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Theorem fprodrev 27401
Description: Reversal of a finite product. (Contributed by Scott Fenton, 5-Jan-2018.)
Hypotheses
Ref Expression
fprodshft.1  |-  ( ph  ->  K  e.  ZZ )
fprodshft.2  |-  ( ph  ->  M  e.  ZZ )
fprodshft.3  |-  ( ph  ->  N  e.  ZZ )
fprodshft.4  |-  ( (
ph  /\  j  e.  ( M ... N ) )  ->  A  e.  CC )
fprodrev.5  |-  ( j  =  ( K  -  k )  ->  A  =  B )
Assertion
Ref Expression
fprodrev  |-  ( ph  ->  prod_ j  e.  ( M ... N ) A  =  prod_ k  e.  ( ( K  -  N ) ... ( K  -  M )
) B )
Distinct variable groups:    A, k    B, j    j, k, ph    j, K, k    ph, k    j, M, k    j, N, k
Allowed substitution hints:    A( j)    B( k)

Proof of Theorem fprodrev
StepHypRef Expression
1 fprodrev.5 . 2  |-  ( j  =  ( K  -  k )  ->  A  =  B )
2 fzfid 11791 . 2  |-  ( ph  ->  ( ( K  -  N ) ... ( K  -  M )
)  e.  Fin )
3 ovex 6115 . . . . 5  |-  ( K  -  j )  e. 
_V
4 eqid 2441 . . . . 5  |-  ( j  e.  ( ( K  -  N ) ... ( K  -  M
) )  |->  ( K  -  j ) )  =  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  |->  ( K  -  j ) )
53, 4fnmpti 5536 . . . 4  |-  ( j  e.  ( ( K  -  N ) ... ( K  -  M
) )  |->  ( K  -  j ) )  Fn  ( ( K  -  N ) ... ( K  -  M
) )
65a1i 11 . . 3  |-  ( ph  ->  ( j  e.  ( ( K  -  N
) ... ( K  -  M ) )  |->  ( K  -  j ) )  Fn  ( ( K  -  N ) ... ( K  -  M ) ) )
7 ovex 6115 . . . . 5  |-  ( K  -  k )  e. 
_V
8 eqid 2441 . . . . 5  |-  ( k  e.  ( M ... N )  |->  ( K  -  k ) )  =  ( k  e.  ( M ... N
)  |->  ( K  -  k ) )
97, 8fnmpti 5536 . . . 4  |-  ( k  e.  ( M ... N )  |->  ( K  -  k ) )  Fn  ( M ... N )
10 simprr 751 . . . . . . . . . 10  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  -> 
k  =  ( K  -  j ) )
11 simprl 750 . . . . . . . . . . 11  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  -> 
j  e.  ( ( K  -  N ) ... ( K  -  M ) ) )
12 fprodshft.2 . . . . . . . . . . . . 13  |-  ( ph  ->  M  e.  ZZ )
1312adantr 462 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  ->  M  e.  ZZ )
14 fprodshft.3 . . . . . . . . . . . . 13  |-  ( ph  ->  N  e.  ZZ )
1514adantr 462 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  ->  N  e.  ZZ )
16 fprodshft.1 . . . . . . . . . . . . 13  |-  ( ph  ->  K  e.  ZZ )
1716adantr 462 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  ->  K  e.  ZZ )
18 elfzelz 11449 . . . . . . . . . . . . 13  |-  ( j  e.  ( ( K  -  N ) ... ( K  -  M
) )  ->  j  e.  ZZ )
1918ad2antrl 722 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  -> 
j  e.  ZZ )
20 fzrev 11515 . . . . . . . . . . . 12  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  e.  ZZ  /\  j  e.  ZZ ) )  -> 
( j  e.  ( ( K  -  N
) ... ( K  -  M ) )  <->  ( K  -  j )  e.  ( M ... N
) ) )
2113, 15, 17, 19, 20syl22anc 1214 . . . . . . . . . . 11  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  -> 
( j  e.  ( ( K  -  N
) ... ( K  -  M ) )  <->  ( K  -  j )  e.  ( M ... N
) ) )
2211, 21mpbid 210 . . . . . . . . . 10  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  -> 
( K  -  j
)  e.  ( M ... N ) )
2310, 22eqeltrd 2515 . . . . . . . . 9  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  -> 
k  e.  ( M ... N ) )
24 oveq2 6098 . . . . . . . . . . 11  |-  ( k  =  ( K  -  j )  ->  ( K  -  k )  =  ( K  -  ( K  -  j
) ) )
2524ad2antll 723 . . . . . . . . . 10  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  -> 
