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Theorem bj-finsumval0 32588
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 6099 . 2  |-  ( A FinSum  B )  =  ( FinSum  `  <. A ,  B >. )
2 df-bj-finsum 32587 . . . 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 5700 . . . . . . . . 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 5955 . . . . . . . . . . . 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 6594 . . . . . . . . . 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 2475 . . . . . . . 8  |-  ( (
ph  /\  x  =  <. A ,  B >. )  ->  ( 1st `  x
)  =  A )
164fveq2d 5700 . . . . . . . . 9  |-  ( (
ph  /\  x  =  <. A ,  B >. )  ->  ( 2nd `  x
)  =  ( 2nd `  <. A ,  B >. ) )
17 op2ndg 6595 . . . . . . . . . 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 2475 . . . . . . . 8  |-  ( (
ph  /\  x  =  <. A ,  B >. )  ->  ( 2nd `  x
)  =  B )
2019dmeqd 5047 . . . . . . . . 9  |-  ( (
ph  /\  x  =  <. A ,  B >. )  ->  dom  ( 2nd `  x )  =  dom  B )
21 fdm 5568 . . . . . . . . . . 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 2475 . . . . . . . 8  |-  ( (
ph  /\  x  =  <. A ,  B >. )  ->  dom  ( 2nd `  x )  =  I )
25 f1oeq3 5639 . . . . . . . . . . . . . . 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 2444 . . . . . . . . . . . . . . . . 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 5700 . . . . . . . . . . . . . . . . . 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 5698 . . . . . . . . . . . . . . . . . . 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 4383 . . . . . . . . . . . . . . . . . 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 11820 . . . . . . . . . . . . . . . 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 12122 . . . . . . . . . . . . . . . . . . . . 21  |-  ( m  e.  NN0  ->  ( # `  ( 1 ... m
) )  =  m )
4544eqcomd 2448 . . . . . . . . . . . . . . . . . . . 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 11800 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x )  /\  m  e.  NN0 )  -> 
( 1 ... m
)  e.  Fin )
48 f1ofo 5653 . . . . . . . . . . . . . . . . . . . . . . . 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 6551 . . . . . . . . . . . . . . . . . . . . . . 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 1793 . . . . . . . . . . . . . . . . . . . . 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 12127 . . . . . . . . . . . . . . . . . . . 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 2475 . . . . . . . . . . . . . . . . . 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 5700 . . . . . . . . . . . . . . . . . 18  |-  ( ( f : ( 1 ... m ) -1-1-onto-> dom  ( 2nd `  x )  /\  ( m  e.  NN0  /\ 
dom  ( 2nd `  x
)  =  I ) )  ->  ( # `  dom  ( 2nd `  x ) )  =  ( # `  I ) )
6158, 60eqtrd 2475 . . . . . . . . . . . . . . . . 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 5702 . . . . . . . . . . . . . . 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 2454 . . . . . . . . . . . . . 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 2444 . . . . . . . . . . . . . . . . 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 1373 . . . . . . . . . . . . . . . . . . . . 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 2448 . . . . . . . . . . . . . . . . . . 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 5700 . . . . . . . . . . . . . . . . 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 2448 . . . . . . . . . . . . . . . . . . . . . 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 5698 . . . . . . . . . . . . . . . . . . 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 4383 . . . . . . . . . . . . . . . . 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 11820 . . . . . . . . . . . . . . . 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 1216 . . . . . . . . . . . . . . . . 17  |-  ( ( f : ( 1 ... m ) -1-1-onto-> I  /\  ( ( ( 1st `  x )  =  A  /\  ( ( 2nd `  x )  =  B  /\  dom  ( 2nd `  x )  =  I ) )  /\  m  e.  NN0 ) )  ->  m  =  ( # `  I
) )
9594eqcomd 2448 . . . . . . . . . . . . . . . 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 5702 . . . . . . . . . . . . . . 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 2454 . . . . . . . . . . . . . 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 1216 . . . . . . 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 1680 . . . . 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 2737 . . . 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 5407 . . 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 2503 . . . . . . . . . 10  |-  ( t  =  I  ->  (
t  e.  Fin  <->  I  e.  Fin ) )
110 feq2 5548 . . . . . . . . . 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 3097 . . . . . . . 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 913 . . . . . 6  |-  ( ph  ->  E. t ( t  =  I  /\  (
t  e.  Fin  /\  B : t --> ( Base `  A ) ) ) )
115 exsimpr 1645 . . . . . 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 2726 . . . . 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 2503 . . . . . . 7  |-  ( y  =  A  ->  (
y  e. CMnd  <->  A  e. CMnd ) )
120 eqidd 2444 . . . . . . . . 9  |-  ( y  =  A  ->  t  =  t )
121 fveq2 5696 . . . . . . . . 9  |-  ( y  =  A  ->  ( Base `  y )  =  ( Base `  A
) )
122120, 121feq23d 5559 . . . . . . . 8  |-  ( y  =  A  ->  (
z : t --> (
Base `  y )  <->  z : t --> ( Base `  A ) ) )
123122rexbidv 2741 . . . . . . 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 5547 . . . . . . . 8  |-  ( z  =  B  ->  (
z : t --> (
Base `  A )  <->  B : t --> ( Base `  A ) ) )
126125rexbidv 2741 . . . . . . 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 4612 . . . . 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 913 . . 3  |-  ( ph  -> 
<. A ,  B >.  e. 
{ <. y ,  z
>.  |  ( y  e. CMnd  /\  E. t  e. 
Fin  z : t --> ( Base `  y
) ) } )
131 iotaex 5403 . . . 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 5784 . 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 2487 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 1369   T. wtru 1370   E.wex 1586    e. wcel 1756   E.wrex 2721   _Vcvv 2977   <.cop 3888   {copab 4354    e. cmpt 4355   dom cdm 4845   iotacio 5384   -->wf 5419   -onto->wfo 5421   -1-1-onto->wf1o 5422   ` cfv 5423  (class class class)co 6096   1stc1st 6580   2ndc2nd 6581   Fincfn 7315   1c1 9288   NNcn 10327   NN0cn0 10584   ...cfz 11442    seqcseq 11811   #chash 12108   Basecbs 14179   +g cplusg 14243  CMndccmn 16282   FinSum cfinsum 32586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-card 8114  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-n0 10585  df-z 10652  df-uz 10867  df-fz 11443  df-seq 11812  df-hash 12109  df-bj-finsum 32587
This theorem is referenced by: (None)
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