv 3113   <.cop 4033   {copab 4504    |-> cmpt 4505   dom cdm 4999   iotacio 5547   -->wf 5582   -onto->wfo 5584   -1-1-onto->wf1o 5585   ` cfv 5586  (class class class)co 6282   1stc1st 6779   2ndc2nd 6780   Fincfn 7513   1c1 9489   NNcn 10532   NN0cn0 10791   ...cfz 11668    seqcseq 12071   #chash 12369   Basecbs 14486   +g cplusg 14551  CMndccmn 16594   FinSum cfinsum 33733
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-seq 12072  df-hash 12370  df-bj-finsum 33734
This theorem is referenced by: (None)
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