( K  -  k
)  =  ( K  -  ( K  -  j ) ) )
2616zcnd 10744 . . . . . . . . . . . 12  |-  ( ph  ->  K  e.  CC )
2726adantr 462 . . . . . . . . . . 11  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  ->  K  e.  CC )
2818zcnd 10744 . . . . . . . . . . . 12  |-  ( j  e.  ( ( K  -  N ) ... ( K  -  M
) )  ->  j  e.  CC )
2928ad2antrl 722 . . . . . . . . . . 11  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  -> 
j  e.  CC )
3027, 29nncand 9720 . . . . . . . . . 10  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  -> 
( K  -  ( K  -  j )
)  =  j )
3125, 30eqtr2d 2474 . . . . . . . . 9  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  -> 
j  =  ( K  -  k ) )
3223, 31jca 529 . . . . . . . 8  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  -> 
( k  e.  ( M ... N )  /\  j  =  ( K  -  k ) ) )
33 simprr 751 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  -> 
j  =  ( K  -  k ) )
34 simprl 750 . . . . . . . . . . 11  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  -> 
k  e.  ( M ... N ) )
3512adantr 462 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  ->  M  e.  ZZ )
3614adantr 462 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  ->  N  e.  ZZ )
3716adantr 462 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  ->  K  e.  ZZ )
38 elfzelz 11449 . . . . . . . . . . . . 13  |-  ( k  e.  ( M ... N )  ->  k  e.  ZZ )
3938ad2antrl 722 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  -> 
k  e.  ZZ )
40 fzrev2 11516 . . . . . . . . . . . 12  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  e.  ZZ  /\  k  e.  ZZ ) )  -> 
( k  e.  ( M ... N )  <-> 
( K  -  k
)  e.  ( ( K  -  N ) ... ( K  -  M ) ) ) )
4135, 36, 37, 39, 40syl22anc 1214 . . . . . . . . . . 11  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  -> 
( k  e.  ( M ... N )  <-> 
( K  -  k
)  e.  ( ( K  -  N ) ... ( K  -  M ) ) ) )
4234, 41mpbid 210 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  -> 
( K  -  k
)  e.  ( ( K  -  N ) ... ( K  -  M ) ) )
4333, 42eqeltrd 2515 . . . . . . . . 9  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  -> 
j  e.  ( ( K  -  N ) ... ( K  -  M ) ) )
44 oveq2 6098 . . . . . . . . . . 11  |-  ( j  =  ( K  -  k )  ->  ( K  -  j )  =  ( K  -  ( K  -  k
) ) )
4544ad2antll 723 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  -> 
( K  -  j
)  =  ( K  -  ( K  -  k ) ) )
4626adantr 462 . . . . . . . . . . 11  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  ->  K  e.  CC )
4738zcnd 10744 . . . . . . . . . . . 12  |-  ( k  e.  ( M ... N )  ->  k  e.  CC )
4847ad2antrl 722 . . . . . . . . . . 11  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  -> 
k  e.  CC )
4946, 48nncand 9720 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  -> 
( K  -  ( K  -  k )
)  =  k )
5045, 49eqtr2d 2474 . . . . . . . . 9  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  -> 
k  =  ( K  -  j ) )
5143, 50jca 529 . . . . . . . 8  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  -> 
( j  e.  ( ( K  -  N
) ... ( K  -  M ) )  /\  k  =  ( K  -  j ) ) )
5232, 51impbida 823 . . . . . . 7  |-  ( ph  ->  ( ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) )  <->  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) ) )
5352opabbidv 4352 . . . . . 6  |-  ( ph  ->  { <. k ,  j
>.  |  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) }  =  { <. k ,  j
