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Theorem fsumcnv 11146
Description: Transform a region of summation by using the converse operation. (Contributed by Mario Carneiro, 23-Apr-2014.)
Hypotheses
Ref Expression
fsumcnv.1  |-  ( x  =  <. j ,  k
>.  ->  B  =  D )
fsumcnv.2  |-  ( y  =  <. k ,  j
>.  ->  C  =  D )
fsumcnv.3  |-  ( ph  ->  A  e.  Fin )
fsumcnv.4  |-  ( ph  ->  Rel  A )
fsumcnv.5  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  CC )
Assertion
Ref Expression
fsumcnv  |-  ( ph  -> 
sum_ x  e.  A  B  =  sum_ y  e.  `'  A C )
Distinct variable groups:    x, y, A   
j, k, y, B   
x, j, C, k    ph, x, y    x, D, y
Allowed substitution hints:    ph( j, k)    A( j, k)    B( x)    C( y)    D( j, k)

Proof of Theorem fsumcnv
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 csbeq1a 2981 . . . 4  |-  ( x  =  <. ( 2nd `  y
) ,  ( 1st `  y ) >.  ->  B  =  [_ <. ( 2nd `  y
) ,  ( 1st `  y ) >.  /  x ]_ B )
2 2ndexg 6032 . . . . . 6  |-  ( y  e.  _V  ->  ( 2nd `  y )  e. 
_V )
32elv 2662 . . . . 5  |-  ( 2nd `  y )  e.  _V
4 1stexg 6031 . . . . . 6  |-  ( y  e.  _V  ->  ( 1st `  y )  e. 
_V )
54elv 2662 . . . . 5  |-  ( 1st `  y )  e.  _V
6 vex 2661 . . . . . . . 8  |-  j  e. 
_V
7 vex 2661 . . . . . . . 8  |-  k  e. 
_V
86, 7opex 4119 . . . . . . 7  |-  <. j ,  k >.  e.  _V
9 fsumcnv.1 . . . . . . 7  |-  ( x  =  <. j ,  k
>.  ->  B  =  D )
108, 9csbie 3013 . . . . . 6  |-  [_ <. j ,  k >.  /  x ]_ B  =  D
11 opeq12 3675 . . . . . . 7  |-  ( ( j  =  ( 2nd `  y )  /\  k  =  ( 1st `  y
) )  ->  <. j ,  k >.  =  <. ( 2nd `  y ) ,  ( 1st `  y
) >. )
1211csbeq1d 2979 . . . . . 6  |-  ( ( j  =  ( 2nd `  y )  /\  k  =  ( 1st `  y
) )  ->  [_ <. j ,  k >.  /  x ]_ B  =  [_ <. ( 2nd `  y ) ,  ( 1st `  y
) >.  /  x ]_ B )
1310, 12syl5eqr 2162 . . . . 5  |-  ( ( j  =  ( 2nd `  y )  /\  k  =  ( 1st `  y
) )  ->  D  =  [_ <. ( 2nd `  y
) ,  ( 1st `  y ) >.  /  x ]_ B )
143, 5, 13csbie2 3017 . . . 4  |-  [_ ( 2nd `  y )  / 
j ]_ [_ ( 1st `  y )  /  k ]_ D  =  [_ <. ( 2nd `  y ) ,  ( 1st `  y
) >.  /  x ]_ B
151, 14syl6eqr 2166 . . 3  |-  ( x  =  <. ( 2nd `  y
) ,  ( 1st `  y ) >.  ->  B  =  [_ ( 2nd `  y
)  /  j ]_ [_ ( 1st `  y
)  /  k ]_ D )
16 fsumcnv.4 . . . 4  |-  ( ph  ->  Rel  A )
17 fsumcnv.3 . . . 4  |-  ( ph  ->  A  e.  Fin )
18 relcnvfi 6795 . . . 4  |-  ( ( Rel  A  /\  A  e.  Fin )  ->  `' A  e.  Fin )
1916, 17, 18syl2anc 406 . . 3  |-  ( ph  ->  `' A  e.  Fin )
