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Theorem fprodshft 29033
Description: Shift the index 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 )
fprodshft.5  |-  ( j  =  ( k  -  K )  ->  A  =  B )
Assertion
Ref Expression
fprodshft  |-  ( ph  ->  prod_ j  e.  ( M ... N ) A  =  prod_ k  e.  ( ( M  +  K ) ... ( N  +  K )
) 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 fprodshft
StepHypRef Expression
1 fprodshft.5 . 2  |-  ( j  =  ( k  -  K )  ->  A  =  B )
2 fzfid 12063 . 2  |-  ( ph  ->  ( ( M  +  K ) ... ( N  +  K )
)  e.  Fin )
3 ovex 6320 . . . . 5  |-  ( j  -  K )  e. 
_V
4 eqid 2467 . . . . 5  |-  ( j  e.  ( ( M  +  K ) ... ( N  +  K
) )  |->  ( j  -  K ) )  =  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  |->  ( j  -  K ) )
53, 4fnmpti 5715 . . . 4  |-  ( j  e.  ( ( M  +  K ) ... ( N  +  K
) )  |->  ( j  -  K ) )  Fn  ( ( M  +  K ) ... ( N  +  K
) )
65a1i 11 . . 3  |-  ( ph  ->  ( j  e.  ( ( M  +  K
) ... ( N  +  K ) )  |->  ( j  -  K ) )  Fn  ( ( M  +  K ) ... ( N  +  K ) ) )
7 ovex 6320 . . . . 5  |-  ( k  +  K )  e. 
_V
8 eqid 2467 . . . . 5  |-  ( k  e.  ( M ... N )  |->  ( k  +  K ) )  =  ( k  e.  ( M ... N
)  |->  ( k  +  K ) )
97, 8fnmpti 5715 . . . 4  |-  ( k  e.  ( M ... N )  |->  ( k  +  K ) )  Fn  ( M ... N )
10 oveq1 6302 . . . . . . . . . . . . 13  |-  ( k  =  ( j  -  K )  ->  (
k  +  K )  =  ( ( j  -  K )  +  K ) )
1110ad2antll 728 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
( k  +  K
)  =  ( ( j  -  K )  +  K ) )
12 elfzelz 11700 . . . . . . . . . . . . . . 15  |-  ( j  e.  ( ( M  +  K ) ... ( N  +  K
) )  ->  j  e.  ZZ )
1312zcnd 10979 . . . . . . . . . . . . . 14  |-  ( j  e.  ( ( M  +  K ) ... ( N  +  K
) )  ->  j  e.  CC )
1413ad2antrl 727 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
j  e.  CC )
15 fprodshft.1 . . . . . . . . . . . . . . 15  |-  ( ph  ->  K  e.  ZZ )
1615zcnd 10979 . . . . . . . . . . . . . 14  |-  ( ph  ->  K  e.  CC )
1716adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  ->  K  e.  CC )
1814, 17npcand 9946 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
( ( j  -  K )  +  K
)  =  j )
1911, 18eqtr2d 2509 . . . . . . . . . . 11  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
j  =  ( k  +  K ) )
20 simprl 755 . . . . . . . . . . 11  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
j  e.  ( ( M  +  K ) ... ( N  +  K ) ) )
2119, 20eqeltrrd 2556 . . . . . . . . . 10  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
( k  +  K
)  e.  ( ( M  +  K ) ... ( N  +  K ) ) )
22 fprodshft.2 . . . . . . . . . . . 12  |-  ( ph  ->  M  e.  ZZ )
2322adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  ->  M  e.  ZZ )
24 fprodshft.3 . . . . . . . . . . . 12  |-  ( ph  ->  N  e.  ZZ )
2524adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  ->  N  e.  ZZ )
26 simprr 756 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
k  =  ( j  -  K ) )
2712ad2antrl 727 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
j  e.  ZZ )
2815adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  ->  K  e.  ZZ )
2927, 28zsubcld 10983 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
( j  -  K
)  e.  ZZ )
