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Theorem fprodshft 27509
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 11816 . 2  |-  ( ph  ->  ( ( M  +  K ) ... ( N  +  K )
)  e.  Fin )
3 ovex 6137 . . . . 5  |-  ( j  -  K )  e. 
_V
4 eqid 2443 . . . . 5  |-  ( j  e.  ( ( M  +  K ) ... ( N  +  K
) )  |->  ( j  -  K ) )  =  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  |->  ( j  -  K ) )
53, 4fnmpti 5560 . . . 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 6137 . . . . 5  |-  ( k  +  K )  e. 
_V
8 eqid 2443 . . . . 5  |-  ( k  e.  ( M ... N )  |->  ( k  +  K ) )  =  ( k  e.  ( M ... N
)  |->  ( k  +  K ) )
97, 8fnmpti 5560 . . . 4  |-  ( k  e.  ( M ... N )  |->  ( k  +  K ) )  Fn  ( M ... N )
10 oveq1 6119 . . . . . . . . . . . . 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 11474 . . . . . . . . . . . . . . 15  |-  ( j  e.  ( ( M  +  K ) ... ( N  +  K
) )  ->  j  e.  ZZ )
1312zcnd 10769 . . . . . . . . . . . . . 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 10769 . . . . . . . . . . . . . 14  |-  ( ph  ->  K  e.  CC )
1716adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  ->  K  e.  CC )
1814, 17npcand 9744 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
( ( j  -  K )  +  K
)  =  j )
1911, 18eqtr2d 2476 . . . . . . . . . . 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 2518 . . . . . . . . . 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 10773 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
( j  -  K
)  e.  ZZ )
3026, 29eqeltrd 2517 . . . . . . . . . . 11  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
k  e.  ZZ )
31 fzaddel 11514 . . . . . . . . . . 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 1219 . . . . . . . . . 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 11474 . . . . . . . . . . . . 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 1219 . . . . . . . . . . 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 2517 . . . . . . . . 9  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  -> 
j  e.  ( ( M  +  K ) ... ( N  +  K ) ) )
45 oveq1 6119 . . . . . . . . . . 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 10769 . . . . . . . . . . . 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 9741 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  -> 
( ( k  +  K )  -  K
)  =  k )
5146, 50eqtr2d 2476 . . . . . . . . 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 828 . . . . . . 7  |-  ( ph  ->  ( ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) )  <->  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) ) )
5453opabbidv 4376 . . . . . 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 4373 . . . . . . . 8  |-  ( j  e.  ( ( M  +  K ) ... ( N  +  K
) )  |->  ( j  -  K ) )  =  { <. j ,  k >.  |  ( j  e.  ( ( M  +  K ) ... ( N  +  K ) )  /\  k  =  ( j  -  K ) ) }
5655cnveqi 5035 . . . . . . 7  |-  `' ( j  e.  ( ( M  +  K ) ... ( N  +  K ) )  |->  ( j  -  K ) )  =  `' { <. j ,  k >.  |  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) }
57 cnvopab 5259 . . . . . . 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 2463 . . . . . 6  |-  `' ( j  e.  ( ( M  +  K ) ... ( N  +  K ) )  |->  ( j  -  K ) )  =  { <. k ,  j >.  |  ( j  e.  ( ( M  +  K ) ... ( N  +  K ) )  /\  k  =  ( j  -  K ) ) }
59 df-mpt 4373 . . . . . 6  |-  ( k  e.  ( M ... N )  |->  ( k  +  K ) )  =  { <. k ,  j >.  |  ( k  e.  ( M ... N )  /\  j  =  ( k  +  K ) ) }
6054, 58, 593eqtr4g 2500 . . . . 5  |-  ( ph  ->  `' ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  |->  ( j  -  K ) )  =  ( k  e.  ( M ... N ) 
|->  ( k  +  K
) ) )
6160fneq1d 5522 . . . 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 5670 . . 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 6119 . . . 4  |-  ( j  =  k  ->  (
j  -  K )  =  ( k  -  K ) )
66 ovex 6137 . . . 4  |-  ( k  -  K )  e. 
_V
6765, 4, 66fvmpt 5795 . . 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 27481 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 1369    e. wcel 1756   {copab 4370    e. cmpt 4371   `'ccnv 4860    Fn wfn 5434   -1-1-onto->wf1o 5438   ` cfv 5439  (class class class)co 6112   CCcc 9301    + caddc 9306    - cmin 9616   ZZcz 10667   ...cfz 11458   prod_cprod 27440
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-inf2 7868  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380  ax-pre-sup 9381
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-se 4701  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-isom 5448  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-recs 6853  df-rdg 6887  df-1o 6941  df-oadd 6945  df-er 7122  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-sup 7712  df-oi 7745  df-card 8130  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-div 10015  df-nn 10344  df-2 10401  df-3 10402  df-n0 10601  df-z 10668  df-uz 10883  df-rp 11013  df-fz 11459  df-fzo 11570  df-seq 11828  df-exp 11887  df-hash 12125  df-cj 12609  df-re 12610  df-im 12611  df-sqr 12745  df-abs 12746  df-clim 12987  df-prod 27441
This theorem is referenced by:  risefacval2  27535  fallfacval2  27536
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