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Theorem fsumcnv 12562
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 3261 . . . 4  |-  ( x  =  <. ( 2nd `  y
) ,  ( 1st `  y ) >.  ->  B  =  [_ <. ( 2nd `  y
) ,  ( 1st `  y ) >.  /  x ]_ B )
2 fvex 5745 . . . . 5  |-  ( 2nd `  y )  e.  _V
3 fvex 5745 . . . . 5  |-  ( 1st `  y )  e.  _V
4 opex 4430 . . . . . . 7  |-  <. j ,  k >.  e.  _V
5 nfcv 2574 . . . . . . 7  |-  F/_ x D
6 fsumcnv.1 . . . . . . 7  |-  ( x  =  <. j ,  k
>.  ->  B  =  D )
74, 5, 6csbief 3294 . . . . . 6  |-  [_ <. j ,  k >.  /  x ]_ B  =  D
8 opeq12 3988 . . . . . . 7  |-  ( ( j  =  ( 2nd `  y )  /\  k  =  ( 1st `  y
) )  ->  <. j ,  k >.  =  <. ( 2nd `  y ) ,  ( 1st `  y
) >. )
98csbeq1d 3259 . . . . . 6  |-  ( ( j  =  ( 2nd `  y )  /\  k  =  ( 1st `  y
) )  ->  [_ <. j ,  k >.  /  x ]_ B  =  [_ <. ( 2nd `  y ) ,  ( 1st `  y
) >.  /  x ]_ B )
107, 9syl5eqr 2484 . . . . 5  |-  ( ( j  =  ( 2nd `  y )  /\  k  =  ( 1st `  y
) )  ->  D  =  [_ <. ( 2nd `  y
) ,  ( 1st `  y ) >.  /  x ]_ B )
112, 3, 10csbie2 3298 . . . 4  |-  [_ ( 2nd `  y )  / 
j ]_ [_ ( 1st `  y )  /  k ]_ D  =  [_ <. ( 2nd `  y ) ,  ( 1st `  y
) >.  /  x ]_ B
121, 11syl6eqr 2488 . . 3  |-  ( x  =  <. ( 2nd `  y
) ,  ( 1st `  y ) >.  ->  B  =  [_ ( 2nd `  y
)  /  j ]_ [_ ( 1st `  y
)  /  k ]_ D )
13 fsumcnv.3 . . . 4  |-  ( ph  ->  A  e.  Fin )
14 cnvfi 7393 . . . 4  |-  ( A  e.  Fin  ->  `' A  e.  Fin )
1513, 14syl 16 . . 3  |-  ( ph  ->  `' A  e.  Fin )
16 relcnv 5245 . . . . 5  |-  Rel  `' A
17 cnvf1o 6448 . . . . 5  |-  ( Rel  `' A  ->  ( z  e.  `' A  |->  U. `' { z } ) : `' A -1-1-onto-> `' `' A )
1816, 17ax-mp 5 . . . 4  |-  ( z  e.  `' A  |->  U. `' { z } ) : `' A -1-1-onto-> `' `' A
19 fsumcnv.4 . . . . . 6  |-  ( ph  ->  Rel  A )
20 dfrel2 5324 . . . . . 6  |-  ( Rel 
A  <->  `' `' A  =  A
)
2119, 20sylib 190 . . . . 5  |-  ( ph  ->  `' `' A  =  A
)
22 f1oeq3 5670 . . . . 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 16 . . . 4  |-  ( ph  ->  ( ( z  e.  `' A  |->  U. `' { z } ) : `' A -1-1-onto-> `' `' A 
<->  ( z  e.  `' A  |->  U. `' { z } ) : `' A
-1-1-onto-> A ) )
2418, 23mpbii 204 . . 3  |-  ( ph  ->  ( z  e.  `' A  |->  U. `' { z } ) : `' A
-1-1-onto-> A )
25 1st2nd 6396 . . . . . . 7  |-  ( ( Rel  `' A  /\  y  e.  `' A
)  ->  y  =  <. ( 1st `  y
) ,  ( 2nd `  y ) >. )
2616, 25mpan 653 . . . . . 6  |-  ( y  e.  `' A  -> 
y  =  <. ( 1st `  y ) ,  ( 2nd `  y
) >. )
2726fveq2d 5735 . . . . 5  |-  ( y  e.  `' A  -> 
( ( z  e.  `' A  |->  U. `' { z } ) `
 y )  =  ( ( z  e.  `' A  |->  U. `' { z } ) `
 <. ( 1st `  y
) ,  ( 2nd `  y ) >. )
)
28 id 21 . . . . . . 7  |-  ( y  e.  `' A  -> 
y  e.  `' A
)
2926, 28eqeltrrd 2513 . . . . . 6  |-  ( y  e.  `' A  ->  <. ( 1st `  y
) ,  ( 2nd `  y ) >.  e.  `' A )
30 sneq 3827 . . . . . . . . . 10  |-  ( z  =  <. ( 1st `  y
) ,  ( 2nd `  y ) >.  ->  { z }  =  { <. ( 1st `  y ) ,  ( 2nd `  y
) >. } )
3130cnveqd 5051 . . . . . . . . 9  |-  ( z  =  <. ( 1st `  y
) ,  ( 2nd `  y ) >.  ->  `' { z }  =  `' { <. ( 1st `  y
) ,  ( 2nd `  y ) >. } )
3231unieqd 4028 . . . . . . . 8  |-  ( z  =  <. ( 1st `  y
) ,  ( 2nd `  y ) >.  ->  U. `' { z }  =  U. `' { <. ( 1st `  y
) ,  ( 2nd `  y ) >. } )
33 opswap 5359 . . . . . . . 8  |-  U. `' { <. ( 1st `  y
) ,  ( 2nd `  y ) >. }  =  <. ( 2nd `  y
) ,  ( 1st `  y ) >.