>.  |  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) } )
54 df-mpt 4349 . . . . . . . 8  |-  ( j  e.  ( ( K  -  N ) ... ( K  -  M
) )  |->  ( K  -  j ) )  =  { <. j ,  k >.  |  ( j  e.  ( ( K  -  N ) ... ( K  -  M ) )  /\  k  =  ( K  -  j ) ) }
5554cnveqi 5010 . . . . . . 7  |-  `' ( j  e.  ( ( K  -  N ) ... ( K  -  M ) )  |->  ( K  -  j ) )  =  `' { <. j ,  k >.  |  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) }
56 cnvopab 5235 . . . . . . 7  |-  `' { <. j ,  k >.  |  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) }  =  { <. k ,  j
>.  |  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) }
5755, 56eqtri 2461 . . . . . 6  |-  `' ( j  e.  ( ( K  -  N ) ... ( K  -  M ) )  |->  ( K  -  j ) )  =  { <. k ,  j >.  |  ( j  e.  ( ( K  -  N ) ... ( K  -  M ) )  /\  k  =  ( K  -  j ) ) }
58 df-mpt 4349 . . . . . 6  |-  ( k  e.  ( M ... N )  |->  ( K  -  k ) )  =  { <. k ,  j >.  |  ( k  e.  ( M ... N )  /\  j  =  ( K  -  k ) ) }
5953, 57, 583eqtr4g 2498 . . . . 5  |-  ( ph  ->  `' ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  |->  ( K  -  j ) )  =  ( k  e.  ( M ... N ) 
|->  ( K  -  k
) ) )
6059fneq1d 5498 . . . 4  |-  ( ph  ->  ( `' ( j  e.  ( ( K  -  N ) ... ( K  -  M
) )  |->  ( K  -  j ) )  Fn  ( M ... N )  <->  ( k  e.  ( M ... N
)  |->  ( K  -  k ) )  Fn  ( M ... N
) ) )
619, 60mpbiri 233 . . 3  |-  ( ph  ->  `' ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  |->  ( K  -  j ) )  Fn  ( M ... N
) )
62 dff1o4 5646 . . 3  |-  ( ( j  e.  ( ( K  -  N ) ... ( K  -  M ) )  |->  ( K  -  j ) ) : ( ( K  -  N ) ... ( K  -  M ) ) -1-1-onto-> ( M ... N )  <->  ( (
j  e.  ( ( K  -  N ) ... ( K  -  M ) )  |->  ( K  -  j ) )  Fn  ( ( K  -  N ) ... ( K  -  M ) )  /\  `' ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  |->  ( K  -  j ) )  Fn  ( M ... N
) ) )
636, 61, 62sylanbrc 659 . 2  |-  ( ph  ->  ( j  e.  ( ( K  -  N
) ... ( K  -  M ) )  |->  ( K  -  j ) ) : ( ( K  -  N ) ... ( K  -  M ) ) -1-1-onto-> ( M ... N ) )
64 oveq2 6098 . . . 4  |-  ( j  =  k  ->  ( K  -  j )  =  ( K  -  k ) )
6564, 4, 7fvmpt 5771 . . 3  |-  ( k  e.  ( ( K  -  N ) ... ( K  -  M
) )  ->  (
( j  e.  ( ( K  -  N
) ... ( K  -  M ) )  |->  ( K  -  j ) ) `  k )  =  ( K  -  k ) )
6665adantl 463 . 2  |-  ( (
ph  /\  k  e.  ( ( K  -  N ) ... ( K  -  M )
) )  ->  (
( j  e.  ( ( K  -  N
) ... ( K  -  M ) )  |->  ( K  -  j ) ) `  k )  =  ( K  -  k ) )
67 fprodshft.4 . 2  |-  ( (
ph  /\  j  e.  ( M ... N ) )  ->  A  e.  CC )
681, 2, 63, 66, 67fprodf1o 27372 1  |-  ( ph  ->  prod_ j  e.  ( M ... N ) A  =  prod_ k  e.  ( ( K  -  N ) ... ( K  -  M )
) B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1364    e. wcel 1761   {copab 4346    e. cmpt 4347   `'ccnv 4835    Fn wfn 5410   -1-1-onto->wf1o 5414   ` cfv 5415  (class class class)co 6090   CCcc 9276    - cmin 9591   ZZcz 10642   ...cfz 11433   prod_cprod 27331
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-sup 7687  df-oi 7720  df-card 8105  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-n0 10576  df-z 10643  df-uz 10858  df-rp 10988  df-fz 11434  df-fzo 11545  df-seq 11803  df-exp 11862  df-hash 12100  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-clim 12962  df-prod 27332
This theorem is referenced by:  fallfacval3  27428
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