20 relcnv 4885 . . . . 5  |-  Rel  `' A
21 cnvf1o 6088 . . . . 5  |-  ( Rel  `' A  ->  ( z  e.  `' A  |->  U. `' { z } ) : `' A -1-1-onto-> `' `' A )
2220, 21ax-mp 5 . . . 4  |-  ( z  e.  `' A  |->  U. `' { z } ) : `' A -1-1-onto-> `' `' A
23 dfrel2 4957 . . . . . 6  |-  ( Rel 
A  <->  `' `' A  =  A
)
2416, 23sylib 121 . . . . 5  |-  ( ph  ->  `' `' A  =  A
)
25 f1oeq3 5326 . . . . 5  |-  ( `' `' A  =  A  ->  ( ( z  e.  `' A  |->  U. `' { z } ) : `' A -1-1-onto-> `' `' A 
<->  ( z  e.  `' A  |->  U. `' { z } ) : `' A
-1-1-onto-> A ) )
2624, 25syl 14 . . . 4  |-  ( ph  ->  ( ( z  e.  `' A  |->  U. `' { z } ) : `' A -1-1-onto-> `' `' A 
<->  ( z  e.  `' A  |->  U. `' { z } ) : `' A
-1-1-onto-> A ) )
2722, 26mpbii 147 . . 3  |-  ( ph  ->  ( z  e.  `' A  |->  U. `' { z } ) : `' A
-1-1-onto-> A )
28 1st2nd 6045 . . . . . . 7  |-  ( ( Rel  `' A  /\  y  e.  `' A
)  ->  y  =  <. ( 1st `  y
) ,  ( 2nd `  y ) >. )
2920, 28mpan 418 . . . . . 6  |-  ( y  e.  `' A  -> 
y  =  <. ( 1st `  y ) ,  ( 2nd `  y
) >. )
3029fveq2d 5391 . . . . 5  |-  ( y  e.  `' A  -> 
( ( z  e.  `' A  |->  U. `' { z } ) `
 y )  =  ( ( z  e.  `' A  |->  U. `' { z } ) `
 <. ( 1st `  y
) ,  ( 2nd `  y ) >. )
)
31 id 19 . . . . . . 7  |-  ( y  e.  `' A  -> 
y  e.  `' A
)
3229, 31eqeltrrd 2193 . . . . . 6  |-  ( y  e.  `' A  ->  <. ( 1st `  y
) ,  ( 2nd `  y ) >.  e.  `' A )
33 sneq 3506 . . . . . . . . . 10  |-  ( z  =  <. ( 1st `  y
) ,  ( 2nd `  y ) >.  ->  { z }  =  { <. ( 1st `  y ) ,  ( 2nd `  y
) >. } )
3433cnveqd 4683 . . . . . . . . 9  |-  ( z  =  <. ( 1st `  y
) ,  ( 2nd `  y ) >.  ->  `' { z }  =  `' { <. ( 1st `  y
) ,  ( 2nd `  y ) >. } )
3534unieqd 3715 . . . . . . . 8  |-  ( z  =  <. ( 1st `  y
) ,  ( 2nd `  y ) >.  ->  U. `' { z }  =  U. `' { <. ( 1st `  y
) ,  ( 2nd `  y ) >. } )
36 opswapg 4993 . . . . . . . . 9  |-  ( ( ( 1st `  y
)  e.  _V  /\  ( 2nd `  y )  e.  _V )  ->  U. `' { <. ( 1st `  y
) ,  ( 2nd `  y ) >. }  =  <. ( 2nd `  y
) ,  ( 1st `  y ) >. )
375, 3, 36mp2an 420 . . . . . . . 8  |-  U. `' { <. ( 1st `  y
) ,  ( 2nd `  y ) >. }  =  <. ( 2nd `  y
) ,  ( 1st `  y ) >.
3835, 37syl6eq 2164 . . . . . . 7  |-  ( z  =  <. ( 1st `  y
) ,  ( 2nd `  y ) >.  ->  U. `' { z }  =  <. ( 2nd `  y
) ,  ( 1st `  y ) >. )
39 eqid 2115 . . . . . . 7  |-  ( z  e.  `' A  |->  U. `' { z } )  =  ( z  e.  `' A  |->  U. `' { z } )
403, 5opex 4119 . . . . . . 7  |-  <. ( 2nd `  y ) ,  ( 1st `  y
) >.  e.  _V
4138, 39, 40fvmpt 5464 . . . . . 6  |-  ( <.
