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Theorem fprodshft 27333
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 11778 . 2  |-  ( ph  ->  ( ( M  +  K ) ... ( N  +  K )
)  e.  Fin )
3 ovex 6105 . . . . 5  |-  ( j  -  K )  e. 
_V
4 eqid 2433 . . . . 5  |-  ( j  e.  ( ( M  +  K ) ... ( N  +  K
) )  |->  ( j  -  K ) )  =  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  |->  ( j  -  K ) )
53, 4fnmpti 5527 . . . 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 6105 . . . . 5  |-  ( k  +  K )  e. 
_V
8 eqid 2433 . . . . 5  |-  ( k  e.  ( M ... N )  |->  ( k  +  K ) )  =  ( k  e.  ( M ... N
)  |->  ( k  +  K ) )
97, 8fnmpti 5527 . . . 4  |-  ( k  e.  ( M ... N )  |->  ( k  +  K ) )  Fn  ( M ... N )
10 oveq1 6087 . . . . . . . . . . . . 13  |-  ( k  =  ( j  -  K )  ->  (
k  +  K )  =  ( ( j  -  K )  +  K ) )
1110ad2antll 721 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
( k  +  K
)  =  ( ( j  -  K )  +  K ) )
12 elfzelz 11439 . . . . . . . . . . . . . . 15  |-  ( j  e.  ( ( M  +  K ) ... ( N  +  K
) )  ->  j  e.  ZZ )
1312zcnd 10735 . . . . . . . . . . . . . 14  |-  ( j  e.  ( ( M  +  K ) ... ( N  +  K
) )  ->  j  e.  CC )
1413ad2antrl 720 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
j  e.  CC )
15 fprodshft.1 . . . . . . . . . . . . . . 15  |-  ( ph  ->  K  e.  ZZ )
1615zcnd 10735 . . . . . . . . . . . . . 14  |-  ( ph  ->  K  e.  CC )
1716adantr 462 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  ->  K  e.  CC )
1814, 17npcand 9710 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
( ( j  -  K )  +  K
)  =  j )
1911, 18eqtr2d 2466 . . . . . . . . . . 11  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
j  =  ( k  +  K ) )
20 simprl 748 . . . . . . . . . . 11  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
j  e.  ( ( M  +  K ) ... ( N  +  K ) ) )
2119, 20eqeltrrd 2508 . . . . . . . . . 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 462 . . . . . . . . . . 11  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  ->  M  e.  ZZ )
24 fprodshft.3 . . . . . . . . . . . 12  |-  ( ph  ->  N  e.  ZZ )
2524adantr 462 . . . . . . . . . . 11  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  ->  N  e.  ZZ )
26 simprr 749 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
k  =  ( j  -  K ) )
2712ad2antrl 720 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
j  e.  ZZ )
2815adantr 462 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  ->  K  e.  ZZ )
2927, 28zsubcld 10739 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
( j  -  K
)  e.  ZZ )
3026, 29eqeltrd 2507 . . . . . . . . . . 11  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
k  e.  ZZ )
31 fzaddel 11479 . . . . . . . . . . 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 1212 . . . . . . . . . 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 529 . . . . . . . 8  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
( k  e.  ( M ... N )  /\  j  =  ( k  +  K ) ) )
35 simprr 749 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  -> 
j  =  ( k  +  K ) )
36 simprl 748 . . . . . . . . . . 11  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  -> 
k  e.  ( M ... N ) )
3722adantr 462 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  ->  M  e.  ZZ )
3824adantr 462 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  ->  N  e.  ZZ )
39 elfzelz 11439 . . . . . . . . . . . . 13  |-  ( k  e.  ( M ... N )  ->  k  e.  ZZ )
4039ad2antrl 720 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  -> 
k  e.  ZZ )
4115adantr 462 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  ->  K  e.  ZZ )
4237, 38, 40, 41, 31syl22anc 1212 . . . . . . . . . . 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 2507 . . . . . . . . 9  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  -> 
j  e.  ( ( M  +  K ) ... ( N  +  K ) ) )
45 oveq1 6087 . . . . . . . . . . 11  |-  ( j  =  ( k  +  K )  ->  (
j  -  K )  =  ( ( k  +  K )  -  K ) )
4645ad2antll 721 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  -> 
( j  -  K
)  =  ( ( k  +  K )  -  K ) )