3026, 29eqeltrd 2555 . . . . . . . . . . 11  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
k  e.  ZZ )
31 fzaddel 11730 . . . . . . . . . . 11  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( k  e.  ZZ  /\  K  e.  ZZ ) )  -> 
( k  e.  ( M ... N )  <-> 
( k  +  K
)  e.  ( ( M  +  K ) ... ( N  +  K ) ) ) )
3223, 25, 30, 28, 31syl22anc 1229 . . . . . . . . . 10  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
( k  e.  ( M ... N )  <-> 
( k  +  K
)  e.  ( ( M  +  K ) ... ( N  +  K ) ) ) )
3321, 32mpbird 232 . . . . . . . . 9  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
k  e.  ( M ... N ) )
3433, 19jca 532 . . . . . . . 8  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
( k  e.  ( M ... N )  /\  j  =  ( k  +  K ) ) )
35 simprr 756 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  -> 
j  =  ( k  +  K ) )
36 simprl 755 . . . . . . . . . . 11  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  -> 
k  e.  ( M ... N ) )
3722adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  ->  M  e.  ZZ )
3824adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  ->  N  e.  ZZ )
39 elfzelz 11700 . . . . . . . . . . . . 13  |-  ( k  e.  ( M ... N )  ->  k  e.  ZZ )
4039ad2antrl 727 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  -> 
k  e.  ZZ )
4115adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  ->  K  e.  ZZ )
4237, 38, 40, 41, 31syl22anc 1229 . . . . . . . . . . 11  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  -> 
( k  e.  ( M ... N )  <-> 
( k  +  K
)  e.  ( ( M  +  K ) ... ( N  +  K ) ) ) )
4336, 42mpbid 210 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  -> 
( k  +  K
)  e.  ( ( M  +  K ) ... ( N  +  K ) ) )
4435, 43eqeltrd 2555 . . . . . . . . 9  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  -> 
j  e.  ( ( M  +  K ) ... ( N  +  K ) ) )
45 oveq1 6302 . . . . . . . . . . 11  |-  ( j  =  ( k  +  K )  ->  (
j  -  K )  =  ( ( k  +  K )  -  K ) )
4645ad2antll 728 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  -> 
( j  -  K
)  =  ( ( k  +  K )  -  K ) )
4739zcnd 10979 . . . . . . . . . . . 12  |-  ( k  e.  ( M ... N )  ->  k  e.  CC )
4847ad2antrl 727 . . . . . . . . . . 11  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  -> 
k  e.  CC )
4916adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  ->  K  e.  CC )
5048, 49pncand 9943 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  -> 
( ( k  +  K )  -  K
)  =  k )
5146, 50eqtr2d 2509 . . . . . . . . 9  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  -> 
k  =  ( j  -  K ) )
5244, 51jca 532 . . . . . . . 8  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  -> 
( j  e.  ( ( M  +  K
) ... ( N  +  K ) )  /\  k  =  ( j  -  K ) ) )
5334, 52impbida 830 . . . . . . 7  |-  ( ph  ->  ( ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) )  <->  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) ) )
5453opabbidv 4516 . . . . . 6  |-  ( ph  ->  { <. k ,  j
>.  |  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) }  =  { <. k ,  j
>.  |  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) } )
55 df-mpt 4513 . . . . . . . 8  |-  ( j  e.  ( ( M  +  K ) ... ( N  +  K