3432, 33syl6eq 2486 . . . . . . 7  |-  ( z  =  <. ( 1st `  y
) ,  ( 2nd `  y ) >.  ->  U. `' { z }  =  <. ( 2nd `  y
) ,  ( 1st `  y ) >. )
35 eqid 2438 . . . . . . 7  |-  ( z  e.  `' A  |->  U. `' { z } )  =  ( z  e.  `' A  |->  U. `' { z } )
36 opex 4430 . . . . . . 7  |-  <. ( 2nd `  y ) ,  ( 1st `  y
) >.  e.  _V
3734, 35, 36fvmpt 5809 . . . . . 6  |-  ( <.
( 1st `  y
) ,  ( 2nd `  y ) >.  e.  `' A  ->  ( ( z  e.  `' A  |->  U. `' { z } ) `
 <. ( 1st `  y
) ,  ( 2nd `  y ) >. )  =  <. ( 2nd `  y
) ,  ( 1st `  y ) >. )
3829, 37syl 16 . . . . 5  |-  ( y  e.  `' A  -> 
( ( z  e.  `' A  |->  U. `' { z } ) `
 <. ( 1st `  y
) ,  ( 2nd `  y ) >. )  =  <. ( 2nd `  y
) ,  ( 1st `  y ) >. )
3927, 38eqtrd 2470 . . . 4  |-  ( y  e.  `' A  -> 
( ( z  e.  `' A  |->  U. `' { z } ) `
 y )  = 
<. ( 2nd `  y
) ,  ( 1st `  y ) >. )
4039adantl 454 . . 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 12522 . 2  |-  ( ph  -> 
sum_ x  e.  A  B  =  sum_ y  e.  `'  A [_ ( 2nd `  y )  /  j ]_ [_ ( 1st `  y
)  /  k ]_ D )
43 csbeq1a 3261 . . . . 5  |-  ( y  =  <. ( 1st `  y
) ,  ( 2nd `  y ) >.  ->  C  =  [_ <. ( 1st `  y
) ,  ( 2nd `  y ) >.  /  y ]_ C )
4426, 43syl 16 . . . 4  |-  ( y  e.  `' A  ->  C  =  [_ <. ( 1st `  y ) ,  ( 2nd `  y
) >.  /  y ]_ C )
45 opex 4430 . . . . . . 7  |-  <. k ,  j >.  e.  _V
46 nfcv 2574 . . . . . . 7  |-  F/_ y D
47 fsumcnv.2 . . . . . . 7  |-  ( y  =  <. k ,  j
>.  ->  C  =  D )
4845, 46, 47csbief 3294 . . . . . 6  |-  [_ <. k ,  j >.  /  y ]_ C  =  D
49 opeq12 3988 . . . . . . . 8  |-  ( ( k  =  ( 1st `  y )  /\  j  =  ( 2nd `  y
) )  ->  <. k ,  j >.  =  <. ( 1st `  y ) ,  ( 2nd `  y
) >. )
5049ancoms 441 . . . . . . 7  |-  ( ( j  =  ( 2nd `  y )  /\  k  =  ( 1st `  y
) )  ->  <. k ,  j >.  =  <. ( 1st `  y ) ,  ( 2nd `  y
) >. )
5150csbeq1d 3259 . . . . . 6  |-  ( ( j  =  ( 2nd `  y )  /\  k  =  ( 1st `  y
) )  ->  [_ <. k ,  j >.  /  y ]_ C  =  [_ <. ( 1st `  y ) ,  ( 2nd `  y
) >.  /  y ]_ C )
5248, 51syl5eqr 2484 . . . . 5  |-  ( ( j  =  ( 2nd `  y )  /\  k  =  ( 1st `  y
) )  ->  D  =  [_ <. ( 1st `  y
) ,  ( 2nd `  y ) >.  /  y ]_ C )
532, 3, 52csbie2 3298 . . . 4  |-  [_ ( 2nd `  y )  / 
j ]_ [_ ( 1st `  y )  /  k ]_ D  =  [_ <. ( 1st `  y ) ,  ( 2nd `  y
) >.  /  y ]_ C
5444, 53syl6eqr 2488 . . 3  |-  ( y  e.  `' A  ->  C  =  [_ ( 2nd `  y )  /  j ]_ [_ ( 1st `  y
)  /  k ]_ D )
5554sumeq2i 12498 . 2  |-  sum_ y  e.  `'  A C  =  sum_ y  e.  `'  A [_ ( 2nd `  y
)  /  j ]_ [_ ( 1st `  y
)  /  k ]_ D
5642, 55syl6eqr 2488 1  |-  ( ph  -> 
sum_ x  e.  A  B  =  sum_ y  e.  `'  A C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   [_csb 3253   {csn 3816   <.cop 3819   U.cuni 4017    e. cmpt 4269   `'ccnv 4880   Rel wrel 4886   -1-1-onto->wf1o 5456   ` cfv 5457   1stc1st 6350   2ndc2nd 6351   Fincfn 7112   CCcc 8993   sum_csu 12484
This theorem is referenced by:  fsumcom2  12563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-oadd 6731  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-sup 7449  df-oi 7482  df-card 7831  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-3 10064  df-n0 10227  df-z 10288  df-uz 10494  df-rp 10618  df-fz 11049  df-fzo 11141  df-seq 11329  df-exp 11388  df-hash 11624  df-cj 11909  df-re 11910  df-im 11911  df-sqr 12045  df-abs 12046  df-clim 12287  df-sum 12485
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