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Theorem fprodrev 13998
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 12172 . 2  |-  ( ph  ->  ( ( K  -  N ) ... ( K  -  M )
)  e.  Fin )
3 ovex 6324 . . . . 5  |-  ( K  -  j )  e. 
_V
4 eqid 2420 . . . . 5  |-  ( j  e.  ( ( K  -  N ) ... ( K  -  M
) )  |->  ( K  -  j ) )  =  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  |->  ( K  -  j ) )
53, 4fnmpti 5715 . . . 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 6324 . . . . 5  |-  ( K  -  k )  e. 
_V
8 eqid 2420 . . . . 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 simprr 764 . . . . . . . . 9  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  -> 
k  =  ( K  -  j ) )
11 simprl 762 . . . . . . . . . 10  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  -> 
j  e.  ( ( K  -  N ) ... ( K  -  M ) ) )
12 fprodshft.2 . . . . . . . . . . . 12  |-  ( ph  ->  M  e.  ZZ )
1312adantr 466 . . . . . . . . . . 11  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  ->  M  e.  ZZ )
14 fprodshft.3 . . . . . . . . . . . 12  |-  ( ph  ->  N  e.  ZZ )
1514adantr 466 . . . . . . . . . . 11  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  ->  N  e.  ZZ )
16 fprodshft.1 . . . . . . . . . . . 12  |-  ( ph  ->  K  e.  ZZ )
1716adantr 466 . . . . . . . . . . 11  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  ->  K  e.  ZZ )
18 elfzelz 11787 . . . . . . . . . . . 12  |-  ( j  e.  ( ( K  -  N ) ... ( K  -  M
) )  ->  j  e.  ZZ )
1918ad2antrl 732 . . . . . . . . . . 11  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  -> 
j  e.  ZZ )
20 fzrev 11845 . . . . . . . . . . 11  |-  ( ( ( 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 1265 . . . . . . . . . 10  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  -> 
( j  e.  ( ( K  -  N
) ... ( K  -  M ) )  <->  ( K  -  j )  e.  ( M ... N
) ) )
2211, 21mpbid 213 . . . . . . . . 9  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  -> 
( K  -  j
)  e.  ( M ... N ) )
2310, 22eqeltrd 2508 . . . . . . . 8  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  -> 
k  e.  ( M ... N ) )
24 oveq2 6304 . . . . . . . . . 10  |-  ( k  =  ( K  -  j )  ->  ( K  -  k )  =  ( K  -  ( K  -  j
) ) )
2524ad2antll 733 . . . . . . . . 9  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  -> 
( K  -  k
)  =  ( K  -  ( K  -  j ) ) )
2616zcnd 11030 . . . . . . . . . . 11  |-  ( ph  ->  K  e.  CC )
2726adantr 466 . . . . . . . . . 10  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  ->  K  e.  CC )
2818zcnd 11030 . . . . . . . . . . 11  |-  ( j  e.  ( ( K  -  N ) ... ( K  -  M
) )  ->  j  e.  CC )
2928ad2antrl 732 . . . . . . . . . 10  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  -> 
j  e.  CC )
3027, 29nncand 9980 . . . . . . . . 9  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  -> 
( K  -  ( K  -  j )
)  =  j )
3125, 30eqtr2d 2462 . . . . . . . 8  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  -> 
j  =  ( K  -  k ) )
3223, 31jca 534 . . . . . . 7  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  -> 
( k  e.  ( M ... N )  /\  j  =  ( K  -  k ) ) )
33 simprr 764 . . . . . . . . 9  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  -> 
j  =  ( K  -  k ) )
34 simprl 762 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  -> 
k  e.  ( M ... N ) )
3512adantr 466 . . . . . . . . . . 11  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  ->  M  e.  ZZ )
3614adantr 466 . . . . . . . . . . 11  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  ->  N  e.  ZZ )
3716adantr 466 . . . . . . . . . . 11  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  ->  K  e.  ZZ )
38 elfzelz 11787 . . . . . . . . . . . 12  |-  ( k  e.  ( M ... N )  ->  k  e.  ZZ )
3938ad2antrl 732 . . . . . . . . . . 11  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  -> 
k  e.  ZZ )
40 fzrev2 11846 . . . . . . . . . . 11  |-  ( ( ( 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 1265 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  -> 
