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Theorem fsumcnv 12333
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 3165 . . . 4  |-  ( x  =  <. ( 2nd `  y
) ,  ( 1st `  y ) >.  ->  B  =  [_ <. ( 2nd `  y
) ,  ( 1st `  y ) >.  /  x ]_ B )
2 fvex 5622 . . . . 5  |-  ( 2nd `  y )  e.  _V
3 fvex 5622 . . . . 5  |-  ( 1st `  y )  e.  _V
4 opex 4319 . . . . . . 7  |-  <. j ,  k >.  e.  _V
5 nfcv 2494 . . . . . . 7  |-  F/_ x D
6 fsumcnv.1 . . . . . . 7  |-  ( x  =  <. j ,  k
>.  ->  B  =  D )
74, 5, 6csbief 3198 . . . . . 6  |-  [_ <. j ,  k >.  /  x ]_ B  =  D
8 opeq12 3879 . . . . . . 7  |-  ( ( j  =  ( 2nd `  y )  /\  k  =  ( 1st `  y
) )  ->  <. j ,  k >.  =  <. ( 2nd `  y ) ,  ( 1st `  y
) >. )
98csbeq1d 3163 . . . . . 6  |-  ( ( j  =  ( 2nd `  y )  /\  k  =  ( 1st `  y
) )  ->  [_ <. j ,  k >.  /  x ]_ B  =  [_ <. ( 2nd `  y ) ,  ( 1st `  y
) >.  /  x ]_ B )
107, 9syl5eqr 2404 . . . . 5  |-  ( ( j  =  ( 2nd `  y )  /\  k  =  ( 1st `  y
) )  ->  D  =  [_ <. ( 2nd `  y
) ,  ( 1st `  y ) >.  /  x ]_ B )
112, 3, 10csbie2 3202 . . . 4  |-  [_ ( 2nd `  y )  / 
j ]_ [_ ( 1st `  y )  /  k ]_ D  =  [_ <. ( 2nd `  y ) ,  ( 1st `  y
) >.  /  x ]_ B
121, 11syl6eqr 2408 . . 3  |-  ( x  =  <. ( 2nd `  y
) ,  ( 1st `  y ) >.  ->  B  =  [_ ( 2nd `  y
)  /  j ]_ [_ ( 1st `  y
)  /  k ]_ D )
13 fsumcnv.3 . . . 4  |-  ( ph  ->  A  e.  Fin )
14 cnvfi 7230 . . . 4  |-  ( A  e.  Fin  ->  `' A  e.  Fin )
1513, 14syl 15 . . 3  |-  ( ph  ->  `' A  e.  Fin )
16 relcnv 5133 . . . . 5  |-  Rel  `' A
17 cnvf1o 6304 . . . . 5  |-  ( Rel  `' A  ->  ( z  e.  `' A  |->  U. `' { z } ) : `' A -1-1-onto-> `' `' A )
1816, 17ax-mp 8 . . . 4  |-  ( z  e.  `' A  |->  U. `' { z } ) : `' A -1-1-onto-> `' `' A
19 fsumcnv.4 . . . . . 6  |-  ( ph  ->  Rel  A )
20 dfrel2 5206 . . . . . 6  |-  ( Rel 
A  <->  `' `' A  =  A
)
2119, 20sylib 188 . . . . 5  |-  ( ph  ->  `' `' A  =  A
)
22 f1oeq3 5548 . . . . 5  |-  ( `' `' A  =  A  ->  ( ( z  e.  `' A  |->  U. `' { z } ) : `' A -1-1-onto-> `' `' A 
<->  ( z  e.  `' A  |->  U. `' { z } ) : `' A
-1-1-onto-> A ) )
2321, 22syl 15 . . . 4  |-  ( ph  ->  ( ( z  e.  `' A  |->  U. `' { z } ) : `' A -1-1-onto-> `' `' A 
<->  ( z  e.  `' A  |->  U. `' { z } ) : `' A
-1-1-onto-> A ) )
2418, 23mpbii 202 . . 3  |-  ( ph  ->  ( z  e.  `' A  |->  U. `' { z } ) : `' A
-1-1-onto-> A )
25 1st2nd 6253 . . . . . . 7  |-  ( ( Rel  `' A  /\  y  e.  `' A
)  ->  y  =  <. ( 1st `  y
) ,  ( 2nd `  y ) >. )