) )  |->  ( j  -  K ) )  =  { <. j ,  k >.  |  ( j  e.  ( ( M  +  K ) ... ( N  +  K ) )  /\  k  =  ( j  -  K ) ) }
5655cnveqi 5183 . . . . . . 7  |-  `' ( j  e.  ( ( M  +  K ) ... ( N  +  K ) )  |->  ( j  -  K ) )  =  `' { <. j ,  k >.  |  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) }
57 cnvopab 5413 . . . . . . 7  |-  `' { <. j ,  k >.  |  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) }  =  { <. k ,  j
>.  |  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) }
5856, 57eqtri 2496 . . . . . 6  |-  `' ( j  e.  ( ( M  +  K ) ... ( N  +  K ) )  |->  ( j  -  K ) )  =  { <. k ,  j >.  |  ( j  e.  ( ( M  +  K ) ... ( N  +  K ) )  /\  k  =  ( j  -  K ) ) }
59 df-mpt 4513 . . . . . 6  |-  ( k  e.  ( M ... N )  |->  ( k  +  K ) )  =  { <. k ,  j >.  |  ( k  e.  ( M ... N )  /\  j  =  ( k  +  K ) ) }
6054, 58, 593eqtr4g 2533 . . . . 5  |-  ( ph  ->  `' ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  |->  ( j  -  K ) )  =  ( k  e.  ( M ... N ) 
|->  ( k  +  K
) ) )
6160fneq1d 5677 . . . 4  |-  ( ph  ->  ( `' ( j  e.  ( ( M  +  K ) ... ( N  +  K
) )  |->  ( j  -  K ) )  Fn  ( M ... N )  <->  ( k  e.  ( M ... N
)  |->  ( k  +  K ) )  Fn  ( M ... N
) ) )
629, 61mpbiri 233 . . 3  |-  ( ph  ->  `' ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  |->  ( j  -  K ) )  Fn  ( M ... N
) )
63 dff1o4 5830 . . 3  |-  ( ( j  e.  ( ( M  +  K ) ... ( N  +  K ) )  |->  ( j  -  K ) ) : ( ( M  +  K ) ... ( N  +  K ) ) -1-1-onto-> ( M ... N )  <->  ( (
j  e.  ( ( M  +  K ) ... ( N  +  K ) )  |->  ( j  -  K ) )  Fn  ( ( M  +  K ) ... ( N  +  K ) )  /\  `' ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  |->  ( j  -  K ) )  Fn  ( M ... N
) ) )
646, 62, 63sylanbrc 664 . 2  |-  ( ph  ->  ( j  e.  ( ( M  +  K
) ... ( N  +  K ) )  |->  ( j  -  K ) ) : ( ( M  +  K ) ... ( N  +  K ) ) -1-1-onto-> ( M ... N ) )
65 oveq1 6302 . . . 4  |-  ( j  =  k  ->  (
j  -  K )  =  ( k  -  K ) )
66 ovex 6320 . . . 4  |-  ( k  -  K )  e. 
_V
6765, 4, 66fvmpt 5957 . . 3  |-  ( k  e.  ( ( M  +  K ) ... ( N  +  K
) )  ->  (
( j  e.  ( ( M  +  K
) ... ( N  +  K ) )  |->  ( j  -  K ) ) `  k )  =  ( k  -  K ) )
6867adantl 466 . 2  |-  ( (
ph  /\  k  e.  ( ( M  +  K ) ... ( N  +  K )
) )  ->  (
( j  e.  ( ( M  +  K
) ... ( N  +  K ) )  |->  ( j  -  K ) ) `  k )  =  ( k  -  K ) )
69 fprodshft.4 . 2  |-  ( (
ph  /\  j  e.  ( M ... N ) )  ->  A  e.  CC )
701, 2, 64, 68, 69fprodf1o 29005 1  |-  ( ph  ->  prod_ j  e.  ( M ... N ) A  =  prod_ k  e.  ( ( M  +  K ) ... ( N  +  K )
) B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   {copab 4510    |-> cmpt 4511   `'ccnv 5004    Fn wfn 5589   -1-1-onto->wf1o 5593   ` cfv 5594  (class class class)co 6295   CCcc 9502    + caddc 9507    - cmin 9817   ZZcz 10876   ...cfz 11684   prod_cprod 28964
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-sup 7913  df-oi 7947  df-card 8332  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-n0 10808  df-z 10877  df-uz 11095  df-rp 11233  df-fz 11685  df-fzo 11805  df-seq 12088  df-exp 12147  df-hash 12386  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-clim 13291  df-prod 28965
This theorem is referenced by:  risefacval2  29059  fallfacval2  29060
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