( k  e.  ( M ... N )  <-> 
( K  -  k
)  e.  ( ( K  -  N ) ... ( K  -  M ) ) ) )
4234, 41mpbid 213 . . . . . . . . 9  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  -> 
( K  -  k
)  e.  ( ( K  -  N ) ... ( K  -  M ) ) )
4333, 42eqeltrd 2508 . . . . . . . 8  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  -> 
j  e.  ( ( K  -  N ) ... ( K  -  M ) ) )
44 oveq2 6304 . . . . . . . . . 10  |-  ( j  =  ( K  -  k )  ->  ( K  -  j )  =  ( K  -  ( K  -  k
) ) )
4544ad2antll 733 . . . . . . . . 9  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  -> 
( K  -  j
)  =  ( K  -  ( K  -  k ) ) )
4626adantr 466 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  ->  K  e.  CC )
4738zcnd 11030 . . . . . . . . . . 11  |-  ( k  e.  ( M ... N )  ->  k  e.  CC )
4847ad2antrl 732 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  -> 
k  e.  CC )
4946, 48nncand 9980 . . . . . . . . 9  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  -> 
( K  -  ( K  -  k )
)  =  k )
5045, 49eqtr2d 2462 . . . . . . . 8  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  -> 
k  =  ( K  -  j ) )
5143, 50jca 534 . . . . . . 7  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  -> 
( j  e.  ( ( K  -  N
) ... ( K  -  M ) )  /\  k  =  ( K  -  j ) ) )
5232, 51impbida 840 . . . . . 6  |-  ( ph  ->  ( ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) )  <->  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) ) )
5352mptcnv 5249 . . . . 5  |-  ( ph  ->  `' ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  |->  ( K  -  j ) )  =  ( k  e.  ( M ... N ) 
|->  ( K  -  k
) ) )
5453fneq1d 5675 . . . 4  |-  ( ph  ->  ( `' ( j  e.  ( ( K  -  N ) ... ( K  -  M
) )  |->  ( K  -  j ) )  Fn  ( M ... N )  <->  ( k  e.  ( M ... N
)  |->  ( K  -  k ) )  Fn  ( M ... N
) ) )
559, 54mpbiri 236 . . 3  |-  ( ph  ->  `' ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  |->  ( K  -  j ) )  Fn  ( M ... N
) )
56 dff1o4 5830 . . 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
) ) )
576, 55, 56sylanbrc 668 . 2  |-  ( ph  ->  ( j  e.  ( ( K  -  N
) ... ( K  -  M ) )  |->  ( K  -  j ) ) : ( ( K  -  N ) ... ( K  -  M ) ) -1-1-onto-> ( M ... N ) )
58 oveq2 6304 . . . 4  |-  ( j  =  k  ->  ( K  -  j )  =  ( K  -  k ) )
5958, 4, 7fvmpt 5955 . . 3  |-  ( k  e.  ( ( K  -  N ) ... ( K  -  M
) )  ->  (
( j  e.  ( ( K  -  N
) ... ( K  -  M ) )  |->  ( K  -  j ) ) `  k )  =  ( K  -  k ) )
6059adantl 467 . 2  |-  ( (
ph  /\  k  e.  ( ( K  -  N ) ... ( K  -  M )
) )  ->  (
( j  e.  ( ( K  -  N
) ... ( K  -  M ) )  |->  ( K  -  j ) ) `  k )  =  ( K  -  k ) )
61 fprodshft.4 . 2  |-  ( (
ph  /\  j  e.  ( M ... N ) )  ->  A  e.  CC )
621, 2, 57, 60, 61fprodf1o 13967 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 187    /\ wa 370    = wceq 1437    e. wcel 1867    |-> cmpt 4475   `'ccnv 4844    Fn wfn 5587   -1-1-onto->wf1o 5591   ` cfv 5592  (class class class)co 6296   CCcc 9526    - cmin 9849   ZZcz 10926   ...cfz 11771   prod_cprod 13926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-inf2 8137  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605  ax-pre-sup 9606
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-se 4805  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-isom 5601  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-1o 7181  df-oadd 7185  df-er 7362  df-en 7569  df-dom 7570  df-sdom 7571  df-fin 7572  df-sup 7953  df-oi 8016  df-card 8363  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-div 10259  df-nn 10599  df-2 10657  df-3 10658  df-n0 10859  df-z 10927  df-uz 11149  df-rp 11292  df-fz 11772  df-fzo 11903  df-seq 12200  df-exp 12259  df-hash 12502  df-cj 13130  df-re 13131  df-im 13132  df-sqrt 13266  df-abs 13267  df-clim 13519  df-prod 13927
This theorem is referenced by:  fallfacval3  14032  bcprod  30158
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