2616, 25mpan 651 . . . . . 6  |-  ( y  e.  `' A  -> 
y  =  <. ( 1st `  y ) ,  ( 2nd `  y
) >. )
2726fveq2d 5612 . . . . 5  |-  ( y  e.  `' A  -> 
( ( z  e.  `' A  |->  U. `' { z } ) `
 y )  =  ( ( z  e.  `' A  |->  U. `' { z } ) `
 <. ( 1st `  y
) ,  ( 2nd `  y ) >. )
)
28 id 19 . . . . . . 7  |-  ( y  e.  `' A  -> 
y  e.  `' A
)
2926, 28eqeltrrd 2433 . . . . . 6  |-  ( y  e.  `' A  ->  <. ( 1st `  y
) ,  ( 2nd `  y ) >.  e.  `' A )
30 sneq 3727 . . . . . . . . . 10  |-  ( z  =  <. ( 1st `  y
) ,  ( 2nd `  y ) >.  ->  { z }  =  { <. ( 1st `  y ) ,  ( 2nd `  y
) >. } )
3130cnveqd 4939 . . . . . . . . 9  |-  ( z  =  <. ( 1st `  y
) ,  ( 2nd `  y ) >.  ->  `' { z }  =  `' { <. ( 1st `  y
) ,  ( 2nd `  y ) >. } )
3231unieqd 3919 . . . . . . . 8  |-  ( z  =  <. ( 1st `  y
) ,  ( 2nd `  y ) >.  ->  U. `' { z }  =  U. `' { <. ( 1st `  y
) ,  ( 2nd `  y ) >. } )
33 opswap 5241 . . . . . . . 8  |-  U. `' { <. ( 1st `  y
) ,  ( 2nd `  y ) >. }  =  <. ( 2nd `  y
) ,  ( 1st `  y ) >.
3432, 33syl6eq 2406 . . . . . . 7  |-  ( z  =  <. ( 1st `  y
) ,  ( 2nd `  y ) >.  ->  U. `' { z }  =  <. ( 2nd `  y
) ,  ( 1st `  y ) >. )
35 eqid 2358 . . . . . . 7  |-  ( z  e.  `' A  |->  U. `' { z } )  =  ( z  e.  `' A  |->  U. `' { z } )
36 opex 4319 . . . . . . 7  |-  <. ( 2nd `  y ) ,  ( 1st `  y
) >.  e.  _V
3734, 35, 36fvmpt 5685 . . . . . 6  |-  ( <.
( 1st `  y
) ,  ( 2nd `  y ) >.  e.  `' A  ->  ( ( z  e.  `' A  |->  U. `' { z } ) `
 <. ( 1st `  y
) ,  ( 2nd `  y ) >. )  =  <. ( 2nd `  y
) ,  ( 1st `  y ) >. )
3829, 37syl 15 . . . . 5  |-  ( y  e.  `' A  -> 
( ( z  e.  `' A  |->  U. `' { z } ) `
 <. ( 1st `  y
) ,  ( 2nd `  y ) >. )  =  <. ( 2nd `  y
) ,  ( 1st `  y ) >. )
3927, 38eqtrd 2390 . . . 4  |-  ( y  e.  `' A  -> 
( ( z  e.  `' A  |->  U. `' { z } ) `
 y )  = 
<. ( 2nd `  y
) ,  ( 1st `  y ) >. )
4039adantl 452 . . 3  |-  ( (
ph  /\  y  e.  `' A )  ->  (
( z  e.  `' A  |->  U. `' { z } ) `  y
)  =  <. ( 2nd `  y ) ,  ( 1st `  y
) >. )
41 fsumcnv.5 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  CC )
4212, 15, 24, 40, 41fsumf1o 12293 . 2  |-  ( ph  -> 
sum_ x  e.  A  B  =  sum_ y  e.  `'  A [_ ( 2nd `  y )  /  j ]_ [_ ( 1st `  y
)  /  k ]_ D )
43 csbeq1a 3165 . . . . 5  |-  ( y  =  <. ( 1st `  y
) ,  ( 2nd `  y ) >.  ->  C  =  [_ <. ( 1st `  y
) ,  ( 2nd `  y ) >.  /  y ]_ C )
4426, 43syl 15 . . . 4  |-  ( y  e.  `' A  ->  C  =  [_ <. ( 1st `  y ) ,  ( 2nd `  y
) >.  /  y ]_ C )
45 opex 4319 . . . . . . 7  |-  <. k ,  j >.  e.  _V