4739zcnd 10735 . . . . . . . . . . . 12  |-  ( k  e.  ( M ... N )  ->  k  e.  CC )
4847ad2antrl 720 . . . . . . . . . . 11  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  -> 
k  e.  CC )
4916adantr 462 . . . . . . . . . . 11  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  ->  K  e.  CC )
5048, 49pncand 9707 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  -> 
( ( k  +  K )  -  K
)  =  k )
5146, 50eqtr2d 2466 . . . . . . . . 9  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  -> 
k  =  ( j  -  K ) )
5244, 51jca 529 . . . . . . . 8  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  -> 
( j  e.  ( ( M  +  K
) ... ( N  +  K ) )  /\  k  =  ( j  -  K ) ) )
5334, 52impbida 821 . . . . . . 7  |-  ( ph  ->  ( ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) )  <->  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) ) )
5453opabbidv 4343 . . . . . 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 4340 . . . . . . . 8  |-  ( j  e.  ( ( M  +  K ) ... ( N  +  K
) )  |->  ( j  -  K ) )  =  { <. j ,  k >.  |  ( j  e.  ( ( M  +  K ) ... ( N  +  K ) )  /\  k  =  ( j  -  K ) ) }
5655cnveqi 5001 . . . . . . 7  |-  `' ( j  e.  ( ( M  +  K ) ... ( N  +  K ) )  |->  ( j  -  K ) )  =  `' { <. j ,  k >.  |  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) }
57 cnvopab 5226 . . . . . . 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 2453 . . . . . 6  |-  `' ( j  e.  ( ( M  +  K ) ... ( N  +  K ) )  |->  ( j  -  K ) )  =  { <. k ,  j >.  |  ( j  e.  ( ( M  +  K ) ... ( N  +  K ) )  /\  k  =  ( j  -  K ) ) }
59 df-mpt 4340 . . . . . 6  |-  ( k  e.  ( M ... N )  |->  ( k  +  K ) )  =  { <. k ,  j >.  |  ( k  e.  ( M ... N )  /\  j  =  ( k  +  K ) ) }
6054, 58, 593eqtr4g 2490 . . . . 5  |-  ( ph  ->  `' ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  |->  ( j  -  K ) )  =  ( k  e.  ( M ... N ) 
|->  ( k  +  K
) ) )
6160fneq1d 5489 . . . 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 5637 . . 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 657 . 2  |-  ( ph  ->  ( j  e.  ( ( M  +  K
) ... ( N  +  K ) )  |->  ( j  -  K ) ) : ( ( M  +  K ) ... ( N  +  K ) ) -1-1-onto-> ( M ... N ) )
65 oveq1 6087 . . . 4  |-  ( j  =  k  ->  (
j  -  K )  =  ( k  -  K ) )
66 ovex 6105 . . . 4  |-  ( k  -  K )  e. 
_V
6765, 4, 66fvmpt 5762 . . 3  |-  ( k  e.  ( ( M  +  K ) ... ( N  +  K
) )  ->  (
( j  e.  ( ( M  +  K
) ... ( N  +  K ) )  |->  ( j  -  K ) ) `  k )  =  ( k  -  K ) )
6867adantl 463 . 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 27305 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 1362    e. wcel 1755   {copab 4337    e. cmpt 4338   `'ccnv 4826    Fn wfn 5401   -1-1-onto->wf1o 5405   ` cfv 5406  (class class class)co 6080   CCcc 9267    + caddc 9272    - cmin 9582   ZZcz 10633   ...cfz 11423   prod_cprod 27264
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-inf2 7835  ax-cnex 9325  ax-resscn 9326  ax-1cn 9327  ax-icn 9328  ax-addcl 9329  ax-addrcl 9330  ax-mulcl 9331  ax-mulrcl 9332  ax-mulcom 9333  ax-addass 9334  ax-mulass 9335  ax-distr 9336  ax-i2m1 9337  ax-1ne0 9338  ax-1rid 9339  ax-rnegex 9340  ax-rrecex 9341  ax-cnre 9342  ax-pre-lttri 9343  ax-pre-lttrn 9344  ax-pre-ltadd 9345  ax-pre-mulgt0 9346  ax-pre-sup 9347
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-fal 1368  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-se 4667  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-1st 6566  df-2nd 6567  df-recs 6818  df-rdg 6852  df-1o 6908  df-oadd 6912  df-er 7089  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-sup 7679  df-oi 7712  df-card 8097  df-pnf 9407  df-mnf 9408  df-xr 9409  df-ltxr 9410  df-le 9411  df-sub 9584  df-neg 9585  df-div 9981  df-nn 10310  df-2 10367  df-3 10368  df-n0 10567  df-z 10634  df-uz 10849  df-rp 10979  df-fz 11424  df-fzo 11532  df-seq 11790  df-exp 11849  df-hash 12087  df-cj 12571  df-re 12572  df-im 12573  df-sqr 12707  df-abs 12708  df-clim 12949  df-prod 27265
This theorem is referenced by:  risefacval2  27359  fallfacval2  27360
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