( 1st `  y
) ,  ( 2nd `  y ) >.  e.  `' A  ->  ( ( z  e.  `' A  |->  U. `' { z } ) `
 <. ( 1st `  y
) ,  ( 2nd `  y ) >. )  =  <. ( 2nd `  y
) ,  ( 1st `  y ) >. )
4232, 41syl 14 . . . . 5  |-  ( y  e.  `' A  -> 
( ( z  e.  `' A  |->  U. `' { z } ) `
 <. ( 1st `  y
) ,  ( 2nd `  y ) >. )  =  <. ( 2nd `  y
) ,  ( 1st `  y ) >. )
4330, 42eqtrd 2148 . . . 4  |-  ( y  e.  `' A  -> 
( ( z  e.  `' A  |->  U. `' { z } ) `
 y )  = 
<. ( 2nd `  y
) ,  ( 1st `  y ) >. )
4443adantl 273 . . 3  |-  ( (
ph  /\  y  e.  `' A )  ->  (
( z  e.  `' A  |->  U. `' { z } ) `  y
)  =  <. ( 2nd `  y ) ,  ( 1st `  y
) >. )
45 fsumcnv.5 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  CC )
4615, 19, 27, 44, 45fsumf1o 11099 . 2  |-  ( ph  -> 
sum_ x  e.  A  B  =  sum_ y  e.  `'  A [_ ( 2nd `  y )  /  j ]_ [_ ( 1st `  y
)  /  k ]_ D )
47 csbeq1a 2981 . . . . 5  |-  ( y  =  <. ( 1st `  y
) ,  ( 2nd `  y ) >.  ->  C  =  [_ <. ( 1st `  y
) ,  ( 2nd `  y ) >.  /  y ]_ C )
4829, 47syl 14 . . . 4  |-  ( y  e.  `' A  ->  C  =  [_ <. ( 1st `  y ) ,  ( 2nd `  y
) >.  /  y ]_ C )
497, 6opex 4119 . . . . . . 7  |-  <. k ,  j >.  e.  _V
50 fsumcnv.2 . . . . . . 7  |-  ( y  =  <. k ,  j
>.  ->  C  =  D )
5149, 50csbie 3013 . . . . . 6  |-  [_ <. k ,  j >.  /  y ]_ C  =  D
52 opeq12 3675 . . . . . . . 8  |-  ( ( k  =  ( 1st `  y )  /\  j  =  ( 2nd `  y
) )  ->  <. k ,  j >.  =  <. ( 1st `  y ) ,  ( 2nd `  y
) >. )
5352ancoms 266 . . . . . . 7  |-  ( ( j  =  ( 2nd `  y )  /\  k  =  ( 1st `  y
) )  ->  <. k ,  j >.  =  <. ( 1st `  y ) ,  ( 2nd `  y
) >. )
5453csbeq1d 2979 . . . . . 6  |-  ( ( j  =  ( 2nd `  y )  /\  k  =  ( 1st `  y
) )  ->  [_ <. k ,  j >.  /  y ]_ C  =  [_ <. ( 1st `  y ) ,  ( 2nd `  y
) >.  /  y ]_ C )
5551, 54syl5eqr 2162 . . . . 5  |-  ( ( j  =  ( 2nd `  y )  /\  k  =  ( 1st `  y
) )  ->  D  =  [_ <. ( 1st `  y
) ,  ( 2nd `  y ) >.  /  y ]_ C )
563, 5, 55csbie2 3017 . . . 4  |-  [_ ( 2nd `  y )  / 
j ]_ [_ ( 1st `  y )  /  k ]_ D  =  [_ <. ( 1st `  y ) ,  ( 2nd `  y
) >.  /  y ]_ C
5748, 56syl6eqr 2166 . . 3  |-  ( y  e.  `' A  ->  C  =  [_ ( 2nd `  y )  /  j ]_ [_ ( 1st `  y
)  /  k ]_ D )
5857sumeq2i 11073 . 2  |-  sum_ y  e.  `'  A C  =  sum_ y  e.  `'  A [_ ( 2nd `  y
)  /  j ]_ [_ ( 1st `  y
)  /  k ]_ D
5946, 58syl6eqr 2166 1  |-  ( ph  -> 
sum_ x  e.  A  B  =  sum_ y  e.  `'  A C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1314    e. wcel 1463   _Vcvv 2658   [_csb 2973   {csn 3495   <.cop 3498   U.cuni 3704    |-> cmpt 3957   `'ccnv 4506   Rel wrel 4512   -1-1-onto->wf1o 5090   ` cfv 5091   1stc1st 6002   2ndc2nd 6003   Fincfn 6600   CCcc 7582   sum_csu 11062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-mulrcl 7683  ax-addcom 7684  ax-mulcom 7685  ax-addass 7686  ax-mulass 7687  ax-distr 7688  ax-i2m1 7689  ax-0lt1 7690  ax-1rid 7691  ax-0id 7692  ax-rnegex 7693  ax-precex 7694  ax-cnre 7695  ax-pre-ltirr 7696  ax-pre-ltwlin 7697  ax-pre-lttrn 7698  ax-pre-apti 7699  ax-pre-ltadd 7700  ax-pre-mulgt0 7701  ax-pre-mulext 7702  ax-arch 7703  ax-caucvg 7704
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-reu 2398  df-rmo 2399  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-if 3443  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-ilim 4259  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-isom 5100  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-irdg 6233  df-frec 6254  df-1o 6279  df-oadd 6283  df-er 6395  df-en 6601  df-dom 6602  df-fin 6603  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769  df-le 7770  df-sub 7899  df-neg 7900  df-reap 8300  df-ap 8307  df-div 8393  df-inn 8678  df-2 8736  df-3 8737  df-4 8738  df-n0 8929  df-z 9006  df-uz 9276  df-q 9361  df-rp 9391  df-fz 9731  df-fzo 9860  df-seqfrec 10159  df-exp 10233  df-ihash 10462  df-cj 10554  df-re 10555  df-im 10556  df-rsqrt 10710  df-abs 10711  df-clim 10988  df-sumdc 11063
This theorem is referenced by:  fisumcom2  11147
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