46 nfcv 2494 . . . . . . 7  |-  F/_ y D
47 fsumcnv.2 . . . . . . 7  |-  ( y  =  <. k ,  j
>.  ->  C  =  D )
4845, 46, 47csbief 3198 . . . . . 6  |-  [_ <. k ,  j >.  /  y ]_ C  =  D
49 opeq12 3879 . . . . . . . 8  |-  ( ( k  =  ( 1st `  y )  /\  j  =  ( 2nd `  y
) )  ->  <. k ,  j >.  =  <. ( 1st `  y ) ,  ( 2nd `  y
) >. )
5049ancoms 439 . . . . . . 7  |-  ( ( j  =  ( 2nd `  y )  /\  k  =  ( 1st `  y
) )  ->  <. k ,  j >.  =  <. ( 1st `  y ) ,  ( 2nd `  y
) >. )
5150csbeq1d 3163 . . . . . 6  |-  ( ( j  =  ( 2nd `  y )  /\  k  =  ( 1st `  y
) )  ->  [_ <. k ,  j >.  /  y ]_ C  =  [_ <. ( 1st `  y ) ,  ( 2nd `  y
) >.  /  y ]_ C )
5248, 51syl5eqr 2404 . . . . 5  |-  ( ( j  =  ( 2nd `  y )  /\  k  =  ( 1st `  y
) )  ->  D  =  [_ <. ( 1st `  y
) ,  ( 2nd `  y ) >.  /  y ]_ C )
532, 3, 52csbie2 3202 . . . 4  |-  [_ ( 2nd `  y )  / 
j ]_ [_ ( 1st `  y )  /  k ]_ D  =  [_ <. ( 1st `  y ) ,  ( 2nd `  y
) >.  /  y ]_ C
5444, 53syl6eqr 2408 . . 3  |-  ( y  e.  `' A  ->  C  =  [_ ( 2nd `  y )  /  j ]_ [_ ( 1st `  y
)  /  k ]_ D )
5554sumeq2i 12269 . 2  |-  sum_ y  e.  `'  A C  =  sum_ y  e.  `'  A [_ ( 2nd `  y
)  /  j ]_ [_ ( 1st `  y
)  /  k ]_ D
5642, 55syl6eqr 2408 1  |-  ( ph  -> 
sum_ x  e.  A  B  =  sum_ y  e.  `'  A C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1642    e. wcel 1710   [_csb 3157   {csn 3716   <.cop 3719   U.cuni 3908    e. cmpt 4158   `'ccnv 4770   Rel wrel 4776   -1-1-onto->wf1o 5336   ` cfv 5337   1stc1st 6207   2ndc2nd 6208   Fincfn 6951   CCcc 8825   sum_csu 12255
This theorem is referenced by:  fsumcom2  12334
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-inf2 7432  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904  ax-pre-sup 8905
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-se 4435  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-isom 5346  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-riota 6391  df-recs 6475  df-rdg 6510  df-1o 6566  df-oadd 6570  df-er 6747  df-en 6952  df-dom 6953  df-sdom 6954  df-fin 6955  df-sup 7284  df-oi 7315  df-card 7662  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-div 9514  df-nn 9837  df-2 9894  df-3 9895  df-n0 10058  df-z 10117  df-uz 10323  df-rp 10447  df-fz 10875  df-fzo 10963  df-seq 11139  df-exp 11198  df-hash 11431  df-cj 11680  df-re 11681  df-im 11682  df-sqr 11816  df-abs 11817  df-clim 12058  df-sum 12256
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